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Title Nonlinear optical signals and spectroscopy with quantum light

Permalink https://escholarship.org/uc/item/70r4b4pz

Journal Reviews of Modern Physics, 88(4)

ISSN 0034-6861

Authors Dorfman, KE Schlawin, F Mukamel, S

Publication Date 2016-12-28

DOI 10.1103/RevModPhys.88.045008

License https://creativecommons.org/licenses/by/4.0/ 4.0

Peer reviewed

eScholarship.org Powered by the California Digital Library University of California 1 Nonlinear optical signals and spectroscopy with quantum light

1 1 Konstantin E. Dorfman ,∗ Frank Schlawin ,† and Shaul Mukamel‡ Department of Chemistry, University of California, Irvine, California 92697, USA

(Dated: May 24, 2016)

Conventional nonlinear spectroscopy uses classical light to detect matter properties through the variation of its response with frequencies or time delays. Quantum light opens up new avenues for spectroscopy by utilizing parameters of the quantum state of light as novel control knobs and through the variation of photon statistics by coupling to matter. We present an intuitive diagrammatic approach for calculating ultrafast spectroscopy signals induced by quantum light, focusing on applications involving entangled photons with nonclassical bandwidth properties - known as “time-energy entanglement”. Nonlinear optical signals induced by quantized light ﬁelds are expressed using time ordered multipoint correlation functions of superoperators. These are different from Glauber’s g- functions for photon counting which use normally ordered products of ordinary operators. Entangled photon pairs are not subjected to the classical Fourier limitations on the joint temporal and spectral resolution. After a brief survey of properties of entangled photon pairs relevant to their spectroscopic applications, different optical signals, and photon counting setups are discussed and illustrated for simple multi-level model systems.

PACS numbers: 123

CONTENTS A. Stationary Nonlinear signals 20 B. Fluorescence detection of nonlinear signals 21 I. Introduction 2 1. Two-photon absorption vs. two-photon-induced A. Liouville space superoperator notation 3 fluorescence 21 B. Diagram construction 4 2. Two-photon induced transparency and entangled-photon virtual-state spectroscopy 21 II. Quantum light 5 3. Fluorescence from multi-level systems 22 A. Classical vs. Quantum Light 6 4. Multidimensional signals 22 B. Single mode quantum states 7 5. Loop (LOP) vs ladder (LAP) delay scanning protocols for C. Photon Entanglement in multimode states 7 multidimensional fluorescence detected signals 23 D. Entangled photons generation by parametric down conversion 8 C. Heterodyne detection of nonlinear signals 27 E. Narrowband pump 9 1. Heterodyne Intensity measurements - Raman vs. TPA F. Broadband pump 10 pathways 27 G. Shaping of entangled photons 11 2. Heterodyne detected four-wave mixing; the H. Polarization entanglement 12 double-quantum-coherence technique 29 I. Matter correlations in noninteracting two-level atoms induced by D. Multiple photon counting detection 31 quantum light 12 1. Photon correlation measurements using gated photon number 1. Collective resonances induced by entangled light 13 operators 32 2. Excited state populations generated by nonclassical light 13 2. Photon counting and matter dipole correlation functions 32 3. Classical vs entangled light 14 3. Connection to the physical spectrum 33 4. Collective two-body resonances generated by illumination E. Interferometric detection of photon coincidence signals 34 with entangled light 15 1. Coincidence detection of linear absorption 35 arXiv:1605.06746v1 [quant-ph] 22 May 2016 J. Quantum light induced correlations between two-level systems 2. Coincidence detection of pump-probe signals 36 with dipole-dipole coupling 16 3. Coincidence detection of Femtosecond Stimulated Raman 1. Model system 16 Signals 37 2. Control of energy transfer 18 3. Scaling of two photon absorption with pump intensity 19 IV. Entangled light generation via nonlinear light-matter interactions; nonclassical response functions 43 III. Nonlinear optical signals obtained with entangled light 20 A. Superoperator description of n-wave mixing 43 B. Connection to nonlinear fluctuation-dissipation relations 44 C. Heterodyne-detected sum and difference frequency generation with classical light 45 1 These authors contributed equally to this manuscript D. Difference frequency generation (DFG) 46 ∗ [email protected] E. Sum Frequency Generation (SFG) 46 † also at Physikalisches Institut, Albert-Ludwigs-Universitat¨ Freiburg F. Two-photon-induced fluorescence (TPIF) vs. Hermann-Herder-Straße 3, 79104 Freiburg, Germany; email: homodyne-detected SFG 47 [email protected] 1. Two-photon-induced fluorescence (TPIF). 47 ‡ [email protected] 2. Homodyne-detected SFG 48 2

G. Two-photon-emitted fluorescence (TPEF) vs. type-I parametric 2012), or strong focussing (Faez et al., 2014; Pototschnig down conversion (PDC). 49 et al., 2011; Rezus et al., 2012) all provide possible means 1. TPEF 49 to observe and control nonlinear optical processes on a funda- 2. Type-I PDC. 50 H. Type II PDC; polarization entanglement. 50 mental quantum level. Besides possible technological appli- I. Time-and-frequency resolved type-I PDC 51 cations such as all-optical transistors (Shomroni et al., 2014) 1. The bare PCC rate 52 or photonic quantum information processing (Braunstein and 2. Simulations of typical PDC signals 54 Kimble, 2000; Franson, 1989; Jennewein et al., 2000; Knill 3. Spectral diffusion 54 J. Generation and entanglement control of photons produced by et al., 2001; Kok et al., 2007; O’Brien et al., 2003; U’Ren two independent molecules by time-and-frequency gated photon et al., 2003), these also show great promise as novel appli- coincidence counting (PCC). 55 cations as spectroscopic tools. Parameters of the photon field 1. PCC of single photons generated by two remote emitters. 56 wavefunction can serve as control knobs that supplement clas- 2. Time-and-frequency gated PCC 58 3. Signatures of gating and spectral diffusion in the HOM dip. 58 sical parameters such as frequencies and time delays. This review surveys these emerging applications, and introduces a V. Summary and Outlook 58 systematic diagrammatic perturbative approach to their theo- VI. Acknowledgements 59 retical description. One of the striking features of quantum light is photon en- A. Diagram construction 59 tanglement. This occurs between two beams of light (field 1. Loop diagrams 59 amplitudes (Van Enk, 2005)) when the quantum state of each 2. Ladder diagrams 60 field cannot be described in the individual parameter space B. Analytical expressions for the Schmidt decomposition 60 of that field. Different degrees of freedom can become entangled. Most common types of entanglement are: their spin C. Green’s functions of matter 61 (Dolde et al., 2013), polarization (Shih and Alley, 1988), po- D. Intensity measurements: TPA vs Raman 63 sition and momentum (Howell et al., 2004), time and energy (Tittel et al., 1999), etc. Entangled photon pairs constitute E. Time-and-frequency gating 63 1. Simultaneous time-and-frequency gating 63 an invaluable tool in fundamental tests of quantum mechanics 2. The bare signal 64 - most famously in the violation of Bell’s inequalities (As- 3. Spectrogram-overlap representation for detected signal 65 pect et al., 1982a, 1981, 1982b) or in Hong, Ou and Man- 4. Multiple detections 66 del’s photon correlation experiments (Hong et al., 1987; Ou References 66 and Mandel, 1988; Shih and Alley, 1988). Besides, their nonclassical bandwidth properties have long been recognized as a potential resource in various “quantum-enhanced” appli- I. INTRODUCTION cations, where the quantum correlations shared between the photon pairs are thought to offer an advantage. For exam- Nonlinear optics is most commonly and successfully for- ple when one photon from an entangled pair is sent through mulated using a semiclassical approach whereby the matter a dispersive medium, the leading-order dispersion is compen- degrees of freedom are treated quantum mechanically, but the sated in photon coincidence measurements - an effect called radiation field is classical (Boyd, 2003; Scully and Zubairy, dispersion cancellation (Abouraddy et al., 2002; Franson, 1997). Spectroscopic signals are then obtained by comput- 1992; Larchuk et al., 1995; Minaeva et al., 2009; Steinberg ing the polarization induced in the medium and expanding et al., 1992a,b). In the field of quantum-enhanced measure- it perturbatively in the impinging field(s) (Ginsberg et al., ments, entanglement may be employed to enhance the reso- 2009; Hamm and Zanni, 2011; Mukamel, 1995). This level lution beyond the Heisenberg limit (Giovannetti et al., 2004, of theory is well justified in many applications, owing to the 2006; Mitchell et al., 2004). Similarly, the spatial resolution typically large intensities required to generate a nonlinear re- may be enhanced in quantum imaging applications (Bennink sponse from the optical medium, which can only be reached et al., 2004; Pittman et al., 1995), quantum-optical coherence with lasers. Incidentally, it was shortly after Maiman’s devel- tomography (Abouraddy et al., 2002; Esposito, 2016; Nasr opment of the ruby laser that the first nonlinear optical effect et al., 2003), as well as in quantum lithographic applications was observed (Franken et al., 1961). (Boto et al., 2000; D’Angelo et al., 2001). Recent advances in quantum optics extend nonlinear sig- However, it is now recognized that many of these applica- nals down to the few-photon level where the quantum nature tions may also be created in purely classical settings: Some of the field is manifested, and must be taken into account: The two-photon interference effects originally believed to be a enhanced light-matter coupling in cavities (Raimond et al., hallmark of quantum entanglement can be simulated by post- 2001; Schwartz et al., 2011; Walther et al., 2006), the en- selecting the signal (Kaltenbaek et al., 2008, 2009). This hancement of the medium’s nonlinearity by additional driving had enabled quantum-optical coherence tomography studies fields (Chen et al., 2013; Peyronel et al., 2012), large dipoles with classical light (Lavoie et al., 2009). Similarly, quan- in highly excited Rdberg states (Gorniaczyk et al., 2014; He tum imaging can be carried out with thermal light (Valencia et al., 2014), molecular design (Castet et al., 2013; Loo et al., et al., 2005), albeit with reduced signal-to-noise ratio. When (a)

3 proposing applications of quantum light, it is thus imperative to carefully distinguish genuine entanglement from classical a)(b) correlation effects. A first clear signature of the quantumness of light is the scaling of optical signals with light intensities: Classical heterodyne χ(3) signals such as two photon absorption scale quadratically with the intensity, and therefore require a high intensity to be visible against lower-order linear-scaling processes. With entangled photons, such signals scale linearly (Dayan et al., 2004, 2005; Friberg et al., 1985; Georgiades et al., 1995; Javanainen and Gould, 1990). This allows to carry out microscopy (Teich and Saleh, 1998) and lithogra- phy (Boto et al., 2000) applications at much lower photon fluxes. The different intensity scaling with entangled photons has been first demonstrated in atomic systems by (Georgiades b) et al., 1995) and later by (Dayan et al., 2004, 2005), as well as in organic molecules (Lee and Goodson, 2006). An entangled two-photon absorption (TPA) experiment performed in a porphyrin dendrimer is shown in Fig. 1a). The linear scaling can be rationalized as follows: entangled photons come in pairs, as they are generated simultaneously. At low light intensity, the different photon pairs are temporally well separated, and the two photon absorption process involves two entangled photons of the same pair. It thus behaves as a linear spectroscopy with respect to the pair. At higher intensities, it becomes statistically more plausible for the two photons to come from different pairs, which are not entangled, and the FIG. 1 a) Linear entangled TPA rate and quadratic nonlinear ran- classical quadratic scaling is recovered [Fig. 1a)]. This is re- dom TPA rate in a porphyrin dendrimer at different entanglement lated to the Poisson distribution of photons. times. The inset shows the dominant effect of the entangled TPA The presence of strong time and frequency correlations of at low input flux of correlated photons. b) Two-photon absorption entangled photons is a second important feature, which we 5S1/2 → 4D5/2,3/2 in atomic Rubidium (Dayan et al., 2004) vs. the shall discuss extensively in the course of this review. Fig. 1b) time delay between the two photons (left panel), and vs. the pump shows the two-photon absorption signal of entangled photons frequency (right panel). in Rubidium vapor (Dayan et al., 2004). In the left panel, a delay stage is placed into one of the two photon beams, cre- properties of entangled photons, and present their impact on ating a narrow resonance as if the TPA resonance was cre- excited state distributions upon their absorption in a complex ated by a 23 fs pulse. However, as the frequency of the pump quantum system. In section III we provide general superoper- pulse which creates the photons is varied in the right panel, ator expressions for nonlinear optical signals, and review the the resonance is also narrow in the frequency domain, as if it proposed available setups. Finally, section IV presents a gen- was created by a ns pulse. This simultaneous time and fre- eral classification of nonlinear optical processes induced by quency resolution along non Fourier conjugate axes is a hall- quantum light, clarifying under which conditions its quantum mark of the time-energy entanglement, and its exploitation as nature may play a role. a spectroscopic tool offers novel control knobs to manipulate the excited state distribution, and thereby enhance or suppress features in nonlinear spectroscopic signals. A. Liouville space superoperator notation In a different line of research, the seminal photon coincidence counting experiments were turned into a spectroscopic The calculation of nonlinear optical signals becomes most tool by placing a sample into the beam line of one of the two transparent by working in Liouville space (Mukamel, 1995; entangled photons, and recording the change of the coinci- Mukamel and Nagata, 2011), i.e. the space of bounded op- dence count rate (Kalachev et al., 2007, 2008; Kalashnikov erators on the combined matter-field Hilbert space. It offers et al., 2014; Yabushita and Kobayashi, 2004). We will ex- a convenient bookkeeping device for matter-light interactions amine related schemes for utilizing entanglement in nonlinear where signals are described as time-ordered products of su- spectroscopy. peroperators. This review is structured as follows: In section I.A, we Here, we introduce the basic notation. With each operator briefly give a the background and introduce the superopera- A acting on the Hilbert space, we associate two superoperators tor notation used in the following. In section II, we discuss (Harbola and Mukamel, 2008; Roslyak and Mukamel, 2009b, 4

2010) B. Diagram construction A X AX, (1) L ≡ We adopt the diagram representation of nonlinear spectro- which represents the action from the left, and scopic signals which can be found, for instance, in (Mukamel and Rahav, 2010). It bears close similarity to analogous meth- A X XA, (2) R ≡ ods in quantum electrodynamics (Cohen-Tannoudji et al., the action from the right. We further introduce the linear com- 1992). We will employ two types of diagrams which are as- binations, the commutator superoperator sociated with calculating the expectation value of an operator A(t) using the density matrix or the wavefunction. For de- A AL AR, (3) tails see appendix A. First, we evaluate it by propagating the − ≡ − and the anti-commutator density matrix ρ(t), 1 A (A + A ) . (4) A(t) DM tr A(t)%(t) (10) + ≡ 2 L R ≡ { } This notation allows to derive compact expressions for spec- Eq. (10) can be best analyzed in Liouville space (3): We troscopic signals. At the end of the calculation, after the write the time evolution of the joint matter plus field den- time-ordering is taken care of, we can switch back to ordinary sity matrix using a time-ordered exponential which can be ex- Hilbert space operators. panded as a Dyson series, The total Hamiltonian of the field-matter system is given by i t Htot = H + Hfield + Hint. (5) %(t) = exp dτHint, (τ) %(t0), (11) 0 T − − ~ Zt0 H0 describes the matter. The radiation field Hamiltonian is given by where Hint is a superoperator (3) that corresponds to the interaction Hamiltonian− (7). The time-ordering operator is T Hfield = ~ ωs as†(ωs)as(ωs). (6) defined by its action on a product of arbitrary superoperators s A(t) and B(t), X Here we introduced the creation (annihilation) operators A(t )B(t ) θ(t t )A(t )B(t ) for mode s which satisfy bosonic commutation relations T 1 2 ≡ 1 − 2 1 2 [as, a† ] = δs,s , [as, as ] = [a†, a† ] = 0. In many situations, + θ(t2 t1)B(t2)A(t1). (12) s0 0 0 s s0 − we will replace the discrete sum over modes by a continu- V ous integral 3 dω D˜(ω ) with D˜(ω ) being the orders the following products of superoperators so that time s → (2π) s s s T density of states that we assume to be flat within the relevant increases from right to left. The perturbative expansion of P R Eq. (11) to n-th order in H generates a number of pathways bandwidths D˜(ωs) D˜. int We assume a dipole' light-matter interaction Hamiltonian, - successions of excitations and deexcitations on both the bra or the ket part of the density matrix. These pathways are repre- Hint = , (7) sented by double-sided ladder diagrams, which represent con- Vν ν volutions of fully time-ordered superoperator nonequilibrium X Green’s functions (SNGF) of the form where ν = Vν + Vν† the dipole operator of molecule ν with the summationV running over all the molecules contained in the sample. ( ) is the excitation lowering (raising) part of the ∞ ∞ (n) V V † dt1 dtnVνn...ν1 (t tn . . . , t tn t1) 0 ··· 0 − − · · · − dipole. = E + E† denotes the electric field operator and E Z Z (n) (E ) its positive (negative) frequency components which can E (t tn . . . , t tn t1) . (13) † × νn...ν1 − − · · · − be written in the interaction picture with respect to the field Hamiltonian (Loudon, 2000) as The field SNGF in Eq. (13)

dω iωt (n) E = e− a(ω) (8) Eνn...ν1 (t tn . . . , t tn t1) 2π − − · · · − Z = Eνn (t tn) Eν1 (t tn t1) (14) which is written in the slowly-varying envelope approxima- T − ··· − · · · − tion. Unless specified otherwise, E will denote the sum of all is evaluated with respect to the initial quantum state of the relevant field modes. light. The indices are νj = L, R or νj = +, . When In the following applications, we will neglect rapidly os- replacing the field operators by classical amplitudes,− we re- cillating terms, by employing the dipole Hamiltonian in the cover the standard semiclassical formalism (Mukamel, 1995). rotating wave approximation, Similarly, we may also convert the field operators into classical random variables to describe spectroscopic signals with H = EV + E V, (9) int, RWA † † stochastic light (Asaka et al., 1984; Beach and Hartmann, where 1984; Morita and Yajima, 1984; Turner et al., 2013). 5

The material SNGF’s in Eq. (13) are similarly defined: propagating of both the bra and the ket, we then place the entire burden of the time evolution on the ket, and write (n) V (t tn . . . , t tn t1) νn...ν1 − − · · · − ( ) ( ) A(t) = ψ(t0) ψ˜(t) , (19) = V † (t ) (t )V † (15) νn G n ···G 1 ν1 h i h | i where where t i i 1 i ψ˜(t) = − exp dτH0 (τ) A(t) (t) = θ(t) exp H0t (16) int G − − | i T ~ t0 ~ ~ Z i t denotes the propagator of the free evolution of the matter sys- exp dτHint0 (τ) ψ(t0) , (20) × T −~ t0 | i tem, νj = L, R. Similarly SNGF may be obtained by replac- Z ing some V † by V and E by E†. This representation further 1 and − denotes the ati-time-ordering operator. Here the ket allows for reduced descriptions of open systems where bath firstT evolves forward and then backward in time, eventually re- degrees of freedom are eliminated. turning to the initial time (Keldysh, 1965; Schwinger, 1961). As an example, the set of fourth-order pathways for the Back propagation of the ket is equivalent to forward propa- population of state f in a three-level scheme shown in Fig. 8 gation of the bra. The resulting terms are represented by loop are given in Fig. 2b. We will refer to these pathways repeat- diagrams, as is commonly done in many-body theory (Hansen edly in the course of this review. This ladder diagram repre- and Pullerits, 2012; Mukamel, 2008; Rahav and Mukamel, sentation is most suitable for impulsive experiments involving 2010; Rammer, 2007). sequences of short, temporally well-separated pulses ranging As can be seen from Fig. 2, this representation yields a more from NMR to the X-ray regimes (Hamm and Zanni, 2011; compact description (fewer pathways) of the signals, since the Mukamel, 1995). In such multidimensional experiments, the relative time ordering of ket and bra interactions is not main- time variables used to represent the delays between succes- tained. In this example, the f-state population is given by the sive pulses (Abramavicius et al., 2009) , , , ... serve as t1 t2 t3 single loop diagram, which is the sum of the three ladder di- the primary control parameters. Spectra are displayed vs. the agrams (as well as their complex conjugates). It is harder to Fourier conjugates ˜ , ˜ , ˜ , ... of these time variables. Ω1 Ω2 Ω3 visualize short pulse experiments in this representation, and As indicated in Fig. 2b), each interaction with a field also due to the backward time propagation the elimination of bath imprints the phase φ onto the signal. Filtering the possi- degrees of freedom in an open system is not possible. Nev- ble phase combinations φ φ φ φ of the sig- ± 1 ± 2 ± 3 ± 4 ertheless, this representation proves most useful and compact nal, known as phase cycling, allows for the selective inves- for frequency domain techniques involving long pulses where et al. tigation of specific material properties (Abramavicius , time ordering is not maintained anyhow (Rahav and Mukamel, et al. et al. 2009; Keusters , 1999; Krcmˇ a´rˇ , 2013; Scheurer and 2010) and for many-body simulations that are usually carried et al. et al. Mukamel, 2001; Tan, 2008; Tian , 2003; Zhang , out in Hilbert space (Dalibard et al., 1992). 2012a,b). Acousto-optical modulation discussed in Fig. 15 The double sided (ladder) and the loop diagrams are two offers the possibility to achieve this selectivity even at the book keeping devices for field-matter interactions. The single-photon level. Phase cycling techniques have been suc- loop diagrams suggest various wavefunction based simula- cessfully demonstrated as control tools for the selection of tion strategies for signals (Dorfman et al., 2013a). The first fixed-phase components of optical signals generated by mul- is based on the numerical propagation of the wavefunction, tiwave mixing (Keusters et al., 1999; Tan, 2008; Tian et al., which includes all relevant electronic and nuclear (including 2003; Zhang et al., 2012a,b). Phase cycling can be easily im- bath) degrees of freedom explicitly. A second protocol uses a plemented by varying the relative inter-pulse phases using a Sum Over States (SOS) expansion of the signals. In the third, pulse shaper, which is cycled over 2π radians in a number of semiclassical approach a small subsystem is treated quantum equally spaced steps (Keusters et al., 1999; Tian et al., 2003). mechanically and is coupled to a classical bath, which causes Alternatively, rather than propagating the density matrix a time dependent modulation of the system Hamiltonian. The Eq. (10), we can follow the evolution of the wave function third approach for a wave function is equivalent to the stochas- by expanding tic Liouville equation for the density matrix (Tanimura, 2006), which is based on ladder diagram book keeping. A(t) ψ(t) A(t) ψ(t) , (17) WF ≡ h | | i with II. QUANTUM LIGHT i t ψ(t) = exp dτHint0 (τ) ψ(t0) . (18) In this section, we first discuss classical light which is the | i −~ t0 | i Z basis for the semiclassical approximation of nonlinear spec- Keeping track of the wave function results in different path- troscopy. We then briefly survey the main concepts from ways which can be represented by loop diagrams. Rather than Glauber’s photon counting theory for a single mode of the 6

FIG. 2 Diagrammatic construction of the double excitation probability %ff in a three level model a). b) The single loop diagram representing the evolution of the wavefunction. On the left hand side, the vertical line represents ket and the right hand it corresponds to the bra. The wave function in this closed time path loop diagram propagates forward in a loop from the bottom branch of the diagram along the ket branch to the top of the diagram. It then evolves backward in time from the top to the bottom of the right branch (bra). This forward and backward propagation is similar to the Keldysh contour diagram rules (Keldysh, 1965). Horizontal arrows represent field-matter interactions. c) The corresponding three ladder diagrams for the evolution of the density matrix. Here the density matrix evolves forward in time upward from the bottom to the top of the diagram. In order to achieve doubly excited state population starting from the ground state the density matrix has to undergo four interactions represented by the absorption of two photons represented by four inwardly directed arrows. The three diagrams represent different time orderings of ket vs bra interactions. These are lumped together in the single loop diagram b) electromagnetic field. Finally, we discuss in detail the multi- into our formalism through its multipoint correlation function mode entangled states of light, which will be repeatedly used (n) Eνn...ν1 (t tn . . . , t tn t1). These correlation func- in this review. tions can always− be rewritten− · · ·as − the sum of a normally ordered term and lower order normal correlation functions multiplied with commutator terms A. Classical vs. Quantum Light (n) E (t tn . . . , t tn t1) νn...ν1 − − · · · − Clearly, our world is governed by quantum mechanics, so = : Eνn (t tn) Eν1 (t tn t1): on a fundamental level light is always quantum. Yet, in many − ··· − · · · − + [Eνn (t tn),Eνn 1 (t tn tn 1)] situations a classical description of the light field may be suf- − − − − − (n) ficient (Boyd, 2003; Scully and Zubairy, 1997). For pedagog- Eνn 2...ν1 (t tn tn 1 tn 2 . . . , t tn t1) + × − − − − − − − · · · − ··· ical reasons, before we describe properties of quantum light, (24) we first discuss in some detail how the classical description of For the semiclassical limit to hold, we must be able to neglect the field emerges from quantum mechanics, and - more impor- all the terms containing field commutators. This is typically tantly - under which circumstances this approximation may the case when the intensity of the coherent state, i.e. its mean break down. photon number is very large. The coherent state of the field, i.e. the eigenstate of the pho- More generally, probability distributions of coherent ton annihilation operator at frequency ω, is generally consid- states are also classical (Mandel and Wolf, 1995): States, ered as “classical”. In this state the annihilation operator, and whose density matrix has a diagonal Glauber-Sudarshan P- hence the electric field operator - has a non-vanishing expecta- representation, are considered classical. This includes clas- tion value, a(ω) = 0. In general, we can write a multimode sical stochastic fields, which show non-factorizing field cor- coherent stateh as i 6 relation functions (for instance, for Gaussian statistics with

αω (t)a†(ω) α∗ (t)a(ω) E = 0 and E†E = 0). Such classical correlations can φcoh(t) = dω e − ω 0 , (21) h i h i 6 | i | i also be employed in spectroscopy (Turner et al., 2013), but Z are not the focus of the present review. where the mode amplitudes α are given by According to this strict criterion, any other quantum state iωt αω(t) = αωe− . (22) must therefore be considered nonclassical. However, that does not mean that such state will also show nonclassical features, In a normally ordered correlation function (all a† are to the and the elucidation of genuine quantum effects in a given right of all a), we may then simply replace the field operators experiment is often very involved; see e.g. the discussion in the correlation functions by by classical amplitudes (Man- around dispersion cancellation with entangled photons (Fran- del and Wolf, 1995), son, 1992; Steinberg et al., 1992b), or quantum imaging (Ben- nink et al., 2004; Valencia et al., 2005). ν iωt E (t) dω α e± . (23) The interest in quantum light for spectroscopy is twofold: ν → ω Z As stated in (Walmsley, 2015), “the critical features of quan- But a coherent state is not yet sufficient for the field to be tum light [...] are exceptionally low noise and strong corre- classical: As we have seen in Eq. (13), the light field enters lations”. In the following section, we first discuss the noise 7 properties of quantum light, i.e. quantum correlations within Thus, a squeezed vacuum state is also a minimum uncertainty an electromagnetic mode. The rest of the chapter is devoted to state. the strong time-frequency correlations, i.e. quantum correla- These fluctuations also show up in nonlinear signals, as we tions between different modes, which is the main focus of this will discuss in section III.A. In the remainder of this section, review. we are interested in multimode analogues of such squeezed states, and show how new correlations arise due to the multimode structure of the field. This is known as time-energy B. Single mode quantum states entanglement.

In confined geometries, the density of states of the electromagnetic field can be altered dramatically. In a cavity, it C. Photon Entanglement in multimode states is often sufficient to describe the field by a single field mode with a sharp frequency ω0, which may be strongly coupled Before presenting some specific models for quantum light, to dipoles inside the cavity. In this section, we shall be con- we first discuss time-energy entanglement in a more general cerned with this situation, where we can write the cavity field setting. An exhaustive review of continuous variable entan- Hamiltonian simply as glement theory can be found in a number of specialized reviews (Kok et al., 2007; Lvovsky and Raymer, 2009). Here, Hcav = ~ω0a†a, (25) we briefly introduce some important quantities of entangled photon pairs, in order to later distinguish genuine entangle- where a (a†) describes the photon annihilation (creation) opment from other bandwidth properties, which could also be erator of the cavity mode. We introduce the dimensionless encountered with classical light sources. cavity field quadratures We shall be concerned with pairs of distinguishable pho- (1) (2) 1 tons that live in separate Hilbert spaces field and field, re- X = a + a† , (26) H H √2 spectively. They may be distinguished, e.g., by their polariza- 1 tion, their frequencies, or their wavevectors and propagation P = a a† , (27) direction. A two-photon state may be generally expanded as √2i − which represent the real and imaginary part of the electric ψ = dω dω Φ(˜ ω , ω )a†(ω )a†(ω ) 0 0 , (33) | i a b a b 1 a 2 b | i1| i2 field, respectively. A whole family of quadratures X(θ) Z Z and P (θ) may be obtained by rotating the field operators as where Φ(˜ ωa, ωb) is the two-photon amplitude. iθ a a e− . → To analyze the properties of this state, it is convenient to Quantum mechanics dictates the Heisenberg uncertainty map it into onto a discrete basis. This can be achieved by 1 the singular value decomposition of the two-photon ampli- ∆X2 ∆P 2 (28) tude (Law and Eberly, 2004; Law et al., 2000; McKinstrie ≥ 4 and Karlsson, 2013; McKinstrie et al., 2013) for any quantum state. Coherent states form minimal uncertainty states, in which ∞ Φ(˜ ω , ω ) = r˜ ψ∗(ω )φ∗(ω ), (34) the lower bound in Eq. (28) applies a b k k a k b kX=1 2 2 1 where ψ and φ form orthonormal bases, known as the ∆X = ∆P = . (29) { k} { k} 2 Schmidt modes, and r˜k are the mode weights. These are ob- This coincides with the quantum fluctuations of the vacuum tained by solving the eigenvalue equations state. Any quantum state that features fluctuations in one of 2 its quadratures below this fundamental limit is therefore dω0κ1(ω, ω0)ψk(ω0) r˜kψk(ω), (35) 1/2 ≡ said to be squeezed. Note that due to the Heisenberg uncer- Z tainty (28) the conjugate quadrature must show larger fluctua- with tions. ˜ ˜ The simplest example thereof is the squeezed vacuum state κ1(ω, ω0) = dω00Φ∗(ω, ω00)Φ(ω0, ω00), (36) Z 1 2 1 2 ξa ξ∗a† ξ = e 2 − 2 0 , (30) and | i | i 2 where we have dω0κ (ω, ω0)φ (ω0) r˜ φ (ω), (37) 2 k ≡ k k Z 2 1 2 ξ ∆X = e− | |, (31) with 2 2 1 2 ξ ˜ ˜ ∆P = e | |. (32) κ2(ω, ω0) = dω00Φ∗(ω00, ω)Φ(ω00, ω0). (38) 2 Z

We then recast Eq. (33 using Eq. (34) the field modes are initially in the vacuum state, and only become populated through the PDC process (see also Eqs. (251) ∞ - (255)). ψ = r˜ dω dω ψ∗(ω )φ∗(ω )a†(ω ) 0 0 . | i k a b k a k b 2 b | i1| i2 In the PDC setup, a photon from a strong pump pulse is k=1 Z Z X (39) converted into an entangled photon pair by the interaction with the optical nonlinearity in the crystal (as will be discussed in The Schmidt decomposition (34) may be used to quantify detail in section IV). A birefringent crystal features two ordi- the degree of entanglement between photon pairs. Rewriting nary optical axes (o), and an extraordinary optical axis (e) in which the group velocity of optical light is different. The PDC the singular values as r˜k = √Bλk, where B denotes the process can be triggered in different geometries: One distin- amplification factor of the signal and λk the normalized set guishes type-I (e oo) and type-II (o eo) downconversion of singular values of the normalized two-photon state, with → → 2 (Shih, 2003), and more recently type-0 (Abolghasem et al., k λk = 1. A useful measure is provided by the entanglement entropy (Law et al., 2000) 2010; Lerch et al., 2013). The two photons show strong time P and frequency correlations stemming from the conservation E(ψ) = λ2 ln(λ2 ). (40) of energy and momentum: Since they are created simultane- − k k ously, the two photons are strongly correlated in their arrival k X time at the sample or detector, such that each individual pho- In quantum information applications, E(ψ) represents the ef- ton wavepacket has a very broad bandwidth. At the same time, fective dimensionality available to store information in the the sum of the two photon frequencies has to match the energy state. of the annihilated pump photon, which may be more sharply We call the state (33) separable, if the two-photon ampli- defined than the individual photons (pump photon bandwidth tude factorizes into the product of single-photon amplitudes is typically below 100 MHz in visible range). (1) (2) Φ(˜ ωa, ωb) = Φ˜ (ωa)Φ˜ (ωb). This implies that λ1 = 1, The created photon pairs are entangled in their frequency, and the entanglement entropy is E = 0. In this situation, position and momentum degree of freedom (Walborn et al., no correlations exist between the two photons: Measuring the 2010). The setup may be exploited to control the central fre- frequency of photon 1 will not alter the wavefunction of the quencies of the involved fields (Grice and Walmsley, 1997). other photon. Otherwise, the state is entangled. We then have For simplicity, we consider a collinear geometry of all fields. λi < 1, and correspondingly E > 0; measuring the frequency This greatly simplifies our notation, while retaining the time- of photon 1 now reduces photon 2 to a mixed state described frequency correlations, which are most relevant in spectro- by the density matrix, scopic applications. After passing through the nonlinear crystal, the state of the light field is given by (Christ et al., 2013, % ψ (ω(0)) 2 2011) 2 ∼ | k a | k X i (42) dω dω0 φ∗(ω )φ (ω0 )a†(ω ) 0 0 a (ω0 ), (41) ψout = exp HPDC 0 1 0 2 UPDC 0 1 0 2. × b b k b k b 2 b | ih | 2 b | i −~ | i | i ≡ | i | i Z Z (0) The propagator UPDC is given by the effective Hamiltonian the measurement outcome ωa thus influences the quantum state of photon 2. HPDC = dωa dωb Φ(ωa, ωb)a1†(ωa)a2†(ωb) + h.c. (43) Z Z D. Entangled photons generation by parametric down which creates or annihilates pairs of photons, whose joint conversion bandwidth properties are determined by the two-photon amplitude Entangled photon pairs are routinely created, manipulated, and detected in a variety of experimental scenarios. These in- ∆k(ω , ω )L Φ(ω , ω ) = αA (ω + ω ) sinc a b ei∆kL/2. clude decay of the doubly excited states in semiconductors a b p a b 2 (Edamatsu et al., 2004; Stevenson et al., 2007), four-wave (44) mixing in optical fibers (Garay-Palmett et al., 2007, 2008) or cold atomic gases (Balic´ et al., 2005; Cho et al., 2014a). Here, Ap is the normalized pump pulse envelope, Here, we focus on the oldest and most established method for sinc(∆kL/2) denotes the phase-matching function, which their production - parametric downconversion (PDC) in bire- originates from wavevector mismatch inside the nonlinear fringent crystals (Kwiat et al., 1995; Wu et al., 1986). crystal, ∆k(ωa, ωb) = kp(ωa +ωb) k1(ωa) k2(ωb). ki(ω) We will introduce the basic Hamiltonian which governs the denotes the wavevector of either the− pump or− beams 1 or 2 PDC generation process, and discuss the output fields it cre- at frequency ω. The prefactor α which is proportional to the ates. These will then be applied to calculate spectroscopic pump amplitude determines the strength of the PDC process signals. We restrict our attention to squeezed vacua, where (and hence the mean photon number). Hence, in contrast to 9

