Mastering Quantum Light Pulses with Nonlinear Waveguide Interactions

Mastering quantum light pulses with nonlinear waveguide interactions

Kontrolle über Quantenlichtpulse durch nichtlineare Interaktion in Wellenleitern

Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von Andreas Eckstein aus Altdorf b. Nürnberg Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät der Universität Erlangen-Nürnberg

Tag der mündlichen Prüfung: 1.3.2012 Vorsitzender der Promotionskommission: Prof. Dr. Rainer Fink Erstberichterstatterin: Prof. Dr. Christine Silberhorn Zweitberichterstatter: Prof. Dr. Uwe Morgner Contents

1 Introduction 1 1.1 The EPR paradox and entangled quantum states ...... 2 1.2 Nonlinear medium polarization and three-wave mixing ...... 3 1.2.1 Sum- and difference frequency generation ...... 4 1.2.2 Spontaneous parametric downconversion ...... 4 1.3 Quantum light pulses ...... 5 1.4 A quantum pulse source and a quantum pulse gate ...... 7

2 Basic concepts 9 2.1 Electromagnetic waves ...... 9 2.2 Electromagnetic ﬁeld quantization ...... 10 2.3 Field quadratures and squeezed light ...... 11 2.4 Important classes of light states and their properties ...... 12 2.4.1 Coherent states of light ...... 12 2.4.2 Single mode squeezed vacuum states ...... 13 2.4.3 Two-mode squeezed vacuum states ...... 14 2.5 Ultrafast pulses ...... 14 2.5.1 Broadband mode operators ...... 15 2.5.2 Functional orthogonality interval ...... 16 2.5.3 Broadband modes in the temporal domain ...... 17 2.5.4 Pulse propagation and quantum mechanical phase ...... 17 2.6 Nonlinear optical interactions and three-wave-mixing ...... 18 2.6.1 Emergence of frequency- and phase-matching conditions ...... 19 2.6.2 SPDC in a channel waveguide with discrete spatial mode spectrum . . . . 20 2.6.3 Time evolution of the SPDC output state ...... 22 2.6.4 Quasi-Phasematching ...... 23 2.6.5 Classical undepleted SPDC pump ...... 23 2.6.6 Broadband mode structure and Schmidt decomposition ...... 24 2.6.7 Effective mode number and spectral entanglement of a photon pair . . . . 25 2.6.8 Multiple squeezer excitation ...... 26 2.7 Modeling photon detection with binary detectors ...... 27 2.7.1 Measurement operator ...... 27 2.7.2 Measuring the joint spectrum of a photon pair ...... 29

3 Spectral engineering 31 3.1 Pure heralded single photons and the two-mode squeezer ...... 32 3.2 The phasematching distribution Φ and group velocity matching ...... 33 3.3 Critical phasematching through backward-wave SPDC ...... 36

i 3.4 Type I SPDC ...... 37 3.5 Type II SPDC ...... 38 3.6 Survey of nonlinear waveguide materials for group velocity matching ...... 38 3.6.1 Lithium niobate ...... 40 3.6.2 Lithium tantalate ...... 41 3.6.3 Potassium niobate ...... 42 3.6.4 Potassium titanyl phosphate ...... 43 3.7 Conclusion ...... 43

4 A PP-KTP waveguide as parametric downconversion source 45 4.1 Single photon detectors ...... 46 4.2 The parametric downconversion source ...... 48 4.3 Phasematching contour ...... 52 4.4 Conclusion ...... 53

5 Fiber spectrometer 55 5.1 Functional principle ...... 55 5.2 Experimental setup for photon pair spectrum measurement ...... 57 5.3 Calibration ...... 58 5.4 Spectral resolution ...... 60 5.5 The joint spectral intensity of photon pairs from the KTP source ...... 62 5.6 Measurements beyond the perturbative limit hˆni 1 ...... 63 5.7 Conclusion ...... 67

6 Two-mode squeezed vacuum source 69 6.1 Mode-number and photon statistics of broadband squeezed vacuum states . . . . 69 6.2 The second order correlation function g(2) ...... 71 6.3 g(2) forbroadbandinputstates ...... 72 6.4 g(2) for the ultrafast multimode squeezer ...... 73 6.5 g(2) measurement ...... 74 6.6 Background event suppression and correction ...... 75 6.7 Mean photon number ...... 77 6.8 Photon collection efﬁciency ...... 80 6.9 Conclusion ...... 80

7 Quantum pulse manipulation 83 7.1 Beam-splitters, spectral ﬁlters and broadband mode selective ﬁlters ...... 85 7.2 Broadband mode SFG ...... 87 7.3 Spectral engineering and the Quantum Pulse Gate ...... 88 7.4 Critical group velocity matching and QPG mode-switching ...... 89 7.5 Experimental feasibility ...... 93 7.6 The Quantum Pulse Shaper ...... 94 7.7 Time ordering and strongly coupled three-wave-mixing ...... 94 7.8 Conclusion ...... 98

8 Conclusion and outlook 99

Summary

Ultrafast quantum light pulses with durations of 1 ps and below show great promise as information carriers in quantum communication and computation. In future they may also be used to probe physical processes at ultrashort timescales with a resolution beyond the limits of Heisenberg uncertainty. This thesis focuses on the creation and manipulation of quantum light pulses with second order nonlinear optical processes in optical waveguides. In chapter “1. Introduction”, we lead the reader towards this work’s topic by giving a brief qualitative overview over the EPR paradox, quantum entanglement, three-wave-mixing and quantum light pulses. Chapter “2. Basic concepts” familiarizes the reader with the physical concepts and mathematical tools underlying this thesis. In chapter “3. Spectral engineering”, we discuss the requirements to produce separable photon pair states with group velocity matching and examine several nonlinear materials widely used for optical waveguide inscription for their suitability to group velocity matching in the telecom wavelength regime. In chapter “4. A PP-KTP waveguide as parametric downconversion source”, we give the basic spontaneous parametric downconversion source setup, as well as some initial measurements to characterize the single photon detectors, to demonstrate the production of correlated photons, and to determine the phasematching properties of the PP-KTP source. In the following chapter, “5. Fiber spectrometer”, we present the single-photon fiber spectrometer[8]. We discuss the experimental setup and calibration of the device, and measure the joint spectrum of photon pairs from our PP-KTP source. Finally, we investigate the behavior of a joint spectrum measurement of a high mean photon number source with binary detectors. In “6. Two-mode squeezed vacuum source”, we characterize spectral correlations of an ultrafast SPDC source with the second order correlation function g(2) . We present the g(2) measurement results[42] and background substraction technique[43] and show that we can control the spectral correlations to produce a two-mode squeezed vacuum state of light. We then determine the mean photon number and gain of the source. “7. Quantum pulse gate”: While the previous chapters focus on the creation and characterization of ultrafast quantum pulses of light, we now propose a way to manipulate the mode structure of a given quantum light state. We discuss the concept of an active optical filter sensitive to spectral/temporal pulse shape: The quantum pulse gate[41], and its reverse process, the quantum pulse shaper[21]. In the final chapter “8. Conclusion and outlook”, we recapitulate the main results of this thesis and provide a few pointers towards possibilities for future research building on it.

Zusammenfassung

Ultrakurze Quantenlichtpulse mit einer Pulsdauer von 1 ps und darunter sind vielversprechende Kandidaten als Informationsträger in der Quantenkommunikation und im Quantencomputing. In Zukunft könnten sie auch dazu genutzt werden, physikalische Prozesse auf ultrakurzen Zeitskalen jenseits der Grenzen der Heisenberg-Unschärfe zu untersuchen. Diese Dissertation konzentriert sich auf die Erzeugung und Manipulation von Quantenlichtpulsen durch nichtlineare optische Prozesse zweiter Ordnung in optischen Wellenleitern. In Kaptitel “1. Introduction” führen wir den Leser an das Thema der Arbeit heran, indem wir einen kurzen, qualitativen Überblick über das EPR-Paradoxon, Quantenverschränkung, Dreiwel- lenmischung und Quantenlichtpulse geben. Kapitel “2. Basic concepts” macht den Leser mit den physikalischen Konzepten und mathemati- schen Werkzeugen vertraut, die dieser Arbeit zugrunde liegen. In Kaptitel “3. Spectral engineering” diskutieren wir die Voraussetzungen, um separable Pho- tonenpaarzustände durch Gruppengeschwindigkeitsanpassung zu produzieren und wir untersuchen einige nichtlineare Materialien, die häuﬁg zur Produktion optischer Wellenleiter verwendet werden, auf ihre Eignung für die Gruppengeschwindigkeitsanpassung im Bereich der Telekommu- nikationswellenlängen. In Kaptitel “4. A PP-KTP waveguide as parametric downconversion source” zeigen wir den grundlegenden Aufbau unserer parametrischen Fluoreszenz-Quelle im PP-KTP Wellenleiter auf, sowie einige vorbereitende Messungen, die die Einzelphoton-Detektorn charakterisieren, die Erzeugung korrelierter Photonen demonstrieren und die Phasenanpassungs-Eigenschaften der PP-KTP-Quelle bestimmen. Im anschließenden Kapitel “5. Fiber spectrometer” präsentieren wir das Einzelphotonen- Faserspektrometer[8]. Wir erörtern den experimentellen Aufbau und die Kalibration des Geräts und messen das Koinzidenz-Spektrum von Photonenpaaren aus unserer PP-KTP Quelle. Zuletzt untersuchen wir das Verhalten einer Messsung eines Koinzidenz-Spektrums einer Photonenpaar- Quelle mit hoher mittlerer Photonenzahl mit binären Detektoren. In “6. Two-mode squeezed vacuum source” charakterisieren wir die spektralen Korrelationen einer ultraschnell gepumpten SPDC-Quelle durch die Korrelationsfunktion zweiter Ordnung g(2) . Wir präsentieren die Ergebnisse der g(2) -Messung[42] und der Untergrund-Subtraktion[43] und zeigen, dass wir durch Kontrolle der spektralen Korrelationen einen zweimodigen gequetschten Vakuumszustand erzeugen können. Dann bestimmen wir die mittlere Photonenzahl und die Konversionsefﬁzienz unserer Quelle. “7. Quantum pulse gate”: Während sich die vorangehenden Kapitel mit der Erzeugung und Charakterisierung ultraschneller Quantenlichtpulse mit einer auseinandersetzen, schlagen wir hier eine Methode vor, um die modale Struktur eines gegebenen Licht-Quantenzustands zu manipulieren. Wir erörtern das Konzept eines aktiven optischen Filters, der auf spektrale oder zeitliche Pulsformen sensitiv ist: Das Quantum Pulse Gate[41], und dessen Umkehrprozess, der Quantum Pulse Shaper[21].

vii Im letzten Kapitel “8. Conclusion and outlook” rekapitulieren wir die Hauptergebnisse dieser Dissertation und zeigen einige mögliche zukünftige Forschungsrichtungen auf. 1 Introduction

The notion of a corpuscular nature of light was first put forward by Isaac Newton, who conjectured that different colors were the result of different sized light particles[94]. While this was a statement that – with the benefit of hindsight – bears a remarkable similarity to modern optical concepts like photons as fundamental excitations of the electromagnetic field and their optical wavelength, his contemporary scientific peers favored a wave-like description of light, as supported by diffraction experiments. The idea was not unearthed again until 1900 when Max Planck succeeded to explain the radiation emission spectrum of a black body by postulating light emission in discrete quanta. Together with Einstein’s discovery of the photoelectric effect it triggered the development of the theory of quantum mechanics, unifying the wave and the particle aspects of light. Although quantum mechanics has been around since the 1920s through the works of Dirac, Schrödinger, Born, Heisenberg, and others, and has been an extraordinarily successful theory, actual, direct technical applications of quantum effects were slower to emerge. One of the earliest (and most visible) of those comes from the field of optics: Lasers, developed in the late 1950s are an ubiquitous source of coherent radiation today, and are used in such diverse fields as optical communication networks, medical applications and entertainment devices. But also today’s micro- electronics and computer technology would be unthinkable but for the understanding of electric conduction in solids afforded by quantum mechanics, which led to semiconductor diodes and transistors, the basic building blocks of every microchip. And as a rather infamous example of quantum mechanical applications there is nuclear technology, both in military and civilian applications. Arguably, during the last sixty years the world has been profoundly shaped by quantum mechanics, mostly by way of the nuclear arms race and information technology; but on the horizon there are already new, potentially disruptive quantum technologies: Quantum communication, which holds the promise of unconditionally secure secret information exchange and authentica- tion between remote parties without complete physical control of the communication channels. Quantum computation, which can solve certain hard mathematical problems faster than any given classical algorithm, an example being the factorization of an arbitrary integer number, useful for breaking classical asymmetric encryption schemes like the widely-used RSA protocol. And finally akin to computation is quantum simulation, the accurate physical simulation of large quantum 2 1 Introduction mechanical systems such as unordered solids or macro-molecules, which is sure to trigger great advances in material sciences, biophysics, and medicine. The main reason those new applications have not (quite) materialized yet is decoherence: In general, after an interaction, two interacting quantum systems will share mutual information, called quantum entanglement. If an observer is able to monitor only one of the systems, he perceives the resulting loss of information as decoherence, as the gradual loss of all genuinely quantum mechanical properties of the system. This problem exists in any experiment: To observe quantum effects, we have to isolate our observed system, the experiment, from the environment, be it by setting it up in a vacuum, by working at cryogenic temperatures, or by observing at time scales short enough for interactions with the environment to be negligible. This aspect of system isolation is why quantum optics, the study of the properties and interactions of quantum states of light, is at the forefront of the implementation of new quantum technologies: A quantum light state interacts weakly with transparent materials and exists at time scales that make thermal interaction with the environment much less of a problem than for e. g. an electronic state in a solid, preserving its quantum nature.

1.1 The EPR paradox and entangled quantum states

It is a well known principle of quantum mechanics that the values of any two non-commuting observables Xˆ and Yˆ cannot be measured for a system at a given time with arbitrary precision. The product of their statistical variances will always have a lower boundary. For two conjugate h i observables, i. e. observables with Xˆ, Yˆ = 1, the lower boundary is a state-independent constant: 1 h∆Xˆ 2i h∆Yˆ 2i ≥ (1.1) 2 One such pair of conjugate observables are the position operator ˆx and the linear momentum operator ˆp of a massive particle. In 1934, Einstein, Podolsky and Rosen (EPR) published a gedanken-experiment[46] to highlight a fundamental flaw in the then-young field of quantum mechanics. Let us consider the decay of a resting particle into two moving daughter particles 1 and 2. The direction in which the daughter particles will move after the decay event is completely undetermined until measurement, but momentum conservation causes the daughter particles to assume quantum mechanical states such that the result of a measurement of the linear momentum vectors gives results with opposite signs such that hˆp1i + hˆp2i = 0: The measurement values, or more precisely their fluctuations around the mean value, are always exactly anti-correlated. Both particles are said to be entangled. Measuring the momentum observable ˆp1 thus gives the exact value of the measurement of ˆp2. It is on the other hand possible to measure the position observable ˆx2 to arbitrary precision when we don’t actually measure ˆp2, but rather infer it from the momentum-correlated daughter particle. So by measuring the observables ˆp1 and ˆx2, we gain knowledge of the values hˆp2i and hˆx2i, and their variances are not subject to the uncertainty relation. From this surprising, seemingly paradoxical result EPR argued that the description of reality according to quantum mechanics was incomplete, since the momentum values were undetermined until measured, yet strictly anti-correlated between both particles irregardless of distance or temporal order of measurement. This would imply what Einstein famously called a “spooky action at a distance” in order to ensure the correlated measurement outcomes. However, the terminus entanglement was chosen over Einstein’s for this phenomenon. 1.2 Nonlinear medium polarization and three-wave mixing 3

Despite the initial skepticism, notably by EPR themselves as well as proponents of alternative hidden variable theories[15], it is well-accepted in the physics community today that entanglement exists and is a fundamental property of quantum mechanics, rather than a theoretical quirk explained by hidden variables, a larger deterministic theory of which quantum mechanics is but a statistical approximation. This has to be attributed to John Bell[13] as well as Clauser et al.[36], who formulated quantitative boundaries for the validity of certain hidden variables theories, which were ﬁrst shown to be violated in the pioneering experiments of Freedman et al.[51] and Aspect et al.[5, 6, 4].

1.2 Nonlinear medium polarization and three-wave mixing

Typically, optically transparent media are electric insulators, so electrons inside dielectric medium, e. g. glass, are not freely mobile as they are in metals. Yet applying a weak electric field E~ can displace the electrons from their rest positions, giving the medium an overall polarization into the opposite direction in response: P~ = 0χE~ . The material-dependent polarizability constant χ reflects a linear dependency between electric field and charge carrier displacement. However, the electrons’ potential inside the medium can be considered harmonic for small displacements only. For higher field intensities, non-harmonic terms in the electrons’ potentials start to make themselves felt, and we have to generalize the polarization response to (1) (2) P~ = 0χ E~ + 0χ E~ E~ + ... (1.2) The third rank tensor χ(2) describes the second order nonlinear polarization response P~ (2) = (2) 0χ E~ E~ which is quadratic in the electric field. Through polarization of the medium’s charge carriers, an opposed electrical field is built up, and there is a back-action on the original electric field. The associated potential energy V ∝ P~ (2) ◦ E~ is already cubic in E~ , and thus an optical medium’s second order nonlinearity mediates a third order self-interaction of an electric field. This self-interaction is widely exploited for the frequency conversion of electromagnetic waves. In the simplest case, the incident field is just a harmonically oscillating wave E~ (~x,t) = E~0cos ~k~x − ωt . Now, the second order polarization term is also time-dependent 2 1 1 P~ (2)(~x,t) ∝ χ(2)E~ (~x,t) E~ (~x,t) ∝ cos ~k~x − ωt = cos 2~k~x − 2ωt + . (1.3) 2 2 and the quadratic dependency on the electric field gives rise to oscillations at the doubled frequency, the second harmonic of the incident wave. The process is consequently named second harmonic generation (SHG). The constant term corresponds to an additional static polarization term. The oscillating polarization of the medium constitutes a continuous distribution of radiation emitting dipoles at frequency 2ω, but they emit at different phases, varying with position ~r proportional to the doubled wave vector 2~k. In general, this will cancel out any effective radiation output of the second harmonic by destructive interference, unless the phase variation of the polarization vector P~ is exactly the same as the in-medium phase variation of the second harmonic frequency, so that constructive interference allows the coherent build-up of a wave. This condition is known as phase-matching, and can be expressed in terms of the associated wave vectors: ~k(2ω) = 2~k(ω). The monochromatic field self-interacting through second order nonlinearity is just a special case though. Let the incident electric field E~ be composed of two oscillating fields E~1 and E~2 at different frequencies: E~ (~x,t) = E~1 (~x,t) + E~2 (~x,t) = E~1cos ~k1~x − ω1t + E~2cos ~k2~x − ω2t (1.4) 4 1 Introduction

Besides the SHG waves, the quadratic dependency on the electric ﬁeld gives rise to oscillations at additional frequencies

(2) + + − − P~ (t) ∝ 2 + cos 2~k1~x − 2ω1t + cos 2~k2~x − 2ω2t + cos ~k ~x − ω t + cos ~k ~x − ω t (1.5) ± ± + with ω = ω1 ± ω2 and ~k = ~k1 ± ~k2. The process generating ω is called sum frequency generation (SFG) and the corresponding process generating ω− is called difference frequency generation (DFG). Both are in principle always present as soon as two light beams at different frequencies overlap inside a χ(2) -nonlinear medium. But again, the coherent build-up of an ~ ~ ~ output wave occurs only if the phase-matching condition kω± = k1 ± k2 is fulﬁlled. SHG is a special case of SFG with ω2 = ω1. The constant polarization contribution is the remnant of the DFG process with ω− → 0.

1.2.1 Sum- and difference frequency generation Through canonical ﬁeld quantization, we can express classical waves as superpositions of elementary excitations of the electromagnetic ﬁeld. Each light mode is modeled as a quantum mechanical harmonic oscillator, and the modal excitations are photons. Fig. 1.1 is a representation of SFG and DFG on the single photon level, reminiscent of a Feynman diagram. In the SFG case depicted in Fig. 1.1(a), two photons “fuse” to form a third photon at the sum frequency. Its frequency matching condition of SFG can, after multiplication with Planck’s constant, be read as energy conservation between two input photons and one output photon:

+ ~ω1 + ~ω2 = ~ω (1.6)

Similarly, phase-matching can be re-interpreted as conservation of the photons’ crystal pseudo- momentum ~~k: ~ ~ ~ + ~k1 + ~k2 = ~k (1.7)

For DFG, depicted in Fig. 1.1, the situation is more complicated: A photon with frequency ω1 is stimulated by the presence of light at a lower frequency ω2 to distribute its energy between − another photon of the same frequency ω2 and a third photon at the difference frequency ω = ω1 − ω2. DFG not only generates the difference frequency output wave, but also ampliﬁes the lower frequency input wave.

1.2.2 Spontaneous parametric downconversion The second order polarization term that gives rise to DFG scales like the product of the input (2) waves P ∝ E1E2. Classically this means that when reducing the amplitude E2 to zero, no DFG will take place. But the quantum vacuum still has non-zero vacuum field fluctuations, even if the − mean value of the electrical field is zero. If the DFG phasematching conditions ~k1 − ~k2 = ~k are fulfilled, those vacuum fluctuations are sufficient to excite the spontaneous decay of some of − the high energy input photons into photons at ω2 and ω . This purely quantum effect is called spontaneous parametric downconversion (SPDC or PDC). One “pump” photon decays into one “signal” and one “idler” photon (Fig. 1.2). Since vacuum fluctuations for any frequency ω2 exist, − PDC will in principle occur for all energy conserving combinations of ω2 and ω at once. But only phase-matched processes will build up an output wave, all others will be suppressed by destructive interference. But the creation of one photon pair is not the only possibility; rather, 1.3 Quantum light pulses 5

Figure 1.1: Three-wave-mixing on the single photon level. (a) SFG: Two photons “fuse” to a new photon at the sum frequency. (b) DFG: One photon is stimulated by a lower energy photon to distribute its energy between the stimulating photon mode and a difference mode.

as Fig. 1.3 illustrates, it must be considered to be a coherent superposition of the case of no interaction, of one-pair-creation, two-pair-creation, and so on. This superposition is called a squeezed vacuum state. If signal and idler photons carry different polarizations, it implements an EPR state in continuous variables. Only for small pair creation probabilities p 1 we can consider the probability to produce a pair of pairs p2 p negligible in comparison and speak of a SPDC process as a probabilistic photon pair source. The output field amplitude of SFG and DFG scales with the product of both input field amplitudes, but SPDC has only one input. Thus the output field amplitudes are a linear function of the input field, making it a parametric process. While SFG and DFG efficiencies can be high enough −10 to deplete the energy of one of its input beams, for SPDC values of the order ηPDC ≈ 10 are typical.

1.3 Quantum light pulses

The problem of localizing photons or, more generally speaking, localizing quantum states of light, has a long history[95] and stems from to the fact that standard field quantization as introduced by Dirac in 1927[39] is based on monochromatic electromagnetic fields. This intrinsically implies completely de-localized plane wave solutions for the quantum fields. In 1966, Titulaer and Glauber introduced broadband modes, polychromatic wave packets as single mode states by defining continuous superpositions of monochromatic modes[130]. Since time and frequency domain amplitudes of a wave are connected by the Fourier transformation, a monochromatic state of light wave has a constant probability amplitude in the time domain (c. f. Fig. 1.4a) and is therefore completely de-localized. A wave packet’s amplitude has a finite time duration (Fig. 1.4c) and thus forms a localized pulse. The pulse shape – the envelope function of a wave package – can be decomposed into broadband modes, a complete basis set of orthogonal weighting functions of the monochromatic modes. In this work, we are going to be concerned with pulses with a duration of the order of 1 ps, right on the edge of what is generally 6 1 Introduction

Figure 1.2: Spontaneous parametric downconversion as a special case of difference frequency conversion: The “stimulating” lower frequency ﬁeld is the quantum vacuum.

