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UC Irvine UC Irvine Previously Published Works UC Irvine UC Irvine Previously Published Works 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 ,y and Shaul Mukamelz 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 vari- ation of its response with frequencies or time delays. Quantum light opens up new avenues for spec- troscopy 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 in- volving entangled photons with nonclassical bandwidth properties - known as “time-energy entangle- ment”. 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, dif- ferent 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 y 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 z [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 en- tangled. 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
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