]P˜ hi in the enangled state (33), the two-photon amplitude Φ Eq. (44) may be approximated as is not normalized, but increases with the pump intensity. Typically, the wavevector mismatch ∆k(ω , ω ) depends Ap(ωa + ωb) δ(ωa + ωb ωp). (48) a b ' − very weakly on the frequencies ωa and ωb. It is therefore possible to expand it around the central frequencies ω1 and ω2 of Using Eqs. (46) and (48), the phase mismatch in Eq. (44) can the two downconverted beams. For type-I downconversion, be expressed as (Perinaˇ et al., 1998) the group velocities dk1/dω1 and dk2/dω2 are identical [un- ∆k(ω , ω )L less the PDC process is triggered in a strongly non-degenerate sinc a b ei∆kL/2 2 regime (Kalachev et al., 2008)], and the expansion around the i(ω1 ωa)T/2 central frequencies yields (Joobeur et al., 1994; Wasilewski =sinc ((ω1 ωa)T/2) e − , (49) et al., 2006) − where T T T is the entanglement time, which represents dk dk ≡ 2 − 1 ∆k(ω , ω )L/2 = p L/2 (ω + ω ω ) the maximal time delay between the arrival of the two entan- a b dω − dω a b − p p gled photons. For a cw pump laser, the first photon arrives at 2 1 d kp 2 a completely random time, but the second photon necessarily + 2 L/2 (ωa + ωb ωp) 2 dωp − arrives within the entanglement time. This property has been exploited in several proposals to probe ultrafast material pro- 1 d2k L/2 (ω ω )2 + (ω ω )2 cesses using a cw photon pair source (Raymer et al., 2013; − 2 dω2 a − 1 b − 2 h (45)i Roslyak and Mukamel, 2009a). As mentioned before, entangled photons affect optical sig- The first two terms of Eq. (45) create correlations between the nals via their multi-point correlation functions (Mukamel and two photon frequencies, while the third determines the band- Nagata, 2011; Roslyak et al., 2009b). The relevant field quan- width of the individual photons. tity in most applications discussed below - describing the in- In type-II downconversion, the group velocities of the two teraction of pairs of photons with a sample - is the four-point beams differ, and the wavevector mismatch may be approxi- correlation function. Using Eqs. (48) and (49), for the entan- mated to linear order (Keller and Rubin, 1997; Rubin et al., gled state (47) this correlation function can be factorized as 1994) E†(ωa0 )E†(ωb0 )E(ωb)E(ωa) ∆k(ωa, ωb)L/2 = (ωa ω1)T1/2 + (ωb ω2)T2/2, (46) − − = ψtwin E†(ω0 )E†(ω0 ) 0 0 E(ω )E(ω ) ψtwin , (50) h | a b | ih | b a | i where the two time scales T1 = L(dkp/dωp dk1/dω1) and − Here, T2 = L(dkp/dωp dk2/dω2) denote the maximal time delays the two photon− wavepackets can acquire with respect to 0 E(ω )E(ω ) ψtwin = δ(ω + ω ω ) the pump pulse during their propagation through the crystal. h | b a | i N a b − p i(ω1 ωa)T/2 Without loss of generality, we assume T2 > T1. In the follow- sinc ((ω ω )T/2) e − × 1 − a ing applications we adopt type-II phase matching (46). The i(ω1 ωb)T/2 +sinc ((ω1 ωb)T/2) e − . (51) corresponding wavefunctions may be controlled to maintain − frequency correlations, as we will elaborate in section II.G. is known as the two-photon wavefunction (Rubin et al., 1994). The following applications will focus on the weak down- Upon switching to the time domain, we obtain conversion regime, in which the output light fields are given by entangled photon pairs, i(ωat1+ωbt2) dωa dωb e− 0 E(ωb)E(ωa) ψtwin i h | | i ψtwin HPDC 0 1 0 2. (47) Z Z | i ≈ − | i | i i(ω1t1+ω2t2) t2 t1 ~ = 0e− rect − N T Eq. (47) now takes on the form of the entangled two-photon state, Eq. (33), discussed above, where Φ˜ denotes the nor- i(ω2t1+ω1t2) t1 t2 + 0e− rect − , (52) malized two-photon amplitude. We will review the quan- N T tum correlations of the created fields upon excitation by cw where rect(x)= 1 for 0 x 1 and zero otherwise, and pump, which simplifies the discussion, or finite-bandwith ≤ ≤ pump pulses. The two cases require different tools. , 0 denote the normalization of the two-photon wavefunction.N N The physical significance of the entanglement time is now clear: it sets an upper bound for the arrival of the sec- E. Narrowband pump ond photon, given the absorption of the first one. Note that Eq. (52) is symmetric with respect to t1 and t2 because each One important class of time-frequency entangled photon interaction occurs with the entire field E = E1 + E2. In a pairs is created by pumping the nonlinear crystal with a nar- situation where, e.g., 0 E2(t2)E1(t1) ψtwin is measured the row bandwidth laser, where the pump spectral envelope in two-photon wavefunctionh | is not symmetric.| i 10

a) b) c) 1 T ) 1 ! a ! (

(! ! )T b 2 2 FIG. 3 Absolute value of the two-photon correlation function (63) of entangled photon pairs with a) strong frequency anti-correlations, σp = 0.6 / T2 [entanglement entropy (40) E = 1.9], b) very weak correlations (E = 0.018), σp = 3.5 / T2, and c) strong positive frequency correlations, σp = 50 / T2 (E = 1.7).

F. Broadband pump formation operator UPDC now reads (Christ et al., 2013, 2011)

We next consider entangled light created by a pump pulse † (56) with a normalized Gaussian envelope, UPDC = exp rkA†Bk h.c. . − ! Xk 1 (ω + ω ω )2 a b p The output state ψ is thus a multimode squeezed state with Ap(ωa + ωb) = exp − 2 − . out 2 − 2σp | i 2πσp squeezing parameters rk, q (53) ∞ ∞ 1 nk ψout = (tanh(r )) n n , This is a realistic model in many experimental scenarios. | i k | ki1| ki2 cosh(rk) n =0 kY=1 Xk The analysis of Eq. (42) becomes most transparent by (57) switching to a basis obtained by the Schmidt decomposi- p tion (34) of the two-photon wavefunction. The mode weights in which fluctuations of the collective quadrature of the two r are positive and form a monotonically decreasing series, k states Yk = (Ak† + Bk† Ak Bk)/(2i) may be squeezed such that in practical applications the sum in Eq. (34) may be − − 2 below the coherent state value ( Yk = 1/2), terminated after a finite number of modes (For a cw pump h i defined above, an infinite number of Schmidt modes is re- 2 1 2rk Y = e− . (58) quired to represent the delta-function). In appendix B, we k 2 present approximate analytic expressions for the eigenfunc- The mean photon number in each of the two fields is given by tions ψ and φ . The following analysis is restricted to { k} { k} a PDC regime in which six- (and higher) photon processes 2 n = A† A = B†B = sinh (r ). (59) may be neglected. Such corrections can affect the bandwidth k k k k k k k k properties at very high photon numbers (Christ et al., 2013). X X X The linear to quadratic intensity crossover of signals will be We will be mostly concerned with the time-frequency corre- further discussed in section II.J.3. lations in the weak downconversion regime, i.e. n 1, when We next introduce the Schmidt mode operators the output state is dominated by temporally well≤ separated pairs of time-frequency entangled photons.

Ak = dωa ψk(ωa)a1(ωa), (54) The multi-point correlation functions of state (57), which Z are the relevant quantities in nonlinear spectroscopy, may be and most conveniently evaluated by switching to the Heisenberg picture, in which the Schmidt mode operators become (Christ et al., 2013, 2011) Bk = dωb φk(ωb)a2(ωb), (55) Z out in in Ak = cosh(rk)Ak + sinh(rk)B† , (60) which inherit the bosonic commutation relations from the or- k out in in thonormality of the eigenfunctions ψ and φ . The trans- Bk = cosh(rk)Bk + sinh(rk)Ak† . (61) { k} { k} 11

The four-point correlation function then reads (Schlawin and a) �� Mukamel, 2013) �� E†(ωa0 )E†(ωb0 )E(ωb)E(ωa) | ) �� = (h12∗ (ωa0 , ωb0 ) + h21∗ (ωa0 , ωb0 )) (h12(ωa, ωb) + h21(ωa, ωb)) ! , ! + (g1(ωa, ωa0 ) + g2(ωa, ωa0 )) (g1(ωb, ωb0 ) + g2(ωb, ωb0 )) (

12 �� f + (g1(ωa, ωb0 ) + g2(ωa, ωb0 )) (g1(ωb, ωa0 ) + g2(ωb, ωa0 )) , | (62) �� with -� � � � h12(ωa, ωb) = cosh(rk) sinh(rk)ψk(ωa)φk(ωb), (63) b) �� Xk 2 �� g (ω, ω0) = sinh (r )ψ (ω)ψ∗ (ω0), (64) 1 k k k0

k | X ) �� ! ,

and ! (

1 �� g 2 | g = sinh (r )φ (ω)φ∗ (ω0). (65) 2 k k k0 �� Xk The first line in Eq. (62) shows the same structure as Eq. (50), -� � � � in that the two absorption events at frequencies ωa and ωb (and at ω and ω ) are correlated. Indeed, in the weak Frequency ! T2 a0 b0 ⇥ pump regime, when r 1, Eq. (63) reduces to the k two-photon wavefunction of the pulsed entangled pairs, and FIG. 4 a) The two-photon correlation function h12(ω, ω), Eq. (63), sinh(r ) cosh(r ) r (Schlawin and Mukamel, 2013), plotted vs the frequency ω in units of T2, and for mean photon num- k k ' k bers (with increasing dashing) n¯ ∼ 0.1, 1, 10, and 100. b) the same for the autocorrelation function g1(ω, ω), Eq. (64). h12(ωa, ωb) rk 1 rkψk(ωa)φk(ωb) ⇒ Xk = Φ∗(ω , ω ) = 0 E (ω )E (ω ) ψtwin . a b h | 2 b 1 a | i amplitude (and photon number) the few largest eigenvalues (66) get enhanced nonlinearly compared to the smaller values, and fewer Schmidt modes contribute to Eqs. (63) and (64). This is h12 denotes the two-photon contribution to the correlation function , which should be distinguished from the autocor- shown in Fig. 4 where the two correlation functions are plotted for different mean photon numbers, n¯ = 0.1, 100. As relation contributions g1 and g2. ··· 1 the number of participating Schmidt modes is decreased, the The ratio of the inverse entanglement time T − defined frequency correlations encoded in h12 are weakened, and h12 above and the pump bandwidth σp determines the frequency broadens. Conversely, the bandwidth of the beams g1 is re- correlations in Eq. (63): As shown in Fig. 3a), for σp 1 duced with increasing n¯. T2− we recover the cw regime with strong frequency anti- 1 correlations. In the opposite regime, when σp T2− [panel c)], the two photons show positive frequency correlations, and in between, there is a regime shown in panel b), in which the G. Shaping of entangled photons photonic wavefunction factorizes, and no frequency correlations exist. The field correlation function as depicted in Fig. 3 The ability to manipulate amplitude and phase of the ultra- can be measured experimentally (Kim and Grice, 2005). short pulses allows to coherently control matter information In spectroscopic applications, two-photon events involving in chemical reactions and other dynamical processes (Wol- uncorrelated photons from different pairs become more likely lenhaupt et al., 2007). Pulse shaping allows to drive a quan- at higher pump intensities. These events are described by the tum system from an initial state to a desired final state by ex- functions g and g , which at low intensities scale as r2, ploiting constructive quantum-mechanical interferences that 1 2 ∼ k so that Eq. (63) with h12 rk dominates the signal. As the build up the state amplitude, while avoiding undesirable fi- pump intensity is increased,∼ events involving photons from nal states through destructive interferences (Silberberg, 2009). different pairs must be taken into account as well. The most common experimental pulse shaping configuration The various contributions to the correlation function behave is based on spatial dispersion and is often based on back-to- differently with increasing photon number: They depend non- back optical grating spectrometer which contains two grat- linearly on the mode weights rk which in turn depend lin- ings (see the top part of Fig. 5a). The first grating disperses early on the pump amplitude. Thus, with increasing pump the spectral components of the pulse in space, and the sec- 12 ond grating packs them back together, while a pixelated spatial light modulator (SLM) applies a specific transfer function a) (amplitude, phase, or polarization mask), thereby modifying the amplitudes, phases, or polarization states of the various spectral components. Originally developed for strong laser beams, these pulse shaping techniques have been later extended to single photon regime (Bellini et al., 2003; Carrasco et al., 2006; Defienne et al., 2015; Pe’er et al., 2005; Zah¨ et al., 2008) allowing to control the amplitude and phase modulation of entangled photon pairs, thereby providing additional spectroscopic knobs. An example of a pulse shaping setup is shown in Fig. 5a). b) c) A symmetric phase profile in SLM yields a single unshaped Gaussian pulse [Fig. 5b) and c)]. The phase step results in a shaped pulse that mimics two pulses with 400 fs delay (Fig. 5d,e). By employing more complex functions∼ in the SLM, one may produce a replica of multiple well separated pulses or more complex shapes. This may be also achieved using e.g. the Franson interferometer with variable phases and delays in both arms of the interferometer as proposed in (Raymer et al., 2013) (see section III.B.4). Beam splitters in the two arms can create four pulses, when starting with a single entangled photon pair.

H. Polarization entanglement FIG. 5 (Color online) a) Experimental layout of the entangled pho- So far we have ignored the photon polarization degrees of ton shaper: A computer-controlled spatial light modulator (SLM) freedom. In type-II PDC the two photons are created with is used to manipulate the spectral phase of the entangled photons. orthogonal polarizations. Using a suitable setup, this allows The photon pairs are detected in the inverse process of PDC - sum frequency generation (SFG). The SFG photons are subsequently for the preparation of Bell states of the form (Pan et al., 2012) counted in a single-photon counting module. In order to demonstrate 1 two-photon interference oscillations, a Mach-Zehnder interferome- Bell = ( H 1 V 2 V 1 H 2) , (67) ter is placed between the last prism and the SFG crystal. b) The SFG | i √2 | i | i ± | i | i counts (circles) and the calculated second-order correlation function where ( H , V ) denote the horizontal (vertical) polarization, (line) of the unperturbed wavefunction as a function of the signal- respectively.| i | Thei fidelity for this state preparation is maximal, idler delay. c) the same for the shaped wavefunction. Figure is taken when the two photon wavepackets factorize. Similarly, type-I from (Pe’er et al., 2005). PDC allows for the creation of states of the form ( HH VV )/√2 (Pan et al., 2012). | i ± | Thei polarization degrees of freedom offer additional control knobs which may be used to suppress or enhance the signal from (anti-)parallel or orthogonal dipoles in a sample system - in a quantum mechanical extension of polarized photon echo techniques (Voronine et al., 2006, 2007). nonlinear response is non additive and does not scale as number of atoms N. Such cooperative (nonadditive) response is I. Matter correlations in noninteracting two-level atoms not possible with classical or coherent light fields. Arguments induced by quantum light were made that the cooperatively is induced in two-photon absorption by the manipulation of the interference among path- The entangled photon correlation functions, may be used ways. One consequence of the interference in nonlinear re- to prepare desired distributions of excited states in matter. In sponse is that the fluorescence from one atom can be enhanced an insightful article (Muthukrishnan et al., 2004), which trig- by the presence of a second atom, even if they do not inter- gered other work (Akiba et al., 2006; Das and Agarwal, 2008), act. If true this effect could be an interesting demonstration of it was argued that using time-ordered entangled photon pairs, quantum nonlocality and the Einstein-Podolsky-Rosen (EPR) two-body two-photon resonances, where two noninteracting paradox. In this section, we shall calculate this two photon particles are excited simultaneously, can be observed in two- process with quantum light and show that quantum locality is photon absorption. The surprising consequence is that the not violated. 13

factorizes and atoms A and B remain uncorrelated at all times: ab ρ(t) = ρ (t) ρ (t), (70) A ⊗ B a with b t ! i A a ρA(t) = exp Hint (τ)dτ ρA,0, (71) !b T − − ~ Zt0

g t A B i A ρB(t) = exp Hint (τ)dτ ρB,0. (72) T −~ t − a) Z 0 This result remains valid for quantum fields as long as all rel- a a ab ab evant normally ordered field modes are in a coherent state, and cooperative spontaneous emission is neglected so that all ab field modes behave classically (Glauber, 1963; Marx et al., ↵ 2008). Eq. (70) does not hold for a general quantum state. ↵ We define the reduced matter density matrix in the joint space w = trph(ρ). Upon expanding Eq. (69) order by order in the field operators and tracing over the field modes, we obtain for i ii the matter density matrix b) t t t t

w(t) = dτ1... dτnν dτ10 ... dτm0 ν t0 t0 t0 t0 FIG. 6 (Color online) a) Scheme of two noninteracting two-level ν Z Z Z Z Xν ν atoms A and B and corresponding many-body state diagram where ρA(τ1, ...τnν )ρB(τ10 , ...τm0 ν )Fν (τ1, ...τnν , τ10 , ..., τm0 ν ), ground state g corresponds to both atom in the ground state, a - atom × (73) A is excited and B is in the ground state, b - atom B is in the excited state and atom A is in the ground state, and ab - both atoms are where ν is summed over all possible pathways. Pathway ν ˜ A ˜ B ν ν in the excited state. Arrow directed upward represent two-photon has nν V interactions and mν V interactions. ρA (ρB) excitation. b) Relevant set of diagrams corresponding to atom A are time-ordered products of system A (system B) operators 2 2 being in excited state to ∼ |µA| |µB | in field-matter interactions. and F (τ , ...τ , τ 0 , ..., τ 0 ) are time-ordered field correc- α, β run over a and b to account for all possible permutations in the ν 1 nν 1 mν tion functions. In each order of this perturbative order-by- excitation pathways. order calculation, all the correlation functions are factorized between the three spaces. The factorization Eq. (70) no longer 1. Collective resonances induced by entangled light holds in general, and atoms A and B may become correlated or even entangled. Eq. (73) will be used in the following. We use superoperator formalism (Marx et al., 2008) to in- Note that pathways with nν = 0 or mν = 0 are single-body vestigate how the two-atom excitation cross section depends pathways, where all interactions occur either with system A on the properties of the photon wave function. or with B. Our interest is in the two-body pathways, where Consider two noninteracting two-level atoms A and B cou- both nν and mν contribute to collective response. pled to the radiation field (see Fig. 6)a. We assume that the total field-matter density matrix is initially in a factorizable form: 2. Excited state populations generated by nonclassical light So far, we have not discussed how the overall excitation ρ(t0) = ρA,0 ρB,0 ρph,0, (68) ⊗ ⊗ probability by entangled light sources compares to classi- where the ρA,0 (ρB,0) corresponds to the density matrix of the cal light with similar photon flux. This is relevant to recent atom A (B) and ρph,0 is the density matrix of the field. The demonstration of molecular internal conversion (Oka, 2012) time-dependent density matrix is given by Eq. (11), which in and two-photon absorption with the assistance of plasmonics the present case reads (Richter and Mukamel, 2011) (Oka, 2015). This is a very important practical point, if quantum spectroscopy is to be carried out at these very low photon ρ(t) = fluxes, in the presence of additional noise sources. We now t t i A i B discuss the correlations on the excitation probability induced exp Hint (τ)dτ Hint (τ)dτ ρ(t0). T − − − − by the interaction with nonclassical light for the present of two ~ Zt0 ~ Zt0 (69) noninteracting two-level systems. The doubly excited state population is given - to leading- If the radiation field is classical then the matter density matrix order perturbation theory in the interaction Hamiltonian (9) - 14 by the loop diagram in Fig. 2a) which may be written as a Time t modulus square of the corresponding transition amplitude ↵, | i !↵ p (t) = T (t) 2, (74) ab | ab;ψ0 | ψ | i X0 ! with the transition amplitude between initial state ψ and final | i state ψ0 , | i t t iat1 ibt2 T (t) = dt dt µ µ e− − ψ E(t )E(t ) ψ0 . ab 1 2 A B h | 2 1 | i Zt0 Zt0 (75) FIG. 7 (Color online) The temporal profiles of two photons emitted by a cascade source illustrate time-frequency entanglement: the red In the following, we only consider a unique final state and will curve represents the marginal probability P (τα) Eq. (85), and the drop the subscript ψ0. blue curve corresponds to the conditional probability P (τβ |τα) Eq. (84). The intrinsic time ordering of the photons, α first, followed by β, suppress the excitation pathway where β is absorbed first, fol- 3. Classical vs entangled light lowed by α, inducing joint two-atom excitation.

We first evaluate Eq. (74) for a classical field composed of Here is the lifetime of the upper level of the three-level two modes α and β , which is switched on at t = t0: γα cascade and γβ is the lifetime of the intermediate state. p and E˜(t) = θ(t t ) E eiωαt + E eiωβ t + c.c. (76) q are the wave vectors of different modes in vacuum and g − 0 α β pα and gqβ are coupling constants. ωα is the transition frequency (c) Tab (t) is then given by (Muthukrishnan et al., 2004) from the highest to the intermediate state and ωβ is the transition frequency from the intermediate state to the ground state. (c) (c) (c) Tab (t) = Ta (t)Tb (t), (77) It is important that photon with momentum p comes first and interacts with upper transition whereas photon with q comes where later and interacts with lower β - transition. The two-photon frequency ωp+ωq is narrowly distributed around ωα+ωβ with (c) iAµm(t) Tm (t) = , m = a, b (78) a width γα, the lifetime of the upper level, whereas the single- m ωµ iγ µX=α,β − − photon frequencies ωp, ωq are distributed around ωβ(ωα) with a width of γβ(γα), the lifetime of the intermediate (highest) i(ων m)t0 γt0 i(ων m)t γt Aνm = µmEν e − − e − − and γ − → level. Maximum entanglement occurs for γβ γα. Using 0. A straightforward calculation of Eq. (75) shows sev- Eq. (79), and assuming that the atoms A and B have the same eral terms, where each of the pathway contains single-particle distance from the cascade source, so that tR is the time retar- resonances a,b = ωα,β as well as a collective resonance dation (with tR = rR /c) we obtain (Zheng et al., 2013) a + b = ωα + ωβ. While combining these terms the collec- | | tive resonance disappear due to destructive interference and ent Amα,nβ(t) the final result factorizes as a product of two amplitudes that Tab (t) = (m ωα iγα)(n ωβ iγβ) m=n=a,b contain only single-particle resonances j = ωµ, j = a, b and 6 X − − − − A (t) µ = α, β and no two-photon resonances a + b = ωα + ωβ + TPA , (see (Richter and Mukamel, 2011)). (a + b ωα ωβ iγα iγβ)(m ωβ iγβ) mX=a,b − − − − − − We first describe the properties of the entangled source be- (80) fore discussing the spectroscopy applications. Consider a field made of entangled photon pairs of a cascade state ψent de- where picted in Fig. 7 which was considered by (Muthukrishnan| i iωnν (t tR) γν (t tR) imtR intR et al., 2004). This state can be prepared when an atom is pro- Anν,mµ(t) = µAµBA(e − − − e− − ), moted to the doubly excited state which consequently decays − (81) spontaneously back to the ground state by emitting a cascade of two photons and described by the wave function: iatR ibtR ATPA(t) = µAµBA e− − ψ = φ 1 , 1 ent p,q p q i(a+b)t+i(ωα ωβ γα γβ )(t tR) | i | i e− − − − − . (82) p,q X − i(p+q) rR gpαgqβe · φp,q = . The first term in Eq. (80) contains single-particle reso- (ωp + ωq ωα ωβ + iγα)(ωq ωβ + iγβ) nances, where the two systems are separately excited. The − − − (79) second term represents collective two-photon resonances a + 15

b ωα ωβ which can only be distinguished for non identical pa0 whereas diagram ii corresponds to pab. The exact for- atoms− −= . mulas read off these diagrams are presented in (Richter and a 6 b Under two photon - two atom resonance condition ωa + Mukamel, 2011). Unlike the pa1(t) which contains only nor- ent ωb = ωα + ωβ for γαt 1, γβt 1 the probability Pab (t) mally ordered ﬁeld operators [see Eq. (74)], pa0(t) contains reads also non normally ordered contributions. These can be recast γ as a normally ordered correlation plus a term that includes a ent β (83) pab = p0 2 , commutator: γα∆

E†(t )E(t )E†(t )E(t ) = E†(t )E†(t )E(t )E(t ) where ∆ = b ωβ is a single atom detuning and p0 = h 2 3 4 1 i h 2 4 3 1 i 2 2 −2 2 2 2 µA µB ab/~ 0c S under normalization of gαgβ = + E†(t )[E(t ),E†(t )]E(t ) . (87) | | | | h 2 3 4 1 i 2c√γαγβ/L. Note, that the classical amplitude (77) scales quadratically The second term involves a commutator of the field which is a with the field amplitude Aνm, whereas the entangled ampli- c-number. For certain type of states of the field, the commuta- tude (80) scales linearly. This reflects the fact that at low in- tor remains dependent upon the state of the field and this term tensity (entangled case) two photon absorption is effectively has to be evaluated exactly (for instance, in the case of coher- a linear absorption of the entangled pair as discussed earlier. ent state this effect is responsible for the revival of damped More detailed analysis of the scaling is presented in Section Rabi oscillations discussed below (Rempe et al., 1987). For II.J.3. other types of states of the field, the commutator becomes in- Comparing Eqs. (77) and (80), we see that two-photon res- dependent of the external field (e.g. Fock state) and therefore onances are induced by the lack of time ordering in the pho- represents spontaneous emission. The spontaneous emission tonic field. To explain this, we calculate the marginal proba- pathways introduce a coupling between the two systems, since bility a photon emitted by system B can be absorbed by system A. This coupling has both real (dipole-dipole) and imaginary (su- 2γατα P (τα) = 2γαθ(τα)e− , (84) perradiance) parts. These couplings will obviously result in collective signals which involve several atoms. and the conditional probability In the case of a classical field, one can neglect the spontaneous contributions and only include the stimulated ones 2γβ (τβ τα) P (τβ τα) = 2γβθ(τβ τα)e− − (85) | − which results in pa0(t) = pab(t). The population pa(t) therefore vanishes, and is not− affected by the collective res- for the two photon absorption process displayed in Fig. 7 onances. We thus do not expect any enhanced fluorescence (Muthukrishnan et al., 2004). These probabilities have the from A. The two-photon absorption signal is a product of indi- following meaning: the absorption of α is turned on at τ = 0 α vidual excitation probabilities for atoms A and B, and shows and decays slowly at the rate γ , while the absorption of β α no collective resonances. turns on at τβ = τα and decays rapidly at the rate γβ. Thus, the two photons arrive in strict succession, α followed by β, with the time interval between the absorption going to zero 4. Collective two-body resonances generated by illumination when γβ γα, i.e. in the limit of large frequency entan- with entangled light glement. Another related effect, which is discussed later in this section and which eliminates one-particle observables, is We shall focus on the change of the photon number for the based on the lack of time ordering of the absorption of the two two-body part ∆nph. This will depend on the following prob- systems, while this interference effect originally described in abilities: the excitation probability of pab,which means that (Muthukrishnan et al., 2004) is based on a lack of time order- two photons are absorbed (counts twice), and the excitation ing of the two photons. Only the single-body single-photon probabilities of only system A pa and of only system B pb resonances remain. (we assume that both photons have equal frequency and are We now turn to the spectroscopy carried out using the resonant to the two-photon absorption): model of entangled light given by Eq. (79). To that end we turn to the excited state population of atom A, which is given ∆nph = 2pab(t) pa0(t) pb0(t). (88) − − − by Now, in the stimulated emission and TPA pathway absorption, p (t) tr[ a a ρ(t)] = p (t) + p (t), (86) we know that pab(t) and pa0(t) cancel, and that pab(t) and a ≡ | ih | a0 ab pb0(t) cancel. So for the stimulated pathways, Eq. (88) van- where the two terms correspond to the final state of atom B: ishes. Since the field-matter interaction Hamiltonian connects pa0 and pab represent B being in the ground and excited state, the photon number with the excitation probability number, the 2 2 respectively. We shall only calculate the µAµB contribu- photon number itself is a single-particle observable like the tions to p , which are relevant to our discussion.∼ The relevant population of state a or state b , and therefore vanishes. a | i | i set of diagrams is depicted in Fig. 6b. Diagram i contains However, collective resonances in pab(t) can be revealed in three interactions with ket- and one with bra- and represents two-photon counting (Hanbury-Brown-Twiss measurements) 16

(Brown, 1956). For our entangled photon state, Eq. (79), the The state (92) corresponds to an atomic cascade for which the change in the photon-photon correlation ∆0nph is attributed starting time is random, thereby averaging to zero all the off- to any buildup of probability, that the either atom A, or atom diagonal time-dependent terms in the density matrix. It gives B is excited or both atoms are excited, which will cause a rise to the following transition probability: reduction of the photon-photon correlation: γ γ 1 1 t2 p p α β + . ∆0nph = pab(t) pa0(t) pb0(t). (89) 1 ' 0 δ2 + γ2 (ω ω )2 (ω ω )2 (L/c)2 − − − α 1 − β 2 − β Since pab(t) enters twice [unlike Eq. (88)] the stimulated con- (93) tributions can only cancel with one of the two other contribu- At exact two-photon two-atom resonance, we have p1 tions pa0(t) or pb0(t), and we have 1 2 2 2 ' p0γβγα− c t /(∆L) . Note that P1 depends on time, as can ∆0nph = pab(t). (90) be expected in a situation where the detecting atoms, which The interference mechanism, which caused the cancellation have an infinite lifetime, are submitted to a stationary quan- for the stimulated signal and the photon number, does not tum state, and therefore to cw light. In order to compare p1 lead to a full cancellation, and two-photon absorption involv- to corresponding probability (83) for entangled light, which is ing both systems might be observed. induced by a pulse of light, we need to fix an interaction time t. One then obtains at t = L/c and at exact resonance γ β ent (94) a. Two-photon two-atom problem with non entangled quantum p1 p0 2 pab . ' γα∆ ' states We now discuss whether entangled light is essential to the collective resonances. Whether or not photon entan- We thus find that a correlated-separable state like ρ1 can in- glement is essential for these properties is the subject of cur- duce the two-photon two-atom transition similar to the entan- rent debate. In (Zheng et al., 2013), the authors presented gled cascade state. Note that even though ρ1 is not entangled, a detailed analysis of the cross section created by entangled it has quantum properties, being a mixture of single-photon pulses, and compared this with “correlated-separable” states states which are highly nonclassical. in which the entanglement is replaced by classical frequency In summary, the two-photon absorption probability of non- correlations. They concluded that it is the frequency anti- interacting atoms can be enhanced compared to classical light correlations [see Fig. 4a)], and not the entanglement per se, by using special types of quantum light. This quantum light which is responsible for the enhancement of the cross sec- does not require entanglement, rather it is necessary to have tion. The effect of enhanced two-photon absorption proba- spectral anticorrelations which can be achieved in e.g. corre- bility has been later shown to come from the entangled mat- lated separable states. This is a consequence of the stationary ter/field evolution that occurs with any quantum light (which system of two noninteracting atoms which has no dynamical is not necessarily entangled) (Zheng et al., 2013) which is properties. consistent with earlier demonstrations of (Georgiades et al., In the following section we consider more complex ma- 1997). Starting from entangled pure quantum state hav- terial system that is subject to various relaxation channels, e.g. transport between excited states therefore adding tem- ing a density matrix ρ0 = ψent ψent of matrix elements | ih | poral scale to the problem. We show that in this case, entan- ρkk0qq0 = 1k, 1q ρ0 1k0 1q0 one can construct others that have the sameh mean| energy| andi the same single-photon spec- glement is required in order to achieve both high spectral and trum, and hence that would give the same transition probabil- high temporal correlations which is not possible by classical ities for a single-photon resonance. We choose a special case correlated light. of the states that originates from entangled state (79) that will allow a quantitative evaluation of the role of correlations. It is J. Quantum light induced correlations between two-level defined as systems with dipole-dipole coupling

ρ1 = ρkkqq 1k, 1q 1k, 1q , (91) | ih | 1. Model system Xk,q It is the diagonal part of ρ0. It has lost any temporal ﬁeld The standard calculation of the nonlinear response to clas- coherence and is time-independent. It is actually a correlated sical light assumes that the matter is made up of N noninter- separable state (Duan et al., 2000), in which the quantum cor- acting particles (atoms or molecules) in the active zone, such relations are replaced by a purely classical frequency distri- that the matter Hamiltonian may be written as the sum over bution. It gives rise, however, to correlations between its two the individual particles [see Eq. (7)] parties. Using the entangled state (79), Eq. (91) reads

2 H0 = Hν . (95) 2c γβ γα ρ1 = 2 2 2 2 ν L (ω + γ ) [(ωqβ + ωkα) + γ ] X k,q qβ β α X The individual nonlinear susceptibilities or response functions 1 , 1 1 , 1 . (92) × | k qih k q| of these atoms then add up to give the total response. The 17

f | 2i f 1 % | i | ff0 | e | 2i e | 1i f f 0 | i h |

g % | i | ff0 | FIG. 8 Level scheme of the multilevel model employed in this review: Two singly excited states e1 and e2 with energies 11, 000 cm−1 and 11, 500 cm−1 are coupled to doubly excited states f1 and f2, as indicated. Furthermore, e2 decays to e1 within 1/k ' 30 fs.