Figure 1.3: SPDC process as a superposition of the creation of zero, one, two, three, ... photon pairs. considered the ultrafast regime of femtosecond pulses. Generally we differentiate between classical and non-classical light states. Ultrafast pulsed lasers can be under ideal conditions be considered sources of classical wave packets of light[56, 130]. Experimentally, those were made available through the first mode-locked lasers in the 1960s[78], but only the relatively recent arrival of ultrafast self-mode-locked lasers[119, 103] in 1991 provided a reliable, high quality source of highly coherent femtosecond laser pulses. In particular the titanium sapphire (Ti:Sa) solid state laser has become the “workhorse” laser source for ultrafast optics experiments. In parallel, the first pulsed squeezed vacuum states, which are a class of non-classical two-partite light states, were generated with SPDC[114, 9]. Both developments, self-mode-locked lasers and pulsed wave mixing, paved the way for the first source of ultrafast squeezed vacuum states[3] in 1995. An aspect which sets apart ultrafast classical from the non-classical squeezed vacuum light states is their broadband mode structure. For the classical case, all sets of basis functions are equivalent. In contrast, squeezed vacuum states, or indeed all two-partite quantum states, carry in general an internal, well-defined, discrete bi-partite mode spectrum[49]. Even in the spectral degree of freedom, which one would naturally consider to be continuous, this mode structure can be found[67, 82], and it is determined by the spectral intra-correlations of the bi-partite squeezed vacuum. And since we match the intrinsic timescales of SPDC with ultrafast pulses, the effects of this internal structure will become much more distinct. Those spectral correlations, and at the same time the intrinsic mode structure, can be manipu- 1.4 A quantum pulse source and a quantum pulse gate 7

Figure 1.4: Real spectral amplitudes (gray) and envelope functions (blue) of a monochromatic wave in (a) and (b) and of a wave packet superposition in (c) and (d), in time domain and frequency domain respectively.

lated by spectral engineering, that is by manipulation of the dispersion properties of the nonlinear generation process as well as the spectral or spatial properties of its pump beam[60, 70, 28]. It is even possible to eliminate the correlations altogether[59, 28]. In this special case only one of the bi-partite modes will be well deﬁned, as photon pairs will be emitted into this mode only. All other modes are unpopulated and therefore degenerate. This way, the source emits quantum light pulses, localized, single-mode ultrafast non-classical states of light. In general though, SPDC source produces a whole ensemble of pairwise correlated quantum pulses[20] In the single pair approximation for low pump pulse energies the quantum pulse source has the additional advantage to produce two photons in completely uncorrelated, separable states, so that by loss of one of the photons of the two-partite system its partner does not suffer from decoherence. By heralding one photon event in the signal arm we can generate pure single photons.

1.4 A quantum pulse source and a quantum pulse gate

In the course of this thesis we develop the means to efficiently generate EPR-entangled quantum light pulses and manipulate their spectral mode structure. We implement a pulsed waveguide SPDC source in the telecom wavelength regime and demonstrate its high photon pair output, its efficiency, and our control over its spectral entanglement that allows us to produce both spectrally single mode and multi-mode light pulses. The source advances its predecessor experiment presented by Mosley et al. in 2008[93] considerably. By choosing a waveguide architecture over bulk crystal, we ensure emission into not only into one spectral, but also into one spatial signal and idler mode, implementing a source of genuine quantum pulses. Wave confinement of the pump beam over the waveguide length leads to a far greater interaction length so that both mean photon number and modal brightness of the output light at modest pump pulse energies far exceed prior experiments. Our source produces high quality continuous variable EPR states as well as pure heralded single photons at telecom wavelengths with multiple applications in discrete[75, 74] (i. e. single photon) and continuous variable quantum computing[84, 19] schemes. It is a particularly viable source of resource states for continuous variable entanglement distillation[98, 37, 96], a protocol that can be 8 1 Introduction used to reverse the deleterious effect of decoherence on the security of long distance quantum communication with a quantum repeater[25]. To manipulate the intrinsic spectral mode structure of a quantum state of light, we apply the same spectral engineering techniques we used for designing the SPDC waveguide source to SFG[105] and propose the quantum pulse gate (QPG). It is implemented by a mode-selective SFG process, i. e. a SFG that converts one well-defined broadband mode and transmits all orthogonal modes. The mode to select is determined by the pump pulse form. This novel approach to quantum pulse manipulation allows to pick from an arbitrary SPDC squeezed vacuum state one of its intrinsic quantum pulses and convert it to another wavelength, where it can be easily separated from the rest with standard optical components. It can be used to de-multiplex an optical signal consisting of several independent quantum pulses, thereby boosting the information channel capacity for ultrafast, broadband pulses. These tools, a source for creating genuine quantum pulses, and a device for manipulating them, will no doubt prove valuable to the emerging field of ultrafast quantum communication and computation. 2 Basic concepts

In this chapter, we will brieﬂy introduce the concepts necessary to understand the rationale and the body of the theoretical and experimental work presented in this thesis.

2.1 Electromagnetic waves

In classical electrodynamics, we can describe monochromatic, linearly polarized electromagnetic radiation idealized as a plane wave E~ (~x,t) = E~0 cos ~k~x − ωt + ϕ (2.1)

~ ~ where k is the wave vector, ω = c k is the associated angular frequency, and ϕ is an arbitrary ω phase. Throughout this work we will use angular frequency ω rather than plain frequency f = 2π . The electrical field amplitude vector E~0 = ~eσE0 consists of the polarization direction unit vector ~eσ and the field amplitude modulus E0. It is a solution to the homogeneous Maxwell equations that govern the dynamics of electric and magnetic fields in vacuum, and is completely characterized by the unit vector of the polarization direction eˆ, optical phase ϕ and wave vector ~k. ~k determines ˆ ~k its propagation direction k = k and frequency ω = ck. Any solution for a certain set of boundary conditions is a light mode with respect to them, and the plane waves are monochromatic modes in unbounded vacuum conditions. We can write for the (non-vectorial) electrical field strength

ı(~k~x−ωt) ∗ −ı(~k~x−ωt) E(~x,t) = E0 cos ~k~x − ωt + ϕ ∝ ae + a e (2.2)

ıϕ with the complex, dimensionless amplitude a ∝ E0e absorbing the phase ϕ. We now deﬁne the amplitude quadrature X = Re[a] and the phase quadrature Y = Im[a] as a real representation of the ﬁeld: E(~x,t) ∝ Xcos ~k~x − ωt + Y sin ~k~x − ωt (2.3) 10 2 Basic concepts

Plane waves do not and cannot have an energy content assigned to them, since they expand over the whole of space-time, and are as such unphysical. Nevertheless, superpositions of plane waves can be used to describe finite, physical wave packages with a well defined energy. We can however define an energy density E for a monochromatic plane wave that is constant over space and time

2 2 2 2 E ∝ E0 + B0 ∝ X + Y (2.4) where E0 is the electrical and B0 the magnetic ﬁeld strength of the wave. X and Y are the aforementioned ﬁeld quadratures.

2.2 Electromagnetic ﬁeld quantization

In the canonical quantization of the electromagnetic field, each plane wave light mode ~k is modeled as a quantum mechanical harmonic oscillator. As such, its energy eigenstates will always contain an integer number of field excitation quanta, or photons. The eigenstates span the Hilbert space of all possible states in this particular mode. The field quadratures X and Y take the roles of displacement x and momentum p respectively from a mechanical oscillator. We then promote c-numbers to Hilbert space operators, and in the special case of real-valued physical quantities, to observables. The complex amplitude a and its complex conjugate a∗ are substituted with the non-commuting photon annihilation and creation operators:

a → ˆa (2.5)

a∗ → â† (2.6) h i â, â† = 1 (2.7) For the field quadratures, we find 1 1 X = Re[a] = (a∗ + a) → Xˆ = â† + â (2.8) 2 2 1 ı Y = Im[a] = ı (a∗ − a) → Yˆ = â† − â (2.9) 2 2 h i ı Xˆ, Yˆ = (2.10) 2 Since both Xˆ and Yˆ are Hermitian, they are observables, quantum mechanically measurable h i properties such that X = hXî = Tr ρˆXˆ with ρˆ a general quantum state. However, since they do not commute, they cannot both be measured to arbitrary precision at the same time. Finally, we can express the electrical field observable in terms of the creation/annihilation operators

~ ~ E(ˆ ~x,t) ∝ ˆa†e−ı(k~x−ωt) + ˆa eı(k~x−ωt) (2.11) or alternatively, in terms of the quadrature operators E(ˆ ~x,t) ∝ Xcosˆ ~k~x − ωt + Ysinˆ ~k~x − ωt . (2.12)

As their names suggest, the creation and annihilation operators create and destroy quanta of the electromagnetic ﬁeld √ ˆa |ni = n |n − 1i (2.13) 2.3 Field quadratures and squeezed light 11

√ â† |ni = n + 1 |n + 1i (2.14) where |ni is a pure state containing n quanta in mode ~k, and is called photon number state or Fock state. Their normal-ordered product forms the Hermitian photon number operator for the mode ~k ˆn= â†â |ni = n |ni (2.15) with the eigenstates |ni. They are identical to the energy eigenstates, and the Hamiltonian, i. e. the energy operator for a field mode describing free propagation is closely related to the photon number operator:

1 1 1 Hˆ = ω Xˆ 2 + Yˆ 2 = ω â†â+ ââ† = ω ˆn+ (2.16) 2~ 2~ ~ 2

It may be surprising at first glance that we can define a Hamiltonian operator instead of a Hamilto- nian density, when we could only give an energy density for a plane wave in classical electrodynamics. The reason for this is that in the classical case, we start from constant field amplitudes E0 and try to sum up their energy content over an infinite space, while the canonical quantization implicitly starts with the notion of finite energy quanta in one mode expanding over an arbitrarily large volume. For the quantized plane waves this results in infinitesimally small field amplitudes, and again, only wavepacket superpositions of plane waves describe physical light states with finite amplitudes. These conflicting notions are reconciled by introducing a quantization volume to which the radiation modes are confined, so that their energy is finite. A more modern take on field quantization by Blow et al. [14] avoids this difficulty by using wavepackets from the outset.

2.3 Field quadratures and squeezed light

We have already defined the observables Xˆ and Yˆ as a representation of an electromagnetic mode. In terms of the harmonic oscillator, they can be understood as analogues to displacement and momentum of a harmonic pendulum in classical mechanics. 1 Xˆ = â† + â (2.17) 2 1 Yˆ = ı â† − â (2.18) 2 They are the field quadrature operators[85, 12, 2] of a radiation mode and they are a pair of canonically conjugate observables, so their commutator is a non-zero constant h i ı Xˆ, Yˆ = . (2.19) 2 As a consequence, it is not possible to measure both observables to arbitrary accuracy; therefore, their statistical variances h∆Xˆ 2i , h∆Yˆ 2i must obey a Heisenberg uncertainty relation 1 h∆Xˆ 2i h∆Yˆ 2i ≥ . (2.20) 16 An arbitrary single-mode state of light ρˆ is fully described by its quadrature pseudo-probability distribution in optical phase space, the Wigner function[109] W (X,Y ). This distribution allows for an intuitive illustration of certain properties of a quantum light state. To illustrate a few 12 2 Basic concepts light states, we will make use of phase space diagrams which are a contour plot of the Wigner function W (X,Y ) = const. In classical electrodynamics, a monochromatic light field has a well defined field strength, and consequently sharp quadrature mean values hXî , hYˆ i with no statistical uncertainty, a point in phase space. But owing to the Heisenberg relation 2.20, this is not allowed in a quantum mechanical treatment: To fulfill it, both statistical variances h∆Xˆ 2i and h∆Yˆ 2i must be non-zero, resulting in a distribution that is spread over phase space instead. We can understand this by looking at the so-called coherent light states.

2.4 Important classes of light states and their properties

2.4.1 Coherent states of light Coherent states were first put forward by Erwin Schrödinger in 1928[111, 120] as a quantum mechanical analog of the excitation states of a classical harmonic oscillator according to the correspondence principle. The original terminus for this class of state in Schrödinger’s work is “Wellengruppe” (wave group), which was meant to hint at the superposition of the energy Eigen- states that constitute each state rather than a multi-chromatic wave-packet in the spectroscopic sense. The notion of coherent states of light was introduced by Roy Glauber starting in 1963[56], when he applied Schrödinger’s concept to the experimental findings of Hanbury Brown and Twiss [26, 27] who had shown that light from certain stellar sources exhibited different coherence properties than from then-current (i. e. pre-laser) light sources on earth. Also in 1963, Sudarshan[123] showed that any quantum state of light can be expressed as a superposition of coherent light states. Around the same time, beginning with MASERs[57] in 1954 and later LASERs[88] in 1960, for the first time there were practical sources of coherent light available.

Figure 2.1: Phase space representations of a coherent state (left) and the vacuum state (right)

A coherent state can be expressed as a coherent superposition of all Fock states |ni

2 ∞ n −ı αˆa†+α∗ˆa − |α| X α |αi = D(ˆ α) |0i = e ( ) |0i = e 2 √ |ni . (2.21) n=0 n!

Its mean photon number hˆni = |α|2 therefore exhibits a statistical spread, and so do its ﬁeld strength and quadrature values. The displacement operator D(ˆ α) generates a coherent state with hXˆi = Re[α] and hYˆ i = Im[α] from a vacuum state and thus is a translation in phase space. 2.4 Important classes of light states and their properties 13

Coherent states are Gaussian minimum uncertainty states: Their Wigner function is a two- ˆ 2 ˆ 2 1 dimensional Gaussian distribution and they exactly fulﬁll h∆X i h∆Y i = 16 , respectively. Also, ˆ 2 ˆ 2 1 their quadrature variances are equal h∆X i = h∆Y i = 4 , so that their phase space diagrams are always circular (Fig. 2.1 left). Under time evolution, they rotate counter-clockwise around the phase-space origin. The quantum vacuum |0i (Fig. 2.1 right) can be considered a special case of the coherent state with α = 0, located at the origin.

2.4.2 Single mode squeezed vacuum states

D. Stoler pointed out in 1970[121, 122] that coherent states are not the only class of Gaussian minimum uncertainty states. There are also the squeezed coherent states[29, 140], for which the equality of quadrature variances does not hold: h∆Xˆ 2i= 6 h∆Yˆ 2i. They derive their name from the elliptic shape of their phase space diagrams (Fig. 2.2 left), which can be imagined to be squeezed circular distributions of ordinary coherent states. For an extensive introduction, see e. g. [86]. In 1985, they were experimentally observed for the ﬁrst time by Slusher et al.[115].

Figure 2.2: Left: Phase space diagram of a squeezed vacuum (solid) and the vacuum state (dotted). Right: Two-mode squeezed vacuum at t = 0 (dotted) and t > 0 (solid).

Formally, a squeezed coherent state is created by applying the squeezing operator Sâ(ζ) to a coherent state α: ∗ ζ † 2 ζ 2 −ı 2 (â ) + 2 â |α; ζi = Sâ(ζ) |αi = e |αi (2.22)

Remembering that the vacuum state is a special case of coherent states, we apply Sˆa(ζ) to vacuum to generate the single mode squeezed vacuum (SMSA) state

∞ p q X (2n)! ı n |ζ i = Sˆ (ζ) |0i = 1 − tanh(r)2 − eıϕtanh(r) |2ni (2.23) a a n! 2 n=0 with the squeezing parameter r = |ζ| and an optical phase ϕ = arg(ζ) that determines the orientation of the squeezing in phase space (Fig. 2.2 right). Surprisingly, the mean photon number of the squeezed vacuum is not zero but rather hˆni = sinh(r)2. The quadrature variances for ϕ = 0 ˆ 2 1 ±r ˆ 2 1 ∓r are h∆X i = 4 e and h∆Y i = 4 e . 14 2 Basic concepts

Figure 2.3: Phase space diagram of both modes ˆa and bˆ of the two-mode squeezed vacuum state (dashed black shapes). Shot-for-shot comparison between quadrature measurements for both modes reveals correlated ﬂuctuations (dotted and solid red shapes)

2.4.3 Two-mode squeezed vacuum states Closely related are two-mode squeezed vacuum (TMSV) states q ∞ ˆ −ı(ζâ†bˆ†+ζ∗âbˆ) 2 X ıϕ n |ζâ,bî = Sâ,bˆ(ζ) |0i = e |0i = 1 − tanh(r) (−ıe tanh(r)) |ni ⊗ |ni . n=0 (2.24) A two-mode squeezed vacuum can be generated by mixing two identical SMSV states on a balanced ˆ beamsplitter (represented by the unitary operator UBS): ˆ ˆ ˆ ˆ ˆ ˆ † ˆ Sc(ζ) Sd(ζ) → UBSSc(ζ) Sd(ζ) UBS = Sa,b(ζ) . (2.25) This is reversible: Mixing both modes of a TMSV state on the same beamsplitter will result in two completely separable SMSV states. The TMSV state in contrast is entangled in photon number. Fig. 2.3 illustrates the effect of this entanglement on phase space measurements. For separately measuring the phase space distribution of each mode, the result is a circular phasor around the origin (black, dashed), with a larger radius than the vacuum state, i. e. it is not a minimum uncertainty state any more. It represents a thermal state, the partial trace over one mode of the h i P 2n input state: ρa = Trb |ζâ,bîhζâ,bˆ| ∝ n tanh(r) |nihn|. If each measurement from each mode is compared to the corresponding measurement from its partner mode, a correlation between the measurement results hXˆ ai and hXˆ bi as well as hYˆ ai and hYˆ bi becomes visible: Fluctuations from the mean values are correlated ( solid red shapes and dashed red shapes in 2.3, respectively). This non-classical correlation between the quadratures of both modes of the two-mode squeezed state can be exploited to implement the EPR experiment and explains why in the context of continuous variable quantum optics the state |ζâ,bî is also referred to as the EPR state[46, 9, 99].

2.5 Ultrafast pulses

A truly monochromatic light wave with one sharp frequency value, according to the Fourier relationship to its temporal amplitude, must have an inﬁnite duration in time. It is therefore, just as a plane wave, a theoretical construct and can be only achieved approximately in experiment. Finite light waves therefore exhibit a spectrum of frequency components, and their spectrum grows broader as the pulses grow shorter. A ﬁrst quantum optical description of broadband modes was given by Titulaer and Glauber in [130], and [116] gives a good overview over the theoretical challenges of introducing broadband creation and destruction operators. 2.5 Ultrafast pulses 15

2.5.1 Broadband mode operators Nevertheless, many quantum optical problems are treated in terms of monochromatic photons â†(ω) |0i for simplicity’s sake. In ultrafast optics, with pulse lengths of 1 ps and below, this simplification stops being viable. To account for a coherent continuum of frequencies, one introduces a spectral function ξ0(ω) to integrate over a monochromatic single photon state â†(ω) |0i (c. f. Fig. 2.4 left): Z h † i dω ξ0(ω) â (ω) |0i . (2.26)

The expression can also be read as a broadband operator Aˆ 0 acting on the vacuum state Z † ˆ † dω ξ0(ω) ˆa (ω) |0i = A0 |0i . (2.27)

For the broadband operators to create physical states, we need to impose smoothness and square-

0.5

u0(x) -0.5 u1(x) u2(x) u3(x) -1 -2 0 2

Figure 2.4: Left: An arbitrary square-integrable function ξ0(ω) as weighting function for the ˆ † broadband mode creation operator A0. Right: The ﬁrst four of the orthonormal Hermite function basis {ui} with σ = 1.

integrability on the complex-valued spectral functions ξ0, otherwise we cannot guarantee that the created photons are of ﬁnite duration or carry a ﬁnite amount of energy. We further require normalized spectral functions for our convenience: Z 2 dω |ξ0(ω)| = 1 (2.28)

In this work, unless otherwise stated, all spectral functions are understood as normalized. For a complete set of orthonormal functions {ξj} that forms a basis of the space of smooth, square- integrable functions, broadband operators obey the standard commutator relations for creation and annihilation operators of the harmonic oscillator hˆ ˆ † i Aj, Ak = δj,k. (2.29) ˆ † In general, the broadband mode operators Aj are related to the monochromatic frequency modes â†(ω) simply by a basis transformation between the discrete and the continuous basis: Z ˆ † † Aj = dω ξj(ω) â (ω) (2.30) † X ∗ ˆ † â (ω) = ξj (ω) Aj j 16 2 Basic concepts

We see that creation operators for different, orthogonal broadband modes commute, and consequently photons created by those orthogonal operators live in completely different Hilbert spaces and do not interact at all: ˆ † ˆ † A1A2 |0i = |1i1 ⊗ |1i2 ∈ H1 ⊗ H2 (2.31) h † i Any commutator Aˆ f , Aˆ g between two non-orthogonal spectral modes f and g can be easily P calculated by using the completeness of the functional basis {ξj} and writing f(ω) = j fjξj(ω) ˆ P ∗ ˆ and consequently Af = j fj Aj (likewise for g) such that:

hˆ ˆ † i X ∗ Af , Ag = fj gj (2.32) j

ˆ † A broadband operator applied to the vacuum Af |0i creates a single photon wavepacket[130], localized in time and space, and hence in frequency and momentum as well. Analogous to the monochromatic case, where photons interfere only if they are of the same frequency, broadband photons fully interfere only if they share the same spectral function. Any difference in spectrum will result in diminished visibility of interference, to the point where photons with orthogonal spectral functions will not interfere at all. This applies for both classical and quantum- or Hong- Ou-Mandel-interference[66] between non-classical states.

2.5.2 Functional orthogonality interval

The orthogonality of two spectral functions depends on the interval [ω1, ω2] over which the overlap integral is evaluated: Z ω2 dω f ∗(ω) g(ω) = 0 (2.33) ω1 We usually assume the full range of real numbers R = ]−∞, ∞[. But for frequencies ω, and + likewise photon energies E = ~ω, the correct choice is obviously R0 = [0, ∞[, since negative values for both are ill-deﬁned and unphysical. Nevertheless we will continue to use the “incorrect” approach throughout this work as an approximation:

Z ∞ Z ∞ Z 0 Z ∞ dω f ∗ g = dω f ∗ g − dω f ∗ g ≈ dω f ∗ g (2.34) 0 −∞ −∞ −∞

The approximation works at high frequencies ω0 and relatively low spectral widths σ and because of the square-integrability requirement for our spectra. The former means σ ω0, while the 2 n − (ω−ω0) ω−ω0 2σ2 latter means that every realistic spectral function must vanish at least as fast as σ e (with n an arbitrary non-negative integer) far from the central frequency ω0. Since the polynomial term grows much slower than the exponential term decreases, an overlap integral over the negative − interval R0 = ]−∞, 0] will be negligibly small. Having established the spectral overlap integration interval as R, we now can use the Hermite functions as a popular choice of spectral basis orthogonal on this interval. The Hermite functions uω0,σ,j(ω) are the weighted, normalized Hermite polynomials Hn around the central frequency ω0 with the spectral width σ [82, 143]:

2 1 − (ω−ω0) ω − ω0 uσ,j(ω − ω0) = √ e 2σ2 Hj (2.35) p π2jj!σ σ 2.5 Ultrafast pulses 17

The ﬁrst four Hermite functions are plotted in Fig. 2.4(right). When the spectral width is apparent from the context, or assumed to be constant and its actual value of no consequence, we abbreviate the Hermite functions as uj. They are orthonormal on R Z ∞ ∗ dω uj (ω) uk(ω) = δj,k (2.36) −∞

and u0 is the Gaussian distribution.

2.5.3 Broadband modes in the temporal domain ˆ † It has already been stated that a broadband single photon state |1i = Ai |0i can be considered wavepacket that is localized in both time and space. When we want to consider the time domain ˆ † representation of an arbitrary set of mode operators {Ai } defined as Z ˆ † † Ai = dω ξi(ω) â (ω) (2.37) with orthonormal spectral functions {ξi}, we substitute the monochromatic photon creation operator â†(ω) with its Fourier transform √1 R dτ e−ıωτ â†(τ) and find 2π Z Z † 1 −ıωτ † Aˆ = dω ξi(ω) √ dτe â (τ) i 2π Z Z 1 −ıωτ † = dτ √ dω e ξi(ω) â (τ) (2.38) 2π Z † = dτ ξ˜i(τ) â (τ) .

After switching order of the integrals in Eq. 2.38, we have applied the Fourier transform to the spectral amplitude ξi(ω) and obtained the temporal amplitude function ξ˜j(τ). Like its frequency counterpart, it is square-integrable and must therefore vanish sufﬁciently fast for large or small time values τ, i. e. it is localized around τ = 0 within a temporal standard deviation στ . The result ˆ † is the time domain representation of the broadband mode operator Aj, which is form invariant under Fourier transform with respect to the frequency domain representation. The orthogonality of the function set {ξi} carries over to the Fourier transformed set {ξ˜i}, which then constitutes a ˆ † complete basis of temporal amplitude functions. We can consider the operators {Ai } not only a set of spectral mode operators, but at the same time as operators that create one photon in a textitpulse mode with a localized temporal distribution.

2.5.4 Pulse propagation and quantum mechanical phase For a full description of an optical network of interacting quantum light pulses, one has to take ˆ † into account quantum mechanical phase, yet the creation operator for an ultrafast pulse Ai does not feature an explicit time dependence. One could argue that any phase dependence is already implicitly contained in its complex spectral amplitude function ξi(ω). For considering one mode or one beam path this is sufficient, but for a dynamical treatment concerning multiple light modes/paths, for instance to describe an interferometer, we need an explicit expression. Starting from an arbitrarily determined time t0 = 0, we define a broadband single photon state Z ˆ † † Ai = dωξi(ω) â (ω) (2.39) 18 2 Basic concepts

1 1 t=0.0ps t=0.5ps ] 0.5 t=4.0ps -i ω -i τ

0 (τ-t) 0.5 (ω)e 0 0 ξ t=0.0ps

Re[ξ -0.5 t=0.5ps t=4.0ps -1 0 ω0 -8 -6 -4 -2 0 2 4 6 8 Frequency ω [THz] Temporal walkoﬀ τ [ps]

−iωt Figure 2.5: Left: Spectral function ξ0(ω) e (real part) for a propagating pulse of 1 ps duration, corresponding to 3 nm spectral FWHM at 1550 nm central wavelength. Right: ˜ Corresponding temporal amplitude function ξ0(τ − t)

In the Heisenberg picture, a monochromatic photon creation operator transforms under free propagation in vacuum like â†(ω) → e−ıωtâ†(ω). Substituting this into the above frequency domain definition of a broadband operator, we obtain Z ˆ † −ıωt † Ai → dωe ξi(ω) â (ω) (2.40)

In the time domain, the translation is straightforward: Z Z ˆ † ˜ † ˜ † Ai → dτ ξi(τ) ˆa (τ + t) ≡ dτ ξi(τ − t) ˆa (τ) (2.41)

So the effect of propagation on an optical broadband pulse can be thought of as a transformation of −ıωt the mode itself; a phase term is multiplied with the initial spectral amplitude: ξi(ω) → e ξi(ω). Fig. 2.5 exempliﬁes this with spectra for a picosecond pulse for relative temporal walk-offs of 0 ps, 0.5 ps and 4 ps. On the left, the real part of the frequency spectral amplitude is plotted, on the right hand its Fourier transform, the temporal amplitude. The overlap between spectra is Fourier-invariant, it is the same in both domains. With a 0.5 ps relative temporal walk-off is still considerable, with a walk-off of 4.0 ps it is negligible. In time domain, the reason for decreasing overlap is the relative translation between two peaks, in frequency domain it is the relative oscillation. We will as a rule resort to the implicit time dependency of the optical pulses, unless stated otherwise.

2.6 Nonlinear optical interactions and three-wave-mixing

Since photons do not carry a charge, electromagnetic waves do not interact in vacuum (at low energies), they merely can interfere with each other. In transparent media, an indirect interaction is introduced via the electric susceptibility χ(1), which reflects the mobility of charge carriers inside the medium and their response to an incident field, as we have already explained in section ~ 1.2. In the simplest case this response, the dielectric polarization vector Pˆ, is linear in the electric field ~ (1)~ Pˆ = 0χ Eˆ. (2.42) 2.6 Nonlinear optical interactions and three-wave-mixing 19

However, many dielectric media exhibit higher order susceptibility terms

~ (1)~ (2)~ ~ (3)~ ~ ~ Pˆ = 0χ Eˆ + 0χ EÊˆ + 0χ EÊÊˆ + ... (2.43) where the according interaction Hamiltonian is given by the interaction between the polarization and the field itself over an interaction volume V : Z ~ ~ Hˆ I (t) = d~r P(ˆ ~r, t) ◦ E(ˆ ~r, t) (2.44) V

The χ(n) are the n-th rank susceptibility tensors of a medium and they give rise to n + 1-wave- mixing. Which of their elements are non-zero is determined by the medium’s microscopic symmetries. For instance the three-wave-mixing tensor χ(2) implies a non-isotropic dipole structure with a preferred direction, so for an isotropic medium like glass the elements of χ(2) will vanish.

2.6.1 Emergence of frequency- and phase-matching conditions Crystals with broken rotational symmetries however may exhibit non-zero χ(2) elements and thus can support three-wave-mixing. To illustrate how this interaction comes about, we assume the ~ ﬁeld operator Eˆ consisting of a superposition of three plane wave modes in arbitrary directions ~ki at arbitrary frequencies ωi with arbitrary linear polarization directions σi

3 3 √ ~ ~ ~ˆ X ˆ X ı(ki◦~r−ωit) † −ı(ki◦~r−ωit) E(~r, t) = E~ (~r, t) ∝ ωi ˆa~ e + ˆa e . (2.45) ki,σi ki,σi ~ki,σi i=1 i=1

Considering the interaction Hamiltonian of a medium that features only second order susceptibility χ(2) H(ˆ t) ∝ χ(2)E(ˆ ~r, t) E(ˆ ~r, t) E(ˆ ~r, t) (2.46) we will, after expanding the triple product of the field operator in terms of mode operators, find in the following form ˆ X (2) † † † −ı(Σ~k◦~r−Σωt) H(t) ∝ χijk â~ â~ â~ e + h. c. ki,σi kj ,σj kk,σk i,j,k∈{1,2,3} (2.47) X (2) † † −ı(∆~k◦~r−∆ωt) + χijk â~ â~ â~ e + h. c. ki,σi kj ,σj kk,σk i,j,k∈{1,2,3}

~ ~ ~ ~ ~ ~ ~ ~ with Σk = ki + kj + kk, Σω = ωi + ωj + ωk and ∆k = ki − kj − kk, ∆ω = ωi − ωj − ωk. The ﬁrst sum contains terms that create or destroy three photons in the modes 1, 2, 3, the terms of the second sum destroy one photon and create two photons and vice versa. The time evolution 1 R t1 ˆ t dtH(t) operator of a Hamiltonian process is deﬁned as U(ˆ t1, t0) = e ı~ 0 , so we have to consider the Hamiltonian integrated over the time interval t0, t1. Since Σω is always positive, all terms ~ proportional to e±ı(Σk◦~r−Σωt) are rapidly oscillating, leading to destructive interference under ~ the integration. In contrast, terms proportional to e±ı(∆k◦~r−∆ωt) will interfere constructively, if and only if the phasematching equation

∆~k = 0 (2.48) 20 2 Basic concepts and the frequency matching equation ∆ω = 0 (2.49) hold. Then, the rotating wave approximation is applicable, and the rapidly oscillating terms of the Hamiltonian can be neglected in favor of the slowly varying terms. Equations 2.49 and 2.48 can be rewritten to explicitly show energy and linear momentum conservation between destroyed and created photons by multiplication with the Planck constant ~. While it is certainly not surprising that energy and momentum are preserved in three-wave-mixing – after all, Hamiltonian mechanics is based on assuming overall energy conservation – it is instructive to see the conditions for both frequency- and phase-matching emerge. We note that while energy- and momentum conservation can in principle be violated with the creation or destruction of two or even three non-matched 1 photons, this is possible only on very short time scales of the order of ∆ω or on very small length scales of the order of 1 respectively, in keeping with the time-energy and position-momentum |∆~k| uncertainty relations. Equation 2.47 does contain all possible combinations of polarization for pump, signal and idler wave. For brevity’s sake we will now commit to one configuration with polarization modes â, bˆ and ˆc and drop the sums over the polarization indices σi. After discarding all rapidly oscillating terms of the Hamiltonian, we find

ˆ † ˆ† −ı(∆~k◦~r−∆ωt) H(t) ∝ â~ b~ ˆc~ e + h. c. (2.50) ka kb kc It is however a simplification to assume the k-vectors as given, so we have to integrate over all possible vectors and write more generally ZZZ ˆ 3~ 3~ 3~ ~ ~ ~ −ı∆ω t+ı∆~k ~x † ˆ† H(t) = d ka d kb d kc ζ ka, kb, kc e â~ b~ ˆc~ + h. c. (2.51) ka kb kc where ζ is the nonlinear coupling strength of the three wave mixing. In the following subsections we will concentrate on spontaneous parametric downconversion (SPDC) as a special case of three wave mixing, but all arguments can be made similarly for sum frequency generation (SFG) and difference frequency generation (DFG).

2.6.2 SPDC in a channel waveguide with discrete spatial mode spectrum A waveguide is a guiding structure restricting the electromagnetical wave propagation to one (channel waveguide) or two dimensions (slab waveguide) by introducing a reflecting boundary[108]. This can be a literal channel with a reflective coating, or a zone of higher refractive index inside a block of a transparent dielectric medium. Through total internal reflection, the traveling waves are confined to the waveguide volume. If a waveguide is small enough in diameter, it supports a discrete set of spatial modes whose form depends on the transversal boundary conditions for the electrical field. For a rectangular cross-section area the modes resemble two-dimensional Hermite-Gaussian modes, for waveguides with radial symmetry, like most optical fibers have, one can observe Laguerre-Gaussian modes instead. These modes become more easily observable for tightly confined beams in small waveguides, since it is much more probable to excite a highly pure mode when coupling into the waveguide. Fig. 2.6 shows a schematic drawing of a waveguide chip of the kind used in this thesis. Multiple dielectric waveguides are inscribed on the surface of the nonlinear material. The waveguide volume features a refractive index n2 greater than that of the surrounding material with n1 and of 2.6 Nonlinear optical interactions and three-wave-mixing 21

Figure 2.6: Nonlinear optical chip with multiple dielectric surface waveguides. The left blow-up shows the input facet area, and the right one illustrates wave guiding through total internal reﬂection.

the air boundary with n0, so that total internal reflection can take place. The effective k-vector or propagation constant β = ~k ◦ ~ex of a ~k-mode is the projection of ~k on the propagation direction. ~ ~ ~ k-modes are guided if their transversal momentum vector k⊥ = k − β~ex fulfills the waveguide’s boundary conditions, causing a discretization of the bound transversal mode spectrum. If a mode’s angle of incidence on the boundary is greater than the critical angle of total internal reflection θ = arcsin n1 , then only part of the mode is reflected. Such leaky modes are subject to critical n2 exponential decay during propagation through the waveguide. If the critical angle allows exactly one mode to be guided with low losses, the waveguide is single-mode, otherwise it is multi-mode. According to geometrical optics, a waveguide with a quadratic cross-section and perfectly reflecting boundaries of width d enforces as boundary conditions for the transverse k-vector to be π part of the waveguide’s square reciprocal lattice with lattice constant d , that is mπ nπ ~k = ~e + ~e (2.52) ⊥ d y d z with m and n positive integers. We can now write for the propagation constant π2 m2 + n2 k2 = β2 = ~k ◦ ~k − . (2.53) x,m,n m,n d2 2 For high mode numbers such that βm,n < 0, this implies a sudden mode cut-off due to losses, apart from the gradual cut-off caused by the break-down of total internal reflection: An imaginary propagation constant in a propagation term eıβx causes exponential damping. This very simplistic model does not take into account evanescent waves or wavelength-dependent effects, but it is sufficient to demonstrate the impact of waveguide mode confinement on wave dispersion. The canonical approach to dermine the guide’s modal dispersion properties is to solve the Helmholtz equation for the waveguide’s refraction index profile, a partial differential equation. Its solutions determine the modes’ electrical field distribution. Due to the connection between waveguide modes and wave dispersion, in a SPDC process we have to allow for different waveguide modes as an additional degree of freedom for all three photons. Each mode features its own dispersion relation and this influences any phasematched process it is involved in. The SPDC phase-mismatch must be indexed accordingly to account for any triple of modes:

(mp,np,ms,ns,mi,ni) (mp,np) (ms,ns) (mi,ni) ∆k = βp − βs − βi (2.54) 22 2 Basic concepts

The corrections to the phase-matching term eı∆kz for each mode triple cause a shift in the spectra of the output photons. In a multi-mode waveguide, there can be thus several spectrally distinct, concurrent SPDC processes due to waveguide modes[11, 44, 34]. The spectral shift between mode triples can be used to isolate the processes and directly observe the signal and idler modes[92]. We will assume that the waveguide source used in the thesis is perfectly single-mode for pump, signal and idler. Our experimental results, in particular in section 6.5, will vindicate this assumption. We now simplify the model of our three-wave-mixing processes for the single-mode waveguide[61, 132]. We apply the plane wave approximation, i. e. we assume that the sole effect of the waveguide on the transversal field distribution is a confinement to its cross-section, and that the wave fronts of guided light are perfectly plane. Since the waveguide enforces collinear propagation directions for all three fields, we replace the wave vector with a scalar and simplify the Hamiltonian to Z L H(ˆ t) ∝ â†bˆ†ˆc eı∆ωt dz e−ı∆kz + h. c. (2.55) 0 where the waveguide is traversed in z-direction and L is the waveguide length. This effective reduction to a one-dimensional problem does not take into account spatial mode discretization and wavefront distortion at all, but will serve for now to study basic features of waveguided SPDC. Evaluating the integral along the length of the waveguide, we obtain Z L ı∆kz ∆kL ı ∆kL dz e = L sinc e 2 . (2.56) 0 2

2.6.3 Time evolution of the SPDC output state In quantum mechanics, non-dissipative temporal evolution – of a quantum state ρˆin the Schrödinger picture, or of an operator in the Heisenberg picture – is described by application of the unitary time evolution operator 1 Z t1 U(ˆ t1, t0) = T exp dt H(ˆ t) (2.57) ı~ t0 with T the time ordering operator. Time ordering can be neglected for small interactions that can be treated perturbatively. In section 7.7 we consider the case of a strongly coupled three-wave mixing process and the effects of time ordering. The evolution of a quantum state ρˆ from time t0 to time t1 is described by the similarity transformation

ˆ† ρˆ(t1) = U(ˆ t1, t0)ρ ˆ(t0) U (t1, t0) . (2.58)

Since the time evolution operator U(ˆ t1, t0) contains the time integral of the Hamiltonian operator, we will from now on use the effective Hamiltonian

1 Z t1 Hˆ = dt H(ˆ t) (2.59) ~ t0

−ıHˆ −ı∆ω t such that U(ˆ t1, t0) = e . Time integration over e – representing the interaction time inside the waveguide – will give a result similar to the length integration, but since we are working in the ultrafast regime where all involved light pulses are much shorter than the integration time, we assume an inﬁnite integration interval and ﬁnd the result converging towards a delta R ∞ ı∆ωt distribution −∞dt e = δ(∆ω), enforcing an exact frequency matching between signal, idler and pump photon: ωp = ωs + ωi. 2.6 Nonlinear optical interactions and three-wave-mixing 23

2.6.4 Quasi-Phasematching The occurrence of three-wave mixing between three plane waves depends on frequency- and phase-matching, putting severe constraints on the experimentally viable combinations of input und output frequencies and polarizations. While there is basically no way around frequency- matching, since it is equivalent to energy conservation, the phase-matching conditions can be relaxed up to a point, by introducing a periodic inversion into the crystal domain structure. This technique called quasi-phase-matching[108] causes the crystal medium itself to absorb part of the pump photon momentum, or conversely to contribute to the overall momentum of the output photon pair. Physically this can be permanently achieved for some materials like lithium niobate (LiNbO3) by applying a strong electrical, spatially modulated field. The domain structure of other materials, such as potassium titanyl phosphate (KTiOPO4 ) can be manipulated by an ion exchange process. We model this by modulating the second order nonlinear tensor element with a 2πz periodic sign flip function g(z) = sign sin Λ . The effect of the sign flip function can be most readily understood by expressing it by its Fourier series X 1 g(z) = eımkΛz (2.60) m ±m∈odd

2π with kΛ = Λ the quasi-phase-matching vector and m the phase-matching order. When the substitution χ(2) → g(z) χ(2) is introduced into the Hamiltonian, the phase-mismatch ∆k gains an additional term kΛ representing the momentum exchange with the periodically poled crystal structure ∆km = kp − ks − ki − mkΛ (2.61) and phase-matching ∆k = 0 can be achieved for each order m ∈ {1, −1, 3, −3, 5, −5, ...} separately. Therefore, the effective Hamiltonian is a superposition of all possible orders m:

Z X 1 Z L Hˆ ∝ dt ˆa†bˆ†ˆc eı∆ωt dz e−ı∆kmz + h. c.. (2.62) m m 0

1 It must also be observed that each phase-matching order m contributes a factor m to the effective ζ coupling constant ζm = m , so that in the weak coupling regime an mth order process will generate 1 a m2 fraction of the photon pairs that a ﬁrst order process (m = ±1) emits; thus the conversion 1 efﬁciency for higher order phase-matching decreases like m2 .

2.6.5 Classical undepleted SPDC pump As a three-wave-mixing process, SPDC is a coupling of a populated light mode ˆc to two vacuum modes ˆa and bˆ. The initial quantum state then reads

|ψ0i = |0ia ⊗ |0ib ⊗ |Ψic . (2.63)

Applying the Hamiltonian to the SPDC initial state then gives

h † † i Hˆ |ψ0i ∝ α(ω1 + ω2) Φ(ω1, ω2) ˆa (ω1) bˆ (ω2) + h. c. |ψ0i (2.65) allowing us to drop all quantum mechanical operators acting on the pump state, and to treat it like a classical, static object. This is justiﬁed in situations where the pump is undepleted, i. e. where the actual photon down-conversion rate is low, and only a minuscle part of the pump beam is converted. Accordingly, the now static ˆc-mode is usually dropped from all equations, and the initial SPDC state is written |ψ0i = |0ia ⊗ |0ib. Also by now, the two output modes are correlated only in frequency. This correlation is described by the joint spectral amplitude (JSA) 1 f(ω , ω ) = α(ω + ω ) Φ(ω , ω ) , (2.66) 1 2 N 1 2 1 2 RR 2 which we will assume to be normalized: dω1 dω2 |f(ω1, ω2)| = 1. The normalization constant N is absorbed by the Hamiltonian’s coupling constant ζ.

2.6.6 Broadband mode structure and Schmidt decomposition If we want to express our classically pumped three-wave-mixing process in broadband modes, we have to decompose the joint amplitude f(ω1, ω2) into a superposition of separable pairs of pulse forms. We choose two arbitrary spectral basis sets { ξ˜i(ω) } and { ψ˜j(ω) } and calculate the overlap between all mode pairs and the amplitude function: ZZ ˜∗ ˜∗ ˜cij = dω1 dω2 ξi (ω1) ψj (ω2) f(ω1, ω2) (2.67)

Conversely, we can now re-construct the amplitude function from the spectral mode functions and the overlap constants ˜cij: ∞ ∞ X X f(ω1, ω2) = ˜cijξ˜i(ω1) ψ˜j(ω2) (2.68) i=0 j=0 The complex matrix ˜c with an inﬁnite number of rows and columns is a representation of the function f(ω1, ω2) with respect to the chosen basis sets. Among those there is exactly one special case that allows for a simpler representation. We can arrive at this special case by applying a singular value decomposition to ˜c: Any matrix with a non-degenerate system of Eigen-vectors can be expressed as c = U˜cV† (2.69) where U and V are unitary matrices and c is a diagonal matrix. Thus, by deﬁning two new basis sets { ξi } and { ψj } with X ∗ ˜ ξj(ω) = Ukjξk(ω) k (2.70) X ˜ ψj(ω) = Vjkψk(ω) k we gain the diagonal representation of the joint spectrum: ∞ X f(ω1, ω2) = cj ξj(ω1) ψj(ω2) . (2.71) j=0 2.6 Nonlinear optical interactions and three-wave-mixing 25

where cj ≡ cjj. This method allows to express any bivariate, smooth, square integrable function 2 f(ω1, ω2) ∈ L in a diagonal representation in terms of two complete orthonormal function sets { ξj } and { ψj }. The choice of basis functions is unambiguous up to a complex phase. The matrix elements cjj are chosen to be real positive values, as any phase can be moved to the basis functions. P 2 For a normalized function f, they also obey the normalization condition j |cj| = 1. In terms of quantum mechanics this means that any physical, pure bi-partite system that is correlated in one degree of freedom can be decomposed[67, 48, 82] into a superposition of mutually orthogonal, separable bi-partite states:

AB X A B |Ψ i = cj |ξj i ⊗ |ψj i . (2.72) j

This is usually referred to as the Schmidt decomposition[110] with the Schmidt modes { ξj } and { ψj } and the Schmidt coefﬁcients cj.

2.6.7 Effective mode number and spectral entanglement of a photon pair The number of separable superpositions necessary to express the entangled bi-partite state |ΨABi is at most countable inﬁnite. The amount of entanglement between both partite systems A and B can be characterized by the number of superimposed states. For an inﬁnite number of differently weighted superimposed modes, the effective mode number or cooperativity parameter or Schmidt number K is used: Z Z Z Z −1 ∗ ∗ K = dω1 dω2 dω3 dω4 f (ω1, ω2) f (ω3, ω4) f(ω1, ω4) f(ω3, ω2)

X 4 (2.73) = |cj| j

AB A B For a separable state |Ψsep i = |Ψ0 i⊗|Ψ0 i and f(ω1, ω2) = f1(ω1) f2(ω2) it assumes its minimal value K = 1. Bell states for instance have K = 2, and for a (unphysical) state in the limit where all cj are equal and infinitesimally small, it diverges: K → ∞. Bell states are often described as maximally entangled states, but this is true only if there are two dimensions available, as is e. g. the case for polarization entangled photon pairs. A Schmidt decomposition of the spectral degree of freedom yields in general an infinite number of modes, so two-partite systems can contain more entanglement, i. e. values K > 2 are allowed. Here the bi-partite system is a two photon state produced – in first-order approximation – by SPDC in a waveguide with weak coupling |ζ| 1. Both output photons are assumed to be entangled in the frequency degree of freedom only. The joint spectral amplitude is decomposed into two complete sets of basis functions, and the output photon pair is written as a superposition of broadband mode pairs

ˆ |Ψi =e−ıH |0i ≈ 1 − ıHˆ |0i ZZ † † (2.74) = |0i − ıζ dω1 dω2 f(ω1, ω2) ˆa (ω1) bˆ (ω2) |0i X = |0i − ıζ cj |ξji ⊗ |ψji j 26 2 Basic concepts

R † R † where |ξji = dω ξj(ω) ˆa (ω) |0i and |ψji = dω ψj(ω) bˆ (ω) |0i. Alternatively, we can use ˆ † R † ˆ † R ˆ† the broadband mode operators Aj = dω ξj(ω) ˆa (ω) and Bj = dω ψj(ω) b (ω) to write the photon pair state as X ˆ † ˆ † |ψABi = ζ cjAjBj |0i . (2.75) j In Fig. 2.7, we show a graphical representation of such a two-partite state with Hermite modes as Schmidt modes. Product-state pairs of corresponding modes are superimposed to form the spectrally correlated photon pair state.

Figure 2.7: A superposition of Schmidt pairs forming a spectrally correlated two-photon state

2.6.8 Multiple squeezer excitation One can write the effective Hamiltonian of the SPDC process as the sum over all Schmidt mode pairs, and likewise the unitary time evolution operator of the waveguided SPDC can be expressed ˆ † ˆ † ∗ ∗ ˆ ˆ −ı ζcj Aj Bj +ζ cj Aj Bj as a product of broadband two-mode squeezing operators Sˆj(ζ) = e :

ˆ −ı P ζc Aˆ †Bˆ †+ζ∗c∗Aˆ Bˆ ˆ −ıH j j j j j j j Y ˆ UPDC = e = e = Sj(ζcj) . (2.76) j

The additive Schmidt decomposition at single photon level corresponds to a decomposition into a tensor product of the modes in the full description of the phenomenon, and accordingly, we can separate the overall Hilbert space of output states into a tensor product of subspaces of the squeezers Sˆj(ζ) H = H0 ⊗ H1 ⊗ ... (2.77) so that we can view the according two-mode squeezed states as physically independent from each other[20]. This decomposition of the Hilbert space for a multi-mode squeezer is also known as the Bloch-Messiah reduction of the process. 2.7 Modeling photon detection with binary detectors 27

The resulting multi-mode squeezed vacuum state is then a product of orthogonal two-mode squeezed states

ıarg(ζc ) with λj = tanh(|ζcj|) e j . Interestingly, other than the approximated photon pair state in Eq. 2.74, this full multi-mode squeezed vacuum contains no spectral entanglement, since the sub-states of the individual Schmidt modes are in a product state, not in a superposition. Only through the non-Gaussian process of photon detection, and thus removal of the vacuum contribution, we can observe entanglement. Thus, when one speaks of the spectral entanglement of the two-mode squeezed state, one usually means that of the photon pair state. Each squeezing operator ˆ † ˆ † −ıHˆ j −ıζj A B +h. c. Sˆj = e = e j j (2.79)

is parametrized by a coupling constant ζj = cjζ that determines the mean photon number nj of the generated light: 2 nj = hˆnji = sinh(|ζj|) (2.80)

The squared modulus of the parameter λj is connected to the mean photon number of mode j through 2 2 2 2 sinh(|ζj|) sinh(|ζj|) nj |λj| = tanh(|ζj|) = 2 = 2 = (2.81) cosh(|ζj|) sinh(|ζj|) + 1 nj + 1 [85]. p The coupling of each single squeezing operator scales with pump beam power like ζj ∝ Pp. From their super-linear scaling of the mean photon number nj it follows that for high pump power values, the stronger squeezers begin to “out-shine” the weaker ones, because through stronger self-stimulation of the downconversion process their photon production relatively increases.

2.7 Modeling photon detection with binary detectors

We now consider the probability for a photon detection event with binary detectors, such as avalanche photo detectors or photo multiplier tubes. A binary detection event indicates the arrival of one or more photons, but it is impossible to determine how many photons were present without an ensemble measurement.