f 0 f h | nonlinear response becomes then a single-body problem and | i no cooperative resonances are expected. It is not obvious how FIG. 9 a) The density matrix for doubly excited states % (t) = to rationalize the N scaling for noninteracting atoms had ff 0 ∗ ∼ T (t)Tfg(t), Eq. (100), prepared by the absorption of entangled we chosen to perform the calculation in the many-body space. f 0g −1 Massive cancellations of most N(N 1) scaling light- photons with pump frequency ωp = 22, 160 cm . b) the same, with ω = 24, 200 cm−1. [from (Schlawin et al., 2012)] matter pathways recover in the end∼ the final− N signal scal- p ing (Spano and Mukamel, 1989). ∼ When the atoms are coupled, the calculation must be car- four-wave mixing processes, we have truncated the Hamilto- ried out in their direct-product many-body space whose size nian in Eq. (96) at the doubly excited level, such that it N Hν grows exponentially with N ( n dimensions for n-level diagonalization reads atoms). The interatomic coupling∼ can be induced by the ex- change of virtual photons leading to dipole-dipole and coop- Hν = ~ωgν gν gν + ~ ωeν eν eν + ~ ωfν fν fν , | ih | | ih | | ih | erative spontaneous emission, or superradiance (Das et al., e X Xf 2008). In molecular aggregates, the dipole interaction be- (98) tween its constituents creates inherent entanglement on the level of the quasiparticles (Mukamel, 2010), which shifts the where g indexes the ground state, e the singly, and f the dou- doubly excited state energies, and redistributes the dipole mo- bly excited states, which will be the central concern. We will ments, in which the individual Hamiltonians Hν are (Abra- consider processes in which the signals factorize into ones mavicius et al., 2009) stemming from individual constituents, as well as collective signals from two or more constituents. We illustrate basic Hν = ~ εmBmν† Bmν + ~ JmnBmν† Bnν properties of the interaction of entangled photons with com- i m=n plex quantum systems at the hand of the simple multilevel X X6 ∆m model depicted in Fig. 8. It consists of two excited states e1 + ~ B† B† Bmν Bmν , (96) 2 mν mν and e2, a two doubly excited states f1 and f2. The higher- m X energy excited state e2 decays to e1 within few tens of fem- toseconds. We shall focus on doubly-excited states created by where Bm (Bm† ) describes an excitation annihilation (creation) operators for chromophore m. These excitations are the absorption of pairs of photons. hard-core bosons with Pauli commutation rules (Lee et al., Similarly to the noninteracting case (75), the population in 1957): the final state f at time t is given by the loop diagram in Fig. 2, which can be written as the modulus square of a transition amplitude. Tracing out the matter degrees of freedom, the [Bm,Bn† ] = δmn(1 2Bn† Bn). (97) − amplitude can be written as a nonlinear field operator, To describe two level sites, which cannot be doubly excited 1 t τ2 according to the Pauli exclusion we set ∆m . In con- Tfg(t; Γ) = dτ2 dτ1E(τ2)E(τ1) densed matter physics and molecular aggregates→ the ∞ many par- − 2 ~ Zt0 Zt0 ticle delocalized states are called excitons. The model Hamil- f(t) V †(τ2)V †(τ1) g(t0) , (99) tonian (96) can represent e.g. Rydberg atoms in optical lattice × h | i| i or Frenkel excitons in molecular aggregates. For considering which may be evaluated for arbitrary field input states. Specif- 18 ically, using Eq. (52), we obtain form (exp(iωT ) 1)/ω, which, for very short entanglement times is independent− of the frequency: (exp(iωT ) 1)/ω 2 − ' (ent) 1 µgeµef 00Ap i(ω1+ω2)t T + (T ). This implies that the intermediate e-states ef- Tfg (t; Γ) = 2 N e− O ~ T ω1 + ω2 ωf + iγf fectively interact with broadband light, whereas the f-states − i(ω1 ωe+iγe)T i(ω2 ωe+iγe)T interact with cw light. Put differently, thanks to their large e − 1 e − 1 − + − , bandwidth, the entangled photons can induce all possible ex- × ω ω + iγ ω ω + iγ 1 − e e 2 − e e citation pathways through the e-manifold leading to a specific (100) selected f-state (Schlawin et al., 2012). Tuning the pump frequency ωp allows to select the excitation of specific wave where we used = 00Ap which is linear with respect to classical pump amplitude.N N For comparison, the classical prob- packets: Fig. 9 depicts the density matrices for doubly ex- ability governed by excitation by two classical fields with am- cited states induced by the absorption of entangled photon pairs with different pump frequencies. This behavior remains plitudes A1 and A2 is given by the same for pulsed excitation in the strong frequency anti-

i(ω1+ω2)t correlations regime (Schlawin, 2015). (c) 1 µgeµef A1A2e− Tfg (t; Γ) = 2 ~ [ω1 ωeg + iγeg][ω2 ωfe + iγfe] − − + ω ω , (101) 2. Control of energy transfer 1 ↔ 2 which is quadratic in the field amplitudes, unlike Eq. (100). When the material system is coupled to an environment The second line denotes the same quantity, but with the beam which causes relaxation among levels, the distributions of ex- frequencies ω1 and ω2 interchanged. We will further discuss cited states populations may no longer be described by the more on the intensity scaling. wavefunction and transition amplitudes, and the density ma- Eq. (100) reflects the entangled photon structure described trix must be used instead. As shown in Fig. 2, the loop di- earlier: The pump frequency ωp = ω1 + ω2 is sharply defined agram, which represent the transition amplitudes needs to be such that the ωf -resonance is broadened only by the state’s broken up into three fully time-ordered ladder diagrams (and lifetime broadening and pure dephasing rate γf . The strong their complex conjugates). These are given in the time domain time correlations in the arrival time creates a resonance of the (Schlawin et al., 2013),

i 4 t τ4 τ3 τ2 pf,(I)(t; Γ) = dτ4 dτ3 dτ2 dτ1 −~ Z−∞ Z−∞ Z−∞ Z−∞ V (τ )V (τ )V †(τ )V †(τ ) E†(τ )E†(τ )E(τ )E(τ ) , (102) × R 4 R 3 L 2 L 1 3 4 2 1 4 t τ τ τ i 4 3 2 pf,(II)(t; Γ) = dτ4 dτ3 dτ2 dτ1 −~ Z−∞ Z−∞ Z−∞ Z−∞ V (τ )V †(τ )V (τ )V †(τ ) E†(τ )E†(τ )E(τ )E(τ ) , (103) × R 4 L 3 R 2 L 1 2 4 2 1 4 t τ τ τ i 4 3 2 pf,(III)(t; Γ) = dτ4 dτ3 dτ2 dτ1 −~ Z−∞ Z−∞ Z−∞ Z−∞ V †(τ )V (τ )V (τ )V †(τ ) E†(τ )E†(τ )E(τ )E(τ ) , (104) × L 4 R 3 R 2 L 1 2 3 4 1

Here, Γ collectively denotes the set of control parameters of the light ﬁeld. The matter correlation functions in Eqs. (102) - (104) are given by Liouville space superoperator correlation functions, which translate for the integration variables in V (τ )V (τ )V †(τ )V †(τ ) Eqs. (102)-(104) in Hilbert space into R 4 R 3 L 2 L 1 = V (τ3)V (τ4)V †(τ2)V †(τ1) , (105)

VR(τ4)VL†(τ3)VR(τ2)VL†(τ 1) = V (τ2)V (τ4)V †(τ3)V †(τ1) , (106)

VL†(τ4)VR(τ3)VR(τ2)VL†(τ 1) = V (τ2)V (τ3)V †(τ4)V †(τ1) . (107)

1.0 (Schlawin, 2015). At the same time, by varying the entan- f 2 glement time becomes possible to probe subsets of transport 0.8 ) pathways via the selection of speciﬁc f-states in the detected t (

f optical signal. p 0.6 The properties described above may also be observed in more complex systems such as molecular aggregates 0.4 (Schlawin et al., 2013), where the number of excited states is

f1 much higher, but the described physics is essentially the same:

Population 0.2 Ultrafast relaxation processes limit the classically achievable degree of selectivity due to the trade-off between spectral and 21 600 21 800 22 000 22 200 22 400 22 600 22 800 time resolution for every single absorption process. With en- 1 Pump frequency !p (cm ) tangled photons, this trade-off only limits the overall two- photon process, but not each individual transition. This may FIG. 10 Variation of the double-excited state populations pf (t), be understood in the following way: In the absorption of en- Eqs. (102)-(104) with the pump frequency ωp after excitation by entangled photons with strong frequency anti-correlations (solid, red), tangled photons, the light-matter system becomes entangled by laser pulses matching the pump bandwidth σp (blue, dashed), and in between the two absorption events, but it remains separable laser pulses matching the photon bandwidths ∼ 1/T (green, dot- at all times for classical pulses so that the energy-time uncer- dashed). The total f-population is normalized to unity at each fre- tainty only applies to the entire system. P quency, i.e. f pf (t) = 1. Finally it is worth noting an additional advantage of using entangled photons for two-photon excitation in molecules pointed out by (Raymer et al., 2013). In many aggregates the The entangled photon field correlation functions are given by doubly-excited state undergoes rapid nonradiative decay to the the Fourier transform of Eq. (62). We first restrict our atten- singly excited states. This may occur on the same time scale tion to the weak downconversion regime, n¯ 1, in which of the excited-state transport (Van Amerongen et al., 2000). the autocorrelation functions g1,2, Eqs. (64) and (65), may be For weakly anharmonic aggregates where the transition fre- neglected. quencies f e and e g are close, it is hard to discriminate The essential properties of entangled photon absorption between the→ two-photon-excited→ and single photon excited flu- may be illusttrated by using the simple model system intro- orescence. Entangled light can help isolate the doubly-excited duced in section II.J.1 where the correlation functions (102) - state population by monitoring the transmitted single-photon (104) can be written as the sum-over-states expressions given pathways with a high-quantum-efficiency detector, that can in appendix C. To excite state f2 faithfully, the intermediate partially rule out one-photon absorption. decay process e e needs to be blocked (see Fig. 8), and 2 → 1 one needs to select the final state f2 spectrally. This can be achieved with entangled photons as shown in Fig. 10, where 3. Scaling of two photon absorption with pump intensity we depict the relative population in each final state vs. the 2 pump frequency ωp. At each frequency the total population Eq. (101) shows that TPA probability Tfg scales quadrat- 2 2 | | is normalized to unity, i.e. pf1 + pf2 = 1. By choosing a ically with pump intensity A1A2 for classical light but Eq. 1 2 spectrally narrow pump bandwidth σp = 100 cm− in com- (100) shows linear scaling Ap with weak entangled light. bination with a short entanglement time T = 10 fs, almost 90 As the pump pulse intensity is increased, the two-photon % of the total f-population can be deposited in the state f2 at state (33) no longer represents the output state (57), and the 1 1 ωp = 22500 cm− (and 95 % in f1 at ωp = 21800 cm− ). autocorrelation contributions g to Eq. (62) must be taken ∼ 1,2 The same degree of state selectivity may not be achieved into account. To understand how this affects the excited state with classical laser pulses: For comparison, the f-manifold distributions, we first simulate in Fig. 11 the relative contri- populations created by classical laser pulses with bandwidths bution of the two-photon term h12 in Eq. (62) to the popu- 1 σ = 100 cm− (same spectral selectivity) and with σ = lation in f2, when the pump frequency is set at resonance, 1 1 1000 cm− (same time resolution in the manifold of singly ωp = 22500 cm− , and its variation with the mean photon excited states) are shown as well in Fig. 10. In the former number (59). Clearly, in the weakly entangled case this con- case, the intermediate relaxation process limits the maximal tribution drops rapidly, and already at n¯ = 0.5 the autocor- yield in f2 to 35 % (with ca. 65 % population in f1 at relation contribution (i.e. excitations by uncorrelated photons ∼ 1 ωp = 22500 cm− ), in the latter case, the lower spectral from different downconversion events) dominates the signal. resolution limits the achievable degree of control over the f- In contrast, in the strongly entangled case the contribution populations. drops much more slowly, and - as can be seen in the inset - For very short entanglement times, transport between var- still accounts for roughly a third of the total absorption events ious excited states in e-manifold may be neglected, and the even at very high photon number n¯ 100. As described in f-populations may as well be calculated using transition (Dayan et al., 2005), this may be attributed' to the strong time amplitudes - thus greatly reducing the computational cost correlations in the former case: For T = 10 fs, one can fit ten 20

regarded as a crossover from a quantum to a classical regime a) �� 1.0 0.8 (Lee and Goodson, 2006), is not necessarily a good indica- 0.6 tor for this transition after all, as pointed out by (Georgiades �� 0.4 0.2 et al., 1997). The time-frequency correlations of entangled 0.0 �� 20 40 60 80 100 pairs may be harnessed at much higher photon ﬂuxes - as long as the coherent contribution h dominates the incoher- �� 12 ent contributions g1,2. ��

Relative contribution Resonant intermediate state III. NONLINEAR OPTICAL SIGNALS OBTAINED WITH �� ENTANGLED LIGHT Mean photon numbern ¯ We now revisit the excited state populations created in mat- b) 1.0 �� 0.8 ter following the absorption of entangled photon pairs in sec- 0.6 tion II, and present optical signals associated with these dis- �� 0.4 0.2 tributions. We first derive compact superoperator expressions 0.0 �� 0 100 200 300 400 500 for arbitrary field observables, which naturally encompass standard expressions based on a semiclassical treatment of the �� field (Mukamel, 1995). Using Eq. (11), the expectation value of an arbitrary field �� operator A(t) is given by

Relative contribution O↵-resonant intermediate state

�� S(t; Γ) = A+(t) ﬁnal (108) Mean photon numbern ¯ t i = A+(t) exp dτHint, (τ) (109) T − − FIG. 11 a) Relative contribution of the coherent two-photon con- ~ Zt0 t tribution h12, Eq. (63), to the full time-integrated population of f2 i = S0 dτ A+(t)Hint, (τ) plotted vs. the mean photon number n¯, Eq. (59). The dashed line − T − ~ Zt0 shows strong entanglement with entanglement time T = 10 fs, and τ the solid line weaker entanglement with T = 100 fs. b) same for an i exp dτ 0Hint, (τ 0) . (110) intermediate state moved to ω = 18, 500 cm−1. × − − e2 ~ Zt0

Here, the first term describes the expectation value in the absence of any interaction with the sample, which we set to zero. times as many photon pairs into a time period compared to The second term describes the influence of the sample on the T = 100 fs, before different pairs start to overlap temporally. expectation value. It may be represented more compactly, In addition, the strong frequency correlations discussed in the when the interaction Hamiltonian is not written in the RWA previous section imply that, on resonance with the final state approximation (Schlawin, 2015), f2, the coherent contribution is enhanced even further. i t The situation becomes even more striking in panel b when S(t; Γ) = dτ [A(t), (τ)] (τ) , (111) − + V+ final the intermediate state e is shifted away from the entangled ~ Zt0 photon frequencies), where we set it to ω = 18, 500 cm 1, e2 − where we recall (t) = E(t) + E (t) and (t) = V (t) + and only the two-photon transition g f is resonant. In † 2 V (t). V the weakly entangled case (solid line), little→ has changed, and † Eq. (111) may be used to calculate arbitrary optical signals parity between the two-photon and the autocorrelation contri- induced by entangled photons, as will be demonstrated in the bution is reached again for n 2. In the strongly entangled following. regime, however, the coherent∼ contribution remains close to unity, and dominates the signal even at n¯ 500, as can be ' seen in the inset. A. Stationary Nonlinear signals The strong frequency correlations of entangled photon pairs lead to the enhancement of the two-photon contribution h12 Here, we consider the situation described in section II.B, compared to the autocorrelation contributions g1,2 - even for where the matter system is either coupled to a single cavity very high photon fluxes. Consequently, The nonclassical ex- mode, and where only a single quantum mode of the light cited states distributions created by entangled photons can still is relevant. This is most prominently the case in the strong dominate the optical signal even when the mean photon num- coupling regime of cavity quantum electrodynamics (Walther ber in each beam greatly exceeds unity (Schlawin, 2015). The et al., 2006), or when one is interested in steady state solu- linear to a quadratic crossover of the two-photon absorption tions, where only a single (or few) frequency(ies) of the field rate with the pump intensity (Fig. 1a) which is traditionally is(are) relevant. 21

The intricate connection between nonlinear optical signals and photon counting was ﬁrst made explicit by (Mollow, 1968), who connected the two-photon absorption rate of stationary ﬁelds with their G(2)-function2

2 ∞ 2iωf t γf t (2) w = 2 g(ω ) dt e − | |G ( t, t; t, t), (112) 2 | 0 | − − Z−∞ where g(ω0) µegµeg/(ω0 ωe + iγe) is the coupling el-

∼ − Logarithmic cross section ement evaluated at the central frequency ω0 of the stationary light field. The transition rate loosely corresponds to the time-integrated, squared two-photon transition amplitude (99) Energy (eV) we derived earlier. If the lifetime of the final state γf is much smaller than the bandwidth of the field ∆ω, γf FIG. 12 Entangled virtual-state spectroscopy: TPIF signal vs. the ∆ω, one may even neglect the time integration, and replace Fourier transform of an additional time delay between the two pho- G(2)( t, t; t, t) G(2)(0). tons (see text) in hydrogen [taken from (Saleh et al., 1998)]. This− implies,− as∼ pointed out in (Gea-Banacloche, 1989), that strongly bunched light can excite two-photon transitions In a different scenario, the two-excitation state may rapidly more efficiently than classical light with identical mean pho- decay nonradiatively - e.g. via internal conversion - and the ton number. It further implies that the two-photon absorption e g fluorescence is detected, rate scales linearly in the low gain regime, even though the − single mode squeezed state does not show time-energy entan- S˜TPIF(Γ) dt p (t; Γ). (115) glement in the sense of section II.C. An experimental verifi- ∝ f cation thereof is reported in (Georgiades et al., 1995, 1997). Z A more recent proposal (Lopez´ Carreno˜ et al., 2015) aims Fluorescence signals are proportional to excited state popula- to employ nonclassical fluctuations contained in the fluores- tions. Therefore, they are closely related to the excited state cence of a strongly driven two-level atom: As is well known, distributions discussed in sections II.I to II.J.3. This is not the driven two-level atom’s fluorescence develops side peaks necessarily the case for absorption measurements, as will be - known as the Mollow triplet (Scully and Zubairy, 1997). By shown in section III.D. driving polaritons - strongly coupled light-matter states in a cavity - with this light, it is predicted to allow for precise measurements of weak interactions between polaritons “even in 1. Two-photon absorption vs. two-photon-induced strongly dissipative environments”. fluorescence

Before reviewing fluorescence signals induced by entan- B. Fluorescence detection of nonlinear signals gled photons, we comment on some ambiguity in the nomen- clature of nonlinear signals such as fluorescence that depend Fluorescence is given by the (possibly time-integrated) in- on the doubly excited state population pf . Since population is tensity A(t) = E†(t)E(t), when the detected field mode is created by the absorption of two photons, the signal is often initially in the vacuum. It is obtained by expanding Eq. (111) termed two-photon absorption. However, it does not pertain to second order in the interaction Hamiltonian with the vac- to a χ(3)-absorption measurement, which will be discussed in uum field (Mukamel, 1995), section III.D. We will refer to signals measuring pf as two- photon-induced fluorescence and to two-photon resonances in 1 (3) † (113) χ as two-photon absorption. SFLUOR = 2 VL(t)VR(t) final. ~ In section III.C.1, we will further show that for off-resonant

We ﬁrst investigate the time-integrated f e ﬂuorescence intermediate state(s) e - as is the case in most experimental signal (113), following excitation by either→ entangled photons studies to date (Dayan et al., 2004, 2005; Guzman et al., 2010; or classical pulses. Using Eqs. (113), and (102)-(104), we Harpham et al., 2009; Lee and Goodson, 2006; Upton et al., may readily evaluate the signal as 2013) - the two signals carry the same information.

2 STPIF(Γ) = dt µ p (t; Γ). (114) | ef | f 2. Two-photon induced transparency and entangled-photon Z Xe,f virtual-state spectroscopy

Entangled virtual-state spectroscopy, proposed in (Saleh et al., 1998), suggests a means to detect far off-resonant in- 2 We change the notation to match the remainder of this review. termediate states in the excitation of f-states by employing 22 entanglement-induced two-photon transparency (Fei et al., a) �� 1997): The transition amplitude (100) of entangled photons )

�� created by a cw-source oscillates as a function of the entan- ( glement time T . For inﬁnite excited-state lifetimes (γe = 0),

TPIF �� the excitation of f is even completely suppressed whenever S (ω1 ωe)T = n 2π n N - the medium becomes trans- − × ∀ ∈ parent. �� Entangled Saleh et al. had proposed to turn this counterintuitive ef- photons fect into a spectroscopic tool by sending one of the two pho- �� tons through a variable delay stage τ, which simply amounts TPIF signal to evaluating the two-photon wavefunction (52) 0 E(τ2 + �� h | 1 τ)E(τ1) ψtwin . For degenerate downconversion the transition Pump frequency !p (cm ) amplitude| (100)i then reads b) ��

iωpt iωpτ/2 0Ap µgeµef e− e− ) T (τ, t; Γ) = �� N ( fg 2T ω ω + iγ ∆ + iγ ~ e p f f e e X − i(∆e+iγe)(T τ) i(∆e+iγe)(T +τ) TPIF ��

e − + e 2 , (116) S × − �� where we introduced the detuning ∆e = ωp/2 ωe. Fourier 2 − transform of the TPIF signal Tfg(τ) with respect to τ reveals different groups of resonances∝ | shown| in Fig. 12 for �� Laser pulses the TPIF signal in hydrogen: Group A denotes resonances TPIF signal of differences between intermediate states, ∆e ∆e0 , group �� − 1 B of detunings ∆e, and group C of the sum of detunings Laser frequency 2 ! (cm ) ⇥ 0 ∆e + ∆e0 . Such signals were simulated in molecular system (Kojima and Nguyen, 2004), and similar resonances can FIG. 13 a) TPIF action spectrum STPIF(Γ), Eq. (114), induced by be identified in absorption TPA measurements, as we will dis- entangled photon pairs with the same parameters as in Fig. 10. The cuss in section III.C.1. fluorescence created by the state f1 is shown separately as a dashed plot, and the signal from f2 as a dot-dashed line. b) TPIF action However, it turns out that the effect depends crucially on −1 spectrum induced by laser pulses with bandwidth σ0 = 100 cm . the tails of the spectral distribution in Eq. (49) (de J Leon- Montiel et al., 2013). The sinc-function in the two-photon wavefunction (51) has a long Lorentzian tail 1/ω, which the state can hardly be observed, even though it has the same covers extremely far off-resonant intermediate∼ states. When dipole strength as f1. the sinc is replaced by Gaussian tails, the resonances vanish. These nonclassical bandwidth features of entangled photon The effect is thus caused by the long spectral tails, and is not pairs may be further exploited to control vibronic states, as intrinsically connected to entanglement. reported in (Oka, 2011a,b). They were also investigated in semiconductor quantum wells. In this system, the absorption of excited state states competes with the absorption into a con- 3. Fluorescence from multi-level systems tinuum of excited electronic states of free electrons and holes. By interpolating between negative and positive frequency cor- The TPIF signal, Eq. (114), directly reflects the doubly ex- relations (see Fig. 3), it was shown in (Salazar et al., 2012) cited state population distributions created by the absorption how the absorption of the excited states may be either en- of entangled photon pairs. This may be seen in simulations hanced or suppressed, as shown in Fig. 14. of the TPIF signal from Frenkel excitons in a molecular aggregates (Schlawin et al., 2012). In Fig. 13a), we present the TPIF signal resulting from the decay of the f-state dis- 4. Multidimensional signals tributions in Fig. 10. Clearly, the signal peaks whenever a two-excitation state is on resonance with the pump frequency In Fig. 2, we had explained how each light-matter inter- ωp. The signal from the two f-states has approximately the action event also imprints the light phase φ onto the matter same strength. In contrast, the TPIF signal created by classi- response. We now exploit this fact to create multidimensional cal pulses shown in panel b) has a strong resonance only when spectroscopic signals. Phase-cycling is essential for partially the light is on resonance with the state f1, whereas f2 can only non-collinear or collinear geometry 2D spectroscopic exper- be observed as a weak shoulder of the main resonance. As dis- iments (Keusters et al., 1999; Krcmˇ a´rˇ et al., 2013; Scheurer cussed in chapter II.I, the fast decay of the intermediate state and Mukamel, 2001; Tan, 2008; Tian et al., 2003; Yan and e2 limits the excitation of f2 with classical light, and therefore Tan, 2009; Zhang et al., 2012a). In phase-cycling protocol, 23

this strategy in more detail.

5. Loop (LOP) vs ladder (LAP) delay scanning protocols for multidimensional ﬂuorescence detected signals

Since the loop and the ladder diagrams involve different time variables they suggest different multidimensional signals TPA (a.u.) obtained by scanning the corresponding time delays. We consider the TPIF signal created by a train of four pulses centered at times T1, T2, T3, and T4 with phases φ1, φ2, φ3, and φ4 0.5 0.6 0.7 (Pestov et al., 2009; Tekavec et al., 2007). We first analyze Photon energy !/!g signals obtained by the LOP/LAP delay scanning protocols with classical light. The two protocols highlight different res- FIG. 14 Two-photon absorption (TPIF in our notation, see sec- onances and processes. Below we demonstrate what type of tion. III.B.1) in a semiconductor quantum well [taken from (Salazar information can be extracted from each protocol for excited et al., 2012)]: excited state resonances within a continuum of de- electronic states in a model molecular aggregate. We then localized states may be enhanced with frequency anti-correlations (black) or suppressed with positive frequency correlations (red). show some benefits of LOP protocols when applied to entangled light. For the system model shown in Fig. 2a) the signal (114) the desired 2D signals are retrieved by the weighted summa- is given by the single loop diagram in Fig. 2b). a, b, c, d denote the pulse sequence ordered along the loop (but not in real tion of data collected using different interpulse phases, φ21, which are cycled over 2π radians in a number of equally time); a represents “first”. on the loop etc. Chronologically- spaced steps. In the pump-probe configuration, Myers et al. ordered pulses in real time will be denoted 1, 2, 3, 4 which are (Myers et al., 2008) have shown that other than the pure ab- permutations of a, b, c, d, as will be shown below. One can sorptive signal, the rephasing and nonrephasing contributions scan various delays Tα Tβ, α, β = 1, 4 and control the phases φ φ φ − φ . In standard··· multidimensional may also be retrieved. This phase difference detection can be ± 1 ± 2 ± 3 ± 4 defined as a two-step phase-cycling scheme. For the phase techniques the time variables represent the pulses as they in- difference detection, the Ogilvie group needs to collect sig- teract with sample in chronological order (Mukamel, 1995). These are conveniently given by the ladder delays ti shown in nals with interpulse phases of φ21 = 0, π, π/2 and 3π/2. The same signal can be obtained with chopping, but this is only Fig. 16c). The LAP maintains complete time-ordering of all half as intense compared to the phase difference method. In four pulses. The arrival time of the various pulses in chrono- the context of this article, what they have performed is similar logical order is T1 < T2 < T3 < T4. The ladder delays are defined as t = T T , t = T T , t = T T . One can to a four-step phase-cycling scheme, where four sets of data 1 2 − 1 2 3 − 2 3 4 − 3 need to be collected and linearly combined. then use phase cycling to select the rephasing and nonrephasing diagrams shown in Fig. 16c, and read off the signal in a Phase cycling provides a means to post-select signals with sum-over-states expansion a certain phase signature, as will be shown in the next section. Using additional delay stages - like in the entangled-photon (LAP ) SkII (t1, t2, t3) = 1∗ 2∗ 3 4 µge0 µe0f µfe∗ µeg∗ virtual state spectroscopy discussed above - these signals can IE E E E e,eX0,f be spread to create multidimensional frequency correlation (t ) (t ) (t ), (117) maps, which carry information about couplings between dif- × Gef 3 Gee0 2 Geg 1 ferent resonances or relaxation mechanisms (Ginsberg et al., (LAP ) S (t1, t2, t3) = ∗ ∗ 3 4 µge µe f µ∗ µ∗ 2009). kIII IE1 E2 E E 0 0 fe eg (Raymer et al., 2013) had exploited the formal similarity e,eX0,f (t ) (t ) (t ). (118) between the TPIF and a photon coincidence signal to propose × Gef 3 Gee0 2 Gge0 1 the setup shown in Fig. 15a). Here, the successive absorption The LAP signals (117) - (118) factorize into a product of events promoting the molecule into the f-state are modulated several Green’s functions with uncoupled time arguments. by phase cycling and delay stages in both photon beams, cre- This implies that the corresponding frequency domain sig- ating interference effects between absorption events in which nal will also factorize into a product of individual Green’s the photon takes the short path through the interferometer, and functions, each depending on a single frequency argument Ω˜ j, those in which it takes the long path. The Fourier transform j = 1, 2, 3 which yields creates two-dimensional signals which are shown in panel b). (LAP ) ˜ ˜ In contrast to the classical signal shown in the upper panel, SkII (Ω1, t2 = 0, Ω3) the proposed scheme only shows resonances at the cross-peak µge0 µe0f µfe∗ µeg∗ 1∗ 2∗ 3 4 between the two electronic states. This could provide a useful = E E E E , (119) R [Ω3 ωef + iγef ][Ω1 ωeg + iγeg] tool to study conformations of aggregates. We next discuss e,eX0,f − − 24

a) b) (rad / fs) t ! (rad / fs) t !

!⌧ (rad / fs)

FIG. 15 a) Experimental setup for the LOP protocol: By sending each beam through a Franson interferometer, the photon excitation may occur either via a delayed or an “early” photon. b) Top panel: two-dimensional ﬂuorescence spectrum induced by classical light. Bottom panel: the same spectrum induced by entangled photons. [taken from (Raymer et al., 2013)]

(LAP ) (LOP ) S (Ω˜ 1, t2 = 0, Ω˜ 3) S (s1, s2, s3) = ∗ ∗ 3 4 µge µe f µ∗ µ∗ kIII kIII IE1 E2 E E 0 0 fe eg e,e0,f µge0 µe0f µfe∗ µeg∗ 1∗ 2∗ 3 4 X = E E E E . (120) (122) R [Ω ω + iγ ][Ω ω + iγ ] ef (s2) ee0 (s1 s2) ge0 (s2 + s3 s1), 3 fe fe 1 e0g e0g × G G − G − e,eX0,f − − where denotes the imaginary part and αβ(t) = This factorization holds only in the absence of additional cor- I [iωαβ +γαβ ]t G ( i/~)θ(t)e− is the Liouville space Green’s func- relating mechanisms the frequency variables caused by e.g. − tion [see Eq. (16)]. Note that the loop delays sj, j = 1, 2, 3 dephasing (bath) or the state of light. are coupled and enter e.g. in the Green’s function eg in Eq. In the LOP the time ordering of pulses is maintained only (121). Due to the Heaviside-theta function in thisG Green’s on each branch of the loop but not between branches. To real- function, the delays sj are not independent but rather cou- ize the LOP experimentally the indices 1 to 4 are assigned as ple the dynamics of the system during these delay times, follows: first by phase cycling we select a signal with phase which eventually result in cross-resonances in multidimen- φ + φ φ φ . The two pulses with positive phase de- 1 2 − 3 − 4 sional spectra. To see the effect on the mixing of the frequency tection are thus denoted 1, 2 and with negative phase - 3, 4. variables that yield these cross-peaks we take a Fourier trans- In the 1, 2 pair pulse 1 comes first. In the 3, 4 pair pulse 4 form of Eqs. (121) - (122) with respect to loop delay variable comes first. The time variables in Fig. 16b) are s1 = T2 T1, s1 and s3 keeping s2 = 0 and obtain the resonant component s = T T , s = T T . With this choice s and s− are 2 3 − 2 3 3 − 4 1 3 of the signal analogous to the frequency-domain LAP signals positive whereas s2 can be either positive or negative. This (119) - (120) completely defines the LOP experimentally. When the electronic system is coupled to a bath, it cannot (LOP ) be described by a wave function in the reduced space where SkII (Ω1, s2 = 0, Ω3) = the bath is eliminated. As described in section II.J.2, the loop µge0 µe0f µfe∗ µeg∗ 1∗ 2∗ 3 4 diagram must then be broken into several ladder diagrams E E E E , (123) R [Ω1 + Ω3 ωee0 + iγee0 ][Ω1 ωeg + iγeg] shown in Fig. 2b) which represent the density matrix. The e,eX0,f − − full set of diagrams and corresponding signal expressions are given in (Dorfman and Mukamel, 2014b). In Fig. 16c) we (LOP ) SkIII (Ω1, s2 = 0, Ω3) = present simplified expressions for the rephasing kII and non- µge0 µe0f µfe∗ µeg∗ 1∗ 2∗ 3 4 rephasing kIII signals in the limit of well separated pulses: E E E E , R [Ω1 + Ω3 + ωee0 + iγee0 ][Ω3 ωe0g + iγe0g] e,e0,f (LOP ) X − S (s1, s2, s3) = ∗ ∗ 3 4 µge µe f µ∗ µ∗ (124) kII IE1 E2 E E 0 0 fe eg e,e ,f X0 where denotes real part. Eqs. (123) - (124) yield cross- (s ) (s ) (s s s ) (121) R ef 2 ee0 3 eg 1 2 3 peaks Ω + Ω = ω . The time correlations therefore result × G G G − − 1 3 ee0 25

FIG. 16 (Color online) a) Pulse sequence for LOP (top panel) and LAP (bottom panel) scanning protocols. b) Loop diagram for the TPA process with indicated loop delays for the non specified phase cycling that depends on chronological time ordering between pulses 1 to 4 c). Ladder diagrams for the TPA signal with selected phase cycling component corresponding to double quantum coherence term kI : −φd − φc + φb + φa, rephasing kII : −φ4 + φ2 + φ1 − φ3, and nonrephasing kIII : −φ4 + φ2 − φ3 + φ1. Both loop sj and ladder tj delays, j = 1, 2, 3 are indicated. The transformation between two is different for each diagram. Time translation invariance implies ω1 + ω2 − ω3 − ω4 = 0. in the frequency mixing of arguments. lated entangled signals using LOP whereas they used LAP for classical signals. Fig. 17 compares the LOP and LAP signals for the model dimer parameters of (Raymer et al., 2013) under two photon excitation by classical light. The LOP spectra for rephasing kII = 1 + 2 + 3, nonrephasing kIII = + 1 + 2 3 and their−k sum arek shownk in Fig. 17a)-c), respectively.k Thek − cor- k responding LAP spectra are shown in Fig. 17d-f. We see that the scanning protocol makes a significant difference as seen a. Entangled vs classical light So far, we presented the LOP by the two columns. The LOP resonances are narrow and and the LAP delay scanning protocols for classical light. We clearly resolve the e1 and e2 states whereas the correspond- now turn to the LOP protocol with entangled light. Similarly ing LAP signals are broad and featureless. This is a conse- to Eqs. (121) - (122) we calculate the signals for the CW- quence of the display variables chosen in each protocols. LOP pump model (51) discussed earlier, and obtain variables are coupled in a very specific fashion that allows to extract the intraband dephasing rate γee0 in (Ω1, Ω3) plot with higher precision compared to LAP case. Of course, one can extract similar information using LAP if displayed using (Ω˜ , Ω˜ ) or (Ω˜ , Ω˜ ) which we will discuss below in the con- (j) ∞ dωa dωb dωd 1 2 2 3 S (s1, s2, s3) = text of entangled light. This difference in two protocols has LOP I 2π 2π 2π Z−∞ been originally attributed to entanglement by (Raymer et al., (j) Rj(ωa, ωb, ωd)D (s1, s2, s3; ωa, ωb, ωd) 2013). However, it was explicitly demonstrated in (Dorfman × LOP E†(ω )E†(ω + ω ω )E(ω )E(ω ) , (125) and Mukamel, 2014b) supported by Fig. 17 that they calcu- × h d a b − d b a i 26

FIG. 18 (Color online) Left column: LOP signal

SkII +kIII (Ω1, τ2 = 0, Ω3) for a molecular trimer with classical light Eq. (123) - (124) -(a), entangled light (125) with ˜ ˜ Te = 100 fs -(b). Right: column: LAP signal SkII (Ω1, t2 = 0, Ω3) using classical light Eq. (119) -(c), entangled light with Te = 100 γ = 1 fs, Eq. (128) -(d). Intraband dephasing ee0 meV. All other parameters are given in Section 5 of (Dorfman and Mukamel, 2014b).

FIG. 17 (Color online) Left column: SLOP (Ω1, τ2 = 0, Ω3) for the molecular dimer model of Ref. (Raymer et al., 2013) calculated Similarly we obtain for LAP signals Eqs. (117) - (118) using classical light for rephasing kII Eq. (123) - (a), nonrephasing kIII Eq. (124) - (b) and the sum of two kII + kIII - (c). Right (j) ∞ dωa dωb dωd S (Ω˜ , t = 0, Ω˜ ) S (t1, t2, t3) = column: same for LAP 1 2 3 Eq. (119) - (120). Note LAP I 2π 2π 2π that the the sum of rephasing and nonrephasing components for LAP Z−∞ signal we ﬂipped the rephasing component to obtain absorptive peaks (j) Rj(ωa, ωb, ωd)DLAP (t1, t2, t3; ωa, ωb, ωd) as is typically done in standard treatment of photon echo signals. The × E†(ω )E†(ω + ω ω )E(ω )E(ω ) , (128) difference between the two columns stems from the display protocol × h d a b − d b a i and is unrelated to entanglement. where the display function for the rephasing signal is

(kII ) DLAP (t1, t2, t3; ωa, ωb, ωd) = θ(t1)θ(t2)θ(t3)

iωbt3 iωat1+i(ωd ωa)t2 where j = kII , kIII and the display function for both rephas- e − − , ing and nonrephasing contributions reads × (129)

and for nonrephasing signal (j) DLOP (s1, s2, s3; ωa, ωb, ωd) = θ(s1)θj( s2)θ(s3) (kIII ) ± DLAP (t1, t2, t3; ωa, ωb, ωd) = θ(t1)θ(t2)θ(t3) e iωas1+iωds3 i(ωa+ωb)s2 , − − iωbt3+iωdt1+i(ωd ωa)t2 × (126) e − . × (130)

The complete set of signals along with frequency domain sig- and matter responses are nals for entangled light can be found in Section 2 of (Dorfman and Mukamel, 2014b). The Fourier transform of the signal (125) was simulated a b c d R (ω , ω , ω ) = µ µ µ ∗ µ ∗ kII a b d ge0 e0f fe eg using the LOP protocol and compared it with the standard e,eX0,f fully time ordered LAP protocol given by Eq. (128) for a ( ω ) (ω ω ) (ω ), model trimer with parameters discussed in Section 5 of (Dorf- × Gef − b Gee0 a − d Geg a a b c d man and Mukamel, 2014b). Fig. 18 shows the simulated RkIII (ωa, ωb, ωd) = µge µe f µfe∗ µeg∗ 0 0 SLOP (Ω1, τ2 = 0, Ω3) for a trimer using classical light (top e,e ,f X0 row) and entangled light (bottom row). Fig. 18a shows clas- ( ω ) (ω ω ) ( ω ). × Gef − b Gee0 a − d Gge0 − d sical light which gives a diagonal cross peak e = e0 and (127) one pair of weak side peaks parallel to the main diagonal at 27

(e, e0) = (e2, e3). The remaining two pairs of side peaks at (e, e0) = (e1, e2) and (e, e0) = (e1, e3) are too weak to be seen. Fig. 18d) depicts the signal obtained using entangled t t photons where we observe two additional strong side cross ⌧ peak pairs with (e, e ) = (e , e ) and (e, e ) = (e , e ). The 3 0 1 3 0 1 2 ⌧2 weak peak at (e, e0) = (e2, e3) is signiﬁcantly enhanced as ⌧10 well. ⌧2 As has been shown in Fig. 17 the LAP signal displayed ⌧1 vs (Ω˜ 1, Ω˜ 3) does not effectively reveal intraband dephasing. This can, however, be done by the LAP signal displayed vs ⌧1 (Ω˜ 2, Ω˜ 3). Fig. 18c) reveals several Ω˜ 3 = ωfe peaks that (I) (II) overlap in Ω˜ 3 axes due to large dephasing γfe. With entangled light, the LAP protocol yields some enhancement in several peaks around 0.01 eV but overall the LOP yields much t t cleaner result. The± pathways for the density matrix (LAP) and the wavefunction (LOP) are different and suggest differ- ⌧3 ⌧20 ent types of resonances. ⌧1 ⌧2

⌧10 C. Heterodyne detection of nonlinear signals ⌧1 So far we have considered fluorescence (homodyne) de- (III) (IV) tection. Homodyne and heterodyne are two complementary detection schemes for nonlinear optical signals. In terms of FIG. 19 Diagrams representing the third-order contributions to classical fields, if the sample radiation is detected for differ- Eq. (131). ent modes than the incident radiation, the signal is proportional to E 2. This is known as homodyne detection whereby | | the intensity is the square modulus of the emitted field itself. coherent detection modes include fluorescence (Barkai, 2008; If the emitted field coincides with a frequency of the inci- Elf et al., 2007), photoaccoustic (Patel and Tam, 1981), AFM dent radiation Ein, then the signal intensity is proportional (Mamin et al., 2005; Poggio et al., 2009; Rajapaksa and to E + E 2. Consequently, the detected intensity contains Kumar Wickramasinghe, 2011; Rajapaksa et al., 2010) or | in| a mixed interference term, i.e., E∗Ein + c.c.. This is defined photocurrent detection (Cheng and Xie, 2012; Nardin et al., as the heterodyne signal, since the emitted field is mixed with 2013). another field. In terms of quantized fields, the signal denoted homodyne if detected at a field mode that is initially vacant and heterodyne when detected at a field mode that is already occupied. Note that the current definition, which is commonly 1. Heterodyne Intensity measurements - Raman vs. TPA used in nonlinear multidimensional spectroscopy, is different pathways from the definition used in quantum optics and optical en- gineering where homodyne and heterodyne refer to mixing With A(t) = E†(t)E(t), Eq. (111) yields the rate of change with a field with the same or different frequency of the sig- of the transmitted photon number, nal. In this review, we will use the spectroscopy terminology (Potma and Mukamel, 2013). Fluorescence detection is of- 2 ten more sensitive than heterodyne as the latter is limited by S (t; Γ) = E† (t)V (t) . (131) 1 = + + final the pulse duration so there are fewer constraints on the laser ~ system. In addition the low intensity requirements for biolog- The semiclassical signal may be obtained from Eq. (131) by ical samples limit the range of heterodyne detection setups. simply replacing the field operator E (t) by a classical field This have been demonstrated by (Fu et al., 2007; Tekavec † amplitude A (t). Similarly, by spectrally dispersing the time- et al., 2007; Ye et al., 2009) even in single molecule spec- ∗ integrated intensity, which amounts to measuring A(ω) = troscopy (Brinks et al., 2010). Historically, Ramsey fringes E (ω)E(ω), we obtain constitute the first example of incoherent detection (Cohen- † Tannoudji and Guery-Odelin,´ 2011; Ramsey, 1950; Schlawin 2 et al., 2013). Information similar to coherent spectroscopy can S(ω; Γ) = E†(ω)V (ω) . (132) be extracted from the parametric dependence on various pulse ~= final sequences applied prior to the incoherent detection (Mukamel and Richter, 2011a; Rahav and Mukamel, 2010). Possible in- The third-order contribution of the time-integrated absorp- 28

3 t τ3 τ2 tion signal (131) is given by the four loop diagrams in Fig. 19, 1 i ∞ S1,(IV)(ω; Γ) = dt dτ3 dτ2 dτ1 ~= −~ Z−∞ Z−∞ Z−∞ Z−∞ 3 t τ2 t iωt 1 i ∞ e V (t)V †(τ3)V (τ2)V †(τ1) S1,(I)(Γ) = dt dτ2 dτ1 dτ10 × −~= −~ Z−∞ Z−∞ Z−∞ Z−∞ E†(ω)E(τ3)E†(τ2)E(τ1) . (140) V (τ 0 )V (t)V †(τ )V †(τ ) × × 1 2 1

E †(τ 0 )E†(t)E(τ )E(τ ) , (133) It is instructive to relate the four diagrams corresponding to × 1 2 1 Eqs. (D1)-(D4) to the transition amplitudes between the ini- 3 t τ3 τ2 1 i ∞ tial and final matter states to gain some intuition for this sig- S1,(II)(Γ) = dt dτ3 dτ2 dτ1 nal. This is only possible by the total photon counting sig- ~= −~ Z−∞ Z−∞ Z−∞ Z−∞ nal, dω S(ω; Γ)/(2π), which represents the full energy ex- V (t)V (τ )V †(τ )V †(τ ) × 3 2 1 changed between the light field and the matter system. We nowR define the transition amplitude operators E †(t)E†(τ )E(τ )E(τ ) , (134) × 3 2 1 (1) µe0g 3 t t τ 0 T (ω) = E(ω), (141) 1 i ∞ 2 e0g ~ S1,(III)(Γ) = dt dτ1 dτ20 dτ10 ~= −~ Z−∞ Z−∞ Z−∞ Z−∞ 1 dω µ µ E(ω ω )E(ω ) V (τ10 )V †(τ20 )V (t)V †(τ1) (2) a ge ef sum a a × Tfg (ωsum) = 2 − , ~ 2π ωa ωe + iγe e Z − E †(τ 0 )E(τ 0 )E†(t)E(τ ) , (135) X × 1 2 1 (142) 3 t τ3 τ2 1 i ∞ S1,(IV)(Γ) = dt dτ3 dτ2 dτ1 = − (2) 1 dωa µgeµeg E†(ωa ω)E(ωa) ~ ~ T (ω) = 0 − , Z−∞ Z−∞ Z−∞ Z−∞ g0g 2 ~ 2π ωa ωe + iγe V (t)V †(τ3)V (τ2)V †(τ1) e Z − × X (143) E †(t)E(τ )E†(τ )E(τ ) . (136) × 3 2 1 T (3)(ω) = e0g An identical expansion of Eq. (132) yields the frequency- 1 dωa dωx µge µef µfe0 resolved third-order signal (its sum-over-state expansion is 3 ~ 2π 2π ωa ωe + iγe ωx ωf + iγf shown in Appendix D) Xe,f Z Z − − E†(ω ω)E(ω ω )E(ω ), (144) × x − x − a a 3 t τ2 t 1 i ∞ S (I)(ω; Γ) = dt dτ dτ dτ 0 1, 2 1 1 (3) −~= −~ T 0 (ω) = Z−∞ Z−∞ Z−∞ Z−∞ e0g iωt e V (τ10 )V (t)V †(τ2)V †(τ1) 1 dωa dω µge µeg µg e × − 0 0 0 3 ~ 2π 2π ωa ωe + iγe ω ωg0 + iγg E†(τ 0 )E†(ω)E(τ2)E(τ1) , (137) e,g0 Z Z − − − × 1 X E(ω ω )E†(ωa ω )E(ωa), (145) × − − − − When we assume unitary time evolution, we can replace the

3 t τ3 τ2 dephasing rates in Eqs. (D1)-(D4) by inﬁnitesimal imaginary 1 i ∞ S1,(II)(ω; Γ) = dt dτ3 dτ2 dτ1 factors γ . This allows us to use the identity 1/(ω + ~= −~ → Z−∞ Z−∞ Z−∞ Z−∞ i) = PP1/ω+iπδ(ω), and carry out the remaining frequency iωt e V (t)V (τ3)V †(τ2)V †(τ1) integrals. We arrive at ×

E†(ω)E†(τ3)E(τ2)E(τ1) , (138) dω (2) (2) × S1, (I)(ω; Γ) = T †(ωf )T (ωf ) , (146) 2π fg fg Z f X dω (1) (3) S1, (II)(ω; Γ) = Teg †(ωe)Teg (ωe) , (147) 3 t t τ 0 2π 1 i ∞ 2 Z e S (III)(ω; Γ) = dt dτ dτ 0 dτ 0 X 1, = − 1 2 1 dω (2) (2) ~ ~ S1, (III)(ω; Γ) = T †(ωg )T (ωg ) , (148) Z−∞ Z−∞ Z−∞ Z−∞ g0g 0 g0g 0 iωt 2π e V (τ 0 )V †(τ 0 )V (t)V †(τ ) Z g0 × 1 2 1 X dω (1) (3) E†(τ 0 )E(τ 0 )E†(ω)E(τ ) , (139) S1, (IV)(ω; Γ) = Teg †(ωe)Teg0 (ωe) . (149) × 1 2 1 2π Z e X 29

The details of sum-over-state expansion are presented in Ap- We thus have a similar configuration to an impulsive exper- pendix D. These results clarify the statement we made at the iment with four short well separated classical fields. Intro- end of section III.B.1: The χ(3)-absorption signal comprises ducing the LAP delays t = T T , t = T T and 3 4 − 3 2 3 − 2 matter transitions from the ground to the f-state - just like the t1 = T2 T1 we can calculate Eq. (150) by expanding TPIF signal -, but it also contains transitions to e- and g-states. perturbatively− in the dipole field-matter interaction Hamilto- The absorption measurement and the TPIF signal contain the nian (7). The two contributions to the signal are represented same information, only when transitions to the e- and g-states by the ladder diagrams shown in Fig. 20b. The corresponding can be neglected, as is the case when e is off-resonant. signal (150) can be read off these diagrams and is given by (LAP ) (LAP ) (LAP ) SDQC (Γ) = SDQCi (Γ) + SDQCii(Γ) where

2. Heterodyne detected four-wave mixing; the (LAP ) 1 ∞ ∞ ∞ ∞ double-quantum-coherence technique SDQCi (Γ) = 3 Re dt ds1 ds2 ds3 ~ 0 0 0 × Z−∞ Z Z Z Ψ E†(t s )E†(t)E (t s s )E (t s s s ) Ψ Time domain two dimensional (2D) spectroscopic tech- h | 3 − 3 4 2 − 3 − 2 1 − 3 − 2 − 1 | i iξ s3 iξfg s2 iξeg s1 niques (Mukamel, 2000), provide a versatile tool for exploring fe0 Ve0f Vge0 Vef∗ Vge∗ e− − − (151) the properties of molecular systems, such as photosynthetic × ee0f aggregates (Abramavicius et al., 2008; Engel et al., 2007) or X coupled (Hybrid-)nanostructures to semiconductor quantum 1 ∞ ∞ ∞ ∞ wells (Pasenow et al., 2008; Vogel et al., 2009; Yang et al., (LAP ) SDQCii(Γ) = 3 Re dt ds1 ds2 ds3 −~ 0 0 0 × 2008; Zhang et al., 2007). These techniques use sequences of Z−∞ Z Z Z short coherent pulses in comparison to the dephasing times of Ψ E4†(t)E3†(t s3)E2(t s3 s2)E1(t s3 s2 s1) Ψ the system. h | − − − − − − | i iξeg s3 iξfg s2 iξ s1 V V V ∗ V ∗ e− − − e0g (152) Earlier we had presented different delay scanning protocols × e0f ge0 ef ge (LOP and LAP) for multidimensional spectroscopy with en- eeX0f tangled photons. These protocols can be experimentally re- We have introduced the complex frequency variables ξij = alized using entangled photon pulse shaping, using collinear ω + iγ , where ω = ε ε are the transition frequencies ij ij ij i − j geometry and precise control of the phase, phase cycling. and γij are the dephasing rates. The signal may be depicted by These protocols allow to extract inter- and intraband dephas- its variation with various parameters of the field wavefunction. ing with high resolution by exciting doubly excited state dis- These are denoted collectively as Γ. Various choices of Γ lead tributions. We now present a different multidimensional time- to different types of 2D signals. These will be specifiedbelow domain spectroscopy that involve higher electronic states (see Eqs. (162) and (163)). manifold, but does not require excited populations. This is so called double-quantum-coherence (DQC) signals (Kim et al., 2009; Mukamel et al., 2007; Palmieri et al., 2009; Yang and b. The field correlation function for entangled photon pairs We Mukamel, 2008) where the system evolves in the coherence consider the pulsed entangled photon pairs described in sec- between ground and doubly excited state f g rather than tion II.F. Specifically, we here consider the wavefunction in- in a population f f . This technique renders| ih the| energies of troduced in (Keller and Rubin, 1997), where the two-photon | ih | single and doubly excited state energies accessible and reveals wavefunction Φ(i, kj) takes the form the correlations between single and doubly excited states . We i(ω(ki)+ω(kj ))ˆτij Φ(k , k ) = g uˆ((ω(k ) ω(k ))/2)e− show how pulsed entangled photons affect the two photon res- i j j − i i(ω(kj ))τij onances. Some bandwidth limitations of classical beams are e− A (ω(k ) + ω(k ) Ω (153)) · P i j − P removed and selectivity of quantum pathways is possible. iTij /2 ω uˆ(ω) = e · sinc(T /2 ω) (154) ij · where A ( ) is the pulse envelope and Ω the central fre- P ··· p a. The DQC signal In the following we use the LAP delay quency of the pump pulse used to generate the pairs. scanning protocol which monitors the density matrix. A pulse The correlation functions of the entangled fields reads shaper creates a sequence of four well separated chronolog- (Keller and Rubin, 1997) ically ordered pulses described by field operator Ej(t) = Ωp dω iω(t T ) 2 (s1+s2 t1) E (ω)e j , j = 1, 2, 3, 4. The control parameters 0 E1(s1 T1)E2(s2 T2) Ψ = V0e− − 2π j − − h | − − | i are their central times T1 < T2 < T3 < T4 and phases φ1, s1 + s2 t1 R rect(s2 s1 t1)Ap( − ) (155) φ2, φ3, and φ4 (see Fig. 20a). The DQC signal selects those − − 2 contributions with the phase signature φ +φ φ φ . The 1 2 − 3 − 4 signal is defined as the change in the time-integrated transmit- 1 T 0 < t < T ted intensity in component φ4, which is given by rect(t) = (0 otherwise 2 S = dt E†(t)V (t) (150) A (t) = exp( t2/(2σ2)), (156) h 4 i p − ~ Z 30

tions to the signal:

(LAP e) 1 2 2 S (Γ) = Re Ve0f Vge0 Vef∗ Vge∗ V0 Ap(ωfg Ωp) DQCi ~3 | | | − | eeX0f iξeg t1 iξ t3 iξfg t2 e− e− fe0 e− i(ωfg /2 ξeg )T iξfg T/2 (e − 1)e− − i(ω /2 ξ )T fg − eg i(ωfg /2 ξ )T iξfg T/2 (e − fe0 1)e− − (157) i(ω /2 ξ )T fg − fe0

(LAP e) 1 2 2 S (Γ) = Re Vge0 Ve0f Vef∗ Vge∗ V0 Ap(ωfg Ωp) DQCii ~3 | | | − | eeX0f iξeg t1 iξ t3 iξfg t2 e− e− e0g e− i(ωfg /2 ξeg )T iξfg T/2 (e − 1)e− − i(ω /2 ξ )T fg − eg i(ωfg /2 ξ )T iξfg T/2 (e − e0g 1)e− − (158) i(ω /2 ξ )T fg − e0g A (ω) 2 = exp( σ2ω2) (159) | p | − The control parameters Γ now include the delay times (t1, t2, t3) and the entanglement time T . For comparison, we present the same signal obtained with four impulsive classical pulses with envelopes Ej( ) j = · 0 1,..., 4 E2( ), E3( ) and E4( ) and carrier frequency Ωp (Abramavicius· et al.,· 2009): ·

(LAP c) 1 S = Re Ve0f Vge0 Vef∗ Vge∗ DQCi ~3 eeX0f 0 0 E∗(ω Ω )E∗(ω Ω ) 4 fe0 − p 3 e0g − p E (ω Ω0)E (ω Ω0) 2 fe − p 1 eg − p iξeg τ12 iξ τ34 iξfg τd e− e− fe0 e− (160)

(LAP c) 1 SDQCii = 3 Re Vge0 Ve0f Vef∗ Vge∗ ~ ee f (LAP ) ˜ ˜ X0 FIG. 20 (Color online) a) 2D signal SDQC (Ω1, Ω3) Eq. (162) (ab- 0 0 E∗(ω Ω )E∗(ω Ω ) solute value), showing correlation plots with different pump band- 4 e0g − p 3 fe0 − p widths σ as indicated. The bottom panel is multiplied by x6. b) E (ω Ω0)E (ω Ω0) (LAP ) 2 fe p 1 eg p Same as a) but for 2D signal S (Ω˜ 1, Ω˜ 2) Eq. (163). Parameters − − DQC iξeg τ12 iξ τ34 iξfg τd for simulations are given in Ref. (Richter and Mukamel, 2010). e− e− e0g e− (161) With Eqs. (157) and (158), we can compare the entangled photon and classical DQC signals. We ﬁrst note that Eq. (157) where V0 is given by ... It describes the pulsed counterpart scales with the intensity of the generating pump pulse, in con- of the cw-correlation function (52) we had described earlier, trast with the intensity square of the classical case Eq. (160). where the step-function rect (t) is amended by the ﬁnite pulse In the classical case the signal is limited by the bandwidths of amplitude Ap. the four pulses (cf. Eqs. (160-161)), which control the four

transitions (ωeg, ωfe, ωe0g, ωfe0 ) in the two photon transitions inside the pulse bandwidth (Abramavicius et al., 2009). In Eq.(157) bandwidth limitations of the envelopes are only im- c. Simulated 2D Signals Below we present simulated DQC posed through the bandwidth of entangled photon pair Ap( ) signal for a model trimer system with parameters given in Ref. and the limitation is only imposed on the two photon transi-· (Richter and Mukamel, 2010). By inserting Eq. (155) in Eq. tion ωfg, leading to a much broader bandwidth for the ωeg and (151) - (152) and carrying out all integrations, assuming that ωfe transitions, if the ωfg transition is within the generating γ σ, we arrive at the ﬁnal expression for the two contribu- pump pulse bandwidth. 31

We illustrate this effect in Fig. 20a for the following 2D D. Multiple photon counting detection signal: Another class of multidimensional signals is possible by detecting sequences of individual photons emitted by an opti- ∞ ∞ ˜ ˜ (LAP ) ˜ ˜ it1Ω1+it3Ω3 SDQC (Ω1, Ω3) = dt1 dt3S(t1, t3)e (162), cally driven system. With proper gating, each photon j can be 0 0 Z Z characterized by a frequency ωj and a time tj. By detecting N photons we thus obtain a 2N dimensional signal parametrized

ωeg resonances are seen along Ω˜ 1 and ωeg and ωfe on axis by ω1, t1, ..., ωN , tN . Unlike coherent multidimensional het- Ω˜ 3. As the bandwidth is reduced, we only get contributions erodyne signals which are parametrized by delays of the in- from the doubly excited state resonant to the generating pump coming fields, these incoherent signals are parametrized by pulse. This results in four identical patterns along the Ω1 axis. the emitted photons. The photon time tj and frequency ωj are All peaks are connected to the same doubly excited states. not independent and can only be detected to within a Fourier More precisely we get four contributions along the Ω˜ 1 axis uncertainty ∆ωj∆tj > 1. This poses a fundamental limit on the temporal and spectral resolution. We derive these signals connected to the transitions ωf3e1 , ωf3e2 overlapping with and connect them to multipoint 2N dipole correlation func- ωe3g, ωe2g overlapping with ωf3e3 and ωe1g. The remaining transitions are not affected by the reduced pump bandwidth. tions of matter. The Fourier uncertainty is naturally built in Here the narrow bandwidth of the pump can be used to select by a proper description of photon gating and need not be im- contributions in the spectra connected to a specific doubly ex- posed in an ad hoc matter as is commonly done. cited state. In Fig. 20b we display a different signal: A semiclassical formalism for photon counting was first derived by (Mandel, 1958, 1959). The full quantum mechanical description of the field and photon detection was devel- ∞ ∞ ˜ ˜ (LAP ) ˜ ˜ it1Ω1+it2Ω2 oped by Glauber (Glauber, 2007b). The theory of the electro- SDQC (Ω1, Ω2) = dt1 dt2S(t1, t2)e (163), Z0 Z0 magnetic field measurement through photoionization and the resulting photoelectron counting has been developed by Kel- ley and Kleiner (Kelley and Kleiner, 1964). The experimen- ωeg resonances are now seen along Ω˜ 1 and ωfg in Ω˜ 2. This is similar to Fig. 20f except that here we see the singly excited tal application to normal and time ordered intensity correlation measurements was given in the seminal work of Kimble state contributions ωe1g,ωe2g, ωe3g to the selected doubly ex- et al. (Kimble et al., 1977). According to these treatments cited state f3 along a single row. free-field operators, in general, do not commute with source- Bandwidth limitations on the of the singly excited state quantity operators. This is the origin of the fact that the normal transitions ω and ω are only imposed indirectly by the eg fe and time ordering of the measured field correlations, accord- factors in Eqs. (157-158) which depend on the entanglement ing to the Kelley-Kleiner theory (Kelley and Kleiner, 1964), time T . These become largest for ω = 2ω . fg fe are transformed into normal and time ordered source quanti- The factors in Eq.(157-158) which depend on the entan- ties occurring inside the integral representations of the filtered glement time T contain an interference term of the form source-field operators. This constitutes the back reaction of i(ω γ)T (e − 1), where ω is a material frequency (see sec- the detector on the field state (Cohen-Tannoudji et al., 1992). − tions II.J and III.B.2). If we now vary the entanglement time, An ideal photon detector is a device that measures the radia- some resonances will interfere destructively for values of the tion field intensity at a single point in space. The detector size entanglement time which match the period of ω. One can should be much smaller than spatial variations of the field. therefore use the entanglement times to control selected res- The response of an ideal time-domain photon detector is inde- onances. This holds only as long the entanglement times are pendent of the frequency of the radiation. not much bigger than the dephasing time, since in this case the The resolution of simultaneous frequency and time domain signal will be weak. The frequencies ω can be in the contribut- measurements is limited by the Fourier uncertainty ∆ω∆t > ing diagrams ω /2 ω or ω /2 ω (which is differs fg − eg fg − fe 1. A naive calculation of signals without proper time and fre- from the first frequency only by a sign) for different combi- quency gating can work for slowly varying spectrally broad nation of the states e, e0 and f. By varying T , we expect an optical fields but otherwise may yield unphysical negative sig- oscillation of the magnitudes of resonances with different fre- nals (Eberly and Wodkiewicz, 1977). In Ref. (Mukamel et al., quencies.The details of manipulation of entanglement time as 1996), the mixed time-frequency representation for the coher- a control parameters have been studied in Refs. (Richter and ent optical measurements with interferometric or autocorre- Mukamel, 2010; Schlawin et al., 2013) lation detection were calculated in terms of a mixed material Previously, to utilize entangled photons for DQC signal response functions and a Wigner distribution for the incoming we used two pulsed entangled photon pairs (Richter and pulses, the detected field and the gating device. Multidimen- Mukamel, 2010) (k1, k2) and (k3, k4). With the pulse shaping sional gated fluorescence signals for single-molecule spec- described in section II.G, a single shaped entangled photon troscopy have been calculated in Ref. (Mukamel and Richter, pair is sufficient to realize any time-domain four-wave mixing 2011b). signals. The standard Glauber’s theory of photon counting and 32 correlation measurements (Glauber, 2007a; Mollow, 1972; space and contains information about system evolution prior Scully and Zubairy, 1997) is formulated solely in the radi- to the detection (e.g. photon generation process, etc.). Γj, ation field space (matter is not considered explicitly). Sig- j = 1, ...N represents other parameters of the detectors such j j nals are related to the multi-point normally-ordered field cor- as e.g. bandwidth (σT and σω are the time gate, and frequency relation function, convoluted with time and frequency gating gate bandwidths, respectively). The time-and-frequency gated spectrograms of the corresponding detectors. This approach photon number superoperator is given by assumes that the detected field is given. Thus, it does not address the matter information and the way this field has been nˆt,ω = dt0 dτD(t, ω; t0, τ)ˆn(t0, τ). (165) generated. Temporally and spectrally resolved measurements Z Z can reveal important matter information. Recent single photon spectroscopy of single molecules (Fleury et al., 2000; Let- Here D(t, ω, t0, τ) is a detector time-domain spectrogram (the tow et al., 2010; Rezus et al., 2012) call for an adequate mi- ordinary function, not an operator) which takes into account croscopic foundation where joint matter and field information the detector’s parameters, and is given by could be retrieved by a proper description of the detection pro- D(t, ω, t , τ) = cess. 0

A microscopic diagrammatic approach may be used for cal- dω00 iω00τ 2 e− Ff (ω00, ω) F ∗(t0 + τ, t)Ft(t0, t), (166) culating time-and-frequency gated photon counting measure- 2π | | t Z ments (Dorfman and Mukamel, 2012a). The observed signal where and are time and frequency gating functions that can be represented by a convolution of the bare signal and Ft Ff are characterized by central time and frequency and de- a detector spectrogram that contains the time and frequency t ω tection bandwidths and , respectively. Note that in gate functions. The bare signal is given by the product of σT σω Eq. (166) the time gate is applied first, followed by the fre- two transition amplitude superoperators (Mukamel and Ra- quency gate. Similar expression can be written if the order in hav, 2010) (one for bra and one for ket of the matter plus gating is reversed. is a bare photon number superop- field joint density matrix), each creating a coherence in the nˆ(t, t0) erator defined in terms of the bare field operators as field between states with zero and one photon. By combining the transition amplitude superoperators from both branches nˆ(t0, τ) = Eˆ† (t0 + τ)Eˆ (t0)ρ(t0). (167) of the loop diagram we obtain the photon occupation num- sR s0L ber that can be detected. The detection process is described Xs,s0 in the joint field and matter space by a sum over pathways Details may be found in Appendix E. each involving a pair of quantum modes with different time orderings. The signal is recast using time ordered superoperator products of matter and field. In contrast to the Glauber 2. Photon counting and matter dipole correlation functions theory that uses normally ordered field operator, the microscopic approach of (Dorfman and Mukamel, 2012a) is based To connect the photon coincidence counting (PCC) signals on time-ordered superoperators. Ideal frequency domain de- to matter properties one needs to expand the density opera- tection only requires a single mode (Mukamel and Richter, tor in Eq. (167) in perturbative series over field-matter inter- 2011b). However, maintaining any time resolution requires a actions. We first calculate the time-and-frequency resolved superposition of several field modes that contain the pathway emission spectra information. This information is not directly accessible in the standard detection theory that operates in the field space alone n = dt dτD(t, ω; t , τ)n(t , τ), (168) (Glauber, 2007b). t,ω 0 0 0 Z Z

where the bare photon number n(t0, τ) nˆ(t0, τ) T is an 1. Photon correlation measurements using gated photon expectation value of the bare photon number≡ hT operatori with number operators respect to the total density matrix. The leading contribution is coming from the second-order in field matter interactions Time-and-frequency gated N- th order photon correlation with vacuum modes [see diagram in Fig. 21b] measurement performed at N detectors characterized by central time tj and central frequency ωj, j = 1, ..., N is defined 1 t0 t0+τ as n(t0, τ) = dt1 dt2 V †(t2) V (t1) ~2 h h i Z−∞ Z−∞ (N) g (t1, ω1, Γ1; ..., tN ; ωN , ΓN ) Eˆs (t2)Eˆ† (t0 + τ)Eˆs(t0)Eˆ†(t1) v, (169) × h 0 s0 s i nˆ ...nˆ s,s0 = hT t1,ω1 tN ,ωN iT , (164) X nˆ ... nˆ hT t1,ω1 iT hT tN ,ωN iT where we had utilized superoperator time ordering and ... = h i where ... T = Tr[...ρT (t)] and ρT (t) represents the total Tr[...ρ(t)] where ρ(t) is the density operator that excludes vac- densityh matrixi of the entire system in joint field plus matter uum modes and ... = Tr[...ρ (t)] with ρ (t) = 0 0 is the h iv v v | ih | 33

FIG. 21 (Color online) (a) Schematic of time-and-frequency resolved photon coincidence measurement. (b) Loop diagram for the bare signal (E15) in a gated measurement. (c) - Loop diagram for correlated two photons measurement (E26). Dashed lines represent the the dynamics of the system driven by the field modes. τi and τs can be either positive or negative. density matrix of the vacuum modes. Using the bosonic com- After tracing over the vacuum modes we obtain mutation relations introduced in section I.A [see Eq. (6)], and 2 2 moving to the continuous density of states, one can obtain nˆ(t0 , τ1)ˆn(t0 , τ 0 ) T = (ω1) (ω2) hT 1 2 2 i D D 2 V †(t20 + τ2)V †(t10 + τ1)V (t10 )V (t20 ) . (174) n(t0, τ) = (ω) V †(t0 + τ)V (t0) , (170) × h i D h i The gated coincidence signal (172) is finally given by where (ω) = 1 ˜(ω) is a combined density of states eval- D 2π D uated at the central frequency of the detector ω for smooth (2) S (t1, ω1; t2, ω2) nˆt ,ω nˆt ,ω T enough distribution of modes. The corresponding detected ≡ hT 1 1 2 2 i 2 2 (1) signal (165) is given by = (ω ) (ω ) dt0 dτ D (t ω ; t0 , τ ) D 1 D 2 1 1 1 1 1 1 (1) Z Z S (t, ω) nt,ω = (2) ≡ dt20 dτ2D (t2, ω2; t20 , τ2) 2 × dt0 dτD(t, ω; t0, τ) (ω) V †(t0 + τ)V (t0) . Z Z D h i V †(t20 + τ2)V †(t10 + τ1)V (t10 )V (t20 ) . (175) Z Z (171) × h i Therefore, the fundamental material quantity that yields the One can similarly calculate the second-order bare correla- emission spectra (171) is a two-point dipole correlation function function tion given in Eq. (170), and for the coincidence g(2)- (1) measurement (175) it is the four-point dipole correlation func- nˆ nˆ = dt0 dτ D (t ω ; t0 , τ ) hT t1,ω1 t2,ω2 iT 1 1 1 1 1 1 tion in Eq. (174). Sum-over-state expansion of these expres- Z Z sions is given in appendix E. (2) dt0 dτ D (t , ω ; t0 , τ ) × 2 2 2 2 2 2 Z Z nˆ(t0 , τ )ˆn(t0 , τ 0 ) . (172) × hT 1 1 2 2 iT 3. Connection to the physical spectrum The bare PCC rate nˆ(t , τ )ˆn(t , τ ) can be read off the 10 1 20 20 T An early work (Eberly and Wodkiewicz, 1977), had argued diagram shown in Fig.hT 21c. The leadingi contribution requires that detector gating with finite bandwidth must be added to a fourth-order expansion in field-matter interactions describe the real detector. In recent work (Gonzalez-Tudela´ 1 t10 t10 +τ1 et al., 2015; del Valle et al., 2012) used a two-level model de- nˆ(t10 , τ1)ˆn(t20 , τ20 ) T = dt1 dt3 tector with a single parameter Γ that characterize both time- hT i ~4 Z−∞ Z−∞ and-frequency detection is comsidered. This was denoted the t0 t0 +τ 2 2 2 physical spectrum (178) which we shall derive in the follow- dt dt V †(t )V †(t )V (t )V (t ) × 2 4h 4 3 1 2 i ing. It can be recovered from our model by removing the time Z−∞ Z−∞ gate Ft = 1 and using a Lorentzian frequency gate Er (t4)Es (t3)E† (t0 + τ2)E† (t0 + τ1) × h 0 0 r0 2 s0 1 Xs,s0 Xr,r0 i Ff (ω, ω0) = . (176) Es(t0 )Er(t0 )E†(t1)E†(t2) v. (173) ω + ω + iΓ/2 × 1 2 s r i 0 34