2.7.1 Measurement operator For an arbitrary single mode state of light ρˆ, the detection probability is

p = Tr[ˆρµˆ] (2.82) where µˆ is the measurement operator of a binary detector, or in general a convex combination of projectors onto the basis subspaces of the Hilbert space of the problem. As input states we will consider only pure states ρˆ = |ΨihΨ|, since in this work we assume that we produce pure states that degenerate into mixed states through optical losses on the path from the light source to the 28 2 Basic concepts ideal detector. Any such loss can be modeled by introducing a virtual beam-splitter, which in turn can be concatenated with the ideal detector into a lossy detector. If only one single photon at one time is assumed to arrive at the detector, the corresponding measurement µˆ|1i,ω operator is simply a projector on a single photon:

† µˆ|1i,ω = |1i h1| = ˆa (ω) |0ih0| ˆa(ω) . (2.83)

However this definition is specific to a single frequency ω. In order to measure photons of any frequency or frequency distribution, we have to integrate over all frequencies. Since in practice detectors do not function with unit probability, we add as a weighting function its quantum efficiency ηQE(ω) which assumes real values between 0 and 1. If the spectral width of the light states in question are well within the quantum efficiency profile of the SPD, sometimes the quantum efficiency is assumed to be constant, and for ideal detectors one sets ηQE = 1. The measurement operator for a lossy detector in the presence of at most one photon thus reads Z † µˆ|1i = dω ηQE(ω) â (ω) |0ih0| â(ω) . (2.84)

In a matrix representation, this operator has only diagonal elements, all coherences – all off- diagonal elements – have been washed out. This is not generally the case, but we here can write the measurement operator like this because we implicitly assume a measurement time much longer than the duration of our photon states[89]. For very short measurement times, the operator would rather have a form like RR dω dω0 F (ω, ω0) ˆa†(ω) |0ih0| ˆa(ω0) where off-diagonal elements ω 6= ω0 can be different from zero. Sometimes it is more convenient, when dealing with ultrafast pulses, to express the measurement operator in a broadband mode basis {ξj(ω)}. We make use of the basis transform in Eq. 2.30 that allows us to rewrite the measurement operator for an ideal SPD as

X ˆ † ˆ µˆ|1i = Aj |0ih0| Aj. (2.85) j

This simple single-photon approach however falls short when there may be more than one photon at a time impinging on the detector. Surely if there are two photons |2ih2|, the probability to detect either or both of them must be higher than for detecting one single photon. Instead, since Fock states are mutually orthogonal, we ﬁnd p = Tr |2ih2| µˆ|1i = 0. So we have to add projectors for every Fock state except the vacuum state |0ih0| to catch any number of photons possible that can trigger an detection event. The identity operator in photon number basis is just the sum of all 1 P∞ Fock state projectors = n=0 |nihn|, and so we can write ∞ X µˆ = |nihn| = 1 − |0ih0| (2.86) n=1 This term is easily generalized for non-ideal SPDs in the multi-photon case by multiplying each term of the sum with the probability of a detection event. If n photons arrive at the detector, the n probability of a detection event and the probability of detecting no photon at all (1 − ηQE) must add up to unity, hence[113]

∞ ∞ X n X n µˆ = (1 − (1 − ηQE) ) |nihn| = 1 − (1 − ηQE) |nihn| (2.87) n=1 n=0 2.7 Modeling photon detection with binary detectors 29

If one wants to measure the detector click probability in the broadband regime where the quantum efficiencies ηQE(ω) for different photons vary significantly, one would have to allow for distinct frequency variables per photon, so that the probability to detect no photon at all was rather Qn i=1 (1 − ηQE(ωi)). However for sufficiently narrow-band light pulses we can consider ηQE(ω) a slowly varying function around their central frequency and approximate it as a constant.

2.7.2 Measuring the joint spectrum of a photon pair

The measurement operator µˆc for coincidence events between two detectors is simply the product of two single detector operators in mode 1 and 2:

µˆc =µ ˆ1 ⊗ µˆ2 (2.88) In order to spectrally resolve a photon pair created in a weakly pumped, ultrafast type II SPDC process we assume frequency dependent, lossy detectors, and introduce very narrow frequency ﬁlters with δ-peak transmissions at ω1 into the signal arm in front of the detector, and at ω2 into the idler arm. For the detector efﬁciencies, we can therefore write

ηi(ω) = ηiδ(ωi − ω) (2.89) with i ∈ {1, 2}. For low coupling strength ζ 1, we neglect multiple pair creation and therefore

The inﬁnitesimal probability to detect a coincidence event at frequencies ω1 and ω2 is thus

For low squeezing ζ or low mean photon number, the inﬁnitesimal probability p(ω1, ω2) to detect a coincidence event with signal frequency ω1 and idler frequency ω2 is proportional to the joint 2 spectral intensity value |f(ω1, ω2)| . Conversely, this means that by measuring spectrally resolved coincidence clicks, we can reconstruct the JSI of a PDC process. For higher photon numbers, the relationship between detection probability and joint spectrum is less straight-forward. We will investigate this matter in section 5.6. To calculate an actual detection probability P , we have to deﬁne a frequency range for both signal and idler photon and integrate over them:

0 0 Z ωs Z ωi P = dω1 dω2 p(ω1, ω2) (2.93) ωs ωi In an actual measurement of the joint spectrum, one obtains the probability P for a range of 0 0 “pixels” with size [ωs, ωs] × [ωi, ωi], and for a sufﬁciently small pixel size one can approximate 2 P 0 0 ∝ |f(ω , ω )| . [ωs,ωs]×[ωi,ωi] s i

3 Spectral engineering

The purpose of spectral engineering of a three-wave mixing process is to shape the spectral correlations between the participating light beams. In the SPDC case, these determine the joint spectral amplitude of the output photon pair. For frequency-degenerate SPDC sources, using ultrafast pump beams with a broadband spectrum can increase indistinguishably between signal and idler photon[60, 70], making for higher visibility in signal-idler quantum interference[38, 58, 22]. In contrast, the idler photons from two identical but separate photon pair sources are necessarily completely identical in their spectral and temporal properties. Yet in general, they do not interfere with full visibility; quantum interference between two photons depends not only on spectral and temporal overlap, but also on the coherence of the single photons. Since the mutual information between entangled bi-photons introduces decoherence in the reduced systems of both single photons, we find optimal interference visibility between photons from two sources if those emit uncorrelated, separable pairs. To generate spectrally separable photon pairs from a ultrafast pumped SPDC source, one chooses a nonlinear optical material where the pump photon’s group velocity value is between signal and idler group velocity. For a pump photon with the right spectral – and thus temporal – variance, the output photon pair’s temporal – and thus spectral – inter-correlation is washed out. Since typically the group velocity is a monotonous function of wavelength in the visible regime and beyond, we need a birefringent crystal and a wave mixing process involving both polarizations to fulfill this condition called group velocity matching[59] (GVM). It is well suited for separable photon pair generation in nonlinear waveguides[133], as it does not utilize the spatial distribution of the pump beam to shape the output spectrum[28, 136]. In the near infrared range, the nonlinear crystal KH2PO4 (potassium dihydrogen phosphate, KDP) has been proposed[134] as a candidate for a group velocity matched photon pair source, and also has been implemented[93] in bulk crystal. For separable telecom wavelength photon pairs, KTiOPO4 (potassium titanyl phosphate, KTP) has been proposed[134]. GVM is also applicable to four-wave-mixing in χ(3)-nonlinear media such as photonic crystal fibers[63, 117] or even standard optical fibers[118]. 32 3 Spectral engineering

3.1 Pure heralded single photons and the two-mode squeezer

One goal of this thesis was the realization of an ultrafast SPDC source of pure heralded single photons. Heralding single photons is achieved by ﬁrst employing a weakly coupled PDC process (|ζ| 1) to probabilistically produce photon pairs

−ıH ||Ψii = e ≈ |0i − ıζ |Ψs,ii (3.1)

In order to herald a single idler photon, we detect its corresponding signal photon with an ideal single photon detector (SPD) module. The corresponding measurement operator µˆs is simply a projector on a single signal photon: Z † X ˆ † ˆ µˆs = |1i h1| = dω ˆa (ω) |0ih0| ˆa(ω) = Aj |0ih0| Aj (3.3) j

To calculate the resulting reduced quantum state, we take the normalized, partial trace of the input state: Trs[ˆµs |ΨihΨ|] X ρˆ = = |c |2 Bˆ † |0ih0| Bˆ . (3.4) i Tr[ˆµ |ΨihΨ|] j j j s j

The signal photon state ρˆi is a statistical mixture of the spectrally orthogonal pure single photon ˆ † ˆ states Bj |0ih0| Bj. This is problematic if the heralded single photon is to be used for any kind of quantum optical experiment involving Hong-Ou-Mandel (HOM) interference[66, 85]. For HOM interference, two single photons are overlapped at a balanced beam splitter. If their wave functions overlap perfectly, both will leave the beamsplitter through the same output port, an effect known as bunching. For their wave functions to overlap, all their physical properties such as frequency distribution, arrival time and transverse spatial distribution must be the same, such that, if only one photon travels through the beam splitter, it would be impossible for an observer to decide from which input port it had entered. The core of the problem here is that a state that can be described by one wave function is necessarily a pure state, while mixed states have to be described by an incoherent sum of pure states. As an example, we consider a beam splitter where each input port is supplied with an identical 2 ˆ † ˆ photon ρˆs. As has been already stated, the broadband mode components |cj| Bj |0ih0| Bj are mutually orthogonal, and thus do not overlap at all. The probability for two photons to overlap and bunch is then simply the sum probability of two equal pure state photons coinciding, which is at the same time the visibility V of the quantum interference and the inverse of the initial P 4 1 photon pairs’ Schmidt number K (Eq. 2.73): pHOM = V = j |cj| = K . The cooperativity parameter is a measure for the spectral entanglement in the initial photon pair |Ψs,ii. A high 3.2 The phasematching distribution Φ and group velocity matching 33

amount of entanglement leads to a high value for K, a highly mixed heralded single photon state ρˆi and a low HOM interference probability because of poor mode overlap. On the other hand, the absence of spectral entanglement means that the Schmidt decomposition of the photon pair state yields c0 = 1 and for j > 0 : cj = 0, meaning that the photon pair is separable ˆ † ˆ |Ψs,ii = |Ψsi ⊗ |Ψii, and the resulting heralded single photon is in a pure state ρˆi = B0 |0ih0| B0. The cooperativity parameter assumes its minimum value at K = 1, and photon bunching occurs with unit probability pHOM = 1 due to perfect mode overlap.

3.2 The phasematching distribution Φ and group velocity matching

In the previous chapter we examined the connection between bi-photon spectral entanglement and heralded single photon purity. This entanglement is the consequence of the spectral correlations in the underlying SPDC effective Hamiltonian’s joint spectral amplitude f(ω1, ω2), and photon pair separability translates directly into the separability of the amplitude function: f(ω1, ω2) = ψ0(ω1) ϕ0(ω2). It is also, up to normalization, the product of the pump spectral amplitude and the phasematching function f(ω1, ω2) = α(ω1 + ω2) Φ(ω1, ω2). When we make a Gaussian ∆kL 2 ∆kL −γ( 2 ) approximation for the phasematching Φ(ω1, ω2) ∝ sinc 2 ≈ e , the numerical constant γ = 0.193 adapts the width of the Gaussian curve to that of the central peak of the sinc function. We then do a Taylor expansion of the phase-mismatch ∆k(ω1, ω2) = kp(ω1 + ω2) − ks(ω1) − ki(ω2) − kΛ around the central frequencies for signal and idler beams, and find that the 0th order vanishes, since we assume phase-matching. The first order is affine linear in signal and idler frequency

0 0 0 ∆k(ω1, ω2) ≈ kp(¯ω1 +ω ¯2) (¯ω1 − ω1 +ω ¯2 − ω2) − ks(¯ω1) (¯ω1 − ω1) − ki(¯ω2) (¯ω2 − ω2) (3.5)

0 ∂ with kµ(ω) = ∂ω kµ(ω). Now the approximated Φ(ω1, ω2) is a two-dimensional Gaussian function, just as the pump spectrum in terms of signal and idler frequency α(ω1 + ω2) for an ultrafast pulsed laser, and one can ﬁnd a graphic understanding of bi-photon entanglement and separability by plotting the contours of both functions and of their product in the (ω1, ω2)-plane.

Figure 3.1: Frequency- and phase-matching plots for a SPDC process with negative phasematching slope.

In Fig. 3.1, the multiplication of a negatively correlated pump function α and an also negatively correlated phasematching function Φ results in an elliptic shape with a negative correlation. This is very typical for SPDC bi-photon amplitude distributions, since α is always negatively correlated 34 3 Spectral engineering

◦ with an angle of −45 due to frequency matching or energy conservation ω¯p =ω ¯1 +ω ¯2, and the phasematching curve normally has a negative slope as well: The phasematching angle Θpm of the 2 −γ( ∆kL ) phasematching function Φ(ωo, ωi) ≈ e 2 can be easily calculated: Φ is maximal on the line ∆k = 0, and the slope of this line at central frequencies ω¯s, ω¯i is according to Eq. 3.5

0 0 kp(¯ωs +ω ¯i) − ki(¯ωi) tan Θpm = − 0 0 . (3.6) kp(¯ωs +ω ¯i) − ks(¯ωs)

∂ω 1 The group velocity v of a wave is related to its wave vector through v = ∂k ≈ k0 , an approximation that is exact for a wave in a bulk medium and still reasonable for a narrow waveguide. Without discussing its accuracy here, it can provide us with a better qualitative understanding of the phasematching slope’s physical signiﬁcance. We can now express it alternatively as

−1 −1 vp − vi tan Θpm = − −1 −1 . (3.7) vp − vs meaning that we get a positive slope only if the pump group velocity has a value between signal and idler group velocities. We can reduce the spectral correlation by choosing a SPDC process

Figure 3.2: SPDC process with positive phasematching slope.

Figure 3.3: SPDC process with positive phasematching slope and matched pump width. with a positively sloped phasematching curve, as depicted in Fig. 3.2. By additionally adjusting the spectral width of the pump beam in Fig. 3.3, we can ﬁnd an uncorrelated spectrum. It is important to notice that this is only possible for a non-negative phasematching slope, or 3.2 The phasematching distribution Φ and group velocity matching 35

◦ ◦ 0 ≤ Θpm ≤ 90 . With a negative slope, there is always residual correlation. In the corner cases of a horizontal or vertical slope, pump width has to go to infinity to create a completely separable two-photon spectrum. Only for a positive slope, it is possible to find a finite pump width for spectral separability. In terms of frequency detunings νi =ω ¯µ − ωµ we can find a formula to determine if this is possible: " # (ν + ν )2 ∆k(ν , ν )2 L2 f(ν , ν ) = α(ω − ω ) Φ(ω , ω ) ≈ exp − 1 2 − γ 1 2 (3.8) 1 2 o i o i 2σ2 4

In order for f to be separable, any term of the exponential expression proportional to the product ν1ν2, and therefore frequency correlated, has to vanish: ν ν L2 − 1 2 − γ k0 − k0 k0 − k0 ν ν = 0 (3.9) σ2 4 p s p i 1 2 1 h γ i− 1 ⇒σ = − k0 − k0 k0 − k0 2 (3.10) L 2 p s p i Since γ is a positive constant, the expression Eq. 3.10 has a real solution only if the outcomes of 0 0 0 0 kp − ks and kp − ki have different signs. This condition is equivalent to a positive phasematching slope, so that a spectral pump width for separability can only be found if the pump group velocity is between signal and idler group velocity

vs ≤ vp ≤ vi (3.11)

and is therefore known as group velocity matching (GVM), even if exact matching is not necessary. For a given conﬁguration, the product of crystal length and separability width is constant: If a spectral width σ for two-photon separability can be found for a crystal length L, then a crystal σ with length 2L has a separability width 2 . We distinguish a special case[77] of solution to Eq. 3.9, where

vs = vp < vi. (3.12)

This condition we will refer to as critical group velocity matching. Its phasematching angle Θpm is either zero or 90◦, depending on the interchangeable signal/idler labeling of the output beams. For a pump width σ significantly greater than the phasematching width σpm, the resulting joint amplitude shows only minimal correlations, as demonstrated in Fig. 3.4. The KDP bulk crystal source implemented by Mosley et al.[93] utilized critical GVM. It has to be remembered that we applied two approximations here, namely linearizing the phase-mismatch ∆k around the SPDC central frequencies and approximating the “sinc”-profile of the phasematching function with a Gaussian curve. For the exact case, the phasematching curve ∆k(ωs, ωi) = 0 is curved instead of strictly linear, and the sinc profile causes additional maxima, so called side lobes, in addition to the main maximum of the bi-photon spectrum, potentially introducing additional frequency correlation. Fig. 3.5 (left) illustrates the approximation of the sinc function by a Gaussian, and its impact on the two-photon spectral intensity (middle and left). However, if one is using QPM to achieve phase-matching, it is possible to introduce a Gaussian modulation to the periodic poling in order to suppress the sinc beatings and physically approximate the phasematching function to a Gaussian distribution[17]. The effects of the curved phasematching contour can be largely avoided by choosing longer crystals, so that the spectral width of signal and idler photons, which scales with the inverse 36 3 Spectral engineering

Figure 3.4: SPDC process with zero phasematching slope, narrow phasematching width and a spectrally broad pump.

0.5

0 Idler frequency Idler frequency

-0.5 Signal frequency Signal frequency -30 -20 -10 0 10 20 30

2 Figure 3.5: Left: The function sinc(x) (blue) and the Gaussian e−γx (violet) to approximate its central peak. Middle: A joint spectral intensity function with the Gaussian approximation for the phasematching Φ. Right: An exact joint spectral intensity function with the ﬁrst pair of “sinc” side lobes visible.

1 interaction length L , is small when compared to the radius of curvature. The sinc side lobes are approximately perpendicular to the phasematching curve’s orientation, so for critical GVM the frequency correlation they cause is mitigated for a spectrally broad pump.

3.3 Critical phasematching through backward-wave SPDC

We speciﬁcally chose as a source of squeezed light a nonlinear waveguide to enforce exactly one propagation direction of the output states. There is however one more possible direction: In what is sometimes called backward-wave SPDC[64], either one or both of the daughter photons travel into the opposite propagation direction of the pump photon. The energy conservation ωp − ωs − ωi = 0 still holds, but the phasematching equation – and every equation depending on the photon pseudo momenta – has to be generalized from the collinear case to

kp ∓ ks ∓ ki − kΛ = 0 (3.13) where we introduced a sign-ﬂip for ks or ki, depending on whether either photon or both travel backwards. Considering a degenerate type I SPDC process where the signal beam is generated as a 3.4 Type I SPDC 37

backward wave, we write down the adapted expression for the phasematching slope

0 0 kp(¯ωs +ω ¯i) − ki(¯ωi) tan Θpm = − 0 0 . (3.14) kp(¯ωs +ω ¯i) + ks(¯ωs)

0 0 where we substituted −ks(¯ωs) → +ks(¯ωs). The phasematching slope for collinear SPDC is exactly −1, since for type I we have signal and idler in the same polarization mode and thus k(ω) = ks(ω) = ki(ω). For the backward signal beam case, the slope is the quotient of the difference and the sum of two numbers that are of the same order of magnitude:

0 0 kp(¯ωs +ω ¯i) − k (¯ωi) tan Θpm = − 0 0 . (3.15) kp(¯ωs +ω ¯i) + k (¯ωs)

0 0 Consequently, the smaller the difference between kp(¯ωs +ω ¯i) and k (¯ωi) is, the smaller the phasematching slope tan Θpm becomes. For small angles, we approach critical GVM, so that with a spectrally wide pump, we expect the generation of near-separable photon pairs from backward-wave SPDC, in a basically material-independent way[102, 32]. The drawback in this 2πm scheme is that the phase-mismatch kp + ks − ki = kΛ = Λ is signiﬁcantly higher than in the collinear case. To compensate for it, one either has to produce very small QPM poling periods Λ, or use a high QPM order m. The former is just beyond the reach of today’s technology, while the 1 latter diminishes the produced photon pair ﬂux by m2 .

3.4 Type I SPDC

We have seen that in order to generate separable photon pairs from SPDC, we need to achieve GVM: A configuration where the pump beam’s group velocity is between signal and idler group velocity. For the materials considered here, the group velocity typically is a monotonous function of wavelength or frequency in the visible to telecom regime. They exhibit an extremal group velocity at the first “zero dispersion wavelength” (i. e. group velocity dispersion and the related k00 vanish). Since signal and idler wavelengths are always greater than the pump wavelength, GVM is simply not possible at wavelengths lower than the first zero dispersion wavelength, if all three beams follow the same chromatic dispersion function k(ω). In Fig. 3.6, this situation is illustrated for lithium niobate and a degenerate type Ia downconversion process: Pump, signal and idler are linearly polarized along the “fast” axis, and their group velocities are plotted against wavelength for signal/ idler and against double wavelength for the pump beam. Their intersection marks a degenerate type Ia SPDC process for which exact group velocity matching vp = vs = vi is possible:

1300 nm → 2600 nm + 2600 nm (3.16)

But if evaluated naively, the expression for the phasematching slope from Eq. 3.7 yields exact 0 tan ϕpm = 0 and in order to calculate a well-deﬁned value, we have to apply the theorem of l’Hospital and once more derive both numerator and denominator with respect to frequency:

00 00 exact kp (¯ωs +ω ¯i) − ki (¯ωi) tan Θpm = − 00 00 (3.17) kp (¯ωs +ω ¯i) − ks (¯ωs)

00 Assuming neither beam is at a zero dispersion point where k{p,s,i} ω{p,s,i} = 0, we then have exact thanks to ks = ki a slope of tan Θpm = −1. This is the case for every wavelength-degenerate 38 3 Spectral engineering

0.5

0.4

Group velocity [c] velocity Group 0.3 signal/idler (fast) pump (fast, x-axis x2)

500 1500 2500 3500 4500 Wavelength [nm]

Figure 3.6: Group velocity matching in lithium niobate for type-I SPDC type I SPDC process, independently of phasematching. We can understand this from simple symmetry considerations: The phasematching curve ∆k(ωs, ωi) = 0 for a type I process must be symmetric in its arguments ωs, ωi; after all, “signal” and “idler” are just labels for two photons with the same polarization, and thus the same material dispersion. Also, the curve must be continuously differentiable. For both conditions to hold around the point ∆k(ω, ω) = 0, it must intersect with the symmetry axis ωs = ωi at a right angle, and thus it must have slope −1 in its vicinity. As a physical consequence, neither degenerate type Ia nor type Ib SPDC processes, even if group velocity matched, can generate spectrally separable photon pairs, as the intersection of pump and phasematching distribution will always result in a negatively correlated joint spectral distribution. Separability is however in principle achievable for frequency non-degenerate type I processes, but since the goal of this work was a degenerate source for the telecom regime, this direction is not further pursued here.

3.5 Type II SPDC

Birefringent materials do not only exhibit polarization-dependent refractive indices, but in turn also polarization dependent group velocities. The existence of one “fast” and one “slow” polarization axis (see Fig. 3.7) gives rise to new possibilities for GVM: In Fig. 3.8, we see that the group velocity curve for the fast pump beam (blue) – again plotted against double wavelength for convenience – intersects with the slow output beam’s curve (green) at lower wavelengths than with the fast output beam (red). Both figures plot the group velocity of a fast- and a slow-polarized light wave in lithium niobate. Between the intersecting points, the GVM condition Eq. 3.11 is fulfilled, the phasematching slope is positive, and we can find a finite spectral width for the pump beam to create spectrally separable SPDC photon pairs, where one “fast” photon is converted to a “fast” and a “slow” one.

3.6 Survey of nonlinear waveguide materials for group velocity matching

Choosing one nonlinear crystal of a length L, a polarization type, and an operating temperature, it is now possible to plot the separability width for a SPDC process in dependence of signal and 3.6 Survey of nonlinear waveguide materials for group velocity matching 39

0.5

0.45

0.4

0.35

Group velocity [c] velocity Group 0.3 fast axis slow axis

500 1000 1500 2000 2500 3000 3500 4000 Wavelength [nm]

Figure 3.7: Group velocities of the fast (red) and slow (green) polarization in lithium niobate

0.5

0.4

signal/idler (fast) signal/idler (slow)

Group velocity [c] velocity Group 0.3 pump (fast, x-axis x2) pump (slow, x-axis x2)

500 1500 2500 3500 4500 Wavelength [nm]

Figure 3.8: Group velocity matching in lithium niobate for type-II SPDC

idler central wavelengths. Since it is our goal to implement a waveguide SPDC source for separable photon pairs, we can restrict ourselves to collinear processes. Also, among the nonlinear crystals widely used for frequency conversion, not all can be used to produce wave-guiding structures. The most common material for nonlinear waveguide devices by far is lithium niobate (LiNbO3), but also the structurally similar lithium tantalate (LiTaO3) is used. In recent years, potassium titanyl phosphate (KTiOPO4 or KTP for short) has seen increased adoption, not the least due to favorable group velocity matching properties for infrared applications. In the following sections, we will provide an overview of the possibility of group velocity matched, frequency degenerate and collinear type II processes by solving equation 3.10 for different crystals with varying working parameters. The Sellmeier equations to determine crystal dispersion are the standard versions for bulk crystal at room temperature. As interaction length we assumed L = 10 mm throughout. If a real-valued solution σ exists, a PDC process pumped with a Gaussian spectrum with variance σ or with FWHM 2.35σ will, in the low power limit, produce photon pairs with an approximately separable joint spectral amplitude f(ωs, ωi). Each process is characterized by the nonlinear coefficient dxy, since besides giving the coupling strength, it unambiguously identifies the polarizations of pump, signal and idler. Owing to the crystal properties of each material, we can discard some processes with dxy = 0 right away. Our objective is to identify a suitable low-loss nonlinear crystal to produce waveguides for spectrally separable photon pairs with feasible working parameters, such as pump spectral width, crystal temperature and QPM poling period. The degenerate signal/idler wavelength should be 40 3 Spectral engineering in the telecom regime, specifically at 1550 nm. Operating at this wavelength allows using and integrating with existing technologies: low-loss optical standard fibers are commercially available, as well as efficient single photon detectors and photo diodes, and a large array of optical devices.

3.6.1 Lithium niobate

100 d24 d34

40 Pump spectrum FWHM [nm] spectrum FWHM Pump

0 1500 2500 3500 4500 Signal/Idler wavelength [nm]

Figure 3.9: Left: GVM in lithium niobate (LiNbO3). Right: Comparison between d24 and d34 process.

One of the most common versatile materials in modern optics is the uni-axial crystal lithium niobate. With its strong χ(2) nonlinearity and its electro-optical properties it is used in many three-wave-mixing applications as well as for devices like electro-optical modulators (EOMs). Due to its ferroelectric properties, its domain structure can be manipulated by applying strong electrical fields, which is exploited to create periodic poling structures for quasi-phasematching inside bulk crystal samples. In Fig. 3.9, we calculate the pump FWHM necessary to create separable, frequency degenerate photon pairs at signal/idler wavelength λ for the PDC processes corresponding to the nonlinear coefficient d24 (red) and d34 (green). Both have cross-polarized signal- and idler-photons, but the former’s pump is y-polarized, while the latter’s is z-polarized. For d34, separability becomes possible for a signal/idler wavelength of 1615 nm, which is close enough to the telecom wavelenghts at 1550 nm to be of interest. However, the tensor element d34 is zero due to the crystal symmetry of lithium niobate. For d24 the separability range begins around 2700 nm, which is too far into the infrared for any kind of single photon detector to see. Yet we now investigate wether manipulating experimental parameters can improve the situation. The Sellmeier equations of lithium niobate[45] show a large temperature dependence. In Fig. 3.10 (left), the GVM spectral width for d24 is plotted for temperatures between 193 K and 493 K, or between −80 ◦C and 220 ◦C. The lowest GVM wavelength rises with the temperature. Even with severe cooling, GVM starts only at 2690 nm, assuming that the Sellmeier equations are still valid in this temperature regime. Often lithium niobate is heated to temperatures over 393 ◦C to lower the light absorption and emission from color centers in the crystal structure. Inside a waveguide, the field confinement changes wave dispersion and group velocity. Thus, the lower GVM threshold is also sensitive on waveguide size. In the simplified waveguide model based on geometric optics (c. f. section 2.6.2), we define an effective dispersion relation keff(ω) = q 2 π2 β = k(ω) − 2 b2 , where k is the bulk crystal dispersion and b is the width and height of a rectangular, perfectly internally reflecting waveguide. Substituting keff into equation 3.10 has a 3.6 Survey of nonlinear waveguide materials for group velocity matching 41

100 100 T=193K bulk crystal T=293K 6 µm T=393K 5 µm T=493K 4 µm 80 80 3 µm

60 60

40 40 Pump spectrum FWHM [nm] Pump spectrum FWHM [nm] Pump

20 20

0 0 2680 2690 2700 2710 2720 2000 2200 2400 2600 2800 Signal/Idler wavelength [nm] Signal/Idler wavelength [nm]

Figure 3.10: Left: d24 process GVM in bulk PPLN for different temperatures. Right: d24 process at 200K for different waveguide sizes.

noticeable effect in GVM: In Fig. 3.10 (left), the bulk crystal situation is compared to waveguide sizes down to 3 µm, and the lower GVM threshold drops to 2050 nm for the smallest waveguide size. One should keep in mind that this result though that this result is based on geometric optics, so for small structure in the order of the target wavelength they will quantitatively differ from the results of a rigorous solution of the Maxwell equations to determine waveguide dispersion. The qualitative trend however is clear enough: Small diameter PPLN waveguides can produce separable two photon pairs at signiﬁcantly lower wavelengths than bulk PPLN, yet the wavelengths are still too high to be of practical use to us.

3.6.2 Lithium tantalate

400 d 350 24 300 250 200 150 100 50 0 Pump spectrum FWHM [nm] Pump 2400 2500 2600 2700 Signal/Idler wavelength [nm]

Figure 3.11: GVM in lithium tantalate (LiTaO3).

Lithium tantalate (LiTaO3) chemically differs from LiNbO3 by only one atom from the same group of elements in the negative ion, and shares the same crystal symmetry group (trigonal 3m), so their mechanical, chemical, and electrical properties are similar. However there is a marked difference in optical dispersion, which impacts on the GVM situation. In Fig. 3.11, we see that GVM is possible only in a small small spectral range around 2500 nm. Fig. 3.12 shows that while cooling and heating the crystal does little to improve the situation, waveguide size has a large impact on GVM and shifts the threshold down to 1900 nm for 3 µm side length. This still falls short of the goal of telecom wavelengths, and along with the enormous 42 3 Spectral engineering

400 400 T=193K bulk crystal 350 350 T=293K 6 µm 300 T=393K 300 5 µm 250 T=493K 250 4 µm 3 µm 200 200 150 150 100 100 50 50 0 0 Pump spectrum FWHM [nm] spectrum Pump [nm] spectrum FWHM Pump 2450 2500 2550 2600 1800 2000 2200 2400 2600 Signal/Idler wavelength [nm] Signal/Idler wavelength [nm]

Figure 3.12: Temperature dependence (left) and waveguide size dependence (right) of GVM in lithium tantalate (LiTaO3). spectral pump width of around 50 nm to 100 nm needed for photon pair separability, it makes LiTaO3 unsuitable for our purposes.

3.6.3 Potassium niobate

20 d24 15

0 Pump spectrum FWHM [nm] Pump 1500 2000 2500 3000 Signal/Idler wavelength [nm]

Figure 3.13: GVM in bulk potassium niobate (KNbO3).

Potassium niobate (KNbO3) is a bi-axial nonlinear crystal that has not been in wide use in integrated quantum optics, although the bulk crystal is used as a PDC source[100] and waveguides for SHG have been produced[50]. Care must be taken when handling this material, as mechanical stress or even vibration can lead to the formation of unwanted crystal domains. With Fig. 3.13, we identiﬁed one type-II processes of interest: d24 with propagation direction x, with GVM starting to be possible at roughly 1500 nm. We investigate the temperature and waveguide size dependence of the dispersion in Fig. 3.14. As before, the GVM threshold rises with both temperature and waveguide size, so that we can expect a waveguide at room temperature to emit spectrally separable photon pairs with a pump width of 2 nm. 3.7 Conclusion 43

20 20 T=193K bulk crystal T=293K 6 µm 15 T=393K 15 5 µm T=493K 4 µm 3 µm 10 10

5 5

0 0 Pump spectrum FWHM [nm] spectrum Pump [nm] spectrum FWHM Pump 1300 1500 1700 1900 2100 2300 1400 1450 1500 1550 1600 Signal/Idler wavelength [nm] Signal/Idler wavelength [nm]

Figure 3.14: Temperature dependence (left) and waveguide size dependence (right) of GVM in potassium niobate (KNbO3).

20 20 d24 bulk crystal 6 µm 15 15 5 µm 4 µm 3 µm 10 10

5 5

0 0 Pump spectrum FWHM [nm] spectrum Pump [nm] spectrum FWHM Pump 1250 1550 1850 2150 2450 1150 1350 1550 Signal/Idler wavelength [nm] Signal/Idler wavelength [nm]

Figure 3.15: Left: GVM in bulk potassium titanyl phosphate (KTiOPO4). Right: GVM with waveguide dispersion

3.6.4 Potassium titanyl phosphate

Finally we come to potassium titanyl phosphate (KTiOPO4, or KTP), a mechanically robust, bi- axial nonlinear crystal, and a popular choice for nonlinear and quantum optics applications, often as a source for SPDC photon pairs (see e. g. [3, 135, 79, 44, 17]). As we see in Fig. 3.15 for the d24 process in bulk crystal, GVM occurs at room temperature from 1200 nm up to almost 2500 nm, and has indeed been shown for photon pairs at 1580 nm[79]. From Fig. 3.15 (right), we ascertain that waveguide dispersion does not introduce drastic changes for the telecom wavelengths around 1550 nm. The necessary pump width for separability ranges, depending on waveguide size, from 2 nm to 2.5 nm at 775 nm central wavelength, which is equivalent to pulse lengths of about 1 ps.

3.7 Conclusion

We have discussed spectral engineering by group velocity matching in collinear geometries and generalized it to anti-collinear geometries, and have investigated the suitability of several popular nonlinear materials for the implementation of a source of frequency-uncorrelated photon pairs at telecom wavelengths. To generate spectrally separable photon pairs in the telecom wavelength regime, both KNbO3 and KTP work as bulk crystal and waveguide sources at room temperature. 44 3 Spectral engineering

LiNbO3 shows promise for small waveguide sizes, if such waveguides can be produced and are still able to guide light modes at 1550 nm without high losses, while LiTaO3 is completely unsuitable since GVM is possible only around signal and idler wavelengths of 2500 nm. We decided to build a KTP waveguide source based on the d24 process mainly due the wide spectral range in which separable photon pair production is possible, but also due to the commercial availability of waveguide chips, and prior experience with the material[44]. 4 A PP-KTP waveguide as parametric downconversion source

For the generation of photon pairs, PDC sources are an established standard. Recent works have shown that source engineering[59, 134, 79] is capable of producing separable two-photon states |ψi = |ψsi ⊗ |ψii from PDC [93]. This allows the preparation of pure heralded single photons needed in single photon linear optical quantum computing schemes[74]. Going beyond the single photon pair approximation, we find that in general PDC can be understood as a source of squeezed states of light[144, 141]. First observed in a four-wave-mixing process by Slusher et al.[115], squeezed states originally garnered interest due to the noise reduction in one of their quadrature observables Xˆ, Yˆ below the classical shot noise level, and found application in quantum-enhanced interferometry[29]. The availability of mode locked laser systems allowed the generation of pulsed squeezed states[114], albeit multimode ones[97], and a strong trend towards miniaturization and integration of PDC squeezing sources can be seen[3, 69, 104]. In more recent developments, the non-classical character of squeezed states has been harnessed as the basic resource in continuous variable (CV) quantum information processing protocols such as CV teleportation[52, 16], CV entanglement swapping[125] as well as advanced metrology applications[131]. A long standing goal of CV quantum information processing was entanglement distillation[98, 37, 96] which can overcome transmission losses in wide area quantum communication networks, as it is an essential building block of quantum repeaters[25], and has been recently demonstrated in experiment[124]. It has been shown that distillation needs non-Gaussian operations[47, 54, 30], most commonly implemented by photon counting with avalanche photo diodes (APDs). Their inability to discriminate between “neighboring” spectral modes introduces mixedness and masks the quantum characteristics of a multimode squeezed state, which can never be completely compensated by narrow spectral filtering, as it cannot restore phase coherence but always introduces additional losses[106]. The straight-forward solution is to use a two-mode squeezed state in the first place. The outlined applications for a two-mode squeezer call for a compact, robust, and very bright source, as the mean photon number generated is related to the amount of usable squeezing. We find these favorable properties in nonlinear waveguide sources[3, 128], which are superior 46 4 A PP-KTP waveguide as parametric downconversion source to bulk crystal sources in several ways: All downconverted light is emitted in tightly defined waveguide modes, mode confinement allows the whole waveguide length to contribute to photon pair generation (pump is “always in focus”), and gain induced diffraction[72] is suppressed[3]. In this chapter we present an ultrafast waveguided type II PDC-based two-mode squeezer source in the telecom wavelength regime. We perform a g(2) correlation function measurement[129, 7] to characterize the phase coherence of our source’s output beams, demonstrating near thermal photon statistics and thus full two-photon wave function separability. We show that our source emits two single mode light pulses with an extraordinarily high mean photon number and moreover features the gain signature of a true two-mode squeezer. It generates separable two- photon states in the low pump power regime or generally two mode squeezed vacuum states by combining the high nonlinearity of χ(2) processes with the enormous gains possible in nonlinear waveguides and the modal control made possible by spectral engineering. It advances modal source brightness over several orders of magnitude with respect to prior experiments[93, 79]. In general, we can describe a type II PDC process as a multimode squeezer in terms of broadband frequency modes with its interaction Hamiltonian

ˆ X ˆ X ˆ † ˆ † ˆ ˆ HPDC = Hj = ζ cj AjBj + AjBj (4.1) j j

Aˆ j and Bˆ j are two sets of orthogonal broadband modes, and the Hˆ j describe a set of non- interacting two-mode squeezers with coupling strength ζcj. For the special case where all coefﬁ- cients cj but c0 vanish. We now produce exactly one mode in each output beam, and therefore the effective mode number of the source is minimal with K = 1, so we have a perfect broadband two-mode squeezer: ˆ ˆ † ˆ † ˆ ˆ H0 = ζc0 A0B0 + A0B0 (4.2)

4.1 Single photon detectors

As single photon detectors we use the idQuantique id201 InGaAs avalanche photo diodes, and since they are crucial components for all of our experiments, their properties and idiosyncrasies bear closer inspection[33]. Avalanche photo diodes can be thought of as the solid state advancement of classic photo- multiplier tubes; instead of a vacuum tube, a semiconductor diode is the underlying device. Depending on the target operation wavelength, different diode materials are used, from silicon for visible light and germanium at 1064 nm to indium gallim arsenide (InGaAs) at telecom wavelengths. To this diode a high reverse bias voltage (smaller than its breakthrough voltage), is applied. If an electrons is freed within the diode’s depletion layer by an impinging photon via the photo-electric effect, the bias voltage accelerates it and causes it to release further electrons via impact ionization. Those electrons are also accelerated and free even more electrons; an electron avalanche is excited. The resulting current ﬂowing in the direction of the bias voltage indicates an impinging photon. There is always the possibility of a thermally excited election giving rise to an avalanche, causing a “dark count” event, therefore most APDs are actively cooled. Also after an avalanche event, the APD’s detection current must be quenched, that is the diode’s depletion layer must be emptied of free charge carriers. Imperfect quenching leads to an increased probability for dark counts due to remaining electrons from the previous avalanche, an effect known as after-pulsing. While a higher bias voltage promises higher quantum efﬁciency of the APD, it will 4.1 Single photon detectors 47

also increase both thermal dark counts and after-pulsing, so that one has to ﬁnd a balance between detector efﬁciency and detector noise for each experiment. To reduce noise from dark counts, one can apply gating to the APD, that is to apply a bias voltage only when an incoming photon is expected. To obviate after-pulsing, the time between gating events must be big enough for any free charge carriers to disperse or re-combine. Alternatively, a “dead time” can be introduced, a time interval in which the detector is shut down after each avalanche event.

Figure 4.1: id201 APD quantum efﬁciency.

The id201 by idQuantique is an actively gated APD, and is specified for gating signal frequencies of up to 4 MHz. Gate widths vary from 2.5 ns to 100 ns. Although the bias voltage can be freely chosen, as a rule it is configured with four pre-calibrated settings for the quantum efficiency of 10%, 15%, 20%and 25%, to be understood for the telecom wavelength of 1550 nm. Fig. 4.1 shows the full quantum efficiency curve as specified by the maker. The id201 electrical inputs and outputs can be given and are available respectively both as TTL and NIM pulses, and an incoming gating pulse can be electronically delayed for up to 25 ns. First we tested the dark count level depending on the trigger rate for the two lowest gate widths of 2.5 ns and 5 ns and different dead times, as depicted in Fig. 4.2. The APD’s light input is blocked, and an external trigger signal from a pulse generator supplies trigger pulses with a variable frequency, and the gain is set to the 25%level. On the left hand, with a 2.5 ns gate, we can observe an approximately linear rise in dark counts, which for gating frequencies lower than 1.2 MHz is dead time independent. Above 1.2 MHz, the low dead time measurement curves rise faster, suggesting an onset of after-pulsing. On the right hand, the gating width is 5 ns. Here we see for gating frequencies over 1 MHz a dramatic increase of detection events when a low dead time is used. Since we use in our PDC experiments a mode-locked pump laser with virtually no timing jitter as a source, and the down-converted light pulse lengths are in the order of magnitude of 1 ps, we can easily use the lowest APD gate widths and no dead time, as long as we stay with low enough laser pulse rates. We therefore employ a pulse picker after our laser source to emit pump pulses at a 1 MHz frequency, and can expect an APD dark count rate of about 50 Hz. To examine the APDs’ after-pulsing characteristics, we coupled a heavily attenuated CW beam from a 1550 nm laser diode into the APD, and monitored the detection event rate in dependence 48 4 A PP-KTP waveguide as parametric downconversion source

Gating width: 2.5 ns Detector efficiency 25% 0.14 Deadtime: 0 µs 0.12 Deadtime: 1 µs Deadtime: 2 µs Deadtime: 5 µs 0.1 Deadtime: 10 µs 0.08

0.06

0.04 kcounts / second 0.02

0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 trigger frequency [MHz]

Gating width: 5 ns Detector efficiency 25% 4 Deadtime: 0 µs 3.5 Deadtime: 1 µs Deadtime: 2 µs 3 Deadtime: 5 µs Deadtime: 10 µs 2.5 2 1.5 1 kcounts / second 0.5 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 trigger frequency [MHz]

Figure 4.2: Dark counts of id201 APDs with no incident light. of the gating frequency. Since photons are impinging on the detector at every time, the count rate should in first order approximation be proportional to the overall open gate time in absence of after-pulsing. Fig. 4.3 confirms this for a 0 ns dead time and a trigger frequency lower than 1 MHz. Above this level, a slight super-linear growth can be observed which we attribute to after-pulsing. For higher dead times, we can observe a drop in detection rate as soon as the time between trigger pulses becomes smaller than the configured dead time value, as the gating pulse after a successful detection is lost. Again, this measurement shows that the APDs’ can be operated at 25%efficiency with 0 ns dead time, if we restrict our experiments to gating frequencies of 1 MHz or lower.

4.2 The parametric downconversion source

The core of all experiments presented here is the waveguide chip employed as the PDC source. It is a 10 mm long KTiOPO4 chip with channel waveguides manufactured by AdvR Inc. In Fig. 4.4, the input facet is shown in 50x magnification. Each waveguide is roughly 4 µm × 6 µm in cross section. Since the waveguides inscribed into the material by a diffusive proton exchange process, the waveguides seem less clearly defined further from the surface, as an influx of particles from the surface decays exponentially with penetration depth. To the left and right, and also at the top due to the material-air-boundary, they are sharply defined. The KTP chip is made from a z-cut wafer (z direction is depth) and contains seven groups of seven waveguides in x direction. They exhibit a periodic domain poling of Λ = 104 µm, designed to support the type-II downconversion process with nonlinear coefficient d24 from a P-polarized pump at 775 nm to P-polarized signal 4.2 The parametric downconversion source 49

Gating width: 2.5 ns Detector efficiency 25% 800 Deadtime: 0 µs 700 Deadtime: 1 µs Deadtime: 2 µs 600 Deadtime: 5 µs Deadtime: 10 µs 500 400 300 200 kcounts / second 100 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 trigger frequency [MHz]

Figure 4.3: After-pulsing of id201 APDs with weak coherent input light at 1550 nm.

Figure 4.4: The waveguide chip’s input facet under 50x magniﬁcation with a light microscope. Three waveguides with a 25 µm spacing are clearly visible.

and S-polarized idler beams at 1550 nm. Production imperfections cause a shift from these ideal telecom wavelength values, but this does not affect our experiments negatively in any way. Two neighboring waveguides of the same group are set 25 µm apart. Between groups the distance is 50 µm. The waveguides show stark differences in quality, dispersion and spatial mode properties; we determined during experimentation – and will discuss later – that for our purposes waveguide #6 gives the best results. Unless stated otherwise, we used this waveguide.

Figure 4.5: An ultrafast pumped KTiOPO4 waveguide chip as PDC source

The basic setup of our PDC experiments is simple: A beam from the pump laser is coupled into the waveguide, where the downconversion takes place, and the three outgoing beams are separated. Signal and idler beam are fed separately into single photon detectors to test if the desired process is actually occurring. In Fig. 4.5, we discuss this in more detail: The pump laser is a Coherent MIRA titanium sapphire mode locked system. In its default configuration, it produces femtosecond laser pulses between 710 nm to 980 nm with an autocorrelation length as short as 200 fs. It features an alternate 50 4 A PP-KTP waveguide as parametric downconversion source cavity configuration that allows for picosecond pulses. The average output power is specified as 650 mW, and the repetition rate is frep = 76 MHz. In order to not over-saturate detectors and data acquisition logic, we use an APE PulseSwitch accusto-optical pulse picker (AOM) to scale down the pulse frequency to 1 MHz. After passing through a power control consisting of half-wave plate HWP1 and the polarizing beam-splitter PBS1, the pump beam’s polarization can be controlled by HWP2. It is then coupled into one of the chip’s waveguides using 10x microscope objective MO, where the P-polarized component of each pump pulse excites a P-polarized signal pulse at 1544 nm and a S-polarized idler pulse at 1528 nm in a quasi-phasematched type-II PDC process. It is mounted on a Elliot Gold Series Professional Workstation which allows besides adjusting waveguide height and z-rotation also a translation in y-direction of 25 mm for easy waveguide selection. The outgoing beams are collimated with lens L1, an NIR-antireflection-coated Q940 lens sold by Thorlabs. The pump beam is split off by a Semrock RazorEdge LP02-808RS dichroic mirror DM and directed to power-meter PM to measure the pump power available inside the waveguide. Signal and idler beam are finally split by PBS2 into separate paths, collimated with NIR-coated lenses L2 and L3 (Thorlabs C430) and coupled into SMF28 single-mode fibers connected to avalanche photo diode (APD) single photon detectors. The detectors are actively gated id201 InGaAs APDs by idQuantique and have been described in the previous section. Their gating signal is supplied by the monitor signal of the AOM that is triggered every time it couples out a laser pulse. To prevent signal degradation and reflexion through a passive T-junction element in the electrical line, it is then multiplexed and by the Stanford Research DG645 pulse generator. The same device is used to delay it in order to synchronize the detectors’ gate time with incoming light pulses from the PDC source. To determine if photon pair generation is actually occurring, the APD detection signals have to be checked for a correlation of detection events. In particular, the level of coincident events has to exceed the expectation value of accidentally occurring coincidences. Important experimental figures in this context are the Klyshko efficiencies[73] that determine the overall fraction of the PDC produced photons being detected by the single photon detectors. Let Rp be the rate of pairs produced in the waveguide chip, and η¯1, η¯2 the absolute single photon detection efficiencies of each arm of the setup, including losses due to maladjustment of the beam-path, transmission losses of optical elements and detector quantum efficiency. If we assume only single photon pairs are produced and R1 and R2 is the rate of detection events of APD1 and APD2 in Fig. 4.5 respectively, and Rc is the rate of coincidence detection events between APD1 and APD2, then they are connected to Rp by: R1 =η ¯1Rp (4.3)

R2 =η ¯2Rp (4.4)

To the rate of coincidences Rc, losses from both setup arms have to be applied:

Rc =η ¯1η¯2Rp (4.5)

Solving for the efﬁciencies yields Rc η¯1 = (4.6) R2

Rc η¯2 = (4.7) R1 We now allow for the production of multiple photon pairs in the PDC and assume as its output a two-mode squeezed vacuum state |Ψ(r)i. For detectors far from saturation, the rate Ri is 4.2 The parametric downconversion source 51

connected to the detection probability pi simply through the pump laser repetition rate frep via Ri = freppi. The Klyshko efﬁciencies, in dependence of the squeezing operator r, now read

p (r) η (r) = c 1 p (r) 2 (4.8) pc(r) η2(r) = . p1(r)

The single click probabilities p1 and p2 can be calculated with the measurement operator µˆi from Eq. 2.87 : 2 η¯itanh r pi = hΨ| µˆi |Ψi = 2 (4.9) 1 − (1 − η¯i) tanh r

The measurement operator µˆc for the coincidence click probability pc is simply the product µˆ1 ⊗µˆ2, and the measurement evaluates to 1 1 1 pc = hΨ| µˆc |Ψi = 1 − 2 − 2 + 2 (4.10) 1 +η ¯1sinh r 1 +η ¯2sinh r 1 + (¯η1 +η ¯2 − η¯1η¯2) sinh r

For low squeezing values r → 0, we use the small angle approximation sinh(r) ≈ tanh(r) ≈ r. Then, we apply the l’Hospital theorem twice to ﬁnd

2 ∂r pc(r) η1(0) ≈ 2 =η ¯1 ∂r p2(r) r=0 2 (4.11) ∂r pc(r) η2(0) ≈ 2 =η ¯2. ∂r p1(r) r=0

For small squeezing values r, the Klyshko efficiencies coincide with the absolute setup efficiencies η¯1, η¯2. For moderate squeezing with r < 1, they scale approximately linearly, and this detection efficiency gain can be simply ascribed to the fact that more photon pairs at once are generated and single photon detectors will detect multiple photons with higher probability than a single photon. When we compare this to the Klyshko efficiencies that arise from any two uncorrelated photon sources with single event detection probabilities q1, q2 and coincidence event probability qc = q1q2, we see the crucial difference illustrated in Fig. 4.6: For low pump powers P , the Klyshko efficiencies η˜ = qc = q for the uncorrelated sources go towards zero for small squeezing (violet 1 q1 2 curve), while the efficiencies for the PDC source approach finite positive values η¯1, η¯2 (red and blue fit curves). Thus we can easily verify the existence of correlated photon events. Also in Fig. 4.6, we see experimentally gained Klyshko efficiencies from a PDC process in our setup according to Fig. 4.5. The laser is set to a 768 nm pump wavelength, and the wave plate HWP1 inside the power control block is rotated to tune pump power. For each pump power P , Klyshko efficiencies for signal and idler beam are recorded with a 15 s measurement time. Above a pump power of 2 µW the efficiencies show a linear growth as expected for a correlated photon pair source. From axis-intercept of the trend lines, we can estimate the absolute photon collection efficiencies for this measurement: η¯1 = 0.068 for the signal beam and η¯2 = 0.035 for idler. Below this power threshold, η1 and η2 drastically change their slope and quickly fall to zero. We can explain this by the presence of a constant background of dark count events in the APDs. As soon as their number is comparable the number of genuine single photon detection events, they will make their presence felt on the Klyshko efficiencies. If both the dark count probability pdark and 52 4 A PP-KTP waveguide as parametric downconversion source the background-free APD click probabilities are small, we can neglect all quadratic terms and write approximately: 0 pc(r) η1 = (4.12) p2(r) + pdark

0 pc(r) η2 = (4.13) p1(r) + pdark These expressions go towards zero for small pump powers and small squeezing parameters r and otherwise show the expected linear behavior, in accordance with the measured efﬁciencies in Fig. 4.6.