Using the physical spectrum, the time-and-frequency resolved delays t1 t2 yields a 2D spectroscopy tool capable of mea- photon coincidence signal is given by suring ultrafast− dynamics. Third, the superoperator algebra allows to connect the gated field correlation function g(2) (ω , ω ; τ) = Γ1Γ2 1 2 (tf) (tf) (tf) (tf) ¯ ¯ † ¯ † ¯ ¯ ¯ AG(¯ω1, ω¯2, t1, t2) = Eω¯1 (t1)Eω¯2 (t2)Eω¯2 (t2)Eω¯1 (t1) Aˆ† (t)Aˆ† (t + τ)A (t + τ)Aˆ (t) h i ω1,Γ1 ω2,Γ2 ω2,Γ2 ω1,Γ1 (181) limt h i , →∞ Aˆ† (t)Aˆω ,Γ (t) Aˆ† (t + τ)Aˆω ,Γ (t + τ) h ω1,Γ1 1 1 ih ω2,Γ2 2 2 i with the bare correlation function (179), with time-and- (177) frequency gates (arbitrary, not necessarily Lorentzian) as well where as material response that precedes the emission and detection of photons. The superoperator expressions require time- t ordering, and can be generalized to correlation functions of ˆ (iω Γ/2)(t t1) ˆ Aω,Γ(t) = dt1e − − E(t1), (178) field operators that are not normally ordered. Superoperators Z−∞ provide an effective bookkeeping tool for field-matter interac- is the gated field. This model provides a simple benchmark for tions prior to the spontaneous emission of photons. We next finite-band detection, which has several limitations. First, the apply it to the detection of photon correlations. Finally, as we time and frequency gating parameters are not independent, un- show in the next section, PCC can be recast in terms of matter like the actual experimental setup, where frequency filters and correlation functions by expanding the total density matrix op- avalanche photodiodes are two independent devices. Second, erator in a perturbation series, and tracing the vacuum modes. this method does not address the generation and photon band- This way, photon counting measurements can be related to the width coming from the emitter, as the analysis is performed matter response which is the standard building block of non- solely in the field space. Finally, the multi-photon correlation linear spectroscopy. function presented in (Gonzalez-Tudela´ et al., 2015) is stationary. For instance, the four-point bare correlation function E. Interferometric detection of photon coincidence signals AB(ω1, ω2, t1, t2) = Eω† 1 (t1)Eω† 2 (t2)Eω2 (t2)Eω1 (t1) h (179)i Photon coincidence signals also known as biphoton signals depends on four times and four frequencies. After gating sug- (Kalachev et al., 2007; Scarcelli et al., 2003; Slattery et al., gested in (Gonzalez-Tudela´ et al., 2015; del Valle et al., 2012) 2013; Yabushita and Kobayashi, 2004) became recently available as a tool for nonlinear spectroscopy. In a typical setup, the correlation function (179) is recast using CB(ω1, ω2, t2 − a pair of entangled photons denoted as E , and E generated t1), which only depends on the time difference t2 t1, which s r is an approximation for stationary fields. This− model also by PDC are split on a beam splitter [see Fig. 22a)]. One 1 photon E is transmitted through the molecular sample and works if t Γ− , which means that Γ cannot approach zero s (perfect reflection in Fabri Perot cavity). It also works when then detected in coincidence with Er. In order to use it as a spectroscopic tool, a frequency filter can be placed in front of Γτ0 1 where τ0 is the scale of change in the field envelope. For comparison, the photon coincidence counting (PCC) one of the detectors which measures the spectrum. This type (164) for N = 2 reads of signal shows a number of interesting features: First, coincidence detection improves the signal-to-noise ratio (Kalash- et al. (2) nˆt1,ω1 nˆt2,ω2 nikov , 2014). Second, it is possible to make the two g (t1, ω1, Γ1; t2; ω2, Γ2) hT i , (180) detectors operate in very different spectral regions and at dif- nˆt1,ω1 nˆt2,ω2 hT ihT i ferent spatial locations (Kalachev et al., 2007). For example, which depends on two time t1, t2 and two frequency ω1, ω2 to measure the spectroscopic properties of a sample in vacuum arguments. The theory summarized above which gives rise ultraviolet (VUV) range, it is not necessary to set a spectrom- to Eq. (180) has several advantages compared to the physi- eter in a vacuum chamber and control it under the vacuum cal spectrum used by (Gonzalez-Tudela´ et al., 2015; del Valle condition. Instead, provided that a entangled photon pair con- et al., 2012). First, independent control of time and frequency sists of a VUV and a visible photon, only the latter is to be gates (with guaranteed Fourier uncertainty for the time and resolved by a spectrometer under usual conditions. Another frequency resolution) along with the fact that the bare pho- advantage is when spectroscopic measurements are to be per- ton number operator depends on two time variables nˆ(t, τ) al- formed in infrared range. The power of the light source must lows to capture any dynamical process down to the very short often be very low to prevent possible damage of a sample, scale dynamics in ultrafast spectroscopy applications. Sec- but an infrared photodetector is usually noisy. Photon coin- ond, the gating (165) provides a unique tool that can capture cidence measurements involve the lowest intensities of light - nonequlibrium and non-stationary states of matter which can single photons, and can overcome the noise. be controlled by gating bandwidths. In this case a series of In the following, we present photon coincidence version of frequency ω1, ω2 correlation plots (keeping the central fre- three signals: linear absorption, pump-probe, and Femtosec- quencies of the spectral gates as variables) for different time ond Stimulated Raman Signals (FSRS). 35

0 In the case of the linear signal, the joint detection of two E entangled photons provides linear absorption information if s Es one of the photons is transmitted through a molecular sample. If the coincidence gate window accepts counts for a time T , PDC then the joint detection counting rate, Rc, between detectors r Er and s is proportional to BS r T T 2 (a) R dt dt 0 a†(t )a†(t ) ψ , (182) c ∝ 1 2|h | s 1 r 2 | i| Z0 Z0 where ψ is a two-photon entangled state (33). The gating window T is typically much larger than the reciprocal bandwidth of the light or the expected dispersive broadening

R dω dω 0 a (ω )a (ω ) ψ 2. (183) c ∝ 1 2|h | s 1 r 2 | i| (b) Z Z Denoting the spectral transfer functions of the sample and monochromator HS(ω) and HM (ω), respectively. In this case, 1 as(ω1) = a˜s(ω1)HS(ω1) (184) √2 and i ar(ω2) = a˜r(ω2)HM (ω2) (185) (c) √2 provided that the signal and idler photons are separated by a 50/50 beam splitter. For narrowband down conversion ω1 ω2 Φ(ω1, ω2) = F −2 δ(ω1 + ω2 2ω0). The coincidence counting rate is then given by −

R dΩ H (ω + Ω)H (ω Ω)F (Ω) 2. (186) c ∝ | S 0 M 0 − | Z

(d) Now, assume that HM (ω) is much narrower than HS(ω) and Φ(˜ ω , ω ), we can set H (ω) = δ(ω ω ), and the fre- 1 2 M − M quency ωM does not exceed the frequency range in which function F (ω) is essentially nonzero. Dividing the coinci- FIG. 22 (Color online) Linear absorption experiment of (Kalachev dence counting rate with a sample, R , by one without et al., 2007) with coincidence detection. The entangled photon pair c,sample the sample, R , we obtain the absorption spectrum in beams Es and Er are split on a beam splitter. Es is employed as c a probe transmitted through the sample, while Er is detected in coincidence - (a).The absorption spectrum of Er3+ ion in YAG crys- Rc,sample 2 SILA(ωM ) = HS(2ω0 ωM ) , (187) tal around 650 nm obtained by single photon counting (blue ﬁlled Rc ∝ | − | squares, left Y-axis) and coincidence counting (red open squares, right Y-axis) at various values of signal-to-noise ratio in the chan- where subscript ILA marks the interferometric nature of the nel with the sample: 1/2 (b), 1/5 (c), and 1/30 (d) photon coincidence detection combined with linear absorption measurement. Thus, the joint detection counting rate reproduces the spectral function of the sample, which is reversed in 1. Coincidence detection of linear absorption frequency with respect of the pump frequency, provided that the line-width of the pumping ﬁeld as well as bandwidth of Linear absorption is the most elementary spectroscopic the monochromator are narrow enough for resolving the ab- measurement. Combined with photon coincidence detection sorption spectrum features. [see Fig. 22a)], it yields interesting results: Below we present In (Kalachev et al., 2007) the coincidence signals were used the simplest intuitive phenomenological approach. In follow- to measure the spectroscopic properties of YAG:Er3+ crystal. ing sections we present more rigorous microscopic derivation In order to demonstrate the advantage of coincidence detec- for pump-probe and Raman signals. tion in the presence of an enhanced background noise. Figs. 36

22b)-d) show the central part of the absorption spectrum mea- state g is dipole-coupled to an electronic excited state e, which sured in two ways: using the coincidence counting as de- is connected to two spin states and undergoing relaxation scribed before, and using the single-photon counting, when (Sandaˇ and Mukamel, 2006). We↑ additionally↓ consider a dou- the sample was placed above the monochromator in the idler bly excited state f, which is dipole-coupled to both e, and channel. The signal-to-noise ratio in the channel was changed e, [see Fig. 23b)]. The electronic states are damped| ↑i by from 1/2 to 1/30 in both cases. It is evident from these ex- a| dephasing↓i rate γ. We assume the low-temperature limit, perimental data that the standard classical method which suf- where only the decay from to is allowed (Dorfman et al., fers from a high noise level does not allow one to obtain any 2013b). The decay is entirely↑ incoherent,↓ such that the de- spectroscopic information, but the coincidence counting mea- scription may be restricted to the two spin populations surement in contrast does not undergo the reduction in resolu- ˆ=(1, 0)T and ˆ=(0, 1)T . The ﬁeld correlation function| ↑ih↑ tion and is resistant to noise. The example of YAG:Er3+ has |(190) is then given| ↓ih↓ by | been later extended to plasmonic nanostructures (Kalashnikov 2 2 γ(t1+2t2) F (t , t ) = µ µ e− et al., 2014). 2 1 | ge| | ef |

iω+t2 2iδ kt1 (k+iω )t2 iω+t2 e− + e− e− − e− , × k + 2iδ − 2. Coincidence detection of pump-probe signals h i(191)

Consider the setup depicted in Fig. 23a). Unlike the linear where δ the energy difference between the two spin states, and absorption case here the probe photon is sent through the sam- ω = ωfe δ. Note that, since we monitor the f e transition, ± ± − ple, which has previously been excited by a classical ultrafast the detected frequency will increase in time, from ω to ω+. laser pulse, and then detected (Schlawin et al., 2015). In the following, we use these results to first simulate− the The interferometric pump probe (IPP) signal with photon classical pump-probe signal, and then the two-photon count- coincidence detection is dispersed spectrally by placing spec- ing signal with entangled photons. For the former case, we tral filters in front of both detectors, and our signal is given by consider a classical Gaussian probe pulse the change to this two-photon counting rate that is governed by a four-point correlation function of the field (Cho et al., 1 2 2 Epr(ω) = exp (ω ω0) /2σ . (192) 2014a; Mosley et al., 2008) √2πσ2 − − We chose the following system parameters: ωfe = 11, 000 Er†(ωr)Es†(ω)Es(ω)Er(ωr) . (188) 1 1 1 1 cm− , δ = 200 cm− , k = 120 cm− , and γ = 100 cm− . The center frequency is fixed at the transition frequency Here, ω/ωr denotes the detected frequency of the respective ω0 spectral filter, and the brackets represent the expecta- ωfe, and we vary the probe bandwidth. Panel a) shows the 1 h· · · i signal for σ = 1, 000 cm− . Two peaks at ω δ corre- tion value with respect to the transmitted fields. To obtain the fe ± desired pump-probe signal in a three-level system, we used a spond to the detected frequency, when the system is either in et al. the upper state (at ωfe δ), or in the lower state (ωfe + δ). third order perturbation theory (Schlawin , 2015). The − first two interactions are with the classical pump pulse, which Due to the spectrally dispersed detection of the signal, the resonance widths are given by the linewidth , and not the much is taken to be impulsive, a(t) = aδ(t), and the third is with γ E E broader probe pulse bandwidth . For very short time delays the probe centered at t = t0. Assuming Es to be far off- σ resonant from the e g transition, we obtain only the single t0, both resonances increase, until the probe pulse has fully diagram shown in Fig.− 23c), which reads passed through the sample. Then the resonance at 10, 800 1 cm− , i.e. the state e, , starts to decay rapidly, while the 3 | ↑i 1 resonance at 11, 200 cm− peaks at longer delay times due 2 i 2 ∞ iω(t t0) S (ω, ω ; t ) = dt e − IPP r 0 − = − |Ea| to its initial population by the upper state. For longer delays, ~ ~ Z0 t both resonance decay due to the additional dephasing. dτF (t τ, τ) E†(ω )E†(ω)E (τ)E (ω ) . (189) The two-photon counting signal with entangled photons of- × − r r s s r r Z0 fers novel control parameters: The dispersed frequency ω of

We have deﬁned the matter correlation function, beam 1, the pump frequency ωp and its bandwidth σp loosely correspond to the classical control parameters, i.e. the central 2 2 frequency ω0 and bandwidth σ. In addition, we may vary the F (t τ, τ) = µge µef Gfe(t τ)Gee(τ) env, (190) − | | | | − entanglement time T and the detected frequency of the refer- where µge and µef denote the dipole moments connecting ence beam ωr. ground state with the singly excited states manifold, as well Fig. 24 depicts the signal (189) obtained from entangled as single with doubly excited state manifold, respectively. photons with T = 90 fs for different time delays t0. For com- env denotes the average with respect to environmental de- parison, we show the classical pump-probe with bandwidth h· · · i 1 1 grees of freedom, obtained from tracing out the bath. Here, σ = 1, 000 cm− in the left column, and with 100 cm− in we employ a stochastic Liouville equation (Tanimura, 2006) the right column. The TPC signals are normalized with re- which represents the two-state jump (TSJ) model: A ground spect to the maximum value of the signal at t0 = 3 fs and 37

FIG. 23 (Color online) a) The IPP setup: the entangled photon pair in beams Es and Er are split on a beam splitter. A classical, actinic, ultrafast pulse Ea excites the sample, and Es is employed as a probe in a pump-probe measurement, while Er is detected in coincidence. b) The level scheme for the two-state jump (TSJ) model considered in this work. The g − e transition is far off-resonant from the spectral range of the entangled photon wavepacket, which only couples to the e−f transition. c) Diagram representing the pump-probe measurement. [taken from (Schlawin et al., 2015)]

1 ωr = 11, 400 cm− . The classical signal is normalized to 2014). The measurement uses a pair of entangled photons, its peak value at zero time delay, and the TPC signals to the one (signal) photon interacts with the molecule, and acts as the 1 signal with ωr = 11, 400 cm− at zero delay. As becomes broadband probe, while the other (idler) provides a reference apparent from the figure, a broadband classical probe pulse for the coincidence measurement. By counting photons, IF- (left column) cannot excite specific states, such that the two SRS can measure separately the gain and loss contributions to resonances merge into one band. A narrowband probe (right the Raman signal (Harbola et al., 2013) which is not possible column), on the other hand, cannot resolve the fast relaxation with classical FSRS signals that only report their sum previ- at all, and only shows the unperturbed resonance at ωfe. Inter- ousy (i.e. the net gain or loss). We showed how the entangled ferometric signals, however, can target the relaxation dynam- twin photon state may be used to manipulate two-photon ab- ics of the individual states. sorption ω1 + ω2 type resonances in aggregates (Dorfman and Mukamel, 2014b; Lee and Goodson, 2006; Saleh et al., 1998; Schlawin et al., 2013) but these ideas do not apply to Raman ω ω resonances. 3. Coincidence detection of Femtosecond Stimulated Raman 1 − 2 Signals In FSRS, an actinic resonant pulse a first creates a vibrational superposition state in an electronicallyE excited state (see So far, we have demonstrated how coincidence detection Fig. 25a,b). After a variable delay τ, the frequency resolved can enhance linear absorption and pump-probe signals. We transmission of a broadband (femtosecond) probe Es in the presence of a narrowband (picosecond) pump shows ex- now demonstrate the power of this interferometric detection Ep for stimulated Raman signals commonly used to probe molec- cited state vibrational resonances generated by an off-resonant ular vibrations. Applications include probing time-resolved stimulated Raman process. The FSRS signal is given by photophysical and photochemical processes (Adamczyk et al., (Dorfman et al., 2013a) 2009; Kukura et al., 2007; Kuramochi et al., 2012; Schreier S (ω, τ) et al., 2007), chemically specific biomedical imaging (Cheng FSRS 2 ∞ i R et al., 2002), remote sensing (Arora et al., 2012; Pestov et al., iω(t τ) H0 (τ)dτ = dte − s∗(ω) p(t)α(t)e− ~ − , 2008). Considerable effort has been devoted to increasing the ~I hT E E i Z−∞ sensitivity and eliminating off-resonant background, thus im- (193) proving the signal-to-noise ratio and enabling the detection where α is the electronic polarizability, denotes the imag- of small samples and even single molecules. Pulse shaping inary part, and = E is expectationI value of the probe (Oron et al., 2002; Pestov et al., 2007) and the combina- s s field operator withE respecth i to classical state of light (hereafter tion of broad and narrow band pulses (technique known as denotes classical fields and E stands for quantum fields). Femtosecond Stimulated Raman Spectroscopy (FSRS) (Di- HE is the Hamiltonian superoperator in the interaction picture etze and Mathies, 2016)) have been employed. Here, we 0 which,− for off resonance Raman processes, can be written as present an Interferometric FSRS (IFSRS) technique that com- bines quantum entangled light with interferometric detection H0(t) = αEs†(t) p(t) + a∗(t)V + H.c., (194) (Kalachev et al., 2007; Scarcelli et al., 2003; Slattery et al., E E 2013; Yabushita and Kobayashi, 2004) in order to enhance the where V is the dipole moment, α is the off resonant polariz- resolution and selectivity of Raman signals (Dorfman et al., ability. Formally, this is a six-wave mixing process. Expand- 38

The two terms correspond to the two diagrams in Fig. 25c). All relevant information is contained in the two four point correlation functions

F (t , t , t ) = VG†(t )αG†(t )αG(t )V † , (197) i 1 2 3 h 1 2 3 i F (t , t , t ) = VG†(t )αG(t )αG(t )V † , (198) ii 1 2 3 h 1 2 3 i where the retarded Green’s function G(t) = iHt ( i/~)θ(t)e− represents forward time evolution with the free-molecule− Hamiltonian H (diagrams (1, 1)a, (1, 1)b) and G† represents backward evolution. Fi involves one forward and two backward evolution periods and Fii contains two forward followed by one backward propagation. Fi and Fii differ by the ﬁnal state of the matter (at the top of each diagram). In Fi (Fii) it is different (the same) as the state after preparation by the actinic pulse.

a. Photon correlation measurements. In IFSRS, the probe pulse Es belongs to a pair of entangled beams generated in degenerate type-II PDC. The polarizing beam splitter (BS) in Fig. 25d) then separates the orthogonally polarized photons. The horizontally polarized beam Es is propagating in the s arm of the interferometer, and interacts with the molecule. The ver- tically polarized beam Er propagates freely in the r arm, and serves as a reference. IFSRS has the following control knobs: −1 FIG. 24 (Color online) a) TPC signals with σp = 2, 000 cm and the time and frequency parameters of the single photon de- −1 −1 T = 90 fs, ωr = 10, 400 cm (blue, dashed) and 11, 400 cm tectors, the frequency of the narrowband classical pump pulse (red, dot-dashed) as well as the classical pump-probe signal (black, ωp, and the time delay T between the actinic pulse a and the σ = 1, 000 −1 E solid) with cm are plotted vs. the dispersed frequency probe Es. ω with a time delay set to t0 = 3 fs. b) Same for t0 = 30 fs, As discussed in section III.D.1, the joint time-and- c) 60 fs, d) 90 fs. e)-h) Same as a)-d), but with classical band- frequency gated detection rate of N photons (N = 0, 1, 2) in width σ = 100 cm−1. The classical signal is normalized, such s s detector s and a single photon in r when both detectors have that its maximum value at t0 = 0 is equal to one. Similarly, the TPC signal are normalized to the maximum value of the signal with narrow spectral gating is given by −1 ωr = 11, 400 cm at t0 = 0.[taken from (Schlawin et al., 2015)] (Ns,1) SIF SRS(¯ωs1 , ..., ω¯sNs , ω¯r, Γi)

Ns 2 2 2 i R ing the signal (193) to sixth order in the fields . we ∞ H0 (τ)dτ s p a = E†(¯ω )E (¯ω ) E†(¯ω )E (¯ω )e− ~ − . ∼ E E E hT r r r r s sj s sj −∞ i obtain the classical FSRS signal j=1 Y (199) t t t0 (i) 2 ∞ SFSRS(ω, τ) = dt dτ1 dt0 dτ2 ~I where Γi stands for the incoming light beam parameters. In Z−∞ Z−∞ Z−∞ Z−∞ iω(t τ) the standard Glauber’s approach (Glauber, 2007a), the corre- e − (t) ∗(t0) ∗(τ ) (τ ) ∗(ω) (t0) × Ep Ep Ea 2 Ea 1 Es Es lation function is calculated in the field space using normally- Fi(t0 τ2, t t0, t τ1), (195) ordered field operators. The present expressions in contrast × − − − are given in the joint field and matter degrees of freedom, and the bookkeeping of the fields is instead solely based on t t t0 (ii) 2 ∞ the time ordering of superoperators. Normal ordering is never S (ω, τ) = dt dτ2 dt0 dτ1 FSRS ~I used. Z−∞ Z−∞ Z−∞ Z−∞ iω(t τ) Expansion of Eq. (199) in the number of the field matter e − p(t) ∗(t0) a(τ1) ∗(τ2) ∗(ω) s(t0) × E Ep E Ea Es E interactions depicted by loop diagrams in Fig. 25e) yields for Fii(t τ2, t t0, t0 τ1). (196) × − − − Ns = 0 - Raman loss (no photon in the molecular arm) 39

FIG. 25 (Color online) Top row: FSRS level scheme for the tunneling model - a, pulse conﬁguration - b, loop diagrams (for diagram rules see Appendix A) - c. d and e - the same as b, and c but for IFSRS. The pairs of indices (0, 1) etc. indicate number of photons registered by detectors s and r in each photon counting signal: (Ns,Nr)

t t0 (0,1) 1 ∞ ∞ S (¯ωr; τ) = dt dt0 dτ1 dτ2 p(t0) p∗(t) a(τ1) a∗(τ2) IF SRS I ~ E E E E Z−∞ Z−∞ Z−∞ Z−∞ E†(t0)E˜†(¯ω )E˜ (¯ω )E (t) F (t0 τ , t t0, t τ ). (200) × hT s r r r r s i i − 2 − − 1 To make sure that there is no photon at detector s we had integrated over its entire bandwidth, thus eliminating the dependence on detector parameters. For the Raman gain Ns = 2 signal (i.e. two photons in the s-arm, one photon in the r-arm), when both s and r detectors have narrow frequency gates, we get

t¯s t t t0 (2,1) 1 ∞ ¯ 1 ¯ iω¯s1 (ts1 τ) S (¯ωs1 , ω¯s2 , ω¯r; τ) = dts1 e − dt dt0 dτ1 dτ2 p(t) p∗(t0) a(τ1) a∗(τ2) IF SRS I ~ E E E E Z−∞ Z−∞ Z−∞ Z−∞ Z−∞ E (t0)E˜†(¯ω )E˜†(¯ω )E˜†(¯ω )E˜ (¯ω )E˜ (¯ω )E˜ (t¯ )E†(t) F (t0 τ , t t0, t τ ). (201) × hT s s s1 s s2 r r r r s s2 s s1 s i i − 2 − − 1 Finally, the Ns = 1 signal (single photon in each arm) is given by

t0 t t0 t0 1 ∞ s s (1,1)a iω¯s(ts0 τ) S (¯ωs, ω¯r; τ) = dts0 e − dt dt0 dτ1 dτ2 p(t) p∗(t0) a(τ1) a∗(τ2) IF SRS −I ~ E E E E Z−∞ Z−∞ Z−∞ Z−∞ Z−∞ E˜†(¯ω )E˜†(¯ω )E˜ (¯ω )E˜ (t0 )E†(t)E (t0) F (t τ , t t0, t0 τ ), (202) × hT s s r r r r s s s s i ii − 2 − − 1

t0 t t0 t0 1 ∞ s s (1,1)b iω¯s(ts0 τ) S (¯ωs, ω¯r; τ) = dts0 e − dt dt0 dτ1 dτ2 p(t0) p∗(t) a(τ1) a∗(τ2) IF SRS −I ~ E E E E Z−∞ Z−∞ Z−∞ Z−∞ Z−∞ E˜†(¯ω )E˜†(¯ω )E˜ (¯ω )E˜ (t0 )E (t)E†(t0) F (t τ , t t0, t0 τ ). (203) × hT s s r r r r s s s s i ii − 2 − − 1 b. Photon counting detection window for the molecular response s- mode and single photon in the r - mode. and is described The input two-photon state has a single photon in each the 40

(1,1) (2,1) FIG. 26 (Color online)Top: (a) - Absorption, (b) - classical FSRS, (c) - SIF SRS , (d) - SIF SRS for a time evolving vibrational mode i vs ω − ωp for slow tunneling rate and narrow dephasing, (e) - (h) are the same as (a) - (d) but for fast tunneling rate and broad dephasing. Bottom left: (i) - time-frequency Wigner spectrogram for classical light, (k) - same as (i) but for entangled twin state given by Eq. (204). Inserts depict ∗ ∗ a 2D projection. (j) - window function for FSRS Es (ω)Es(ω + iγa) -black, and IFSRS Φ (ω, ω¯r)Φ(ω + iγa, ω¯r) different values of T1. (l) - (1,1) spectrum of the eigenvalues λn in the Schmidt decomposition (34) for entangled state with amplitude (205). Bottom right: SIF SRS signal vs (2,1) ω¯s − ωp for correlated - (m) and uncorrelated -(o) separable states. (n), and (p) - same as (m) and (o) but for SIF SRS signal. Parameters for simulations are listed in (Dorfman et al., 2014). 41 by [compare to Eq. (33)] We next compare the different field spectrograms which represent the temporal and spectral windows created by the ∞ ∞ fields. Fig. 26i depicts the time-frequency Wigner spectro- ψ = 0 + dωs dωrΦ(ωs, ωr)aω† s aω† r 0 , (204) | i | i | i gram for the classical probe field s: Z−∞ Z−∞ E where aω† s (aω† r ) is the creation operator of a horizontally (ver- ∞ d∆ i∆t tically) polarized photon and the two-photon amplitude is W (ω, t) = ∗(ω) (ω + ∆)e− . (207) s 2π Es Es given by Z−∞ 2 The Fourier uncertainty ∆ω∆t 1 limits the frequency reso- ≥ Φ(ωs, ωr) = sinc (ωs0Ti/2 + ωr0Tj/2) lution for a given time resolution. The corresponding Wigner i=j=1 6 X two-photon spectrogram for the entangled twin photon state A (ω + ω )eiωs0Ti/2+iωr0Tj /2, (205) × p s r ∞ d∆ i∆t where ω = ω ω , k = s, r, A (ω) = A /[ω ω + iσ ] Wq(ω, t;ω ¯r) = Φ∗(ω, ω¯r)Φ(ω + ∆, ω¯r)e− , k0 k − 0 p 0 − 0 0 2π Z−∞ is the classical pump Tj = (1/vp 1/vj)L, j = 1, 2 is (208) the time delay between the j th −entangled and the classical pump beam after propagation− through the PDC crystal. is depicted in Fig. 26k. For the same temporal resolution T = T2 T1 is the entanglement time. In Ap(ω), the sum- − of the FSRS, the spectral resolution of IFSRS is significantly frequency ωs + ωr is centered around 2ω0 with bandwidth higher since the time and frequency variables for entangled σ0. For a broadband classical pump, the frequency difference 1 light are not Fourier conjugate variables (Schlawin et al., ωs ω0 becomes narrow with bandwidth T − , j = 1, 2. The − j 2013). The high spectral resolution in the entangled case is output state of light in mode s may contain a varying number 1 of photons, depending on the order of the field-matter interac- governed by Tj− , j = 1, 2 which is narrower than the broad- tion. band probe pulse such that ∆ω∆t 0.3. Fig. 26j) demon- ' (Ns,1) As discussed in section II.C, the twin photon state Eq. (204) strates that entangled window function Rq for Ns = 1, 2 is not necessarily entangled. Using the Schmidt decomposi- (see Eqs. (212), (213)) that enters the IFSRS Signal (210) tion (34), we obtain for the two-photon amplitude in Eq. (205) yields a much higher spectral resolution than the classical Rc the rich spectrum of eigenvalues shown in Fig. 26l) shows that in Eq. (215). the state is highly entangled with up to 20 modes contributing The molecular information required for all three measure- with appreciable weight. As will be shown later [Eq. (208)], ment outcomes (Ns = 0, 1, 2) is given by two correlation this entanglement is reflected in the violation of the Fourier functions Fi and Fii (see Fig. 25c,e and Eqs. (197) - (198)), uncertainty ∆ω∆t 1 in two-photon transitions. which are then convoluted with a different detection window ≥ We next turn to how entanglement affects the Raman reso- for FSRS and IFSRS. Fi and Fii may not be separately de- nances. Both FSRS and IFSRS signals are governed by a four- (0,1) tected by FSRS. However, in IFSRS the loss SIF SRS and the point matter correlation function (two polarizabilities αac and (2,1) gain SIF SRS Raman signals probe Fi (the final state c can be two dipole moments Vag as depicted by the loop diagrams different from initial state a) whereas the coincidence count- shown in Fig. 25c) and e), respectively. Depending on the (1,1) ing signal S depends on F (initial and final states are number of photons detected, this four-point matter correlation IF SRS ii identical). Interferometric signals can thus separately measure function is convoluted with different field correlation func- the two matter correlation functions. tions. For Ns = 0, and Nr = 1 Eq. (200) is given by the four-point correlation function for a quantum field [red arrows in Fig. 25e)]. For a twin photon state, we recall from Eq. (50) that the field correlation function can be factorized as c. IFSRS for a vibrational mode in a tunneling system. To demonstrate the effect of entanglement in interferometric ψ Es†(ωa)Er†(ωb)Er(ωc)Es(ωd) ψ = Φ∗(ωa, ωb)Φ(ωc, ωd). h | | i (206) measurements, we show the calculated signals for the three- level model system depicted in Fig. 25a. Once excited by The Ns = 2 signal is given by an eight-point [see Eq. (201)], the actinic pulse, the initial state with vibrational frequency and for Ns = 1 it is governed by a six-point field correla- ω+ = ωac + δ can tunnel through a barrier at a rate k and tion function as shown in Eqs. (202) - (203). The detailed assume a different vibrational frequency ω = ωac δ (see derivation and explicit closed form expressions for multipoint Ref. (Dorfman et al., 2013b)). The probability− to be− in the kt correlation functions of the field are presented in Ref. (Dorf- initial state with ω+ decreases exponentially P+(t) = e− , kt man et al., 2014). All three IFSRS signals with Ns = 0, 1, 2 whereas for ω it grows as P (t) = 1 e− . This model rep- eventually scale linearly with the classical pump intensity resents Kubo’s− two-state jump− model− in the low temperature 2 SIF SRS A0 , similar to classical FSRS even though a limit (Dorfman et al., 2013b; Kubo, 1963) which we had also different number∝ | | of fields contribute to the detection. discussed before in section III.E.2. The absorption lineshape 42 is given by dominant resonance at ω+ which decays with increase of delay T , whereas the ω peak slowly builds up and dominates at 2 − 4 2 µac longer T . This signal contains both blue- and red-shifted Ra- Sl(ω) = (ω) | | −I ~2 |E | k + 2iδ man resonances relative to the narrowband pump frequency: k + iδ iδ ω ωp = ω . If the modulation and dephasing rates are + . (209) − ± ± × ω ω + iγa ω ω+ + i(γa + k) comparable to the level splitting k, γa δ, then the ω res- − − − onances in the absorption [Fig. 26e)] and∼ the classical± FSRS This shows two peaks with combined width governed by de- [Fig. 26f)] broaden, and become less resolved. phasing γa and tunneling rate k. The corresponding IFSRS We next compare this with IFSRS. For small modulation (N ,1) s (1,1) signal SIF SRS with Ns = 0, 1, 2 is given by and long dephasing, SIF SRS is similar to the classical FSRS µ [see Fig. 26c)]. However, both temporal and spectral resolu- S(Ns,1) (¯ω , ω¯ ; ω , τ) = 2 2 α2 µ 2 IF SRS s r p I 4 |Ep| |Ea| ac| ag| tion remains high, even when the modulation is fast and the ~ a,c X dephasing width is large, as seen in Fig. 26g). The same ap- kτ (2,1) 2γaτ (Ns,1) 2iδe− plies to the S signal depicted for slow - Fig. 26d) and e− R (¯ωs, ω¯r, 2γa, νω¯ ν iγa) IF SRS × q − − k + 2iδ fast - Fig. 26h) tunneling. (Ns,1) Apart from the different detection windows, there is another [Rq (¯ωs, ω¯r, 2γa + k, νω¯ ν iγa) × − important distinction between IFSRS (Eq. (210)) and the clas- (Ns,1) R (¯ωs, ω¯r, 2γa + k, νω¯ ν¯ i(γa + k)] , (210) sical FSRS (214) signals. In the latter, the gain and loss con- − q − tributions both contain red- and blue- shifted features relative where ν = for Ns = 0, 2 and ν = + for Ns = 1, µ = to the narrow pump. The FSRS signal can contain both Stokes − − for Ns = 1, 2 and µ = + for Ns = 0. The Raman response and anti Stokes components. FSRS can only distinguish be- (Ns,1) Rq which depends on the window created by the quantum tween red and blue contributions. In contrast, the interfer- (2,1) field for different photon numbers Ns is given by ometric signal can measure separately the gain SIF SRS and (0,1) the loss contributions SIF SRS. (0,1) ∞ dω Φ∗(ω, ω¯r)Φ(ω + iγ, ω¯r) Rq (¯ωs, ω¯r, γ, Ω) = , 2π ω ωp Ω Z−∞ − − (211) d. The role of entanglement. We now show that the enhanced resolution of Raman resonances may not be reproduced by classically shaped light and entanglement is essential. To this Φ∗(¯ωs, ω¯r)Φ(ωp + Ω iγ, ω¯r) R(1,1)(¯ω , ω¯ , γ, Ω) = − , end, we calculate the IFSRS signals (210) for the correlated- q s r ω¯ ω Ω s p separable state (Zheng et al., 2013) described by the density − − (212) matrix

∞ 2 (2,1) Φ∗(¯ωs, ω¯r)Φ(¯ωs + iγ, ω¯r) ρcor = dωsdωr Φ(ωs, ωr) 1ωs , 1ωr 1ωs , 1ωr . Rq (¯ωs, ω¯r, γ, Ω) = . (213) | | | ih | ω¯ ω Ω Z−∞ s − p − (216) For comparison, we give the classical FSRS signal (193) This is a diagonal part of the density matrix corresponding to state Eq. (204) with amplitude Eq. (205), which results (c) 2 2 2 2 2 2γaτ S (ω, τ) = α µ e− FSRS −I 4 |Ep| |Ea| ac| ag| × from the disentanglement of the twin state. This state yields ~ a,c X the same single-photon spectrum and shows strong frequency kτ 2iδe− correlations similar to entangled case, and is typically used as Rc(ω, 2γa, ω iγa) [Rc(ω, 2γa + k, ω iγa) − − − k + 2iδ − − a benchmark to quantify entanglement in quantum information processing (Law et al., 2000). We further compare this Rc(ω, 2γa + k, ω+ i(γa + k)] (ω ω )] , − − − ± ↔ − ∓ (214) with signals from the fully separable uncorrelated Fock state given by Eq. (204) with where Φuncor(ωs, ωr) = Φs(ωs)Φr(ωr) (217) (ω) (ω + iγ) R (ω, γ, Ω) = s∗ s (215) c E E with Φk(ωk) = Φ0/[ωk ω0 +iσ0], k = s, r with parameters ω ωp Ω − − matching the classical probe− pulse used in FSRS. is the Raman response gated by the classical ﬁeld. (1,1) Fig. 26m-p illustrates SIF SRS for these two states of light. Figs. 26a)-h) compare the classical FSRS signal [Eq. The separable correlated state shown in Fig. 26m has high (1,1) (2,1) (214)] with the IFSRS signals SIF SRS and SIF SRS [Eq. spectral and no temporal resolution, as expected from a cw (210)]. For slow modulation and long dephasing time k, γa time-averaged state in which the photons arrive at any time δ the absorption spectrum [Fig. 26a)] has two well-resolved (Zheng et al., 2013). The separable uncorrelated state (see peaks at ω . The classical FSRS shown in Fig. 26b) has one Fig. 26o) yields slightly better resolution than in the classical ± 43

FSRS signal in Fig. 26f. Similar results can be obtained for (2,1) the SIF SRS (see Fig. 26n and p respectively). -k2 -k2 k2 k2 -k -k3 k1 3 k1 -k3 -k3 k1 k1

IV. ENTANGLED LIGHT GENERATION VIA NONLINEAR SFG DFG TPIF TPEF, PDC LIGHT-MATTER INTERACTIONS; NONCLASSICAL RESPONSE FUNCTIONS FIG. 27 Three wave process involuting classical (solid line) and quantum (dashed line) modes. Black lines represent incoming fields, red lines correspond to generated light. The light generation process is typically based on the as- sumption that the light field is given, or that different optical fields that participate in generation of the desired light state A. Superoperator description of n-wave mixing interact with other fields. This way the generation process can be described by an effective Hamiltonian in the field space So far we mostly used L and R representation for de- alone. The material quantities that assist the light conversion scribing signals. Here, we consider various nonlinear signals are typically set to be constant parameters governed by non- which involve two photon resonances (Roslyak and Mukamel, linear semiclassical susceptibilities. 2009b) (see sketch in Fig. 27) and describe them using representation. For coherent optical states, all field superop-± Superoperator non-equilibrium Green’s functions are use- erator nonequilibrium Green’s functions (SNGF) in the L, R ful for calculating nonlinear optical processes involving any representation are identical E = E (the superoperator in- combination of classical and quantum optical modes. Closed L0 R0 dex makes no difference since all operations commute). In correlation-function expressions based on superoperator time- the +, representation ”minus” field indices are not allowed, ordering may be derived for the combined effects of causal since E− = 0. The general m wave mixing signals are given (response) and non-causal (spontaneous fluctuations) correla- 0 by 2m products− of material and corresponding optical field tion functions (Roslyak and Mukamel, 2009b). SNGF’s of mth order: Below we survey several wave-mixing schemes for gener- m ∞ ating quantum light by using a combination of classical and (m) i δm+1,α Sα = ... dtm+1dtm . . . dt1 quantum modes of the radiation field. Homodyne-detected =πm!~m+1 νm ν1 Z sum frequency generation (SFG)(Shen, 1989) and difference X X−∞ frequency generation (DFG)(Dick and Hochstrasser, 1983; (218) Mukamel, 1995) involve two classical and one quantum mode. (m) Θ(tm+1)Vνm+1νm...ν1 (tm+1, tm, . . . , t1) Parametric down conversion (PDC)(Hong and Mandel, 1985; × (m) (t , t , . . . , t ) Klyshko, 1988; Louisell et al., 1961; Mandel and Wolf, 1995) Eνm+1ν¯m...ν¯1 m+1 m 1 involves one classical and two quantum modes and is one where tm+1, tm, . . . , t1 are the light/matter interaction times. of the primary sources of entangled photon pairs (Edamatsu, m 2007; Gerry and Knight, 2005; U’Ren et al., 2006). All of The factor Θ(t ) = θ(t t ) insures that the t m+1 m+1 − i m+1 these are coherent measurements, and scale as N(N 1) for i=1 is the last light-matter interaction. The indexes ν¯ are the con- N active molecules the signals (Marx et al., 2008). − Q j jugates to νj and defined as follows: the conjugate of + is We further consider incoherent N-scaling signals. These and vice versa. Equation (218) implies that is the excita- − include heterodyne detected SFG∼ and DFG, which involve tion in the material are caused by fluctuations in the optical three classical modes, and two types of two photon fluores- field and vice versa. Equation (218) also holds in the L, R cence (Denk et al., 1990): two photon induced fluorescence representation. Here we have ν = L and ν L, R , m+1 j ∈ { } with one classical and two quantum modes (TPIF)(Callis, j = m, . . . , 1. ν¯j. In the L, R representation the conju- 1993; Rehms and Callis, 1993; Xu and Webb, 1996) and two gate of ”left” is ”left” and the conjugate of ”right” is ”right”: photon emitted fluorescence with two classical and one quan- L¯ = L, R¯ = R. The material SNGF (m) rep- VL L...L R...R tum make (TPEF). The list of the different measurement is n m n summarized in the Table I. resent a Liouville space pathway with n + 1 interactions− from the left (i.e. with the ket) and m n interactions| {z } from| {z the} right Finally we present a more detailed microscopic theory of − entangled light generation in two schemes, which describe (i.e. with the bra). light-matter interactions that involve quantum field. The first Field SNGF in Eq. (218) is defined in Eq. (14). Mate- is based on type-I PDC in a cascade three-level scheme, and rial SNGF is defined by Eq. (15). The latter in the form of (m) give causal ordinary response function of mth or- in the second scheme an entangled photon pair is generated by V+ ... − − two remote molecules assisted by an ideal 50:50 beam splitter. m

| {z } 44

Three wave processes Heterodyne Homodyne Incoherent Coherent Technique SFG DFG TPIF TPEF SFG PDC Modes c/c/c c/c/c c/c/q c/q/q c/c/q c/q/q

ω3 ω1 + ω2 ω1 − ω2 - - ≈ ω1 + ω2 ≈ ω1 − ω2 2 2 (2) (2) (5) (5) (5) (2) (2) (2) SNGF’s χ+−− χ+−− (χ++−−−− (χ++−−−− + 2χ+++−−− χ+−− χ+−− + χ++− (5) (5) χ++−−−−)/2 +χ++++−−)/4 Expression eq.(235) eq.(233) eq.(240) eq.(250) eq.(235) Type I eq.(253), Type II eq.(256),(258)

TABLE I SNGF’s of three wave mixing techniques: heterodyne-detected SFG and DFG with all classical modes (c); incoherent TPIF with two classical and one quantum (q) mode and corresponding coherent homodyne-detected SFG; incoherent TPEF with one classical and two quantum modes and Type I PDC; Type II PDC SNGF with one classical and four quantum modes. der. The material SNGF of the form (m) represent by a quasi-probability distribution P (β) V+ + ... +

m mth moment of molecular fluctuations. The material SNGF ρˆ = d2βP (β) β β . (220) of the form (m) indicates changes| {z } in nth mo- | ih | V+ + ... + ... Z − − n m n − It has been suggested (Almand-Hunter et al., 2014; Kira et al., ment of molecular fluctuations up by m n perturbations. We 2011; Mootz et al., 2014) that, since the response of a material | {z } | {z } − next recast the material SNGF (15) in the frequency domain system to a field initially prepared in a coherent state β is by performing a multiple Fourier transform: given by the classical response function CRF, the quantum| i response RQM may be recast as an average of the classical (m) χν ν ...ν ( ωm+1; ωm, . . . , ω1) = (219) response R β with respect to this quasi-probability m+1 m 1 − | i

∞ i(ωmtm+...+ω1t1) dtm+1 . . . dt1Θ(tm+1)e 2 RQM = d βP (β)R β . (221) Z−∞ | i (m) Z δ( ωm+1 + ωm + ... + ω1)Vνm+1νm...ν1 (tm+1, tm, . . . , t1) − By transforming to the response of a nonclassical state given by P (β), in which the fluctuations along one quadrature are The SNGF χ(m) ( ω ; ω , . . . , ω ) (with one + and + ... m+1 m 1 squeezed below the classical limit, this form of data process- − − − m ing can uncover otherwise hidden features in the signal. For the rest indices) are the mth order nonlinear susceptibili- instance, it was put to use in (Almand-Hunter et al., 2014) to − ties or causal response| {z } functions. The rest of SNGF’s in the reveal the “dropleton”, as new kind of quasiparticle excitation frequency domain can be interpreted similarly to their time in semiconductors. domain counterparts (15). While this analysis has proven successful, it misses the multimode nature of the entangled states, which lies at the heart of time-energy entanglement presented here. Based on our analysis of the entangled light state (33), we have shown B. Connection to nonlinear fluctuation-dissipation that optical signals induced by such entangled states cannot relations be reduced to a simple sum over the signal of each quantum mode: The quantum correlations depicted in Fig. 3 are nec- Spontaneous fluctuations and response are uniquely re- essarily missed in a single-mode approach. For instance, the lated in the linear regime (Hashitsume et al., 1991) by the transition amplitude (99) may be written as a sum over the fluctuation-dissipation theorem, but not when they are non- Schmidt modes, linear. Some nonlinear fluctuation-dissipation relations have been proposed for specific models under limited conditions T (t) dω dω R (ω , ω )ψ (ω )φ (ω ), (Bertini et al., 2001; Bochkov and Kuzovlev, 1981; Lippiello fg ∼ a b t a b k a k b k Z Z et al., 2008) but there is no universal relation of this type X (222) (Kryvohuz and Mukamel, 2012).

A different approach for calculating and analyzing quantum where Rt shall denote the material response. Hence, the TPIF 2 spectroscopy signals is based on the Glauber Sudarshan P - signal Tfg cannot be reduced to the signal of the indiv- representation which expresses the ﬁeld density matrix as an dual Schmidt∼ | | modes, T 2 = , and the fg k,k0 k integral over coherent state density matrices β β weighted above transformation fails| | to∼ capture such· · · 6a signal.··· | ih | P P 45

The idea behind the Glauber-Sudarshan quasi-probability (Hashitsume et al., 1991), in relation to quantum and classical light outlined in (Almand- 1 Hunter et al., 2014; Kira et al., 2011; Mootz et al., 2014) C++ = coth(β~ω/2)C+ (ω). (225) 2 − makes an equality between classical light and coherent states. Here This is a potential source of confusion: For instance, the ef- iωτ fect of revival of Rabi oscillations demonstrated by Rempe C+ (ω) = dτ V+(τ)V (0) e (226) − h − i (Rempe et al., 1987) which is observed when a coherent state Z is treated quantum mechanically shows clearly that a coher- is the response function, whereas ent state is generally a quantum state of light. Therefore, iωτ Eq. (220) merely represents a transformation between two dif- C++(ω) = dτ V+(τ)V+(0) e (227) h i ferent basis sets for the quantum state. Of course, in the non- Z linear response case, if the operators are normally ordered as denotes spontaneous fluctuations. The classical response function C+ (ω) thus carries in Eq. (74) the result of the coherent state is equivalent to the − classical case. In Rempe’s example, the nonclassical contribu- all relevant information about linear radiation matter cou- tions arise due to commutator terms in Eq. (87) and the higher pling, including the quantum response. In the nonlinear the order in perturbative expansion - the larger the contribu- regime the CRF is a specific causal combination of mat- tion of the commutator terms. Therefore, in the strong field ter correlation functions given by one “+” and several “- limit one can observe the revival of Rabi oscillations. ” operators. e.g V+V V V for the third order response. However,h the− quantum− −i response may also de- We now use the superoperator notation introduced in sec- th tion I.A to show more broadly why the quantum response is pend on the other combinations. To n order in the ex- different from the classical one so that Eq. (221) is violated. ternal field the CRF V+(ωn+1)V (ωn)...V (ω2)V (ω1) h − − n − i In Liouville space, the time-dependent density matrix is given is one member of a larger family of 2 quantities V+(ωn+1)V (ωn)...V (ω2)V (ω1) representing various by Eq. (69). We now make use of the algebraic relation of h ± ± ± i superoperators (Marx et al., 2008): combinations of spontaneous fluctuations (represented by V+ ) and impulsive excitations (represented by V ). For example, − Hint = E+V + E V+. (223) an ”all +” quantity such as V+V+V+V+ represents purely − − − spontaneous fluctuations. Theh CRF does noti carry enough in- Let us first assume that the electric field operators commute formation to reproduce all 2n possible quantities which are ac- and set E = 0. We then calculate the expectation value of a cessible by quantum spectroscopy. The reason why the CRF − system A operator OA: and QRF are not simply related in is the lack of a fluctuation- dissipation relation in the nonlinear regime (Bertini et al., tr[OAρ(t)] 2001; Bochkov and Kuzovlev, 1981; Kryvohuz and Mukamel, t 2012; Lippiello et al., 2008). i A = tr[OA exp E+(t0)V (t0)dt0 The field commutator E is intrinsically related to vacuum T − − ~ Zt0 − t modes of the field which may induce coupling between non- i B interacting parts of the system. One example where such an exp E+(t0)V (t0)dt0 ρA,0ρB,0ρph,0]. (224) × − − ~ Zt0 effect arising from E is combined with the appearance of collective resonances,− which occur for E , has been recently Since the trace of a commutator vanishes, and since there are + investigated in the context of harmonic systems (Glenn et al., only V B operators for system B, all correlation functions of 2015). The response of classical or quantum harmonic oscil- the form− V BV B...V B = 0. The nonlinear response func- lators coupled linearly to a classical field is strictly linear; all tion is thush − additive,− and− i contains no cooperative terms. The nonlinear response functions vanish identically. We have re- time evolution of two coupled quantum systems and the field cently shown that quantum modes of the radiation field that is generally given by a sum over Feynman paths in their joint mediate interactions between harmonic oscillator resulted in phase space. Order by order in the coupling, dynamical ob- nonlinear susceptibilities. A third-order nonlinear transmis- servables can be factorized into products of correlation func- sion of the optical field yields collective resonances that in- tions defined in the individual spaces of the subsystems. These volve pairs of oscillators, and are missed by the conventional correlation functions represent both causal response and non- quantum master equation treatment (Dorfman and Mukamel, causal spontaneous fluctuations (Cohen and Mukamel, 2003; 2013). Roslyak and Mukamel, 2010). The linear response contains several possible combinations of field superoperators V V . V V represents a com- C. Heterodyne-detected sum and difference frequency h ± ±i h − −i mutator, and thus vanishes since its trace is zero. There- generation with classical light fore V+V+ (two anticommutators) and V+V (commu- tatorh followedi by anticommutator) are theh only− twoi quanti- We first compare two experiments involving three classical ties that contribute to the linear response. These two quanti- modes. The third mode is singled out by the heterodyne detec- ties are related by the universal fluctuation-dissipation relation tion which measures its time-averaged photon flux (photons 46

FIG. 28 Heterodyne detected DFG: (A) phase matching condition k3 = k1 − k2; (B) molecular level scheme; (C) loop diagrams. FIG. 29 Heterodyne detected SFG: (A) phase matching condition k3 = k1 + k2; (B) level scheme; (C) loop diagrams. per unit time). Both techniques represent second order non- (2) linear signals S3 . The initial state of the field is given by a The material SNGF’s are: direct product of coherent states: t = = β 1 β 2 β 3, | −∞i | i | i | i (2) where β α are eigenfunctions of the mode α annihilation op- V+ (t3, t2, t1) = (230) | i −− erator: aα β α = βα β α. Coherent states are the most clas- 3 2, 1, 3 2, 1, V (t)V †(t2)V †(t1) + V (t)V †(t2)V †(t1) sical states| ofi quantum| i light, hence we shall refer to them as hT L L L hT L R L i classical optical fields (Glauber, 2007a). The DFG signal in the frequency domain can be written as: Only one type of optical SNGF contribute to the classical SDFG( ω3, ω2, ω1) = (231) third order signalsignal: − 1 (2) (2) δ(ω1 ω2 ω3)χ+ ( ω3; ω2, ω1) 3∗ 2∗ 1 E (t3, t2, t1) = 0(t3) 0(t2) 0(t1) π~= − − −− − − E E E +++ E E E Utilizing the rules given in Appendix A and the diagrams where is the 0(t) = EL0 (t) = ER0 (t) = βα 2π~ωα/Ω shown in Fig.28(C) we obtain: classicalE field amplitude.h i h i p (2) The conjugate material SNGF’s become: χ+ ( (ω1 ω2); ω2, ω1) = (232) −− − − − (2) (2) (2) 1 (t , t , t ) = (t , t , t ) + (t , t , t ) † V+ 3 2 1 VLLL 3 2 1 VLRL 3 2 1 2 ( g V3G(ωg + ω1 ω2)V2G(ωg + ω1)V1 g −− (228) 2!~ h | − | i − where we have assumed that initially the material system is in g V2G†(ωg + ω1 ω2)V3G(ωg + ω1)V1† g ). (2) h | − | i the ground state, which implies that VLRR (t3, t2, t1) 0. A sum-over-states (SOS) expansion then yields We assume the same three-level system used≡ earlier. (2) The L, R representation plus RWA allows the material χ+ ( (ω1 ω2); ω2, ω1) = (233) −− − − − x x x SNGFs (228) to be represented by the loop diagrams shown 1 µgf µfeµeg in Fig.28(C), Fig.29(C). The rules for constructing these 2 2!~ (ω1 ωgf + i~γgf )(ω1 ω2 ωeg + i~γeg)− partially-time-ordered diagrams are summarized in Appendix − − − 1 µx µx µx A (Roslyak et al., 2009a). The signal is given by the causal gf fe eg 2 (2) 2!~ (ω1 ωgf + i~γgf )(ω2 ωeg i~γeg) χ+ ( ω3, ω2, ω1) response function. This result is not − − − new−− and− can± be obtained± by using various combinations of a Equation (233) indicates that the signal induced by classical classical field from the outset. To calculate equation (228) we optical fields is given by the second order classical response have to specify the phase matching conditions (frequencies function (CRF). and wave vectors of the optical modes). This will be done below. E. Sum Frequency Generation (SFG)

D. Difference frequency generation (DFG) In SFG the first two modes promote the molecule from its ground state g through intermediate state e into the state | i | i In DFG the first mode k promotes the system from its f . The third mode induces stimulated emission from f to 1 | i | i ground state g into state f . The second mode k induces the ground state g as sketched in Fig.29 (B). The signal is | i | i 2 | i stimulated emission from f to an intermediate e ; and the generated in the direction: k3 = k1 + k2 (See Fig.29 (A)). | i | i third mode k3 stimulates the emission from e to g , as The heterodyne-detected SFG signal can be obtained in an sketched in Fig.29 (B). The signal is measured| ini the| phasei analogous manner to the DFG by utilizing the diagrams shown matching direction: k3 = k1 k2 (See Fig.28 (A)). in Fig.29(C): The corresponding loop diagrams− corresponding are shown S (ω , ω ) = (234) in Fig.28(C). The optical field SNGF yields: SFG 1 2 1 (2) (2) δ(ω1 + ω2 ω3)χ+ ( ω3; ω2, ω1) 3∗ 2 1 E (t3, t2, t1) = ∗(t) ∗(t2) 1(t1) (229) π~= − −− − E E E +++ E3 E2 E 47 where the CRF is given by

(2) χ+ ( (ω1 + ω2); ω2, ω1) = (235) −− − 1 g V3G(ωg + ω1 + ω2)V2†G(ωg + ω1)V1† g = 2!~2 h | | i x x x 1 µgeµfeµeg 2 2!~ (ω1 ωeg + i~γeg)(ω1 + ω2 ωgf + i~γgf ) − − In the coming sections equations (235) and (233) will be compared with other techniques involving various combinations of quantum and classical optical ﬁelds. These include-homodyne detected SFG, DFG and PDC where one or more optical modes are spontaneously generated and must be treated quantum mechanically.

F. Two-photon-induced ﬂuorescence (TPIF) vs. homodyne-detected SFG

We now turn to techniques involving two classical and one quantum mode where the initial state of the optical field is: t = = β1 1 β2 2 0 3. The modes interact with the FIG. 30 Three wave process with two classical and one quantum three| −∞i level material| i system| i | (Fig.30(B)).i We assume that ω = mode: a) phase-matching condition; b) molecular level scheme; c) eg 6 ωef . This allows to focus on the resonant SNGF’s and reduces loop for the incoherent TPIF (C1) and coherent homodyne SFG (C2) the number of diagrams. The two classical modes k1 and k2 promote the molecule from its ground state g into the intermediate state e and to interact with different systems at different times. Fig.30(C) the final state f . The| systemi then spontaneously decays| i back shows processes involving three different molecules and six | i light/matter interactions. These modes can either interact with into one of the ground state manifold g0 emitting a photon | i the same system (incoherent process, Fig.30(C1)) or two dif- into the third mode k3 which is initially in the vacuum state ferent systems (coherent process, Fig.30(C2))(Marx et al., (See Fig.30 (B)). The phase-matching condition k1 k1 + k k + k k = 0 is automatically satisfied− for any 2008). 2 − 2 3 − 3 k3. Therefore the spontaneous photons are emitted into a cone (See Fig.30 (A)). 1. Two-photon-induced fluorescence (TPIF). We shall calculate the photon flux in the k3 mode. Since the process involves three different modes and six light/matter interactions. The signal (218) must by expanded to fifth order in This is an incoherent three wave process. Using the identity N (5) Hint. The fifth order signal SCCQ (CCQ implies two classi- exp i (k1 k1 + k2 k2 + k3 k3)(r ri) = N the 5 i=1 − − − − cal and one quantum mode) may be written as sum of 2 terms optical field SNGF yields: each given by a product of molecule/field six point SNGF’s. P Only the corresponding diagrams shown in Fig.30(C). These (5) (t , t , . . . , t ) = (236) satisfy the following conditions: ELR++++ 6 5 1 2π~ω3 N 1(t1) 1∗(t2) 2(t3) 2∗(t4) exp (iω3(t6 t5)) 1. The creation operator of the quantum mode a3† must be E E E E Ω − accompanied by the corresponding annihilation opera- The relevant material SNGF is: tor a3. (5) 2. The quantum modes de-excite the molecule, which im- VLR++++ (t6, t5, . . . , t1) = (237) plies that the annihilation operators must act on the bra 3 3, 2 2, 1 1, VL (t6)VR †(t5)V (t4)V †(t3)V (t2)V †(t1) and the creation operators act on the ket. hT − − − − i 3. The coherent optical fields are tuned off resonance with Utilizing equations (236,237) the frequency domain signal can be written as: respect to the ωfg0 transition. Hence, the signal is not masked by stimulated emission. N 2 2 2π~ω3 S (ω , ω ) = (238) TPIF 1 2 π |E1| |E2| Ω × To address the collective properties of three wave mixing. ~ k X3 We consider a collection of N noninteracting molecules posi- (5) tioned at r so that V = V δ(r r ). The optical field can χLR ( ω3; ω3, ω2, ω2, ω1, ω1) i i i − i = −−−− − − − P 48

Note that the above expression is given in the mixed 2. Homodyne-detected SFG (L/R, +/ ) representation. It can be recast into the +, rep- − (5) (5) (5) − Here the optical field SNGF is given by equation (236), but resentation using χLR = (χ+ + χ++ )/2. The second term (two−−−− ”plus” four−−−−− ”minus” indices)−−−− arises instead of factor N it contains the factor: factor: N N 1 since one of the modes is nonclassical. The frequency do- − i∆k(r ri) i∆k(r rj ) main material SNGF can be calculated from the diagram of e − e− − N(N 1) (242) ≈ − i=1 j=i Fig.30(D1): X X6 1 where ∆k k1 +k2 k3. reflects phase uncertainty given (5) (239) ≈ − ≈ χLR ( ω3; ω3, ω2, ω2, ω1, ω1) = 5 by the reciprocal of the molecular collection length which ef- −−−− − − − 5!~ fectively narrows the optical cone. For large N the coherent g V G†(ω + ω )V G†(ω + ω + ω ) h | 1 g 1 2 g 1 2 part N(N 1) dominates over the incoherent N re- V †G†(ω + ω + ω ω )V ∝ − ∝ × 3 g 1 2 − 3 3 × sponse. For a small sample size, exact calculation of the optical field part of the SNGF is rather lengthy , but it can be G(ωg + ω1 + ω2)V2†G(ωg + ω1)V1† g | i performed in the same fashion as done by Hong and Man- Expanding equation (239) in molecular states gives: del(Hong and Mandel, 1985) for the probability of photon de- (5) x x x 2 tection. χLR ( (ω1 + ω2); ω2, ω1) = µg f µfeµeg −−−− − | 0 | To calculate the matter SNGF, we must work in the joint Xgg0 space of two molecules = interacting with the (240) 1,2 1 2 same field mode. The matter|i SNGF|i |i of the joint system can 1 1 be factorized into product of each molecule SNGF’s: 5 2 2 ×5!~ [(ω1 ωeg) + γeg][ω1 + ω2 ωfg + iγfg] − − (5) (t , t , . . . , t ) = (243) 1 VLR++++ 6 5 1 3 3, 2 2, 1 1, ×[ω1 + ω2 ωfg iγfg ][ω1 + ω2 ω3 ωgg iγgg ] V+(t)V+ †(t5)V (t4)V †(t3)V (t2)V †(t1) 1,2 = − 0 − 0 − − 0 − 0 hT − − − − i 3 2, 1, 3, 2 1 Provided the energy splitting within the ground state manifold V+(t)V †(t3)V †(t1) 1 V+ †(t5)V (t4)V (t2) 2 hT − − i hT − − i is small comparing to the optical transitions the signal can be where we have used the fact that the last interaction must be a recast in the form: ”plus”. 2Nω Since the two molecules are identical the following identity 3 2 2 (241) STPIF (ω1, ω2) = 5 1 2 holds: 5!~ Ω |E | |E | 2 3, 2 1 Tg g(ω1, ω2) δ(ω1 + ω2 ω3 ωgg ) V+ †(t5)V (t4)V (t2) = (244) | 0 | − − 0 hT − − i 3 2, 1, ∗ where V+(t + ∆t)V †(t3 + ∆t)V †(t1 + ∆t) hT − − i µgf µfeµeg wherevg∆t is the optical path length connecting molecules Tg0g(ω1, ω2) = (ω1 ωeg + iγeg)(ω1 + ω2 ωfg + iγfg) r r − − situated at i and j. Using this identity and equation (243) is the transition amplitude. This is similar to the Kramers- The matter SNGF in the frequency domain can be factorized Heisenberg form of ordinary (single-photon) fluorescence into the square of an ordinary (causal) response function (one (Marx et al., 2008). As in single photon fluorescence, for a “+” and several “-”): (5) correct description of the TPIF the ground state must not be χ++ ( ω3; ω3, ω2, ω2, ω1, ω1) = = −−−− − − − degenerate. Otherwise γ = 0 (the degenerate ground state 2 gg (2) the system has infinite life time) and the signal vanishes. χ+ ( (ω1 + ω2); ω2, ω1) −− − The SNGF in equation (239) is commonly called as the flu- The unique factorization of the coherent matter SNGF’s can orescence quantum efficiency (Webb (Xu and Webb, 1996)) be also obtained in the L, R representation using the dia- and two-photon tensor (Callis (Callis, 1993)). Our result is grammatic technique as shown in Fig.30(C2). The classical identical to that of Callis, apart from the δ(ω +ω ω ω ) 1 2− 3− gg0 modes have to excite the molecules and the quantum mode factor. has to de-excite them. Hence, the interaction with the first When the two classical coherent modes are degenerate molecule ket (L) are accompanied by the conjugate interac- ( ) the signal given by equation (241) describes non- ω1 = ω2 tions with the second molecule bra (R). We obtain that the resonant Hyper-Raman scattering (ω = 0) also known as gg0 ordinary, causal response function is give by equation (235) as incoherent second harmonic inelastic scattering6 (Andrews and χ(2) ( (ω + ω ); ω , ω ) = χ(2) ( (ω + ω ); ω , ω ). Allcock, 2002; Callis, 1993). When ω = ω and ω + 1 2 2 1 LLL 1 2 2 1 1 2 gg0 The−− homodyne-detected− SFG signal− is finally given by: 0 (but not equal to) equation (241) describes non-resonant→ Hyper-Rayleigh scattering also known as incoherent second 2 2 2(ω1 + ω2) S = N(N 1) harmonic elastic scattering. Off-resonant hyper-scattering is SFG − |E1| |E2| Ω × 2 a major complicating factor for TPIF microscopy (Xu et al., (2) χ+ ( (ω1 + ω2); ω2, ω1) (245) 1997). −− −

Both homodyne and heterodyne SFG are given by the same (2) causal response function χ+ . The main difference is that the latter satisfies perfect phase−− matching, while for the former this condition is only approximate. For sufficiently large samples the two techniques are identical. To conclude, we present the total signal for the three wave process involving two classical and one quantum field (CCQ)) which includes both an incoherent and a coherent components: 2ω S(5) (ω , ω ) = 2 2 3 (246) CCQ 2 1 |E1| |E2| Ω (5) [N χ++ ( ω3; ω3, ω2, ω2, ω1, ω1) + = −−−− − − − 2 (2) N(N 1) χ+ ( (ω1 + ω2); ω2, ω1) ] − −− −

G. Two-photon-emitted ﬂuorescence (TPEF) vs. type-I parametric down conversion (PDC).

1. TPEF FIG. 31 Three wave process involuting two quantum and one classical mode: (A) phase-matching condition; (B) molecular level We now turn to three wave processes involving one clas- scheme; (C) loop diagrams for the incoherent TPEF (C1) and co- sical and two quantum modes. We start with the incoherent herent Type I PDC (C2) response of N identical molecules initially in their ground state. The initial state of the optical field is t = = | −∞i Diagram (C1) in Fig.31 satisfies the above conditions. The β1 1 0 2 0 3. The classical field k1 pumps the molecule from| i its| i ground| i state g into the excited state f . The sys- non-resonant diagrams have been omitted. Using this diagram | i | i the optical field SNGF yields: tem then spontaneously emits two photons into modes k2, k3 (See Fig.31 (B)) which are initially in the vacuum state. Such (5) incoherent process which involves one classical and two quan- ELLRR++ (t6, t5, . . . , t1) = (247) tum modes will be denoted two-photon-emitted fluorescence 2π~ω3 2π~ω2 N (t ) ∗(t ) (TPEF). To our knowledge there is neither theoretical nor ex- E1 1 E1 2 Ω Ω × perimental work concerning this process. exp (iω3(t6 t5)) exp (iω3(t4 t3)) This process is not phase sensitive since the phase matching − − condition k1 k1+k2 k2+k3 k3 = 0 is automatically sat- The matter SNGF assumes the form: − − − isfied for any k3. Therefore spontaneously generated modes are emitted into two spatial cones (See Fig.31 (A)). For a sin- (5) VLLRR (t6, t5, . . . , t1) = (248) −− gle molecule the cones are collinear, as in Type I paramet- 3 3, 2, 2 1 1, VL (t)VR †(t5)VR †(t4)VL (t3)V (t2)V †(t1) (249) ric down conversion (PDC). Due to this similarity we chose hT − − i beams polarizations as is usually done for PDC of this type: Using equations (247), (248) the incoherent part of the fre- the spontaneously generated photons have the same polariza- quency domain signal is given by: tion along the x axis, and orthogonal to that of the classical mode polarized along the y axis. N 2 2π~(ω2) 2π~ω3 The time-averaged photon flux in the k3 mode is our TPEF STPEF (ω1) = 1 π~ |E | Ω Ω × signal. We again make use of the loop diagrams to identify (5) the relevant SNGF contributing to the signal. Using the ini- χLLRR ( ω3; ω3, ω2, ω2, ω1, ω1) = −− − − − tial state of the field the diagrams must satisfy the following conditions: The corresponding SNGF can be calculated from the diagram in Fig.31(C1): 1. The creation operators of a spontaneously generated

modes a3†, a2† acting on the ”ket” must be accompanied (5) χLLRR ( ω3; ω3, ω2, ω2, ω1, ω1) = by the corresponding annihilation operators a3, a2 act- −− − − − ing on the ”bra”. 1 g V1†G†(ωg + ω1)V2†G†(ωg + ω1 ω2) 5!~5 h | − × 2. The ﬁrst mode ω (k ) is off-resonance with both ω 1 1 eg V3G†(ωg + ω1 ω2 ω3)V3 and ωfe transitions to avoid stimulated emission contri- − − × G(ω + ω ω )V G(ω + ω )V † g butions. g 1 − 2 2 g 1 1 | i 50