0.15

0.1

0.05 Klyshko eﬃciency Klyshko

η1 exp. data η2 exp. data uncorrelated sources 0 0 5 10 15 20 Pump power P [µW]

Figure 4.6: Experimental Klyshko efﬁciencies from our PDC source (red and blue) and theoretical Klyshko efﬁciency for two uncorrelated sources (violet)

In conclusion this measurement proves not only that correlated photon pairs are produced in our setup, but also gives an estimate of its overall photon collection efﬁciency.

4.3 Phasematching contour

Figure 4.7: Experimental setup for measuring PDC signal and idler photon marginal spectra with the MicroHR monochromator.

As has been discussed in section 2.6.1, the PDC phasematching function Φ(ωo, ωi) describes the restrictions by momentum conservation on the distribution of pump photon energy between signal and idler photons. We have shown that it must exhibit a non-negative slope to allow for 4.4 Conclusion 53

group velocity matching (GVM), and thus for the heralding of pure single photons. We will now trace the phasematching contour ∆k = 0 of the phasematching function Φ(ωo, ωi) by measuring marginal spectra of signal and idler photons at different pump wavelengths and plotting their maxima.

400 600

500 300

400 200 300

APD count rate [Hz] APD count 100 APD count rate [Hz] 200

0 100 1520 1560 1600 1480 1520 1560 1600 Signal wavelength [nm] Idler wavelength [nm]

Figure 4.8: Marginal spectra of signal (left) and idler (right) photons. Equal colors label corresponding signal and idler peaks. The left-most peaks correspond to a pump wavelength λp of 750 nm.

The setup in Fig. 4.7 is used to measure marginal spectra of signal and idler. Pump pulses at central wavelength λp are coupled into waveguide #6 , and the output light then travels through some polarization optics before being coupled into a single mode fiber. The half-wave plate HWP is rotated to select one of the type-II PDC output polarizations to be transmitted through the PBS, and then is spectrally filtered by the fiber-coupled Horiba MicroHR. The MicroHR is a motorized monochromator, and features the “510 16 X36NJ” line grating for the spectral range from 1000 nm lines to 2500 nm with 600 mm and a 1500 nm blazing. The filtered output is detected by an id201 APD. For λp between 750 nm to 790 nm, we measured in steps of 5 nm the marginal spectra as presented in Fig. 4.8. Each peak in the signal and idler plots respectively, corresponds to one pump wavelength, increasing from left to right. Plotting the peaks’ central wavelengths against the pump central wavelength λp (Fig. 4.9 left), we are able to determine the waveguide’s degeneracy point: Signal and idler wavelengths are equal for λp = 786 nm. The shape of Φ(ωo, ωi) is obtained by plotting signal against idler central frequencies in Fig. 4.9 (right). We note the positive slope of the measured phasematching contour that in good approximation can be fitted against a linear ◦ function with a positive slope, corresponding to a phasematching angle of Θpm = 59.2 and conclude that successful GVM is indeed possible.

4.4 Conclusion

We have implemented a type-II SPDC source in a KTP waveguide, and proven the production of correlated, cross-polarized photon pairs by measuring non-zero Klyshko efﬁciencies for small pump powers. We then have mapped the phasematching contour ∆k = 0 from the central wavelengths of the marginal spectra for signal and idler measured with a monochromator and 54 4 A PP-KTP waveguide as parametric downconversion source

1.28 1580 1.26 1560

1.24 1540

1.22 1520

1500 [PHz] Idler frequency 1.2 Signal/Idler wavelength [nm] Signal/Idler 1480 1.18 750 760 770 780 790 1.18 1.2 1.22 1.24 1.26 Pump wavelength [nm] Signal frequency [PHz]

Figure 4.9: Left: Central wavelengths for signal (red) and idler (blue) beams against pump central wavelength on the x-axis. Right: Phasematching contour from plotting corresponding signal and idler central frequencies as data points. single photon detectors. As expected, both photons are in the telecom wavelength regime for a pump beam at 775 nm. For 786 nm, the photons are wavelength degenerate at 1572 nm. The presented method of obtaining the marginal spectra of PDC output photons does have major drawbacks though: Due to very high losses within the MicroHR monochromator, the photon fluxes observed are quite small, making measurement either slow or inaccurate. This problem will be greatly exacerbated when we try to adapt it to measure the joint spectrum of signal and idler: The setup’s transmissivity, already much smaller than 1, will be squared when counting coincident photons filtered by a monochromator in each PDC output beam. We therefore we developed the more efficient, cheaper alternative free of moving parts alternative: The fiber spectrometer. 5 Fiber spectrometer

The great majority of spectrometers follows the same basic principle in combining a dispersive optical element with an ideally wavelength-agnostic optical detector[108, 89]. The dispersive element typically introduces a wavelength-spatial mode correlation and uses a simple movable slit to select one spatial mode to be measured. This is usually implemented with prisms or gratings. Instead of a slit and one detector, one can use a detector array, and reach higher detection efficiencies by avoiding the loss of all photons filtered out by the slit. Grating spectrometers with CCD arrays as detectors make use of this method and provide a cheap, sensitive device without moving parts. When spectrally characterizing a single photon source, detection efficiency is of paramount importance; due to typically low photon fluxes, measurements take a relatively long time, and losses would add to that, so an array of detectors would be preferable to a movable-slit-and- detector setup. But single-photon sensitive detectors, such as avalanche photo diodes (APDs) or recent approaches like cryogenic detectors such as nanowire single photon detectors[62], are much more expensive than standard photo diodes or CCD sensors, and an array of them might not be affordable to the physicist on a budget. In order to solve this dilemma, we introduce the fiber spectrometer[8]: We transfer the spectrometer principle from the spatial to the time domain: With the help of chromatic dispersion we create a spectral-temporal correlation in our photons[137, 24, 10], and by measuring their arrival time at the detector, we are able to reconstruct their wavelength.

5.1 Functional principle

The fiber spectrometer setup is straightforward: A beam of single photons travels through a long stretch of dispersive optical fiber, picks up a wavelength-dependent group delay through the fiber’s chromatic dispersion and its arrival time at a single photon detector is measured. A graph depicting the wavelength-time correlation of such a chirped photon – as might be measured with a SPIDER apparatus[68] – can be seen in Fig. 5.1. On the left, a Fourier-limited pulse exhibits no time-wavelength correlation, while on the right, the pulse after has been transmitted through the 56 5 Fiber spectrometer

ﬁber and clearly shows arrival time depending on wavelength. A calibration of the spectrometer gives the exact magnitude of this dependence λ(τ), but in general will need to be modeled with a higher order polynomial, if a larger wavelength range is considered. Any intrinsic temporal bandwidth of the single photons is assumed to be much smaller than the effects of the ﬁber dispersion, and therefore will be neglected.

Figure 5.1: Time-Wavelength correlation graphs of a broadband photon signal before (left) and after (right) ﬁber transmission

We aim to measure the spectra of photons in the telecom wavelength regime. As dispersive medium we use dispersion compensating fiber (DCF) modules originally designed to counter chromatic dispersion of standard telecommunication fibers. Their dispersion is ten times the negative value of SMF28 fiber at 1550 nm, so that for 10 km of this type, 1 km of the DCF is needed to counter the accumulated chirp of the telecom signals. Our detectors are again id201 InGaAs APDs by idQuantique. We have made an implicit assumptions about our single photon source here that is necessary for this kind of spectrometer to work, namely the existence of a clock signal to measure the photon arrival time against. A PDC source pumped by light pulses from a mode-locked laser provides such a time reference naturally, but also a CW-pumped PDC source is capable of this: By heralding the photon whose spectrum is to be measured with its partner that does not travel through a dispersive medium, we also obtain a clock.

Figure 5.2: Fiber spectrometer working principle: The incoming singe photon pulse is elongated inside the DCF ﬁber coil, and its arrival time against a clock reference signal is measured.

Fig. 5.2 illustrates the basic working principle of the fiber spectrometer. From the measured detection events we obtain a single photon arrival time statistic, that with the help of a calibration curve directly translates into a single photon spectral intensity. Timing errors inherent in the setup inherent are restricting the spectrometer’s resolution: Jitter of the clock and the detection signal, as well as the temporal uncertainty of the initial single photon pulses themselves. With initial 5.2 Experimental setup for photon pair spectrum measurement 57 pulse lengths in the order of 1 ps, several nanometers of spectral width, and a fiber dispersion ps of the DCF coils used of −862 nm at 1550 nm, we can neglect this last source of errors for all applications within this work. Great care has to be taken to keep constant (or at the very least to account for changes of) the optical and electrical runtime of all signals involved, as they have a direct impact on the crucial timing difference between clock and photon arrival. While it is relatively trivial to keep all optical elements of the setup fixed to avoid longer or shorter photon paths, there are more subtle ways to introduce timing errors: We found that changing the wavelength of a mode locked laser source offset its output pulses with respect to its electrical trigger signals used as clock. Also, any kind of equipment to delay or multiplex electrical signals is prone to react with altered intrinsic time delays to changes of their configuration, often in unexpected and undocumented ways.

5.2 Experimental setup for photon pair spectrum measurement

Nonlinear optical processes such as PDC can generate photon pairs entangled in, depending on perspective, time, energy or frequency. This frequency entanglement is visible as a correlation in the photon pair’s joint spectral intensity (JSI), but will not feature in either single photon spectrum of the pair’s constituent photons. To correctly measure the JSI, one has to use a dedicated spectrometers for each photon, and additionally post-select only coincident detection events between both spectrometers. Such a setup combines the PDC source from 4.5 with two ﬁber spectrometers and is shown in Fig. 5.3: A type II PDC KTP waveguide is pumped by a Ti:Saph laser at 768 nm to generate a photon pair in the telecom regime. The signal and idler photons are split up by polarization and fed into individual DCF coils, and their arrival times detected separately by the time-to-digital (TDC) module TDC-GPX by acam. But only if two photons arrive in the same clock cycle, the event is considered in the joint arrival time statistic. An early version of this experiment used one DCF coil and separated photons after that. For this to work, the polarization rotation by the DCF has to be compensated before a PDC can be employed to split up signal and idler. In general though, they have different wavelengths, and thus their polarization rotation is different and cannot be undone by the same QWP-HWP combination in all cases, only for special signal/idler wavelength combinations. It would be possible to separate by wavelength instead of polarization if sufﬁciently different signal and idler wavelengths with an appropriate dichroic mirror. Here, we sidestep this problem by simply using one DCF coil per output arm.

Figure 5.3: Fiber spectrometer for measuring the joint spectral intensity of PDC photon pairs

A measured temporal distribution of photon pair detection events from the waveguide source can be seen in Fig. 5.4. Each pixel represents a measurement interval of 30 s during which coincidence events from the APD pair are counted. The APD detection windows of 2.5 ns length are delayed by the x- or the y-value of the pixel respectively. Pump wavelength is 768 nm and 58 5 Fiber spectrometer

725 [ns] i 723

721

Idler arm delay τ Idler arm 719

579 581 583 585

Signal arm delay τs [ns]

Figure 5.4: Statistical distribution of coincident signal and idler photon arrival times pump FWHM is 0.7 nm. Signal and idler central wavelengths are as of yet undetermined, and we cannot extract any information from the central arrival times 581.5 ns and 722 ns without making assumptions about signal and idler beam paths.

5.3 Calibration

Calibration of the two-photon ﬁber spectrometer was done against the commercial Horiba Mi- croHR motorized grating monochromator with the 510 16 X36NJ etched grating for near infrared. A broadband pulse from a OPA system is ﬁltered down to the spectral resolution of the monochromator and then sent through our DCF coils. Against the OPA trigger signal as clock, an arrival time of the pulses at the APD detectors for a range of wavelengths is recorded.

Figure 5.5: Fiber spectrometer calibration setup against MicroHR grating spectrometer

The experimental setup is depicted in Fig. 5.5. The OPA system generates broadband light pulses in the telecom wavelength regime with a repetition rate of 120 kHz. This is filtered to a spectral width of 0.6 nm around wavelength λ by the MicroHR monochromator, coupled into two ps subsequent Lucent WBDK-25 DCF coils with a specified dispersion of −431 nm each. The fiber is directly connected to a idQuantique id201 APD. For each OPA light pulse, a trigger signal is 5.3 Calibration 59

sent to the Stanford Research DG645 pulse generator, where it is delayed for a time τ, and then forwarded as a gating signal to the APD. If the electrical signal delay τ and the optical delay in the ﬁber for the pulse at wavelength λ match, the detector registers photon events, otherwise only dark counts. By scanning through the delay time τ, we measure a peak in detection count rate that is displaced when we tune the wavelength λ also. In Fig. 5.6 we show the results of this calibration measurement for λ = 1400 nm to λ = 1600 nm in 5 nm steps. Each peak corresponds to one wavelength, with wavelengths increasing from right to left.

150

100

50 APD count rate [kHz] 0 3500 3550 3600 3650 Time delay for APD trigger [ns]

Figure 5.6: APD response against OPA trigger delay for Horiba MicroHR monochromator settings from 1400 nm (leftmost peak) to 1600 nm (rightmost peak) in 5 nm steps

Group velocity dispersion 1650 1600 1550 1500 1450 1400 Wavelength [nm] Wavelength 1350 3500 3550 3600 3650 Photon detection time [ns]

Figure 5.7: Each data point (red triangle) corresponds to one measurement from Fig. 5.6; the ﬁt polynomial (blue) serves as calibration curve of the spectrometer

In Fig. 5.7, we map the center of each peak to its associated wavelength. With a least-square method we ﬁt a third-order polynomial to our data. The resulting calibration curve is

λ(τ) 10−7τ 3 10−3τ 2 τ = −5.3056 × + 3.2986 × − 4.5908 × + 0.1732 (5.1) 1 nm 1 ns3 1 ns2 1 ns 60 5 Fiber spectrometer

It is important to keep in mind though that this curve is only applicable if the relative time delay between optical pulse and electrical gating signal in the setup is not changed. Such a temporal offset can be caused by changes to the optical path, by electrical cables of different length, by a different laser conﬁguration or different device settings of the DG645 and the APD. In that case, we have to adjust the calibration curve accordingly by introducing a temporal offset. For our photon pair source we need two time delays τ¯µ with µ ∈ {s, i} for signal and idler photon detection arm, respectively. The calibration function now reads:

λµ(τ) := λ(τ +τ ¯µ) (5.2)

In our further experimental program, we use the same waveguide #6 on our KTP chip at the constant pump wavelength λp = 768 nm at room temperature. We use the phasematching contour obtained independently from spectral measurements with a standard grating spectrometer in Fig. 4.9 (left). The signal and idler temporal maxima τs,max and τi,max of any temporal distribution from the same pump wavelength measured with the ﬁber spectrometer is identiﬁed with λs = 1544 nm for signal and to λi = 1528 nm for idler, according to the phasematching contour. The calibration −1 −1 time delays for each arm are τ¯s = λ (1545 nm) − τs,max and τ¯s = λ (1528 nm) − τs,max. Applying this calibration to the measurement results pictured in Fig. 5.4, we obtain the bi-photon

(a) Pump FWHM 0.70nm 1532 1529 Idler wavelength [nm] 1526 1541 1544 1547 Signal wavelength [nm]

Figure 5.8: Joint spectral intensity of photon pairs generated by a pulsed pump at 768 nm spectrum shown in Fig. 5.8. Each data point for arrival times (τs, τi) has been converted to a data point at (λs(τs) , λi(τi)).

5.4 Spectral resolution

Arguably the most important ﬁgure of merit for any spectrometer is its spectral resolution. The ﬁber spectrometer’s resolution is determined by the steepness of its calibration curve and the timing error of the photon detection events:

∂λ(τ) ∆λ = ∆τ (5.3) ∂τ τ0 5.4 Spectral resolution 61

Source Symbol Value Description PulseSwitch ∆τPulseSwitch 50 ps Jitter of the pump pulse picker DG645 ∆τDG645 80 ps Jitter of the delay generator id201 ∆τid201 500 ps Jitter of the APD detection signal id201 ∆τgate 2.5 ns APD gate width

Figure 5.9: Fiber spectrometer timing error sources

All active elements of the setup can be sources of time jitter, but most prominently the electrical devices are prone to this. The sources of electrical signal jitter in the fiber spectrometer setup with the vendor specified values are compiled in Fig. 5.9. For our MIRA laser system itself we neglected any timing error, because the timing jitter of a mode-locked femto-second laser cavity must be much smaller than the pulse length, or otherwise the laser would fall out of lock. Each of the electrical components in the setup contributes with their vendor-specified signal jitter. The time correlation between a photon detection event and the id201 APD’s electrical output pulse is not documented by the vendor. So in the worst case, we cannot time-resolve photon arrivals within the id201 APD gate width, and have to consider it a timing error as well. Assuming Gaussian error propagation, the squared timing error of a photon detection event is the squared sum of all single timing errors. Using the vendor specified time jitter values and the APD gate width, we make a conservative estimation of the overall time error ∆τ.

2 2 2 2 2 2 ∆τ = ∆τDG645 + ∆τid201 + ∆τgate + ∆τpulse picker = (2.55 ns) (5.4)

As we can see, the id201 gate width ∆τgate is the dominant term here. From the ﬁber calibration curve λ(τ) we can now estimate the wavelength error and thus the resolution of the ﬁber spectrometer at 1550 nm ⇒ ∆τ = 2.55 ns ⇒ ∆λ = 3.0 nm.

600 Signal Idler 500 Signal half-maximum Idler half-maximum

400

300

200

100 Single event count rate [kHz] Single event count rate

0 41 41.5 42 42.5 Arrival time [ns]

Figure 5.10: “Marginal spectra” of the two-photon state measured without DCF coils to determine the temporal jitter of the experimental setup.

This spectral resolution width seems rather high, but then again we used a worst-case assumption for the APD gate width. To get a more realistic idea about the timing uncertainty of 62 5 Fiber spectrometer our setup, we conduct a measurement of the “marginal distributions” of a photon pair with the setup depicted in Fig. 5.3, but with the DCF coils replaced with 1 m long standard telecom fiber patch cords. This way, the optical pulse length of the detected photons is not elongated through chromatic dispersion, but rather of the order of 1 ps. Since all other elements of the setup are in place, the temporal spread of the measured distributions is only due to the overall timing jitter ∆τ. The resulting curves for signal and idler arm are plotted in Fig. 5.10. The idler distribution is wider than the signal distribution, reflecting differences in the electrical and timing properties of the id201 APDs. As an experimental measure of the timing jitter of the fiber spectrometer setup we take the FWHM ∆τ = 1.56 ns of the idler distribution. The resulting spectral resolution at 1550 nm is the significantly lower ∆λ = 1.84 nm. The reason for this discrepancy must come from the highest contribution to the jitter, Dtgate. As the compounded jitter ∆τ is even lower than the specified id201 gate width of ∆τgate = 2.5 ns we used, either the effective gate width is shorter than 1.56 ns, or there is some correlation between the exact time of a photon detection event and the electrical detection signal from the APD.

5.5 The joint spectral intensity of photon pairs from the KTP source Pump intensity [a. u.] Pump

764 766 768 770 772 Pump wavelength [nm]

Figure 5.11: Pump spectra with a FWHM of 0.70 nm (red), 1.95 nm (green) and 4.0 nm (blue), respectively

Having developed and tested the fiber spectrometer, we now use it to demonstrate control over the spectral correlations over photon pairs generated by our source. As experimental setup we continue to use the setup depicted in Fig. 5.3. With the 4f grating setup, we can manipulate the width of the ultrafast pump pulses, and thus the correlation of the photon pair joint spectra[59, 134]. We verified this by measuring the joint spectral intensity of generated photon pairs at different spectral pump widths, as depicted in Fig.5.11. APD efficiency was set to 25%, the AOM released pump pulses at 1 MHz. Fig. 5.12 shows the results for spectral pump FWHM of 0.70 nm and 26 µW, 1.95 nm and 81 muW and lastly 4.0 nm and 51 µW: We observe negative spectral correlation, an uncorrelated spectrum, and a positive spectral correlation between signal and idler photons, respectively. This shows that we can control spectral correlations in our source by spectrally filtering the pump beam, and that we can expect minimal spectral correlation of photon pairs around 1.95 nm pump FWHM. 5.6 Measurements beyond the perturbative limit hˆni 1 63

(a) Pump FWHM 0.70nm (b) Pump FWHM 1.95nm (c) Pump FWHM 4.00nm 1531 1531 1531 1528 1528 1528 Idler wavelength [nm] Idler wavelength 1525 1525 1525 1541 1544 1547 1541 1544 1547 1541 1544 1547 Signal wavelength [nm] Signal wavelength [nm] Signal wavelength [nm]

Figure 5.12: Joint spectra from setup 6.2(b) with pump width above (c), equal to (b) and below (a) separability width at 1.95 nm FWHM.

5.6 Measurements beyond the perturbative limit hˆni 1

If with a non-negligible probability more than one photon pair is produced, then we cannot assume with certainty any more that the coincident detection events are caused by photons in corresponding Schmidt modes. Consequently the spectral correlation apparent in the JSI at low mean photon numbers is weakened, and we arrive at a “rounder”, less correlated spectral shape. It cannot be viewed as a true two-photon spectrum any more, as multiple pairs are involved. This effect will inevitably appear at higher mean photon numbers, and is independent of detector saturation. If the multi-pair state suffers optical losses, there is again no guarantee that the surviving photons are in corresponding Schmidt modes, so attenuation of the SPDC output beams in front of the detectors cannot prevent it.

To illustrate the effect, we pump waveguide #6 on our KTP chip with pulses at 768 nm with 0.35 nm spectral width. We use the ﬁber spectrometer to measure joint spectra for a pump power of 10 µW (Fig. 5.13 left) and for a signiﬁcantly higher power of 160 µW. The result is a matrix of coincidence event counts, and its indices refer to discrete signal and idler photon arrival times at the detectors. With a calibration function C(t), we can translate these arrival time values to photon wavelengths or frequencies. We will however refrain from doing so for now, since the structure of the Schmidt decomposition itself does not change under this kind of transformation, as it is applied to signal and idler separately.