Expansion in the molecular eigenstates brings the response This mixed representation can be recast in +, and L, R rep- function into the Kramers-Heisenberg form: resentations: − (5) χLLRR ( ω3; ω3, ω2, ω2, ω1, ω1) = (250) −− − − − (2) (2) 1 (2) (2) 1 x x y 2 χLL = χLLL = (χ+ + χ++ ) (254) µg eµef µ δ(ω1 ω2 ω3) − 2 −− − 5!~5 | 0 fg| − − Xgg0 1 1 2 Comparing the CRF for heterodyne detected DFG (233) (ω ω )2 + γ2 |ω ω ω + iγ | 1 − fg fg 1 − 2 − eg eg and the SNGF’s for Type I PDC (254) we see that the latter Note that unlike TPIF, the TPEF signal depends on the is described not only by causal response function χ(2) but (5) + (2)−− SNGF other than causal response function χ+ . −−−−− also by the second moment of material fluctuations χ++ . On the other hand, in the L, R representation it singles out one− Li- (2) 2. Type-I PDC. ouville pathway χLLL, while the classical optical fields drive the material system along all possible pathways. We now turn to the coherent response from a collection of An interesting effects arise in type-I PDC if the detection identical molecules which interact with one classical pumping process is included explicitly. The details of these effects are mode and two spontaneously generated quantum modes (See discussed in Section IV.I. Fig.31 (A,B)). This is known as Type I parametric down conversion (PDC), which is widely used for producing entangled In summary of this section we give the signal for the three photon pairs. Hereafter we assume perfect phase matching wave process involving one classical and two quantum fields (CQQ). This contains both an incoherent and a coherent com- ∆k = k1 k2 k3 which is the case for a sufficiently large sample (Gerry− and− Knight, 2005). ponent: The initial condition for PDC are the same as for TPEF and most PDC experiments are well described by the causal (5) 1 2 2π~ω2 2π~ω3 (2) SCQQ(ω2, ω1) = 1 (255) response function χ+ . Therefore one would expect a con- π~ |E | Ω Ω × nection between TPEF−− and PDC, similar to that of TPIF and (5) N (χ++ ( ω3; ω3, ω2, ω2, ω1, ω1)+ homodyne-detected SFG. However, as we are about to demon- = −−−− − − − (5) strate, for a complete description of the PDC process the χ+++ ( ω3; ω3, ω2, ω2, ω1, ω1)+ −−− − − − causal second order response function is not enough. (5) χ++++ ( ω3; ω3, ω2, ω2, ω1, ω1))+ We shall establish the corrections to the second order CRF −− − − − (2) caused by the quantum origin of the spontaneous modes. To N(N 1) χ+ ( (ω1 ω2); ω2, ω1)+ − | −− − − do so we again resort to the CTPL diagrams (See Fig.31 (C2)). (2) 2 χ++ ( (ω1 ω2); ω2, ω1) For the coherent response the optical field SNGF is given by − − − | equation (247) with the factor N replaced by N(N 1). The material SNGF (248) can be factorized as: − (5) VLLRR (t6, t5, . . . , t1) = (251) −− H. Type II PDC; polarization entanglement. 3 2 1, 3, 2, 1 VL (t6)VL (t3)VL †(t1) VR †(t5)VR †(t4)VR(t2) hT ihT (252)i In Type II parametric down-conversion, the two Note that this factorization is unique as spontaneously-generated signals have orthogonal polar- 3 2, 1, V (t)V †(t )V †(t ) = 0 due to the material ini- hT L R 4 L 1 i izations. Because of birefringence, the generated photons tially being in its ground state. The coherent PDC signal in are emitted along two non-collinear spatial cones known the frequency domain can then be written as: as ordinary and extraordinary beams (See Fig.32 (A)). Polarization-entangled light (Gerry and Knight, 2005) is SPDC (ω1) = (253) generated at the intersections of the cones. An x polar- N(N 1) 2 2π~ω2 2π~(ω1 ω2) − 1 − ization filter and a detector are placed at one of the cones 4π~ |E | Ω Ω × intersections. The detector cannot tell from which beam a (2) 2 χLL ( (ω1 ω2); ω2, ω1) photon is obtained. To describe the process we need five | − − − | Here the generalized response function expanded in the eigen- optical modes: one classical 1 y and four quantum modes 2 x , 2 y , 3 x , 3 y |. i| i states has form: {| i| i | i| i | i| i | i| i} (2) Polarization-entangled signal is described by CTPL dia- χLL ( (ω1 ω2); ω2, ω1) = − − − grams shown in Fig.32 (C). The Type II PDC signal consist y x x 1 µgf µfeµeg (5) (5) (5) (5) of two parts S = S3x + S2x . The signal S3x as- 2 PDCII 2!~ (ω1 ωgf + i~γgf )(ω1 ω2 ωeg + i~γeg) − − − sumes the form of equation (253), with the material pathway 51

I. Time-and-frequency resolved type-I PDC

Previously we briefly summarized various wave-mixing signals in a very simple toy model without addressing the detection process in detail. In the following we include the photon counting detection described in Section III.D and demonstrate how does the actual generation process in type-I PDC relate to nonlinear response and how different it is compared to semiclassical theory. The standard calculation of nonlinear wave mixing assumes that all relevant field frequencies are off resonant with matter. It is then possible to adiabatically eliminate all matter degrees of freedom and describe the process by an effective Hamilto- nian for the field that contains a nonlinear cubic coupling of three radiation modes (Mandel and Wolf, 1995). For SFG this reads:

(2) H = drχ E† (ω ω ) E† (ω ) E (ω ) (260) eff − 3 1 − 2 2 2 1 1 Z FIG. 32 Type II PDC: (A) phase matching, (B) molecular level and for DFG and PDC: scheme; (C) CTPL diagrams for the signal from k3 (C1) and k2 (C2) modes polarized along x direction. (2) H = drχ E† (ω + ω ) E (ω ) E (ω ) (261) eff − 3 1 2 2 2 1 1 Z depicted in Fig.32(C1): All matter information is embedded into a coefficient that is proportional to χ(2) (Gerry and Knight, 2005) that is de- (2) χLL ( (ω1 ω2); ω2, ω1) = (256) fined by the semiclassical theory of radiation-matter coupling. − − − − Langevin quantum noise is added to represent vacuum fluctu- 1 (2) (χ+ ( (ω1 ω2); ω2, ω1) + ations caused by other field modes (Avenhaus et al., 2008; 2 −− − − − (2) Glauber, 2007b; Scully and Zubairy, 1997) and account for +χ++ ( (ω1 ω2); ω2, ω1)) = − − − − photon statistics. 1 C The microscopic theory of type I PDC presented below 2 2!~ (ω1 ωgf + i~γgf )(ω1 ω2 ωeg + i~γeg) holds if cascade of two photons in a three level system is − − − generated in both on and off resonance. The resonant case is where the coefficient C is given by the following equation: especially important for potential spectroscopic applications (Mukamel, 1995), where unique information about entangled C2 = µy 2( µx 2 µx 2 + 2µx µx µy µy + (257) | gf | | fe| | eg| fe eg fe eg matter (Lettow et al., 2010) can be revealed. Other exam- x x x y y y y x µfeµegµfeµeg + µfeµegµfeµeg ples are molecular aggregates and photosynthetic complexes or biological imaging (Saleh et al., 1998). Second, it properly (5) The signal S3x is described by the diagram in Fig.32 (C2): takes into account the quantum nature of the generated modes through a generalized susceptibility that has a very different (2) behavior near resonance than the semiclassical (2). (2) is χLL ( ω2; (ω1 ω2), ω1) = (258) χ χ − − − − derived for two classical fields and a single quantum field. 1 (2) (χ+ ( ω2; (ω1 ω2), ω1) + While this is true for the reverse process (sum frequency gen- 2 −− − − − (2) eration) it does not apply for PDC, which couples a single +χ++ ( ω2; (ω1 ω2), ω1)) = − − − − classical and two quantum modes. Third, macroscopic prop- 1 C agation effects are not required for the basic generation of 2 2!~ (ω1 ωgf + i~γgf )(ω2 ωeg + i~γeg) the signal. Fourth, gated detection (Dorfman and Mukamel, − − 2012a) yields the finite temporal and spectral resolution of the The net Type II PDC signal is: coincident photons limited by a Wigner spectrogram. For either time or frequency resolved measurement of the generated (259) SPDCII (ω1) = field, the signal can be expressed as a modulus square of the N(N 1) 2 2π~ω2 2π~(ω1 ω2) transition amplitude that depends on three field modes. This − 1 − 4π~ |E | Ω Ω × is not the case for photon counting. Shwartz et al. recently (2) 2 reported PDC in diamond, where 18keV pump field gener- χLL ( (ω1 ω2); ω2, ω1) + | − − − − | ates two X-ray photons (Shwartz et al., 2012). Calculation (2) 2 χLL ( ω2; (ω1 ω2), ω1) . in (Dorfman and Mukamel, 2012b) reproduces experimental | − − − − | 52 data in the entire frequency range without adding Langevin Eq. (264) has a single commutator and is symmetric to a per- noise. mutation of ωi and ωs = ωp ωi, as it should. Eq. (263) has The nature of entangled light can be revealed by photon two commutators and lacks this− symmetry. correlation measurements that are governed by energy, momentum and/or angular momentum conservation. In PDC, a nonlinear medium is pumped by electromagnetic field of fre- 1. The bare PCC rate quency ωp and some of the pump photons are converted into pairs of (signal and idler) photons with frequencies ωs and ωi, The measured PCC rate of signal and idler photons starts respectively (see Fig. 33a) satisfying ωp = ωs + ωi. with the definition (175). The basis for the bare signal in Eq. To get more detailed description of wave-mixing process in (174) is given by the time-ordered product of Green’s func- PDC we recast an effective semiclassical Hamiltonian in Eq. tions of superoperators in the interaction picture (see (Dorf- (261) as follows man and Mukamel, 2012b)).

dωs dωi dωp (2) (i) (s) (s0) (i0) Hint(t) = i~ χ+ (ωs, ωi, ωp) E† (t0 )E† (t0 + τ )E (t0 )E (t0 τ ) 2π 2π 2π −− hT R i R s s L s L i − i j i R Z ∞ √2H0 (τ)dτ X ~ (s) (i) (p) i∆k(r rj ) i(ωp ωs ωi)t] e− −∞ − ρ( ) . (265) aˆ †aˆ †β e − e− − − + H.c., (262) × −∞ i × (s) (i) which represents four spectral modes arriving at the detectors, where aˆ† and aˆ † are creation operators for signal and idler modes, β(p) is the expectation value of the classical pump where modes s, s0, i, i0 are defined by their frequencies ωs, ωi0, (2) ωi = ωp ωs, ωs0 = ωp0 ωi0. field, ∆k = kp ks ki, j runs over molecules, χ+ , − − − − −− In type I PDC the sample is composed of N identical (normally denoted χ(2)( ω ; ω , ω )) is the second-order s i p molecules initially in their ground state. They interact with nonlinear susceptibility − − one classical pump mode and emit two spontaneously generated quantum modes with the same polarization into two 2 (2) i ∞ ∞ collinear cones. The initial state of the optical field is given χ+ (ωs = ωp ωi, ωi, ωp) = dt2dt1 −− − ~ 0 0 × by 0 s 0 i β p. A classical pump field promotes the molecule Z Z | i | i | i iωi(t2+t1)+iωpt1 from its ground state g to the doubly excited state f (see e [[V (t2 + t1),V (t1)],V (0)] + (i p). | i | i h i ↔ (263) Fig 33b). Due to the quantum nature of the signal and the idler fields, The “ + 00 indices in Eq. (263) signify two commu- −− the interaction of each of these fields with matter must be tators followed by an anti commutator. The bottom line of at least second order to yield a non vanishing signal. The the semiclassical approach is that PDC is represented by 3- leading contribution to Eq. (265) comes from the four dia- point matter-field interaction via the second order suscepti- grams shown in Fig. 33c (for rules see Appendix A (Roslyak (2) bility χ+ that couples the signal, idler and pump modes. et al., 2009a)). The coherent part of the signal represented However,−− it has been realized, that other field modes are by interaction of two spontaneously generated quantum and needed to yield the correct photon statistics. Electromagnetic one classical modes is proportional to the number of pairs field fluctuations are then added as quantum noise (Langevin of sites in the sample N(N 1), which dominates the forces) (Scully and Zubairy, 1997). other, incoherent, N∼response− for large N. Details of the Below we summarize a microscopic calculation of the pho- calculation of the∼ correlation function (265) are presented ton coincidence counting (PCC) rate in type I PDC (Roslyak in Ref. (Dorfman and Mukamel, 2012b). We obtain for the (B) et al., 2009a). We show that PDC is governed by a quantity “bare” frequency domain PCC rate Rc (ω , ω0, ω , ω0 ) s i p p ≡ that resembles but is different from (263). In contrast with i(ωs+ω0 ω0 )t0 +(ωs+ω0 ωp)t0 dt0 dt0 nˆ (t0 , ω0 )ˆn (t0 , ω0) e− i− p s i− i the semiclassical approach, PDC emerges as a 6-mode two- s ihT s s s i i i iT molecule rather than 3-mode matter-field interaction process R and is represented by convolution of two quantum susceptibil- 2π 4 (2) (2) (B) ~ Rc (ωs, ωi0, ωp, ωp0 ) = N(N 1) ities χLL (ωs, ωi, ωp) and χLL∗ (ωs0 , ωi0, ωp0 ) that represent a − V × − − pair of molecules in the sample interacting with many vacuum (p) (p) ∗ (ωp) (ω0 ) (ωs) (ωp ωs) (ω0) (ω0 ω0) modes of the signal (s, s0) and the idler (i, i0). Field fluctua- E E p D D − D i D p − i × tions are included self consistently at the microscopic level. (2) (2) χLL [ (ωp0 ωi0), ωi0, ωp0 ]χLL∗ [ ωs, (ωp ωs), ωp], − − − − − − − − Furthermore the relevant nonlinear susceptibility is different (266) (2) from the semiclassical one χ+ and is given by −− where (p)(ω) E(p)(ω)β(p) is a classical field amplitude, i ∞ ∞ (2) (p) E ≡ ˜ χLL (ωs = ωp ωi, ωi, ωp) = dt2dt1 E (ω) is the pump pulse envelope and (ω) = ω (ω) − − × D D ~ Z0 Z0 where ˜(ω) = V ω2/π2c3 is the density of radiation modes. iωi(t2+t1)+iωpt1 D e [V (t2 + t1)V (t1),V (0)] + (i s). For our level scheme (Fig. 33b) the nonlinear susceptibility h i ↔ (264) (2) χLL (see Eq. (264)) is given by − 53

FIG. 33 (Color online) Left: Schematic of the PDC experiment - (a), the three level model system used in our simulations - (b). One out of four loop diagram (remaining three diagrams are presented in Ref.(Dorfman and Mukamel, 2012b)) for the PCC rate of signal and idler photons generated in type I PDC (Eq. (265)) - (c). The left and right diagrams represent a pair of molecules. Blue (red) arrows represent ﬁeld-matter interaction with the sample (detectors). There are four possible permutations (s/i and s0/i0) which leads to four terms when Eq. (2) (267) is substituted into Eq. (266). Right: Absolute value of semiclassical susceptibility |χ+−−(−(ωp − ωi), −ωi, ωp)| (arb. units) (268) - (2) (d), and quantum the susceptibility |χLL−(−(ωp − ωi), −ωi, ωp)| (267) - (e) vs pump ωp and idler frequency ωi. We used the standard KTP (2) (2) parameters outlined in the text. Left column: real - (f), imaginary part - (h) and absolute value - (j) of χ+−− - red line and χLL− - blue line for off resonant pump ωp − ωgf = 10γgf . Right column: (g,i,k) - same as (f,h,j) but for resonant pump ωp ' ωgf .

we obtain a photon occupation number that can be detected. Thus, the detection process must be described in the joint µ µ µ (2) 1 gf∗ fe eg space of the two molecules and involves the interference of χLL [ (ωp0 ωi0), ωi0, ωp0 ] = − − − − 2~ ω0 ωgf + iγgf × p − four quantum pathways (two with bra and two with ket) with 1 different time orderings. Note that this pathway information + (ω0 ω0 ω0). (267) ω ω ω + iγ ) i ↔ p − i is not explicit in the Langevin approach. p0 − i0 − eg eg

Eq. (266) represents a 6-mode (ωp, ωi, ωs, ωp0 , ωi0, ωs0 ) ﬁeld-matter correlation function factorized into two generalized susceptibilities each representing the interaction of two quantum and one classical mode with a different molecule.

Because of two constraints ωp = ωs + ωi, ωp0 = ωs0 + ωi0 that originate from time translation invariance on each of the two molecules that generate the nonlinear response, Eq. (266) For comparison, if all three fields (signal, idler and pump) only depends on four field modes. Each molecule creates a co- are classical, the number of material-field interactions is re- herence in the field between states with zero and one photon. duced to three - one for each field. Then the leading contribu- By combining the susceptibilities from a pair of molecules tion to the field correlation function yields the semiclassical 54

ωi = ωeg - Fig. 33g,h,k. However, close to resonance - Fig. (a) 24 (c) 6

33d,e the two susceptibilities are very different. The semi- (2) 12 THz 12 classical susceptibility χ vanishes at resonance, where THz

, + , s s −−(2) ω 0 ω 0 the quantum susceptibility χLL reaches its maximum. To put the present ideas into− more practical perspective we show in Fig. 34 the PCC rate for a monochromatic pump ω (b) (d) p 22 22 and mean signal detector frequency ω¯s. The quantum theory

11 11 yields one strong resonant peak at ω¯s = ωp ωeg and two THz THz , − s , s weak peaks at ωp = ωgf and ω¯s = ωeg if the idler detector is ω ω 0 0 resonant with the intermediate state e : ω¯i = ωeg - Fig. 34a. However, if we tune the idler detector| toi a different frequency, 562.996 563.000 563.003 562.996 563.000 563.003 for instance ω ω¯ = 2GHz there is an additional peak at ωp, THz ωp, THz eg i ω¯ = ω ω¯ - Fig.− 34b. Similarly, when the pump consists of s p − i FIG. 34 (Color online) Left column: two dimensional PCC rate (log two monochromatic beams ωp = ωp0 (panels c,d) the number scale in arb. units) calculated using quantum theory Eq. (175) as- of peaks are doubled compare to6 single monochromatic pump. suming a single monochromatic pump with frequency ωp. Idler de- Clearly, one can reproduce the exact same peaks for ωp0 as for tector resonant with intermediate level ω¯i = ωeg = 282THz - (a), ωp. while ωeg − ω¯i = 2GHz for (b). Right column: same as left but for a pump made out of two monochromatic beams with frequencies 0 −6 ωp − ωp = 2 · 10 ωp. 3. Spectral diffusion

(2) nonlinear susceptibility χ+ Spectral diffusion (SD) which results from the stochastic −− modulation of frequencies can manifest itself either as dis- (2) χ+ [ (ωp0 ωi0), ωi0, ωp0 ] = crete random jumps of the emission frequency (Santori et al., −− − − − 1 2010; Siyushev et al., 2009; Walden-Newman et al., 2012) or g VL (ωp0 ωi0)VL (ωp0 )VL† g + as a broadening of a hole burnt in the spectrum by a narrow- 2~2 h |T G − G | i band pulse (Wagie and Geissinger, 2012; Xie and Trautman, 1 1 µgf∗ µfeµeg † 2 g VL †(ωi0)VL (ωp0 )VL g = 2 1998). We focus on the latter case assuming that the electronic ~ h |T G G | i ~ ωp ωgf + iγgf × − states of molecule α = a, b are coupled to a harmonic bath 1 1 ˆ α α α described by the Hamiltonian H = ~ωk(â† aˆ + 1/2) + + (ωi0 ωp ωi0), B k k k ωp ω0 ωeg + iγeg ω0 ωeg iγeg ↔ − (see Fig. 36b). The bath perturbs the energy of state ν. This − i − i − − (268) is represented by the Hamiltonian P where (ω) = 1/(ω H0 /~ + iγ) is the retarded Liouville ˆ α 1 ˆ ˆ α H = ~− να H να = να +q ˆνα + H , (269) G − − (2) ν h | | i B Green’s function, and γ is lifetime broadening. χLL pos- − sesses a permutation symmetry with respect to s i (both where qˆν is a collective bath coordinate have L index). In contrast the semiclassical calculation↔ via (2) 1 ˆ χ++ is non symmetric with respect to s i ( one + and qˆνα = ~− να HSB να = dνανα,k(âk† +a ˆk), (270) one − indexes), which results in PCC rate that↔ depends upon h | | i Xk whether− the signal or idler detector clicks first (Shwartz et al., 2012). dmn,k represents bath-induced fluctuations of the transition energies (m = n) and the intermolecular coupling (m = n). We define the line-shape function 6 2. Simulations of typical PDC signals dω Cν00 ν (ω) g (t) g (t) = α α0 Fig. 33d-k compares both susceptibilities calculated for α ≡ νανα0 2π ω2 a typical KTP crystal (PPKTP) represented by a degenerate Z β~ω three-level system with parameters taken from Ref. (D’auria coth (1 cos ωt) + i sin ωt iωt , (271) × 2 − − et al., 2008; Wu et al., 1986). Fig. 33f,h,j show that far from resonances (ωi = ωeg, ωp = ωgf ) the semiclassical and quan- where the bath spectral density is given by tum susceptibilities6 coincide6 and depend weakly on the frequencies ωp and ωi. This is the regime covered by the semi- 1 ∞ iωt C00 (ω) = dte [ˆqνα(t), qˆν α(0)] , (272) classical theory, where the susceptibility is assumed to be a νανα0 0 2 0 h i constant. Similar agreement between classical and quantum Z susceptibilities can be observed if the pump is resonant with β = kBT with kB being the Boltzmann constant and T is the two-photon transition ω ω but the idler is off resonance ambient temperature. We shall use the overdamped Brownian p ' gf 55 oscillator model for the spectral density, assuming a single regime is relevant to crystals which store information in the nuclear coordinate (να = να0 ) form of reversible notches that are created in their optical absorption spectra at specific frequencies. Long storage times ωΛα C00 (ω) = 2λ , (273) (Longdell et al., 2005), high efficiencies (Hedges et al., 2010), νανα α 2 2 ω + Λα and many photon qubits in each crystal (Shahriar et al., 2002) can be achieved in this limit. The HBL limit is natural for where Λα is the fluctuation relaxation rate and λα is the system-bath coupling strength. The corresponding lineshape long-term quantum memories where entanglement is achieved function then depends on two parameters: the reorganization with telecom photons, proved the possibility of quantum inter- net (Clausen et al., 2011; Saglamyurek et al., 2011). energy λα denoting the strength of the coupling to the bath Time-and-frequency resolved fluorescence is the simplest and the fluctuation relaxation rate Λα = 2λαkBT/~. In the way to observe SD. The molecular transition frequency is cou- high-temperature limit kBT ~Λα we have p pled linearly to an overdamped Brownian oscillator that rep- 2 ∆α λα Λαt resents the bath (see Fig. 36b). This fluorescence signal given g (t) = i e− + Λ t 1 . (274) α Λ2 − Λ α − by Eq. (284) is depicted as a series of the snapshot spectra at α α different times for molecule a in Fig. 35c. It shows a time For a given magnitude of fluctuations ∆ , α = a, b the α dependent frequency redshift ωã(t) and time dependent spec- FWHM of the absorption linewidth (Mukamel, 1995) 0 tral broadening given by σã0(t). Initially ωã(0) = ωa + λa 0 2.355 + 1.76(Λ /∆ ) whereas at long times ωã( ) = ωa λa, where 2λα is the α α (275) ∞ − Γα = 2 ∆α. Stokes shift. Same for molecule b is shown in Fig. 35d. Be- 1 + 0.85(Λα/∆α) + 0.88(Λα/∆α) cause of the different reorganization energies λa, λb and relax- The absorption and emission lineshape functions for a pair of ation rates Λa, Λb the Stokes shift dynamics and dispersion are molecules obtained in the slow nuclear dynamics limit: Λ different. Even when the absorption frequencies are the same α ∆α are given by (Mukamel, 1995) ωa = ωb, the fluorescence can show a different the profile due to SD. This affects the distinguishability of the emitted 0 2 (ω ω λα) − α− 1/2 2∆2 photons as will be demonstrated below. σA(ω) = (2π∆α)− e− α , (276) αX=a,b

0 2 J. Generation and entanglement control of photons (ω ωα+λα) 1/2 − 2 produced by two independent molecules by σ (ω) = (2π∆ ) e− 2∆α , (277) F α − time-and-frequency gated photon coincidence counting αX=a,b (PCC). where 2λ is the Stokes shift and ∆ = 2λ k T/ is α α α B ~ As discussed in section II, entangled photons can be pro- a linewidth parameter. Together with the relaxation rate Λ p α duced by a large variety of χ(2)-processes. Alternatively, pho- (see Eq. (274)) these parameters completely describe the SD tons can be entangled by simultaneous excitation of two re- model and govern the evolution of the emission linewidth be- mote molecules and mixing the spontaneously emitted pho- tween the initial time given by Eq. (276) (see Fig. 35a) and tons on the beam splitter. This entangled photon generation long time given by Eq. (277) (see Fig. 35b). scheme is suitable e.g. for spectroscopic studies of single The corresponding four-point matter correlation function molecules by single photons discussed below. The manipu- may be obtained by the second order cumulant expansion lation of single photon interference by appropriate time-and- (Mukamel, 1995) frequency gating discussed below is presented in details in 4 iωα(t1 t2+t3 t4) Φα(t1,t2,t3,t4) Ref. (Dorfman and Mukamel, 2014a). The generated entan- Fα(t1, t2, t3, t4) = µα e− − − e , | | (278) gled photon pair can be used in logical operations based on optical measurements that utilize interference between indis- where ωα ωe ωg is the absorption fre- tinguishable photons. ≡ α − α quency, Φα(t1, t2, t3, t4) is the four-point lineshape function We examine photon interference in the setup shown in Fig. Φα(t1, t2, t3, t4) = gα(t1 t2) gα(t3 t4) + gα(t1 36a. A pair of photons is generated by two remote two- − − − − − t3) gα(t2 t3) + gα(t2 t4) gα(t1 t4). level molecules and with ground and excited state , − − − − − a b gα eα We focus on the SD in the “hole burning” limit (HBL) α = a, b. These photons then enter a beam splitter and are which holds under two conditions: First, the dephasing is subsequently registered by time-and-frequency gated detec- 1 much faster than the fluctuation timescale, i.e. tk0 Λα− , tors s and r. There are three possible outcomes: two photons 1 k = 1, 2, 3, 4. Second, if excitation pulse duration σp− and registered in detector s, two photons registered in r or coin- j 1 j 1 the inverse spectral (σω)− , and temporal (σT )− , j = r, s cidence where one photon is detected in each. The ratios of gate bandwidths are much shorter than the fluctuation time these outcomes reflects the photon Bose statistics and depends scales, one may neglect the dynamics during the delay be- on their degree of indistinguishability. If the two photons in- tween population evolution and its detection. This parameter cident on the beam splitter are indistinguishable the photon 56

1.0 1.2 1.0 (a) (b) t=0 (c) t=0 (d) 2.0 1.0 0.8 0.8 t=50 ps t=50 ps 0.8 L L L L 1.5 t=150 ps

w t=150 ps 0.6 w w w

0.6 D D D D , , H

H 0.6 t t A F H H a 1.0 b s s 0.4 C 0.4 C 0.4 t=1 ns 0.5 t=1 ns 0.2 0.2 0.2

0.0 0.0 0.0 0.0 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 -2 -1 0 1 2 Dw, kHz Dw, kHz Dw, kHz Dw, kHz 1.2 1.2 1.2 1.2 0 0 (Λ =0.5, Λ =1) MHz σ =50 MHz σ =80 MHz ωb -ωa =150MHz (e) a b (f) ω (g) T (h) 1.0 1.0 1.0 1.0

0.8 0.8 Units Units 0.8 0.8 Units Units . . . .

Arb 0.6 Arb 0.6 Arb Arb 0.6 0.6 , , , ,

0.4 100MHz 0.4 0.4 0.4 Counts Counts Counts Counts σ =120 MHz σ =200 MHz σ =110 MHz σ =150 MHz 0.2 0.2 0.2 ω ω 0.2 T T 50MHz (10,1)MHz (18,1)MHz 0.0 0.0 0.0 0.0 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 -20 -10 0 10 20 t -t , ns t -t , ns s r s r ts-tr, ns ts-tr, ns

1 FIG. 35 (Color online) Top row: Absorption (276) - (a) and fluorescence (277) - (b) line shapes vs displaced frequency ∆¯ω = ω− 2 (ωa +ωb). Frequency dispersed time-resolved fluorescence (284) displayed as a snapshot spectra for molecule a - (c) and b - (d). Bottom row: PCC for 0 different transition energies of molecules excited at ωp = ωb + λb - (e); PCC for different values of the SD time scale Λa and Λb and fixed linewidth Γa, Γb according to Eq. (275) - (f); PCC for different frequency gate bandwidths - (g) at fixed time gate bandwidth σT = 100 MHz and for different time gate bandwidths - (h) at fixed frequency gate bandwidth σω = 100 MHz. Molecules have distinct SD timescales 0 0 Λa = 15 MHz, Λb = 1 MHz and ωb − ωa = 1 MHz. coincidence counting signal (PCC) vanishes. This causes the ferometry for direct-field reconstruction (SPIDER) (Dorrer Hong-Ou-Mandel (HOM) dip when varying the position of et al., 1999) are well established tools for ultrafast metrol- the beam splitter which causes delay T between the two pho- ogy (Walmsley and Dorrer, 2009; Wollenhaupt et al., 2007). tons. The normalized PCC rate varies between 1 for com- Extending these techniques to a single photon time and fre- pletely distinguishable photons (large T ) and 0 when they are quency resolved detection is challenging and may be achieved totally indistinguishable (T = 0). For classical fields and if combined with on-chip tunable detectors (Gustavsson et al., 50:50 beam splitter the PCC rate may not be lower than 1/2. 2007) or upconversion processes (Gu et al., 2010; Ma et al., We denote the photons as indistinguishable (distinguishable) 2011). if the PCC rate is smaller (larger) than 1/2.

PCC is typically measured by time-resolved detection 1. PCC of single photons generated by two remote emitters. (Gerry and Knight, 2005; Glauber, 2007b). Originally performed with entangled photons generated by parametric down The time-and-frequency gated PCC signal is described by conversion (PDC) (Hong et al., 1987) the shape of the dip vs the two pairs of loop diagrams shown in Fig. 36c. Each loop delay is usually related to the two-photon state envelope which represents one molecule (a or b) which undergoes four ﬁeld- is governed by an effective PDC Hamiltonian (Dorfman and matter interactions and each detector interacts twice with the Mukamel, 2012b). Bath effects can become important for re- ﬁeld. Fig. 36c shows that after interacting with the pump mote emitters and have been introduced phenomenologically (with its ket) at time t2 molecule a evolves in the coherence

(Lettow et al., 2010) below. We present a microscopic de- ρeaga during time interval t20 . The second interaction of the scription of PCC with bath fluctuations by formulating the sig- pump with the bra then brings the molecule into a popula- nal in the joint field-matter space measured by simultaneous tion state ρeaea which evolves during interval t1 until the first time-and-frequency resolved detection. This complex mea- interaction with spontaneous emission mode occurs with ket. surement can be achieved using high-speed photodiode which The molecule then evolves into the coherence ρgaea during t10 converts a fast optical signal into a fast electric current, fast until the second bra- interaction of spontaneous mode. During oscilloscopes to observe the waveform, wide bandwidth spec- population and coherence periods, the characteristic timescale trum analyzers and other elements. Short pulse characteri- of the dynamics is governed by population relaxation and de- zation using time-frequency map such as frequency-resolved phasing, respectively. optical gating (FROG) (Trebino, 2000), spectral phase inter- The relevant single particle information for molecule 57

FIG. 36 (Color online) Time-and-frequency resolved measurement of PCC with spectral diffusion. Schematic of the PCC experiment with two indistinguishable source molecules - (a), the two-level model of the molecule with SD used in our simulations - (b). (c) - Loop diagrams for the PCC rate of emitted photons from two molecules (for diagram rules see Appendix A). The left and right branches of each diagram represent interactions with ket- and bra- of the density matrix, respectively. Field-matter interactions with the pump pulses p1 and p2 (blue), spontaneously emitted s, s0, r, r0 photons (red) and detectors (brown).

α is given by the four point dipole correlation function are related by F (t , t , t , t ) = V (t )V † (t )V (t )V † (t ) , where α 1 2 3 4 h ge 1 eg 2 ge 3 eg 4 iα and are the lowering and raising dipole transition op- V V † E1(t) iE2(t + T ) E2(t) iE1(t T ) erators, respectively. Diagrams i in Fig. 36c represent non- E3(t) = − ,E4(t) = − − , √2 √2 interfering terms given by a product of two independent flu- (279) orescence contributions of the individual molecules. Dia- grams ii represent an interference in the joint space of the where cT is small difference of path length in the two arms. two molecules and involves the interference of eight quan- ± tum pathways (four with the bra and four with the ket) with Taking into account that Eq. (175) should be modified to in- different time orderings. Each molecule creates a coherence clude the beam splitter position and absorb it into the gating in the field between states with zero and one photon 0 1 spectrograms, the time-and-frequency resolved PCC in this and 1 0 . By combining the contributions from a pair| ih of| case is given by molecules| ih | we obtain a photon population 1 1 that can be | ih | detected (Dorfman and Mukamel, 2012a,b). For a pair of 34 1 ∞ 2 2 R (Γ , Γ ; T ) = d Γ0 d Γ0 identical molecules, the beam splitter destroys the pathway in- c r s (2π)2 r s Z−∞ formation making the molecules indistinguishable and giving (r) (s) (i) rise to quantum interference. [WD (Γr, Γr0 ; 0)WD (Γs, Γs0 , 0)RB (Γr0 , Γs0 )+ (r) (s) (ii) W (Γ , Γ0 ; T )W (Γ , Γ0 ,T )R (Γ0 , Γ0 )] D r r − D s s B r s + (s r, T T ). (280) ↔ ↔ −

The PCC signal (175) (see Ref. (Dorfman and Mukamel, Here Γ0 = t0 , ω0 represents the set of parameters of the j { j j} 2014a)) has been calculated for the output fields E3 and E4 matter plus field incident on the detector j = r, s. Eq. (280) (i = 3, s = 4) of the beam splitter (see Fig. 36a). The fields is given by the spectral and temporal overlap of the Wigner (s) (r) in the output 3, 4 and input 1, 2 ports of the 50:50 beamsplitter spectrograms of detectors WD , WD and bare signal path- 58

(i) (ii) ways RB and RB given by term (285) generally cannot be recast as a product of two amplitudes (Mukamel and Rahav, 2010). Eqs. (283) - (285) are

(i) ∞ iω0 τs iω0 τr simulated below using the typical parameters of the two pho- R (t0 , ω0 ; t0 , ω0 ) = dτ dτ e− s − r B s s r r s r ton interference experiments (Ates et al., 2012; Bernien et al., u,u0 v,v0 Z−∞ X X 2012; Coolen et al., 2008; Lettow et al., 2010; Patel et al., ˆ ˆ ˆ Eu† R(ts0 + τs, rb)Ev† R(tr0 + τr, ra)EvL(tr0 , ra) et al. et al. et al. × hT 0 0 2010; Sanaka , 2012; Santori , 2002; Sipahigil , i R ∞ Hˆ 0 (T )dT 2012; Trebbia et al., 2010; Wolters et al., 2013). Eˆ (t0 , r )e− ~ − , (281) × uL s b −∞ i

3. Signatures of gating and spectral diffusion in the HOM dip. (ii) ∞ iω0 τs iω0 τr R (t0 , ω0 ; t0 , ω0 ) = dτ dτ e− s − r B s s r r − s r u,uX0 Xv,v0 Z−∞ Photon indistinguishability depends on the molecular tran- Eˆ† (t0 + τ , r )Eˆ† (t0 + τ , r )Eˆ (t0 , r ) u0R s s b v0R r r a uL s a sition frequencies. Fig. 35e shows that for fixed time and fre- × hT j i R ˆ j ∞ H0 (T )dT quency gate bandwidths σω and σT , j = r, s the photons are Eˆ (t0 , r )e− ~ − . (282) × vL r b −∞ i distinguishabe as long as the transition energy offset ω0 ω0 is b − a Eqs. (281) -(282) contain all relevant field matter interactions. larger than the gate bandwidth and are indistinguishable oth- The modified expressions for the gating spectrograms that in- erwise. The SD timescale is a key parameter affecting the clude delay T are given in (Dorfman and Mukamel, 2014a). degree of indistinguishability. Using Eq. (275) we fixed the absorption linewidth Γα and varied Λα and ∆α. The PCC signal (283) depicted in Fig. 35f shows that if the molecules have 2. Time-and-frequency gated PCC nearly degenerate transition energy offset for slower fluctuations the photons are indistinguishable. Increasing the SD rate Under SD and HBL conditions the PCC signal (175) is of one of the molecules increases the photon distinguishabil- given by ity when both time and frequency gates are broader than the difference in transition frequencies. R34(Γ , Γ ; T ) = R Cr(Γ )Cs(Γ ) We further illustrate the effect of frequency and time gating c r s 0 a r b s × r ¯ s ¯ on spectral diffusion. Fig. 35g shows that if two molecules Ia(Γr, ts, T )Ib (tr, Γs,T ) Γ(˜ t¯s t¯r ) 1 − cos U(Γr, Γs; T )e− − have different SD timescales and the frequency gate band- − Cr(Γ )Cs(Γ ) a r b s width is narrow the photons are rendered distinguishable and + (a b, T T ), (283) ↔ ↔ − HOM dip is 0.6. By increasing the σω the photons become indistinguishable and HOM dip is 0.48 for σ = 120 MHz where expressions in the last line represent permutation of the ω and 0.35 for σ = 200 MHz. In all three cases we kept the molecules a and b, Γ = t¯ , ω¯ represents a set of gating ω j j j time gate fixed. Alternatively we change the time gate band- parameters for the detector{j = r,} s. C (Γ = t, ω ) is the α width while keeping the frequency gate fixed. Fig. 35h shows time-and-frequency resolved fluorescence of molecule{ } α = that initially indistinguishable photons at σ = 80 MHz with a, b corresponding to diagram i in Fig. 36c: T HOM dip 0.675 become indistinguishable at σT = 110 MHz 0 2 2 (ωp ω λa) (ω ω˜α(t)) with HOM dip 0.45 and at σT = 150 MHz with HOM dip − α− − j 2˜σ2 j2 j − pα − 2˜σα (t) Cα(t, ω) = Cα0(t)e , (284) 0.275. Thus, if the presence of the bath erodes the HOM dip the manipulation of the detection gating allows to preserve the 0 ωα = ωα λα is the mean absorption and fluorescence fre- quantum interference −j j quency. Iα(Γ1, t2, τ) and Iα(t1, Γ2, τ) with t1 < t2 is the interference contribution α = a, b, j = r, s corresponding to diagram ii in Fig. 36c V. SUMMARY AND OUTLOOK

2 ωab 1 j 2 2 j2 στα(t1,t2) τ j 4σ 4 The term quantum spectroscopy broadly refers to spec- j − T − Iα(Γ1, t2, τ) = Iα0(t1, t2)e troscopy techniques that make use of the quantum nature of j 2 (ωp ω (t ,t )) j 2 − pα 1 2 (ω1 ωα(t1,t2)) light. Photon counting studies obviously belong to this cate- j2 −j2 e− 2σpα(t1,t2) e− 2σα (t1,t2) , (285) × gory. Studies that detect the signal field such as heterodyne detection or fluorescence are obtained by expanding the sig- ¯ ¯ r ¯ ¯ U(Γr, Γs; τ) = ωa(ts tr) + ωτa(tr, ts, ω¯r)τ + nals in powers of the field operators. These depend on mul- ¯ ¯ − ¯ ¯ (λa/Λa)(2[Fa(tr) Fa(ts)]+Fa(ts tr)) (a b, r s), tipoint correlation functions of the incoming fields. Classical ∆−2 − − ↔ ↔ ˜ α Λαt Γ(t) = α=a,b 2 Fα(t) with Fα(t) = e− + Λαt 1, spectroscopy is then recovered when these can be factorized Λα − α = a, b and all the remaining parameters are listed in Ref. into products of field amplitudes. Otherwise the technique is P (Dorfman and Mukamel, 2014a). The contribution of Eq. quantum. Spectroscopy is classical if all fields are in a co- (284) is represented by an amplitude square coming from each herent state and the observable is given by normally ordered molecule in the presence of fluctuations. The interference product. 59

Another important aspect of quantum spectroscopy, which we had only touched briefly in section IV.B, concerns the non- EL(⌧)V †(⌧) ' L classical fluctuations of quantum light and their exploitation as spectroscopic tools (Benatti et al., 2010; Giovannetto et al., 2011). These may also lead to novel features in nonlinear optical signals (Lopez´ Carreno˜ et al., 2015). EL† (⌧)VL(⌧) We now classify and briefly survey the main features of ' quantum spectroscopy. First, the unusual time/frequency windows for homodyne, heterodyne and fluorescence detection FIG. 37 Diagram construction in a QED formulation: The vertical blue lines indicate the evolution of the matter density matrix. An ar- arising due to the quantum nature of the light generation re- row pointing towards (away from) it corresponds to a matter excita- sulting in the enhanced resolution of the signals not accessi- tion V †(τ) (de-excitation V (τ)), which is accompanied by a photon ble by classical light. Second, photon counting and interfer- annihilation E(τ) [creation E†(τ)]. In the Liouville space formu- ometric detection schemes constitute a class of multidimen- lation, interactions on the left side represent left-superoperators (1), sional signals that are based on detection and manipulation of and interaction on the right right-superoperators (2). a single photons and are parametrized by the emitted photons rather by the incoming fields. Third, the quantum nature of VI. ACKNOWLEDGEMENTS light manifest in collective effects in many-body systems by projecting entanglement of the field to matter. This allows to We gratefully acknowledge the support of the National e.g. prepare and control of higher excited states in molecu- Science Foundation through Grant No. CHE- 1361516, the lar aggregates, access dark multi particle states, etc. Fourth, Chemical Sciences, Geosciences and Biosciences Division, due to the lack of fluctuation dissipation relations, quantum Office of Basic Energy Sciences, Office of Science, U.S. De- light can manifest new combinations of field and correspond- partment of Energy. KD was supported by the DOE grant. We ing matter correlation functions not governed by semiclassi- also thank the support and stimulating environment at FRIAS cal response functions such as in parametric down conversion, (Freiburg Institute for Advanced Studies). FS would like to sum- or difference-frequency generation, two-photon-induced thank the German National Academic Foundation for support. fluorescence, etc. Finally, elaborate pulse shaping techniques that have been recently scaled down to single photon level provide an additional tool for multidimensional measurements Appendix A: Diagram construction using delay scanning protocols not available for classical laser experiments. In leading (fourth-) order perturbation theory, the popula- The reason for the potential advantage of quantum spec- tion of a two-excitation state f is given by troscopy was traced back to the strong time-frequency corre- 4 t τ τ τ lations inherent to quantum light, and the back-action of the i 4 3 2 pf(t; Γ) = dτ4 dτ3 dτ2 dτ1 interaction events onto the quantum field’s state. The com- − ~ Zt0 Zt0 Zt0 Zt0 bination of the two effects leads to the excitation of distinct Pf (t)Hint, (τ4)Hint, (τ3)Hint, (τ2)Hint, (τ1)%(t0) , wavepackets, which can be designed to enhance or suppress × − − − − (A1) selected features of the resulting optical signals. We have described the nonlinear signals in terms of convo- where Pf (t) = f(t) f(t) is the projector onto the final | ih | lutions of multi-time correlation functions of the field and the state, the interaction Hamiltonian Hint as given in Eq. (7), and matter. This approach naturally connects to the established Γ denotes the set of control parameters in the light field. framework of quantum optics, where field correlation func- Eq. (A1) contains 44 = 256 terms, of which only very few tions are analyzed (Glauber, 1963), with nonlinear laser spec- contribute to the signal. These contributions can be conve- troscopy, which investigates the information content of matter niently found using a diagrammatic representation. Its basic correlation functions. As such, it provides a flexible platform building blocks are shown in Fig. 37: The evolution of both to explore quantum light interaction with complex systems the bra and the ket side of the density matrix in Eq. (10) [ ψ h | well beyond spectroscopic applications. This could include and ψ in Eq. (17)] are represented by vertical blue lines, and | i coherent control with quantum light (Schlawin and Buchleit- sample excitations (de-excitations) are represented by arrows ner, 2015; Wu et al., 2015), or the manipulation of ultracold pointing towards (away from) the density matrix. atoms with light (Mekhov and Ritsch, 2012). Entangled quantum states with higher photon numbers (Shalm et al., 2013) 1. Loop diagrams promise access to the χ(5)-susceptibility and its additional information content. Combination of quantum light with strong The following rules are used to construct the field and mat- coupling to intense fields in optical cavities (Herrera et al., ter correlation function from the diagrams (Marx et al., 2008): 2014; Hutchison et al., 2012; Schwartz et al., 2013) may possibly result in a new coherent control techniques of chemical 1. Time runs along the loop clockwise from bottom left to reactions. bottom right. 60

f(t) f(t) =tr VR(⌧10 )VR(⌧20 )Pf (t)VL†(⌧2)VL†(⌧1)%(t0) | ih | s2 s3 n o ⌧ tr E3†(⌧20 )E4†(⌧10 )E2(⌧2)E1(⌧1)%field 2 ⌧20 ⇥ E2, 2 E3, 3 n o s1 e e0 s4 | i =tr VR †(s4)VR †(s3)Pf (s2)V † (s1)V †% h | G G G LG L ⌧1 ⌧10 n o tr E3†(t s4)E4†(t s4 s3)E2(t s2)E1(t s2 s1)%field E1, 1 E4, 4 ⇥ tr E†(t s )E†(t s sn)E (t s )E (t s s )% o ⇥ 3 4 4 4 3 2 2 1 2 1 field n o FIG. 38 Loop diagram construction in a QED formulation: Using the diagram rules given in the text, the diagram translates into the Heisenberg picture expression (top) or the Schrodinger¨ picture expression (bottom). The corresponding field correlation functions are reordered as in the loop diagram.

2. The left branch of the loop represents the ”ket”, the right ”earlier” interactions along the loop. Additionally, the represents the ”bra”. ground state frequency is added to all arguments of the propagators. 3. Each interaction with a field mode is represented by an arrow line on either the right (R-operators) or the left 10. The Fourier transform of the time-domain propagators (L-operators). adds an additional factor of i( i) for each retarded (advanced) propagator. − 4. The field is marked by dressing the lines with arrows, where an arrow pointing to the right (left) represents the 11. The overall sign of the SNGF is given by ( 1)NR , field annihilation (creation) operator Eα(t) (Eα† (t)). − where NR stands for the number of R superoperators. 5. Within the RWA, each interaction with the field annihilates the photon Eα(t) and is accompanied by apply- 2. Ladder diagrams ing the operator Vα†(t), which leads to excitation of the state represented by ket and deexcitating of the state represented by the bra, respectively. Arrows pointing The same rules may be applied to evaluate ladder diagrams, ”inwards” (i.e. pointing to the right on the ket and to with the exception of rule 1, which reads the left on the bra) consequently cause absorption of a 1. Time runs from bottom to top. photon by exciting the system, whereas arrows pointing ”outwards” (i.e. pointing to the left on the bra and to the right on the ket) represent deexcitating the system Appendix B: Analytical expressions for the Schmidt by photon emission. decomposition 6. The observation time t, is fixed and is always the last. As a convention, it is chosen to occur from the left. This Here, we point out how the Schmidt decomposition (34) can always be achieved by a reflection of all interac- may be carried out analytically. To this end, we approximate tions through the center line between the ket and the the phase-matching function by a Gaussian, bra, which corresponds to taking the complex conjugate ∆k(ω , ω )L of the original correlation function. sinc a b exp γ (∆k(ω , ω )L)2 , (B1) 2 ≈ − a b 7. The loop translates into an alternating product of inter- h i actions (arrows) and periods of free evolutions (vertical where the factor γ = 0.04822 is chosen, such that the Gaus- solid lines) along the loop. sian reproduces the central peak of the sinc-function. In combination with the Gaussian pump envelope (53), this enables 8. Since the loop time goes clockwise along the loop, peri- us to use the relation for the decomposition of a bipartite ods of free evolution on the left branch amount to prop- Gaussian (Grice et al., 2001; U’Ren et al., 2003) agating forward in real time with the propagator give by the retarded Green’s function . Whereas evolution on iα G exp ax2 2bxy cy2 the right branch corresponds to backward propagation − 2 − − − ~ 2πσp (advanced Green’s function G†). ∞q 9. The frequency arguments of the various propagators = rnHn(k1x)Hn∗(k2y), (B2) are cumulative, i.e. they are given by the sum of all n=0 X 61

FIG. 39 Ladder diagram construction in a QED formulation: Using the diagram rules given in the text, the diagram translates into the Heisenberg picture expression (top) or the Schrodinger¨ picture expression (bottom). The corresponding ﬁeld correlation functions are reordered as in the loop diagram.

where we deﬁned the Hermite functions Hn. Here, we have Appendix C: Green’s functions of matter 2 2 2 a = 1/(2σp) + γT1 , b = 1/(2σp) + γT1T2, and c = 2 2 1/(2σp) + γT2 . The Hermite functions are then given by

3π 2 k i (kix) Hn(kix) = e 8 − hn(kix), (B3) s√π2nn! with the Hermite polynomials hn, and we further deﬁned the parameters

√ac + √ac b2 µ = − − , (B4) b and

α 1 + µ2 n (B5) rn = 2 µ , ~ s 4acσp 2a(1 µ2) k1 = − , (B6) s 1 + µ2 In the case of the model system described in section II.J.1, 2c(1 µ2) (B7) the superoperator correlation functions in Eqs. (102)-(104) k2 = − 2 . s 1 + µ may be rewritten as sum-over-state expressions, 62

4 i ∞ ∞ ∞ ∞ p (I)(t; Γ) = dt dt dt dt f, − 4 3 2 1 ~ Z0 Z0 Z0 Z0 (t )µ (t )µ (t )µ (t )µ × Gff 4 e0f Gfe0 3 ge0 Gfg 2 ef Geg 1 ge Xe,e0 E†(t t t )E†(t t )E(t t t t )E(t t t t t ) , (C1) × − 4 − 3 − 4 − 4 − 3 − 2 − 4 − 3 − 2 − 1 4 i ∞ ∞ ∞ ∞ p (II)(t; Γ) = dt dt dt dt f, − 4 3 2 1 ~ Z0 Z0 Z0 Z0 (t )µ (t )µ (t )µ (t )µ × Gff 4 e00f Gfe00 3 e00f Gee0;e00e00 2 ge0 Geg 1 ge e,eX0,e00 E†(t t t )E†(t t )E(t t t t )E(t t t t t ) , (C2) × − 4 − 3 − 4 − 4 − 3 − 2 − 4 − 3 − 2 − 1 4 i ∞ ∞ ∞ ∞ p (III)(t; Γ) = dt dt dt dt f, − 4 3 2 1 ~ Z0 Z0 Z0 Z0 (t )µ (t )µ (t )µ (t )µ × Gff 4 e00f Ge00f 3 e00f Gee0;e00e000 2 ge0 Geg 1 ge e,eX0,e00 E†(t t t )E†(t t )E(t t t t )E(t t t t t ) . (C3) × − 4 − 3 − 4 − 4 − 3 − 2 − 4 − 3 − 2 − 1

Here, we have changed to the time delay variables t1, t4. and the single-excitation propagator The various propagators are given by ··· i(ω iγ )t (t) = Θ(t) (1 δ )δ δ e− ee0 − ee0 Gee0;e00e000 − ee0 ee00 e0e000 + δ δ exp [Kt] , (C7) ee0 e00e000 ee00 with the transport matrix

2π 1/20 1 K = − , (C8) T0 1/20 1 − ! which transfers transfers the single excited state populations i(ωeg iγeg )t eg(t) = Θ(t) e− − , (C4) to the equilibrium state (with p = 5% and p = 95%) with G e1 e2 i(ωfe iγfe)t the rate 2π/T . fe(t) = Θ(t) e− − , (C5) 0 G In the frequency domain, the sum-over-state expressions i(ωfg iγfg )t (t) = Θ(t) e− − , (C6) Gfg read 63

4 i dωa dωb dωa0 dωb0 i(ω0 +ω0 ωb ωa)t p (I)(t; Γ) = E†(ω0 )E†(ω0 )E(ω )E(ω ) e a b− − f, − 2π 2π 2π 2π a b b a ~ Z Z Z Z

(ω + ω ω0 ω0 )µ (ω + ω ω0 )µ (ω + ω )µ (ω )µ , (C9) × Gff a b − a − b e0f Gfe0 a b − a ge0 Gfg a b ef Geg a ge Xe,e0 4 i dωa dωb dωa0 dωb0 i(ω0 +ω0 ωb ωa)t p (II)(t; Γ) = E†(ω0 )E†(ω0 )E(ω )E(ω ) e a b− − f, − 2π 2π 2π 2π a b b a ~ Z Z Z Z

(ω + ω ω0 ω0 )µ (ω + ω ω0 )µ (ω ω0 )µ (ω )µ , (C10) × Gff a b − a − b e00f Gfe00 a b − a e00f Gee0;e00e00 a − a ge0 Geg a ge e,eX0,e00 4 i dωa dωb dωa0 dωb0 i(ω0 +ω0 ωb ωa)t p (III)(t; Γ) = E†(ω0 )E†(ω0 )E(ω )E(ω ) e a b− − f, − 2π 2π 2π 2π a b b a ~ Z Z Z Z

(ω + ω ω0 ω0 )µ (ω ω0 ω0 )µ (ω ω0 )µ (ω )µ . (C11) × Gff a b − a − b e00f Ge00f a − a − b e00f Gee0;e00e000 a − a ge0 Geg a ge e,eX0,e00

Appendix D: Intensity measurements: TPA vs Raman

Expanding the frequency domain signal (137) - (140) (Roslyak and Mukamel, 2009a) in eigenstates we obtain

2 dωa dωsum S1, (I)(ω; Γ) = E†(ωsum ω)E†(ω)E(ωsum ω )E(ω ) 4 = 2π 2π − − a a ~ Z Z µge µef µfe0 µe0g, (D1) × ωa ωe + iγe ωsum ωf + iγf ωsum ω ωe0 iγe0 e,eX0,f − − − − − 2 dωa dωsum S1, (II)(ω; Γ) = E†(ω)E†(ωsum ω)E(ωsum ω )E(ω ) 4 = 2π 2π − − a a ~ Z Z µge µef µfe0 µe0g. (D2) × ωa ωe + iγe ωsum ωf + iγf ω ωe0 + iγe0 e,eX0,f − − − 2 dωa dωb S1, (III)(ω; Γ) = E†(ω + ω ω)E(ω )E†(ω)E(ω ) 4 = 2π 2π a b − b a ~ Z Z µge µeg0 µg0e0 µe0g, (D3) × ωa ωe + iγe ωa ω ωg0 iγg ωa + ωb ω ωe0 iγe e,eX0,g0 − − − − − − − 2 dωa dωb S1, (IV)(ω; Γ) = E†(ω)E(ω )E†(ω + ω ω)E(ω ) 4 = 2π 2π b a b − a ~ Z Z µge µeg0 µg0e0 µe0g. (D4) × ωa ωe + iγe ω ωb ωg0 + iγg ω ωe0 + iγe e,eX0,g0 − − − −

Using the deﬁnitions of transition operators (141) - (145) we Appendix E: Time-and-frequency gating recast the signal as

(2) (2) 1. Simultaneous time-and-frequency gating dω dωsum Tfg †(ωsum)Tfg (ωsum) S1, (I)(ω; Γ) = , 2π = π ωsum ωf + iγf To a good approximation we can represent an ideal detector Z Z Xf − (D5) by two-level atom that is initially in the ground state b and is (1) (3) promoted to the excited state a by the absorption of a photon dω dω Teg †(ω)Teg (ω) (see Fig. 21a). The detection of a photon brings the ﬁeld from S1, (II)(ω; Γ) = , 2π = π ω ωe + iγe its initial state ψ to a ﬁnal state ψ . The probability amplitude Z Z e − i f X (D6) (2) (2) T †(ω )T (ω ) dω dω g0g g0g S1, (III)(ω; Γ) = − − − , 2π = π ω ω iγ g0 g Z Z Xg0 − − − (D7) (1) (3) dω dω Teg †(ω)Teg0 (ω) S1, (IV)(ω; Γ) = . 2π = π ω ω + iγ Z Z e − e e X (D8) 64

2 1 2 2 (ω0 ω) for photon absorption at time t can be calculated in ﬁrst-order 2 σ˜T (t0 t) −2 iA(ω0 ω)(t0 t) W (t, ω; t , ω ) = N e− − − 2˜σω − − − , perturbation theory, which yields (Glauber, 2007b) D 0 0 D (E8)

ψf E(t) ψi a d b , (E1) h | | i · h | | i where where d is the dipole moment of the atom and E(t) = 1 E†(t) + E(t) is the electric field operator (we omit the spatial 2 2 2 2 2 σ˜ω = σT + σω, σ˜T = σT + 2 2 , dependence). Clearly, only the annihilation part of the electric σω− + σT− field contributes to the photon absorption process. The transi- 1 σ2 N = ,A = T . (E9) tion probability to find the field in state ψf at time t is given D T 2 2 1/2 2 2 σ [σ + σ ] σ + σω by the modulus square of the transition amplitude ω T T

2 Note that σT and σω can be controlled independently, but ψf E(t) ψi = ψi E†(t) ψf ψf E(t) ψi |h | | i| h | | ih | | i the actual time and frequency resolution is controlled by σ˜T ψf ψf X X and σ˜ω, respectively, which always satisfy Fourier uncertainty = ψ E†(t)E(t) ψ . (E2) σ˜ /σ˜ > 1. For lorentzian gates h i| | ii ω T Since the initial state of the field ψi is rarely known with cer- σT (t t0) i tainty, we must trace over all possible initial states as deter- Ft(t0, t) = θ(t t0)e− − ,Ff (ω0, ω) = , − ω0 ω + iσω mined by a density operator ρ. Thus, the output of the ideal- − (E10) ized detector is more generally given by tr ρE†(t)E(t) . Simultaneous time-and-frequency resolved measurement the detector time-domain and Wigner spectrograms are given must involve a frequency (spectral) gate combined with time by gate - a shutter that opens up for very short interval of time. i The combined detector with input located at r is repre- (iω+σω +σT )τ 2σT (t0 t) G D(t, ω, t0, τ) = θ(τ)θ(t0 t)e− − − sented by a time gate Ft centered at t¯ followed by a fre- 2σω − (E11) quency gate Ff centered at ω¯ (Stolz, 1994). First, the time ˆ gate transforms the electric field E(rG, t) = s Es(rG, t) ˆ iωst with Es(rG, t) = E(rG, ωs)e− as follows: 2σT (t0 t) P 1 e− − (t) WD(t, ω; t0, ω0) = θ(t t0) . E (t¯; rG, t) = Ft(t, t¯)E(rG, t). (E3) −2σ − ω ω + i(σ + σ ) ω 0 − T ω (tf) (E12) Then, the frequency gate is applied E (t,¯ ω¯; rG, ω) = (t) Ff (ω, ω¯)E (t¯; rG, ω) to obtain the time-and-frequency- The gated signal is given by gated field. We assume that the time gate is applied first. Therefore, the combined detected field at the position rD can ∞ ˆ(tf) ˆ(tf) nt,¯ ω¯ = dt E †(t,¯ ω¯; rD, t)E (t,¯ ω¯; rD, t) , be written as h sR s0L i Z−∞ Xs,s0 (tf) ∞ (E13) E (t,¯ ω¯; r , t) = dt0F (t t0, ω¯)F (t0, t¯)E(r , t0), D f − t G where the angular brackets denote ... Tr[ρ(t)...]. The Z−∞ h i ≡ (E4) density operator ρ(t) is defined in the joint field-matter space of the entire system. Note, that Eq. (E13) represents the ob- where is the electric field operator (8) in the Heisenberg E(t) servable signal, which is always positive since it can be recast picture. Similarly, one can apply the frequency gate first and as a modulus square of an amplitude (Eq. (E2)).For clarity we obtain frequency-and-time-gated field (ft). E hereafter omit the position dependence in the fields assuming

(ft) ∞ that propagation between rG and rD is included in the spectral E (t,¯ ω¯; r , t) = dt0F (t, t¯)F (t t0, ω¯)E(r , t0). D t f − G gate function. Z−∞ (E5)

The following discussion will be based on Eq. (E4). Eq. (E5) 2. The bare signal can be handled similarly. For gaussian gates The bare signal assumes inﬁnite spectral and temporal res-

(ω ω)2 olution. It is unphysical but carries all necessary information 1 2 2 0− σ (t0 t) 4σ2 Ft(t0, t) = e− 2 T − ,Ff (ω0, ω) = e− ω , (E6) for calculating photon counting measurement. It is given by the closed path time-loop diagram shown in Fig. 21(Harbola the detector time-domain and Wigner spectrograms are given and Mukamel, 2008). We assume an arbitrary ﬁeld-matter by evolution starting from the matter ground state g that promotes 1 2 2 1 2 2 2 σω σ (t0 t) σ˜ τ [σ (t0 t)+iω]τ D(t, ω, t0, τ) = e− 2 T − − 2 ω − T − the system up to some excited state. The system then emits a √ 2π photon with frequency ωs that leaves the matter in the state e. (E7) This photon is later absorbed by the detector. 65

In the absence of dissipation (unitary evolution) the matter Wigner spectrogram correlation function can be further factorized into a product of two amplitudes that correspond to unitary evolution of bra- ∞ dω0 nt,¯ ω¯ = dt0 WD(t,¯ ω¯; t0, ω0)n(t0, ω0). (E18) and ket-. This transition amplitude can be recast in Hilbert 2π Z−∞ space without using superoperators and is given by Eq. (165) can be alternatively recast in terms of Wigner i 2π ω t spectrograms ~ s iωs(t t10 ) iωeg t Teg(t) = dt10 e− − − −~ Ω s Z−∞ dω0 X nˆt,ω = dt0 WD(t, ω; t0, ω0)ˆn(t0, ω0), (E19) i t10 2π e(t) V (t10 ) exp H0(T )dT g (E14) Z Z ×h | T −~ | i Z−∞ ! where WD(t, ω, t0, ω0) is a detector Wigner spectrogram given This gives for the bare Wigner spectrogram by

∞ iω0τ iω0τ W (t, ω, t0, ω0) = dτD(t, ω, t0, τ)e (E20) n(t0, ω0) = dτe− D e Z0 Z X and Wigner spectrogram for the bare photon number operator Teg(t0 τ/2)Teg∗ (t0 + τ/2). (E15) × − is given by

iω0τ 3. Spectrogram-overlap representation for detected nˆ(t0, ω0) = dτe− nˆ(t0, τ). (E21) signal Z In the standard theory (Stolz, 1994), the detected signal is This is the conventional form (Stolz, 1994) introduced orig- given by a convolution of the spectrograms of the detector and inally for the field space alone. Eq. (E15) contains explicitly bare signal. The detector spectrogram is an ordinary function the multiple pairs of radiation modes s and s0 that can be re- of the gating parameters whereas the bare signal is related vealed only in the joint field plus matter space. Eventually this to the field prior to detection. We now show that when the takes into account all the field matter interactions that lead to process is described in the joint matter plus field space the the emission of the detected field modes. All parameters of signal can be brought to the same form, except that now the Ff and Ft can be freely varied. The spectrogram will always bare signal is given by a correlation function of matter that satisfy the Fourier uncertainty ∆t∆ω > 1. further includes a sum over the detected modes. We denote Together with the gated spectrogram (E16) the bare signal this the spectrogram-overlap (SO) representation of the signal. (E15) represents the time and frequency resolved gated signal. Note, that in the presence of a bath, the signal (E15) is no Alternatively one can introduce a spectrogram-superoperator- ˆ overlap (SSO) representation where field modes that interact longer given by a product of two amplitudes. Teg(t) is then an with detector are included in the detector spectrogram, which operator in the space of the bath degrees of freedom. There- fore, one has to replace product of amplitudes in Eq. (E15) becomes a superoperator in the field space. Details of this ˆ ˆ representation are presented in Ref. (Dorfman and Mukamel, by Teg(t0 τ/2)Teg∗ (t0 + τ/2) , where ... corresponds to averagingh over− the bath degrees ofi freedom.h i The convolution 2012a). Below we present the signals in the time domain, ˆ which can be directly read of the diagram (Fig. 21b). We then of two operators Teg reveals the multiple pathways between recast them using Wigner spectrograms, which depict simul- these initial and final states of matter that allows to observe taneously temporal and spectral profiles of the signal. We now them through the simultaneous time and frequency resolution. define the detector Wigner spectrogram We now consider two limiting cases. In the absence of a frequency gate, then Ff (ω, ω¯) = 1 we get WD(¯ω, t¯; t, τ) = ¯ ¯ . For the narrow time gate ∞ ∞ dω i(ω0 ω)τ δ(τ)Ft∗(t + τ/2, t)Ft(t τ/2, t) WD(t,¯ ω¯; t0, ω0) = dτ e − 2 − 2π Ft(t, t¯) = δ(t t¯) we then obtain the time resolved mea- Z−∞ Z−∞ | | − 2 surement F (ω, ω¯) F ∗(t0 + τ/2, t¯)F (t0 τ/2, t¯), (E16) × | f | t t − 2 n¯ = T (t¯) . (E22) if the spectral gate applied first, using Eq. (E5). The detector t | eg | e spectrogram is alternatively given by X In the opposite limit, where there is no time gate, i.e. ∞ iω0τ ∞ WD(t,¯ ω¯; t0, ω) = dτe dt Ft(t, t¯) = 1, and the frequency gate is very narrow, such that √γ iωt¯ γt Z−∞ Z−∞ F (t, ω¯) = e− − θ(t) at γ 0, then W (¯ω, t¯; t, τ) = 2 f π D F (t, t¯) F ∗(t t0 τ/2, ω¯)F (t t0 + τ/2, ω¯). iωτ¯ → × | t | f − − f − e− . In this case we obtain the frequency resolved measure- (E17) ment

Combining Eqs. (E4) - (E16) we obtain that the gated signal is n = T (¯ω) 2, (E23) ω¯ | eg | given by the temporal overlap of the bare signal and detector e X 66

iωt where Teg(ω) = ∞ dte Teg(t). Eqs. (E22) and (E23) in- and for the bra dicate that if the measurement−∞ is either purely time or purely R 2 ts ti frequency resolved, the signal can be expressed in terms of i iωeg ts Teg∗ (ts, ti) = dt1 dt2e the modulus square of a transition amplitude. Interference ~ Z−∞ Z−∞ can then occur only within Teg in Hilbert space but not be- gg Vˆ †(t )Vˆ †(t ) tween the two amplitudes. Simultaneous time and frequency ×hh | R 1 R 2 max[t1,t2] gating also involves interference between the two amplitudes; i ˆ exp HR0 (T )dT e(ts)g . (E28) the pathway is in the joint ket plus bra density matrix space. ×T ~ ! | ii In the presence of a bath, the signal can be written as a corre- Z−∞ lation function in the space of bath coordinates Tˆ∗ (t¯)Tˆ (t¯) h eg eg i for Eq. (E22) and Tˆ∗ (¯ω)Tˆ (¯ω) for Eq. (E23). h eg eg i REFERENCES Abolghasem, P., J. Han, B. J. Bijlani, and A. S. Helmy, 2010 Jun, Opt. Express 18 (12), 12681, http: 4. Multiple detections //www.opticsexpress.org/abstract.cfm?URI= oe-18-12-12681 Abouraddy, A. F., M. B. Nasr, B. E. A. Saleh, A. V. Sergienko, The present formalism is modular and may be easily ex- and M. C. Teich, 2002 May, Phys. Rev. A 65, 053817, http:// tended to any number of detection events. To that end it is link.aps.org/doi/10.1103/PhysRevA.65.053817 more convenient to use the time domain, rather than Wigner Abramavicius, D., B. Palmieri, D. V. Voronine, F. Sanda,ˇ and representation. For coincidence counting of two photons mea- S. Mukamel, 2009, Chem. Rev. 109 (6), 2350, http://pubs. acs.org/doi/abs/10.1021/cr800268n sured by ﬁrst detector with parameters ω¯i, t¯i followed by second detector characterized by ω¯ , t¯ the time-and-frequency Abramavicius, D., D. V. Voronine, and S. Mukamel, 2008, PNAS s s 105 (25), 8525, http://www.pnas.org/content/105/ resolved measurement in SO representation is given by 25/8525.abstract Adamczyk, K., M. Premont-Schwarz,´ D. Pines, E. Pines, and E. T. J. Nibbering, 2009 Dec., Science (New York, N.Y.) ¯ ¯ ∞ ∞ S(ts, ω¯s; ti, ω¯i) = dts0 dτs dti0 dτi 326 (5960), 1690, ISSN 1095-9203, http://www.ncbi. Z−∞ Z−∞ nlm.nih.gov/pubmed/19965381 (s) ¯ (i) ¯ D (ts, ω¯s; ts0 , τs)D (ti, ω¯i; ti0 , τi0)B(ts0 , τs; ti0 , τi0) Akiba, K., D. Akamatsu, and M. Kozuma, 2006, Opt. × (E24) Commun. 259 (2), 789 , ISSN 0030-4018, http: //www.sciencedirect.com/science/article/ pii/S0030401805010850 where the detector spectrogram for mode ν = i, s reads Almand-Hunter, A. E., H. Liu, S. T. Cundiff, M. Mootz, M. Kira, and S. W. Koch, 2014, Nature 506, 471 Andrews, D. L., and P. Allcock, 2002, Wiley-VCH, Weinheim, Ger- ∞ dων iων τν D(t¯ , ω¯ ; t0 , τ ) = e− ν ν ν ν 2π many (2002). 1 Z−∞ Arora, R., G. I. Petrov, V. V. Yakovlev, and M. O. Scully, 2012, Proc. 2 ¯ ¯ Ff (ων , ω¯ν ) Ft∗(tν0 + τν /2, tν )Ft(tν0 τν /2, tν ). Nat. Acad. Sci. doi:“bibinfo doi 10.1073/pnas.1115242108 × | | − (E25) Asaka, S., H. Nakatsuka, M. Fujiwara, and M. Matsuoka, 1984 Apr, Phys. Rev. A 29, 2286, http://link.aps.org/doi/10. 1103/PhysRevA.29.2286 The bare signal is given by the loop diagram in Fig. 21c which Aspect, A., J. Dalibard, and G. Roger, 1982 Deca, Phys. Rev. reads Lett. 49, 1804, http://link.aps.org/doi/10.1103/ PhysRevLett.49.1804 B(t , τ ; t , τ ) = Aspect, A., P. Grangier, and G. Roger, 1981 Aug, Phys. Rev. s0 s i0 i Lett. 47, 460, http://link.aps.org/doi/10.1103/ T (t0 τ /2, t0 τ /2)T ∗ (t0 + τ /2, t0 + τ /2). PhysRevLett.47.460 − eg s − s i − i eg s s i i e Aspect, A., P. Grangier, and G. Roger, 1982 Julb, Phys. X (E26) Rev. Lett. 49, 91, http://link.aps.org/doi/10.1103/ PhysRevLett.49.91 Ates, S., I. Agha, A. Gulinatti, I. Rech, M. T. Rakher, The transition amplitude for the ket reads A. Badolato, and K. Srinivasan, 2012 Oct, Phys. Rev. Lett. 109, 147405, http://link.aps.org/doi/10.1103/ PhysRevLett.109.147405 i 2 ts ti iωeg ts Avenhaus, M., H. B. Coldenstrodt-Ronge, K. Laiho, W. Mauerer, Teg(ts, ti) = dt10 dt20 e− −~ I. A. Walmsley, and C. Silberhorn, 2008, Phys. Rev. Lett. 101 (5), Z−∞ Z−∞ 053601 e(t )g Vˆ (t0 )Vˆ (t0 ) ×hh s | L 1 L 2 Barkai, E., 2008, Theory and Evaluation of Single-molecule max[t10 ,t20 ] Signals (World Scientiﬁc Publishing Company) ISBN i ˆ exp HL0 (T )dT gg , (E27) 9789812793492, http://books.google.com/books? ×T −~ | ii Z−∞ ! id=2y0q8JksRXYC 67

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