We now analyze the effect quantitatively by measuring photon coincidence detection events with a pair of lossy binary detectors with quantum efficiencies η1 and η2. The use of the monochromatic mode operators â(ω1) and b(ˆ ω2) models sharp spectral filtering at the frequencies ω1, ω2, as if we introduced spectral filters with a δ-function shaped transmission spectrum in front of both 64 5 Fiber spectrometer

449 449 [ns] [ns] i i 446 446

443 443 Idler arm delay τ delay Idler arm delay τ Idler arm

440 440 325 328 331 334 325 328 331 334

Signal arm delay τs [ns] Signal arm delay τs [ns]

Figure 5.13: Joint spectra with 10 µW pump power (left) and 160 µW pump power (right) detectors. We apply the measurement operators (c. f. section 2.7)

∞ X 1 µˆ = (1 − (1 − η )n) ˆa†(ω )n |0ih0| ˆa(ω )n 1 n! 1 1 1 n=0 ∞ (5.5) X 1 µˆ = (1 − (1 − η )n) bˆ†(ω )n |0ih0| b(ˆ ω )n 2 n! 2 2 2 n=0 to signal and idler arm of a general multi-mode squeezed vacuum state from type-II SPDC (cf. section 2.6.8)

ıarg(ζc ) with λj = tanh(|ζcj|) e j , the coupling constant ζ and the Schmidt coefﬁcients cj. We rewrite the state |Ψi as a sum over all integer non-negative series {nj}. Each series corresponds to one possible mode occupation conﬁguration. n † † j ∞ λjAˆ Bˆ X Y q 2 j j |Ψi = 1 − |λj| |0i . (5.7) nj! {nj } j=0

The probability differential to detect a coincidence detection event at the sharp frequencies (ω1, ω2) is p(ω1, ω2) = Tr[ˆµ1 ⊗ µˆ2 |ΨihΨ|] (5.8) and the probability to detect a coincidence event in a ﬁnite frequency area is the integral of the differential probability over this area. 5.6 Measurements beyond the perturbative limit hˆni 1 65

The squeezed state is expressed in broadband modes Aˆ j and {Bˆ j}, while the measurement operators are built from monochromatic mode operators ˆa(ω1) and b(ˆ ω2). We can express the latter ones in terms of broadband modes using the completeness theorem for orthogonal functions and write X ˆa(ω) = ξj(ω) Aˆ j j X (5.9) b(ˆ ω) = ψj(ω) Bˆ j j where ξj and ψj are the spectral mode functions corresponding to Aˆ j and Bˆ j, respectively. We then re-write the powers of the monochromatic mode operators as

n   ∞ k n X ˆ X Y ˆ j ˆa(ω) =  ξj(ω) Aj = ξj(ω) Aj (5.10) P j kj =n j=0

PP {k } n where kj =n is the sum over all integer, non-negative series j that total to . The form of both state vector and measurement operators was specifically chosen such that the problem of normal-ordering would not arise in further calculations. Substituting the broadband definitions into Eq. 5.8, we find

p(ω1, ω2) = hΨ| µˆ1 ⊗ µˆ2 |Ψi ∞ ∞  ∞  X X 1 Y = (1 − (1 − η )m) (1 − (1 − η )n) 1 − |λ |2 m!n! 1 2  j  m=0 n=0 j=0 (5.11) 2 ∞ ∗mj ∗ kj ∗ lj λ ξ (ω ) ψ (ω ) mj kj lj X X X Y j 1 2 ˆ ˆ ˆ † ˆ † × h0| AjBj Aj Bj |0i P P mj! {mj } kj =m lj =m j=0 which can be simpliﬁed considerably. m k l ˆ ˆ j ˆ † j ˆ † j The vacuum expectation value h0| AjBj Aj Bj |0i evaluates almost trivially to Q∞ 2 j=0 (mj!) δmj ,kj δmj ,lj so that the series {mj}, {kj} and {lj} have to be identical. Since we P P required j kj = m and j lj = n, this infers m = n and we can write for the product of all Q∞ 2 vacuum expectation values δm,n j=0 (mj!) δmj ,kj δmj ,lj . Three of the ﬁve sums in Eq. 5.11 thus collapse and we are left with the rather more compact expression

∞ X 1 n n p(ω1, ω2) = 2 (1 − (1 − η1) ) (1 − (1 − η2) ) n=0 (n!) 2 (5.12) ∞ X Y nj × nj!(λjξj(ω1) ψj(ω2)) P nj =n j=0

ıarg(ζc ) For low coupling constants ζ 1, we have λj = tanh(|ζcj|) e j ≈ ζcj. Also, we can approximate λj λjλk ≈ 0 and neglect all terms with photon number n > 1. The infinitesimal 66 5 Fiber spectrometer coincidence event detection probability thus simplifies to the expression for a single photon pair we already encountered in Eq. 2.92 2 1 ∞ X 1 n n X p(ω1, ω2) = (1 − (1 − η1) ) (1 − (1 − η2) ) ζ cjξj(ω1) ψj(ω2) (n!)2 (5.13) n=0 j=0 2 2 =η1η2 |ζ| |f(ω1, ω2)| where f(ω1, ω2) is the normalized joint spectral amplitude function of the SPDC process. Here, the coincidence click probability is proportional to the photon pair joint spectral amplitude. But in the general case it is not so easy any more to find a physical property of the squeezing source that corresponds with p(ω1, ω2) from Eq. 5.12, apart from the obvious one of the coincidence click probability. The spectral correlations of a high photon number multi-mode SPDC state are still governed by the amplitude function f(ω1, ω2), but it does not directly map to the joint spectral measurement any more. This is in stark contrast to the situation for classical, coherent pulse spectra. Here, any measured spectrum for any mean photon number is proportional to the 2 modulus squared of the amplitude function |fclassical(ω)| .

r=0.1, η=1.0 r=0.6, η=1.0 r=1.6, η=1.0

r=0.1, η=0.01 r=0.6, η=0.01 r=1.6, η=0.01

Figure 5.14: Calculated coincidence click spectra from Eq. 5.12, with a mode cutoff after j = 9 mode and a multi photon pair cutoff after n = 12.

Fig. 5.14 qualitatively illustrates the impact on the joint spectral measurement of a multi-mode SPDC source states with squeezing values r = 0.1, r = 0.6 and r = 1.6, corresponding to mean photon numbers hˆni of 0.1, 0.4 and 5.6 respectively (roughly, since we have a multi-mode state). The case r = 0.4 is comparable to the measurement for 10 µW in Fig. 5.13 (left), r = 1.6 to the higher powered measurement with 1.6 µW Fig. 5.13 (right). The joint spectra are plotted for perfect detectors in the ﬁrst row, and for detectors with a quantum efﬁciency η = 0.01 in the 5.7 Conclusion 67

second row. In the first column, there is virtually no difference between the plots. Since there is in the vast majority of cases no more than one photon pair present, the coincidence events are triggered by photons in corresponding Schmidt modes. The second column for r = 0.6 is already affected by multi-pair contributions: The central peak for low detection efficiencies is slightly lower and rounder. The trend continues in the third column for r = 1.6. While the upper spectral distribution is slightly narrower than the other plots in the first row, the spectral correlation lossy-detector distribution is noticeably weaker than both in the upper plot and the plots for smaller squeezing. This simulation of the fiber spectrometer coincidence measurement, together with the estimated spectrometer resolution of 1.86 nm at 1550 nm, qualitatively explains the measurement results from Fig. 5.13. It also shows that spectrally resolved coincidence measurements with binary 2 detectors give accurate measurements of the SPDC joint spectral intensity |f(ωs, ωi)| only if the production rate of photon pairs is low. The presence of multiple photon pairs in the same measurement cycle will distort the resulting spectrum noticeably, even if optical losses prevent detector saturation. SPDC sources with an arbitrarily high photon pair output do not and cannot have a well defined two photon spectrum, because of the ambiguity that multiple pairs create. The joint spectral amplitude function f(ωs, ωi) still governs the spectral structure of the two- mode squeezed vacuum, but loses its immediate physical interpretation as an actual, measurable bi-photon spectrum.

5.7 Conclusion

We have described the setup and calibration of an optical fiber based spectrometer to measure the spectrum of single photon states. The spectral spread of a measured photon is translated into a temporal spread by a long dispersive fiber. By measuring the arrival time of the photon at the end of the fiber with a time-resolving binary detector, we gather temporal statistics from which we gain the spectral distribution of the photon. The fiber spectrometer’s spectral resolution is 1.84 nm and is mainly caused by the detection gate length of our binary detectors that limits time resolution. To measure the joint spectrum of a photon pair, we use a fiber spectrometer for each photon and take statistics only from coincident measurement events. Finally we show that SPDC states from the same source but with a higher mean photon number lead to results with seemingly less spectral correlation, and argue that the results, while still indicating spectral correlations in the photon pairs, do not faithfully give the joint spectral intensity |f(ωs, ωi)| any more.

6 Two-mode squeezed vacuum source

For a genuine two-mode squeezing source, exhibiting an uncorrelated joint spectral intensity for photon pairs generated at low pump power is necessary but not sufﬁcient. Since the joint intensity 2 is proportional to the modulus square of the complex joint amplitude |f(ω1, ω2)| of the photon pair, all phase information is lost in an intensity measurement, so that any phase entanglement between signal and idler is undetected. In order to demonstrate full photon pair separability and the resulting two-mode character of our source, we need to measure an additional quantity sensitive to its mode number. In this chapter, we use the second order correlation function g(2) to discriminate between beams with thermal (g(2) = 2) and Poissonian photon statistics (g(2) = 1) from a two-mode squeezing source[129] and ensure that the the power-dependency of the source’s photon pair output coincides with that of a two-mode squeezer.

6.1 Mode-number and photon statistics of broadband squeezed vacuum states

It has been shown early on in the experimental exploration of squeezing that PDC produces squeezed states of light[144]. In photon number representation, a two-mode squeezed vacuum state has the form

ˆ q ıHa,b 2 X n |Ψi = Sâ,b |0i = e |0i = 1 − |λ| λ |n, ni (6.1) n ˆ where â and b are two orthogonal modes, Sâ,b is the two-mode squeezing operator, and Hˆ a,b is its effective Hamiltonian. It is a coherent superposition of strictly photon number correlated Fock state pairs, and exhibits thermal photon statistics in both modes â and bˆ, i. e. the probability to generate n photons in each mode scales like (const)n:

2 2n pn = 1 − |λ| |λ| (6.2)

The photon number correlation between both modes allows for the heralding of pure Fock states using a photon number resolving detector, in the simplest case for pure heralded single photons 70 6 Two-mode squeezed vacuum source

†ˆ† with binary detectors. However, the underlying bilinear effective Hamiltonian Hˆ a,b = ζâ b + h. c. describes only a special case of PDC. In general though, the effective PDC Hamiltonian has a richer spatio-spectral structure; additional to the broadband spectrum there can exist also a contiuous spectrum of ~k-modes, or a discrete spectrum of waveguide modes for pump, signal and idler respectively. The effective Hamiltonian must reflect this by summation over all possible combinations of mode triples, potentially resulting in hyper-entangled states[34]. This most general case cannot be decomposed into a set of independent two-mode squeezers any more, and also goes against our goal to engineer a two-mode squeezer, or separable photon pairs at low powers. Assuming a single mode waveguide, the effective Hamiltonian reads ZZ ˆ † ˆ† HPDC = ζ dω1 dω2 f(ω1, ω2) â (ω1)b (ω2) + h. c.. (6.3)

It generates a generalized version of the two-mode squeezed vacuum in Eq. 6.1; its output beams are spectrally correlated. The coupling constant ζ determines the strength of this interaction, while spectral correlations between photons of the pairs produced are governed by the normalized joint spectral amplitude

f(ω1, ω2) ∝ α(ω1 + ω2) Φ(ω1, ω2) (6.4) where α(ω) is the spectral amplitude of the pump beam and Φ(ω1, ω2) is the phasematching function that depends on the nonlinear medium’s dispersion properties. For low pump powers, PDC is in good approximation a probabilistic source of photon pairs. By P applying a Schmidt decomposition to the pairs’ joint amplitude[82] f(ω1, ω2) = j cjξj(ω1) ψj(ω2), we obtain two orthonormal basis sets ξj(ω1) and ψj(ω2) and a set of weighting coefﬁcients cj P 2 with j |cj| = 1. Now the PDC Hamiltonian can be expressed in terms of broadband modes

ˆ X ˆ X ˆ † ˆ † ˆ ˆ HPDC = Hj = ζ cj AjBj + AjBj . (6.5) j j

Each broadband mode operator Aˆ j, Bˆ j is defined as superposition of monochromatic annihilation ˆ ˆ † R † operators â(ω) , b (ω) weighted with a function from the Schmidt basis: Aj := dωξj(ω) â (ω) ˆ † R ˆ† ˆ and Bj := dωψj(ω) b (ω). Since the effective Hamiltonians Hj do not interact with each h i other (i. e. Hˆ j, Hˆ l = 0), we see that the PDC time evolution operator is in fact an ensemble of ˆ ˆ ˆ ıHPDC ˆ ˆ independent two-mode squeezing operators S = UPDC = e = SA0,B0 ⊗ SA1,B1 ⊗ ... where the coefficients cj determine the relative strength of all squeezers as well as spectral correlation between signal and idler beams[82]. This only an approximation for a small coupling constant |ζ| 1 and low probability of multiple photon pair creation, since it implicitly assumes that multiple photon pairs per pump pulse are created independently. This is not true, since the bosonic character of photons leads to higher photon pair creation probabilities into already populated modes. This interaction leads to a deformation of the broadband mode system, the multimode squeezer system however can still be decomposed into orthogonal two-mode squeezers[20, 141, 87], but with increasing coupling strength their spectral modes will increasingly deviate from the Schmidt modes of the joint spectral amplitude f(ω1, ω2). We will ignore this effect for now, and reserve a deeper investigation for section 7.7. The strength of correlation between the output 1 beams of such a source is characterized by its effective mode number K = P 4 . For c0 = 1 j |cj | 6.2 The second order correlation function g(2) 71

and all other cj = 0, K assumes its minimum value of 1, and the PDC process can be described as a two-mode squeezer according to Eq. 6.1. A multimode SPDC source will not exhibit thermal photon statistics, although each of its constituent two-mode squeezers – each one corresponding to one Schmidt mode pair – will. The overall statistics is a convolution of their thermal statistics and for a high effective mode number K it converges towards Poissonian photon statistics, where the n-photon probability scales like (const)n n! .

6.2 The second order correlation function g(2)

The second order or Glauber correlation function is defined as the normal-ordered correlation of the field intensity (proportional to â†â ) with itself at a different time:

† † (2) hâ (t1) â (t2) â(t1) â(t2)i g (t1, t2) = † † (6.6) hâ (t1) â(t1)i hâ (t2) â(t2)i

The value of g(2)(0, 0) is characteristic for states of light with certain well-defined photon statistics. A coherent state with Poissonian statistics exhibits g(2)(0, 0) = 1, a state with thermal statistics (2) (2) 1 will result in g (0, 0) = 2. A Fock state |ni has g (0, 0) = 1 − n . The difference in second order correlation function values for different photon statistics is maximal for t1 = t2, so we will consider only g(2)(0, 0) here. For light states with low photon number, i. e. hˆni 1, g(2)(0, 0) is readily measured with a Hanbury-Brown-Twiss type experiment[26], with the setup depicted in Fig. 6.1: The input light state ρˆ comprises of an arbitrary state ρâ in mode â and a vacuum state |0ih0| in mode bˆ that are mixed on a balanced beam-splitter. The output modes ˆc and dˆ impinge on a pair of single photon detectors, and single and coincident detection events are recorded. In the low photon number limit, the mean photon number goes linear with the detector click probability: pi ≈ ηi hˆni, where ηi is the detector’s quantum efficiency. Accordingly, we can approximate the detectors’ measurement † operators µˆ1 and µˆ2 with the photon number operators in both beam-splitter output modes, ˆc ˆc and dˆ†dˆ, respectively. The coincidence measurement operator – again, for a low photon number † † approximation – is their product µˆc =µ ˆ1 ⊗ µˆ2 = η1η2ˆc ˆcdˆ dˆ. In each case, the probability pi is given by hµîi = Tr[ˆµiρˆ] with i ∈ {1, 2, c}.

Figure 6.1: Hanbury-Brown-Twiss interferometer to measure g(2)

To understand how this is a measurement of the second order correlation function g(2)(0, 0) , we consider the quotient pc of coincidences over single click events, and apply the beam-splitter p1p2 72 6 Two-mode squeezed vacuum source

transformations ˆc, dˆ → √1 â ± ıbˆ to express the measurement operators in terms of the input 2 modes: † † h â† − ıbˆ† â† + ıbˆ† â+ ıbˆ â − ıbˆ i pc η1η2 hµˆci hˆc dˆ ˆcdî = = = (6.7) p1p2 η1 hµˆ1i η2 hµˆ2i hˆc†ˆci hdˆ†dî h â† − ıbˆ† â+ ıbˆ i h â† + ıbˆ† â − ıbˆ i

Since the statistical average h.i is a linear operation, both numerator and denominator can be expanded into averages of normal ordered products of the mode operators ˆa and bˆ. Mode bˆ is in the vacuum state, and thus all expressions containing bˆ or bˆ† vanish, and we are left with

† † pc Tr â â ââˆρ (2) = 2 ≡ g (0, 0) . (6.8) p1p2 Tr[â†âˆρ]

6.3 g(2) for broadband input states

The derivation of g(2)(0, 0) in the last section implicitly assumes single photon detectors with perfect time resolution. This may be a fair assumption when working with very fast detectors and light states with a long duration, such as a CW laser beam. The situation for our ultrafast SPDC source is very much different however[129]: Our id201 APD modules are periodically triggered with a 2.5 ns gate width, and consequently their time resolution is also of that order of magnitude. We are pumping with ultrafast laser pulses with pulse lengths of about 1 ps, and the SPDC the duration of the output pulses is of the same order. We need to modify the mathematical model of the g(2) measurement procedure to accommodate this experimental situation[35]. The probabilities p1, p2 are in fact time-averaged mean values of the time resolved expressions over a detection window with length 2T 1 Z T pi = dt hµî(t)i (6.9) 2T −T for i ∈ {1, 2}. The coincidence event probability depends on two events and hence needs to be time-averaged twice. 1 Z T Z T pc = 2 dt1 dt2 hµˆc(t1, t2)i (6.10) 4T −T −T In terms of the ultrafast light pulse we set out to measure, the detection window has a duration of the order of 103 temporal variances. As the pulse amplitude decays like a Gaussian function, we can approximate it to be zero at the edges of the detection window. Therefore, it is a good approximation to assume an infinite detection window. For the ultrafast second order correlation function which we will simply refer to g(2) from now on, this leads to R T R T (2) pc −T dt1 −T dt2 hµˆc(t1, t2)i g = lim = lim T T (6.11) T →∞ p1, p2 T →∞ R R −T dt1 hµˆ1(t1)i −T dt2 hµˆ2(t2)i From this point we can apply the same set of transformations that led from Eq. 6.7 to Eq. 6.8 and find R R † † dt1 dt2 hâ (t1) â (t2) â(t1) â(t2)i g(2) = (6.12) R dt hâ†(t) â(t)i2 6.4 g(2) for the ultrafast multimode squeezer 73

Invoking the Fourier transformation â(t) = R dω â(ω) eıωt allows us to switch to a frequency- based view of the problem. R R † † dω1 dω2 hâ (ω1) â (ω2) â(ω1) â(ω2)i g(2) = (6.13) R dω hâ†(ω) â(ω)i2 P ˆ The subsequent basis transform â(ω) = j ξj(ω) Aj gives us the measurement in the arbitrarily chosen spectral basis {ξj(ω)}. P hAˆ †Aˆ † Aˆ Aˆ i (2) j,k j k j k g = 2 (6.14) P ˆ † ˆ j hAjAji

In this form, the g(2) function reveals an alternate view on its physical meaning. Rather than intensity correlations between different times, Eq. 6.14 suggests the measurement of intensity correlations between different broadband modes.

6.4 g(2) for the ultrafast multimode squeezer

ˆ † ˆ † ˆ ˆ ˆ† ˆ †ˆˆ† ˆ † ˆˆ† ˆ ˆˆ† ˆ ˆ hΨ| AjAkAjAk |Ψi = h0| S AjSS AkSS AjSS AkS |0i ˆ † ˆ ˆ † ˆ = h0| cosh(|ζcj|) Aj − sinh(|ζcj|) Bj cosh(|ζcj|) Ak − sinh(|ζcj|) Bk (6.17) ˆ ˆ † ˆ ˆ † × cosh(|ζcj|) Aj − sinh(|ζcj|) Bj cosh(|ζcj|) Ak − sinh(|ζcj|) Bk |0i which is a vacuum expectation value of normal ordered mode operators Aˆ j and anti-normal ordered mode operators Bˆ j. Upon expansion, all terms containing the normal-ordered Aˆ j vanish, leaving only

† † 2 2 † † hΨ| Aˆ Aˆ Aˆ jAˆ k |Ψi = sinh(|ζcj|) sinh(|ζck|) h0| Bˆ jBˆ kBˆ Bˆ |0i j k j k (6.18) 2 = sinh(|ζcj|) sinh(|ζck|) (1 + δj,k). Substituting equations 6.18 and 6.15 into 6.14 ﬁnally results in P 2 (2) j,k sinh(|ζcj|) sinh(|ζck|) (1 + δj,k) g = 2 (6.19) P 2 j sinh(|ζcj|) 74 6 Two-mode squeezed vacuum source

For a small squeezing parameter ζ, we can use the small angle approximation sinh(|ζcj|) ≈ |ζcj| P 2 and the normalization condition of the Schmidt coefﬁcients j |cj| = 1 to simplify Eq. 6.19 to

X 1 g(2) = 1 + |c |4 = 1 + (6.20) j K j where K is the effective mode number, or cooperativity parameter of the two mode squeezer. So apparently for two-mode squeezed vacuum states, the time-averaged g(2) correlation function for large detection times is connected in this very elegant manner to its mode number. We try to understand this mathematical connection in physical terms: As has been noted above, type II PDC can in general be seen as an ensemble of broadband two-mode squeezers, each of them emitting a two mode squeezed vacuum state with thermal photon statistics. All broadband modes Aˆ j or Bˆ j of this decomposition share one polarization mode, ˆa and bˆ respectively. A standard single photon detector cannot resolve them. It “sees” a convolution of the thermal photon statistics of all broadband modes, and in the limit of a large number of modes, this is a Poissonian distribution[7]. If, on the other hand, there is only one mode per polarization to begin with (which is only true for a two-mode squeezer), the detector receives a thermal distribution of photon numbers. Therefore, with the assumption that PDC emits a pure state, we can infer from g(2) = 2 measured in either output beam a two-mode squeezer source. In the presence of additional background events, we measure again a convolution of different photon statistics. This will always reduce the experimentally obtained value of g(2) further towards 1.

6.5 g(2) measurement

In Fig. 6.2 (c) we illustrate the g(2) measurement: The KTP waveguide is pumped with ultrafast laser pulses at 768 nm and a duration of the order of 1 ps. After the SPDC source, the idler beam is discarded, and the signal beam split by a 50/50 beamsplitter. The output modes are fed into id201 APDs with a 2.5 ns gate width. Single (p1, p2) and coincidence (pc) click probabilities for different spectral pump widths are recorded. In the previous sections we have shown that the time averaged second order correlation function g(2) can be reconstructed from these event probability measurements: p g(2) ≈ c . (6.21) p1p2

Since frequency correlations between signal and idler beam and thus squeezer mode number can be controlled by manipulation of the spectral width of the PDC pump beam, we see in Fig. 6.4 (left) measurement results that show a maximum g(2) value at 1 .95 nm pump FWHM, in accordance with the uncorrelated joint spectrum in Fig. 6.3 (left). When departing from the optimum pump width, we see g(2) drop towards 1 as expected. Due to residual background events from waveguide material ﬂuorescence and detector dark counts, we ﬁnd a maximum of g(2) = 1.80, and g(2) = 1.95 after background correction, corresponding to a cooperativity parameter of K = 1.05. This result demonstrates the next-to-perfect two-mode character of our PDC squeezing source, and the degree of control we exact over the mode number and photon statistics of the system. 6.6 Background event suppression and correction 75

Figure 6.2: Experimental setup: (a) Squeezed light source: A PP-KTP waveguide, spatially single mode at 1550 nm, is pumped with a mode locked Ti:Sa laser emitting ultrafast pulses with 8 nm FWHM. An accusto-optic modulator (AOM) reduces full repetition rate of 76 MHz to 1 MHz, a HWP+PBS combination controls pump beam power. We adjust pump spectral width with a 4f spectral ﬁlter setup (SF-4f) and monitor it with a grating spectrometer (GS). Pump light coupled through the waveguide is then separated from the generated signal and idler beams with a dichroic mirror (DM) and its power measured (PM). (b) g(2) measurement: Background light is removed from the signal beam with a 12 nm FWHM spectral ﬁlter (SF12), then split at a 50/50 BS and each output arm fed into APDs. Single, coincidence and trigger event rates are recorded.

6.6 Background event suppression and correction

In the JSI measurements, apart from the main SPDC peak, we detect a low intensity background flux in a wide spectral range. In Fig. 6.4 (right), we plot the marginal spectrum of the signal beam of our source with a pump wavelength of 768 nm with setup 6.2. The blue spectral function was measured with, the red one without a spectral band-pass filter specified at 1550 nm with 12 nm width (SF12) in front of the APD in-coupling. Both spectral measurements ran back-to-back, with identical experimental parameters apart from the filter. The filter transmission at the central wavelength of the peak (which is at 1544 nm rather than 1550 nm) is almost perfect at 98%. The background we see here can originate from several sources. Our waveguide source supports in propagation direction along x more PDC processes than the phase-matched type II process we are utilizing, namely the type I processes d32 and d33. Being not phase-matched in the telecom wavelength range, they produce a non-resonant, spectrally flat background in our detection window. The most significant source of background is PDC coupling to radiation modes[65]: Not all photons propagate in waveguide modes, sometimes either signal or idler are created in an unguided radiation mode. As there are no boundary conditions for those, they form a continuum rather than a discrete spectrum, so that phasematching for such a guided-unguided pair is much easer to obtain. The radiation modes then allow for a wide range of energy distribution in the 76 6 Two-mode squeezed vacuum source

exp Pump FWHM 1.95nm theo. 10000 5000 Intensity [counts/s]

0 1541 1544 1547 1531 Signal wavelength [nm]

exp 1528 theo. 10000 1525 1541 1544 1547 5000 Intensity [counts/s]

Signal wavelength [nm] 0 1525 1528 1531 Idler wavelength [nm]

Figure 6.3: Left: Joint spectrum from setup 6.2(b) with pump width at separability width 1.95 nm FWHM. Right: Marginal spectra. photon pair, such that the guided half of these pairs show up as uncorrelated, spectrally broadband background such as we observe. Other fluorescence processes besides PDC, caused by color centers or faults in the nonlinear crystal, are an unlikely source, since one would expect distinctly different arrival time for their photons, due to long decay times compared to the time frames of our ultrafast PDC process. We were unable, with the equipment at our disposal allowing for timing resolution of ca. 500 ps, to observe any significant arrival time difference between signal beam photons and background photons. Whatever the photon statistics of these processes are, in combination with the thermal statistics of either beam of a perfect two-mode squeezed vacuum state they result in a “less thermal” distribution, i. e. g(2) will drop below a value of 2 for a two-mode squeezer source with background noise. Detector dark counts are most easily dealt with: We apply the least possible bias voltage to our APDs and use the smallest possible gate width of 2.5 ns to arrive at a dark count probability of 5 × 10−5 per measurement cycle. Background photons degrade our experimental results most noticeably. However, most of them can be removed from the signal beam by using a suitable bandpass filter. After introducing spectral filter SF12 in setup 6.2(b), our detection event count dropped by a factor of five and we were able to see g(2) values significantly greater than unity with an event probability of 2 × 10−2. This leaves the background photons that are transmitted by the bandpass filter SF12. Assuming identically efficient single photon detectors (p = p1 = p2), we add the background photons as a statistically independent source of detector events with probability q at each detector. Coincidence detection events from these uncorrelated background photons happen with probability q2. We 2 2 substitute p → p + q − pq and pc → pc + q − pcq and find p + q2 − p q2 g(2) ≈ c c (6.22) (p + q − pq)2

In order to estimate q, we compare the unfiltered spectral distribution of the signal beam to the same distribution with and without a spectral filter SF12 applied (Fig. 6.4 right). We see that the filtered spectrum quickly falls to the level of detector dark counts outside the main peak, while the 6.7 Mean photon number 77

2 20 Filter No ﬁlter

1.9 15 (a) (b) (c)

(2) 1.8 10 g

1.7 5 Detection event rate [kHz]

1.6 0 1 2 3 4 1540 1544 1548 Pump spectrum FWHM [nm] Signal wavelength [nm]

Figure 6.4: Left: g(2) values from setup 6.2(c) for a variable pump FWHM. (red) experimental 1 values. (blue) Theory curve according to Eq. 6.22 with R = 20 . (violet) Background corrected theory curve according to Eq. 6.21. Right: Signal beam spectrum with background events in unﬁltered spectrum (blue) and dark count events only in background-ﬁltered spectrum (red).

unﬁltered beam maintains a background event level 450 counts over the dark count level of 50 counts over a wide range. We can expect those to be present in the main peak in both instances, q and the signal-to-noise ratio R = p of detection events to background events equals the ratio of the areas of the background level “under the main peak” and the main peak itself. From the graph 1 in Fig. 6.4 (right), we estimate R = 20 and thus ﬁnd with Eq. 6.22 the theory curve in Fig. 6.4 (left) in excellent agreement with our experimental data. We use the background corrected curve to predict g(2) = 1.95 in the absence of background at an optimal pump FWHM of 1.95 nm. From the relationship between photon statistics and effective mode number in Eq. 6.20 we determine the mode number K = 1 = 1.05 of our source. In a pure single photon heralding experiment g(2)−1 1 we can therefore expect a photon purity of P = K = 0.95 (c. f. section 3.1).

6.7 Mean photon number

We now investigate the power-dependent photon pair flux of the two-mode squeezer source. We still pump the SPDC process with the optimal pump width of 1.95 nm at 768 nm central wavelength and 1 MHz repetition rate and use the experimental setup from Fig. 6.2 (a). The corresponding separable joint spectrum is pictured in Fig. 6.3 (b). We feed the two-mode squeezed beams into the setup according to Fig. 6.5, where signal or idler are separated with a PBS. Either beam is cleaned of most of its background events with a spectral filter and coupled into the id201 APDs. Single event probabilities ps and pi and the coincidence click probability pc are recorded. For signal we use again the filter SF12. The idler arm is at 1528 nm, and therefore we use as filter SF12b a 12 nm wide band-pass filter at 1530 nm. Its transmission of the idler beam is lower than the SF12’s transmission of the signal beam, which causes a significantly lower count rate. Fig. 78 6 Two-mode squeezed vacuum source

Figure 6.5: Setup for measuring single and coincidence photon detection events.

6.6 (left) holds the measurement results. Pump pulse energies are tuned with the half-wave-plate (HWP) in the SPDC source (c. f. Fig. 6.2a) up to 75 pJ. A source of two-mode squeezed vacuum states with squeezing parameter r has in each arm a mean photon number of hˆni = sinh2r. (6.23) The squeezing parameter r grows linear with pump field amplitude. Therefore it is proportional to the square root of the pump intensity, and also of the pump pulse energy Ep. With a fittingly p defined proportionality constant B such that r = B Ep, we write

2 p hˆni = sinh B Ep. (6.24)

2 For small pump energies there is a regime of linear growth of the photon number: hˆni ≈ B Ep. The mean photon number of a multi-mode SPDC source is simply the sum of the mean photon numbers of all its modes. For a multi-mode state with Schmidt coefﬁcients cj, this means

X 2 p hˆnMMi = sinh cjB Ep. (6.25) j

A highly multi-mode state with K → ∞ has inﬁnitesimally small Schmidt coefﬁcients, so that we can apply the small angle approximation sinh(x) ≈ x:

2 X p X 2 2 2 hˆnK→∞i = cjB Ep = cj B Ep = B Ep. (6.26) j j

In the limit of a very large effective mode number K, the mean photon number is expected to increase linearly with pump pulse energy, since none of its squeezer modes will be pumped strongly enough to leave the linear regime. Already in Fig. 6.6 (left), we can see a departure in linear growth for both signal and idler, as can be expected for a nearly two-mode squeezed vacuum source. However, since we are using APDs, i. e. binary detectors, rather than intensity detectors, we will see the profile of the two-mode squeezer mean photon number only for small squeezing values. For higher squeezing, and higher click probabilities, detector saturation will cause deviations, so that we cannot directly fit sinh(r)2 against the signal and idler measurements, and instead an expression for the APD click probability is needed. 6.8 Photon collection efficiency 79

0.2 15 Signal Signal Idler Idler Coincidences 0.15 10

0.1

5 0.05 Klyshko eﬃciency [%] Klyshko Detection event probability [1] event probability Detection

0 0 0 25 50 75 0 5 10 15 20 25

Pump pulse energy Ep [pJ] Pump pulse energy Ep [pJ]

Figure 6.6: Left: Click event rate of the KTP source pumped with pump spectrum FWHM 1.95 nm for signal (red) and idler (blue) arm and coincidence events (violet). Right: Signal and idler Klyshko efﬁciencies with a linear ﬁt each in the linear regime.

Since we have approximately single-mode signal and idler beams, we can use the result from Eq. 4.9 and write for the single click probabilities

p 2 η¯xtanh B Ep p (E ) = (6.27) x p p 2 1 − (1 − η¯x) tanh B Ep

p where η¯x is the absolute detector efficiency and x ∈ {s, i}. Because of r = B Ep, we have two unknown parameters, η¯x and the constant B, which must be the same for signal and idler. First, we separately determine the efficiencies from the Klyshko efficiencies plotted in Fig. 6.6 (right). As we have explained in section 4.2, η¯s is the Ep = 0 intercept of a fit of the linear region of the signal arm click probability curve, and likewise for idler. In our case, this is the region from 5 pJ pump energy to 25 pJ. For smaller energies, detector dark counts suppress the Klyshko efficiencies towards 0 and for higher energies, the super-linear growth in mean photon number starts to be noticeable. From the linear fit lines in Fig. 6.6 (right), we read η¯s = 7.0% and η¯i = 3.5% and now can fit the parameter B. The fit curves in Fig. 6.6 (left) follow Eq. 6.27 with the measured η¯ , η¯ B = 0.149 √1 efficiencies s i for signal and idler respectively, and pJ for both arms. The fact that both fits return the same value for B further confirms the fit results. 2 hˆni = sinh 0.149 √1 pE In Fig. 6.7 (left), we finally can plot the mean number of photon pairs pJ p generated in the KTP source. For the maximal pump pulse energy of Ep = 75 pJ, we find hˆni = 2.8. This corresponds to a two-mode squeezing parameter r = 1.29 or a logarithmic 20r two-mode squeezing value of ln(10) dB = 11.2 dB. Since one pump pulse contains an average pJ 75 = 75 pJ = 4.63 × 107 2.8 of 2π~ 1.62×10−6 pJ photons, the mean photon number corresponds to a 768 nm conversion efficiency from pump photon to signal and idler photon pair of 6.0 × 10−8. 80 6 Two-mode squeezed vacuum source

3 12

2.5

2 8

1.5

1 4 Two-mode squeezing [dB] Two-mode

Mean photon number Mean photon 0.5 Signal, idler Linear 0 0 0 25 50 75 0 25 50 75

Pump pulse energy Ep [pJ] Pump pulse energy Ep [pJ]

Figure 6.7: Left: Mean photon number for signal and idler beams (red) of the KTP source pumped with pump spectrum FWHM 1.95 nm and a linear curve (violet) for comparison. Right: Corresponding two-mode squeezing values.

6.8 Photon collection efﬁciency

To determine the photon collection efficiency of our experiment, the fraction of photons that are not lost due to optical losses between source and detector, we need to measure the overall quantum efficiency of the setup and correct it against detector losses. After very carefully adjusting our setup (c. f. Fig. 6.5) for optimal coupling of the idler beam to its fiber-coupled id201 APD, we repeat the power-dependent photon counting measurement from the previous section. The pump central wavelength is 768 nm and its spectral FWHM is 1.95 nm. The APD efficiencies are set to 20%. The results can be seen in Fig. 6.8: The intercepts for the signal and idler Klyskho efficiency curves are η¯s = 1.3% and η¯i = 13.7%, respectively. The signal efficiency obviously suffered in comparison to idler, since we optimized the out-coupling lens after the waveguide source for maximal coupling into the idler arm detector. This hints at small differences in numerical aperture at the waveguide out-coupling facet between the p-polarized signal and the s-polarized idler mode, as during simultaneous signal and idler in-coupling optimization we always had to compromise between η¯s and η¯i by maximizing coincidence counts. Here, we opted for one-sided optimization of the idler arm instead to get a measure of the mode quality of the waveguide with respect to coupling into standard telecom single-mode fiber. With a low power Klyshko efficiency η¯i = 13.7% and 20% 13.7% detector efficiency we estimate a photon collection efficiency of 20% = 68.5%.

6.9 Conclusion

In this chapter, we discuss the photon statistics of a twin-beam source, and their connection with the second order correlation function g(2) and effective mode number K. We then present a measured g(2) value of 1.80, or 1.95 after background correction, demonstrating the near-perfect two-mode character of our waveguide source. A mean photon number per pump pulse of up to 2.8 at 75 pJ pulse power is measured, corresponding to 11.2 dB two mode squeezing. The photon collection efﬁciency has been measured with 68%, indicating a good modal overlap between the 6.9 Conclusion 81

0.2 30 Signal Signal Idler Idler Coincidences 25 0.15 20

0.1 15

10 0.05 Klyshko eﬃciency [%] Klyshko 5 Detection event probability [1] event probability Detection

0 0 0 25 0 5 10 15 20 25

Pump pulse energy Ep [pJ] Pump pulse energy Ep [pJ]

Figure 6.8: Left: Click event rate of the KTP source pumped with pump spectrum FWHM 1.95 nm for signal (red) and idler (blue) arm and coincidence events (violet). Right: Signal and idler Klyshko efficiencies with a linear fit each in the linear regime. waveguide output modes and the spatial mode of the SMF28 standard telecom fibers guiding light to the detectors. This is to our knowledge the first waveguide implementation of a two-mode squeezing source[42], and it out-performs its predecessor earlier bulk-crystal based two-mode squeezing sources[136, 93, 79] in terms of modal brightness, that is mean photon number per mode, by several orders of magnitude. In terms of single-modedness, it surpasses a contemporary fiber two-mode squeezing experiments[117, 118].

7 Quantum pulse manipulation

With applications in secure quantum key distribution[40] and high precision positioning and clock synchronization protocols[55], ultrashort pulses of light play an ever-increasing role in modern quantum information and communications. In recent years there has been increased interest in a finer control over the rich temporal and spectral structure of quantum light pulses, for applications such as ultrafast probing of the temporal wave function of photon pairs[101], or efficiently coupling single photons to trapped atoms. Given direct access, this structure could also be utilized to encode more information into or extract more quantum information from one pulse of light. As we have discussed in section 2.6.6, it is possible to decompose any pulse amplitude into any complete set of orthogonal basis functions, or broadband modes[130]. Thus it can be considered to be made up of an infinite number of temporally overlapping but independent pulses. While for classical light all basis sets are equivalent, for quantum light there may be one special, intrinsic basis choice[90]. For photon pair states this choice determined by a Schmidt decomposition of their bi-photon spectral amplitude into two correlated basis sets of broadband pulse forms, the Schmidt modes[82]. Heralding one of those photons by detecting the other with a single photon detector (SPD), this correlation results in the preparation of a photon in a mixed state of all Schmidt modes present[59]. But with a SPD sensitive to a certain Schmidt mode, it opens up the possibility to prepare pure single photons in the correlated Schmidt mode. Typically though, SPDs and optical detectors in general exhibit very broad spectral response and are not able to discern between different pulse forms. To compensate for the detectors’ shortcomings, one needs to include a filter operation sensitive to broadband modes. It has already been shown that ordinary spectral filters cannot fulfill this role[106, 18]: They always transmit part of all impinging broadband modes at once, and thus cannot be matched to a single broadband mode. A sufficiently narrow spectral filter can be used to select a monochromatic mode, however this way, high purity heralded quantum states are impossible[80, 18]. Also, most of the original beam’s brightness as well as its pulse characteristics are lost. The idea of using broadband modes as quantum information carriers is compelling because of their natural occurrence in ultrafast pulses, and their stability in transmission: Centered around 84 7 Quantum pulse manipulation

Figure 7.1: Quantum Pulse Gate schema: Gating with a pulse in spectral broadband mode uj converts only the corresponding mode from the input pulse to a Gaussian wave packet at sum frequency. one frequency within a relatively small bandwidth typically, they allow for optical components that are highly optimized for a small spectral range. Since all broadband modes experience the same chromatic dispersion in optical media, they exhibit the same phase modulation and thus stay exactly orthogonal to each other. So a light pulse’s broadband mode structure is resilient to the effects of chromatic dispersion, making a multi-channel protocol based on them ideal for optical fiber transmission. Additionally they allow for high transmission rates, as they inherit the ultrashort properties of their ’carrier’ pulse, when compared to the ’long’ pulses used for classical, narrow-band frequency multiplexing techniques. However, it is extremely challenging to actually access them in a controlled manner: Ordinary spectral filters and standard optical detectors destroy the mode structure of a beam. A homodyne detector with an ultrafast pulsed local oscillator beam is able to select a single broadband mode by spectral overlap, but only at the cost of consuming the whole input beam.[146, 112] For discrete spatial modes, complete control of a beam’s multi-mode structure can be accom- plished with linear optics, as combining them to synthesize multi-mode beams and separating constituents without losses is possible[83, 131, 81, 145]. In order to exploit the pulse form degree of freedom, we must be able to exact similar control over broadband modes. An important step towards this goal is to selectively target a single broadband mode for in- terconversion into a more accessible channel, for instance to shift it to another frequency with SFG. On the single photon level, in the SFG process two single photons “fuse” into one photon at their sum frequency inside a χ(2)-nonlinear material. Well known in classical nonlinear optics, in recent years it has seen increasing adoption in quantum optics for efficient NIR single photon detection[107, 1, 138, 127], all-optical fast switching[139], super high resolution time tomography of quantum pulses[79], quantum information erasure[126], and for translating non-classical states of light to different frequencies[91]. Moreover, combined with spectral engineering[59, 134, 93], it enables a new type of quantum interference between photons of different color[105]. Here, we introduce the Quantum Pulse Gate (QPG)[41]: A device based on spectrally engineered SFG to extract photons in a well-defined broadband mode from a light beam. We overlap an incoming weak, multi-mode input pulse with a bright, classical gating pulse inside a nonlinear optical material (Fig. 7.1). Spectral engineering ensures that only the fraction of the input pulse 7.1 Beam-splitters, spectral filters and broadband mode selective filters 85 which follows the gating pulse form is converted. The residual pulse, orthogonal to the gating pulse, is ignored. An input quantum light pulse’s quantum properties can be preserved in conversion by mode-matching the gating pulse to its intrinsic mode structure. SFG conversion efficiency can be tuned with gating pulse power, and unit efficiency is in principle reachable. Thus the QPG is able to unconditionally filter broadband modes from arbitrary input states, and to convert them into a well-defined Gaussian wave packet at the sum frequency. By pulse-shaping the gating pulse we are able to switch between different target broadband modes during the experiment. By superimposing gating pulses for two different broadband modes, we create interference between those previously orthogonal pulses. In combination with a standard single photon detector we are able to herald pulsed, pure, single-mode single photons from a multi-mode photon pair source.

7.1 Beam-splitters, spectral ﬁlters and broadband mode selective ﬁl- ters

Our goal here is to develop the notion of a broadband mode selective filter, a filter that ideally transforms one broadband mode from a beam and lets all others pass through unchanged. The simplest and without a doubt most frequently used mode transformer is the standard beam-splitter, operating on spatial rather than spectral modes. Assuming a ideal wavelength-independent beam- splitter, the effective Hamiltonian reads Z ˆ † ∗ † HBS = dω θâ(ω) ˆc (ω) + θ â (ω) ˆc(ω) (7.1) where θ is the beam-splitter angle that governs its transmittivity T = cos2θ and reflectivity 2 R = sin θ. To express the Hamiltonian in an arbitrary broadband mode basis {ψj(ω)} such that Z ˆ ∗ Aj = dω ξj (ω) â(ω) (7.2) Z ˆ ∗ Cj = dω ψj (ω) ˆc(ω) (7.3)

P ∗ 0 0 we once again use the completeness relation j ψj (ω) ψj(ω ) = δ(ω − ω ) and derive the reverse transformation X ˆa(ω) = ξj(ω) Aˆ j (7.4) j X ˆc(ω) = ψj(ω) Cˆ j. (7.5) j

ˆ Substituting this into the beam-splitter Hamiltonian HBS we calculate ˆ X ˆ ˆ † ˆ † ˆ HBS = θ AjCj + AjCj (7.6) j

and see that the beam-splitter transforms each input broadband mode into a mode with the same spectral function ψj(ω) but in a different spatial mode ˆc. It acts on all broadband modes indiscriminately, however. As we required it to act wavelength-independent from the outset, this 86 7 Quantum pulse manipulation result is not uprising. Yet we want to achieve a single-mode transformation with a Hamiltonian that must have the form ˆ ˆ ˆ † ˆ † ˆ Hsinglemode = θ A0C0 + A0C0 (7.7) but the simple model of the ideal beam-splitter does not provide us with any parameters we can tune to engineer this outcome. If we now replace the constant θ in the ideal beam-splitter Hamiltonian in Eq. 7.1 with the frequency dependent real-valued function θ(ω), we can model a standard thin layer spectral ﬁlter with transmissivity function τ(ω) = cos(θ(ω)) and reﬂectivity function ρ(ω) = sin(θ(ω)). The associated Bogoliubov transformations read

ˆa†(ω) =τ(ω) ˆc†(ω) + ρ(ω) dˆ†(ω) (7.8) bˆ†(ω) = −ρ(ω) ˆc†(ω) + τ(ω) dˆ†(ω) . (7.9)

Transmissivity and reﬂectivity obey the usual constraint to ensure energy conservation and positive energy states at the beam-splitter:

|τ(ω)|2 + |ρ(ω)|2 = 1 (7.10)

Note that unlike spectral mode functions, τ and ρ do not have to be square-integrable functions. ˆ The associated unitary operator UBS describing this ﬁlter operation will transform an incoming broadband state of light such that the broadband mode Z ˆ † ˆ† Aj = dω ξj(ω) a (ω) (7.11) will be substituted with Z Z ˆ † † ˆ† Aj → dω τ(ω) ξj(ω) ˆc (ω) + dω ρ(ω) ξj(ω) d (ω) (7.12)

Therefore the spectrum of the transmitted part (in mode ˆc) of a beam will go from initially 2 2 |ξj(ω)| to |τ(ω) ξj(ω)| . If the spectral functions {ξj} constitute an orthonormal basis, we can infer two facts: Firstly, every spectral broadband mode ξj is partly transmitted, and secondly, the transmitted mode functions are not generally orthogonal any more Z 2 ∗ dω |τ(ω)| ξi (ω) ξj(ω) 6= δij, (7.13) so that a new set of spectral basis functions has to be found for the transmitted beam[18]. From this we see that a spectral ﬁlter cannot be used to select one broadband mode from a beam of light, while discarding all orthogonal modes. Mathematically, such an operation must test orthogonality between input mode ξj(ω) and ﬁlter function φ(ω). Also, the transformation must describe the spectral output mode ξ˜j(ω), so in terms of spectral functions we can write down the transformation for the transmitted part of the beam as Z 0 0 ∗ ξj ω → ξ˜j ω dωφ (ω) ξj(ω) (7.14)

This filter will transmit light in the spectral mode φ, and reflect light in any orthogonal mode. So for the special case where the filter function coincides with an element of the input spectral basis 7.2 Broadband mode SFG 87

φ(ω) = ξi(ω), in terms of broadband mode operators we ﬁnd the simple operator transformation rule Z ˆ † ˜ † ˆ † Aj → dω ξj(ω) ˆc (ω) = Cj if j = i (7.15) Z ˆ † ˆ† ˆ † Aj → dω ξj(ω) d (ω) = Dj if j 6= i (7.16)

Formally, we have now defined a mode-selective process that targets exactly one broadband mode from a basis set, disregarding all others. To realize a true broadband mode filter, we must ensure that the transmitted mode Cˆ i can be physically separated from the reflected modes Dˆ j. In order to physically implement the mode-selective filter, we ponder sum frequency generation (SFG) and identify the ’transmitted’ and ’reflected’ modes of our as yet purely theoretical filter with the frequency-converted and the unconverted parts respectively of an optical beam undergoing an SFG process. The frequency gap between them allows us to easily separate them with a dichroic mirror.

7.2 Broadband mode SFG

To demonstrate that a SFG process can be used to generate a mode transformation of a simple form according to Eq. 7.16, we first express its Hamiltonian operator in terms of broadband modes. For a bright classical gating pulse, the effective Hamiltonian of SFG that up-converts a photon in mode ’â’ to mode ’ˆc’ is given by ZZ † Hˆ = θ dωi dωo f(ωi, ωo) â(ωi)ˆc (ωo) + h. c. (7.17) √ Here we introduced the coupling constant θ ∝ χ(2) P with χ(2) denoting the second order nonlinear polarization tensor element of the SFG process and P the SFG pump pulse power. The SFG transfer function f(ωi, ωo) = α(ωo − ωi) × Φ(ωo, ωi) maps the input frequencies ωi to the sum frequencies ωo, where α is the spectral amplitude of the gating pulse and Φ the phase matching distribution of the SFG process. In parametric down-conversion (PDC), the Schmidt decomposition of the joint spectral amplitude of the generated photon pairs reveals their broadband mode structure. Applying the same approach to SFG[105] to decompose the spectral transfer function we find

X f(ωi, ωo) = κj ξj(ωi) ψj(ωo). (7.18) j The decomposition is well-defined and yields two correlated sets of orthonormal spectral amplitude functions {ξj(ω)} and {ψj(ω)} and the real Schmidt coefficients κj which satisfy the relation 2 (ω−ω0) P 2 2σ2 ω−ω0 j κj = 1. If the gating pulse has the form of a Hermite function uj (ω) ∝ e Hj σ with Hj the Hermite polynomials, the basis functions of both sets are in good approximation Hermite functions as well. In the Schmidt-decomposed form, the transfer function describes a mapping between pairs of broadband modes ξj(ω) → ψj(ω). R R By defining broadband mode operators Aˆ j = dω ξj(ω) â(ω) and Cˆ j = dω ψj(ω) ˆc(ω) corresponding to the Schmidt bases, the effective Hamiltonian from Eq. 7.17 can be rewritten as 88 7 Quantum pulse manipulation

ˆ X ˆ ˆ † ˆ † ˆ H = θ κj AjCj + AjCj , (7.19) j ˆ † An optical beam splitter has a Hamiltonian of the form HBS = θ ˆaˆc + h. c.; so with respect to broadband modes, SFG can be formally interpreted as a set of beam splitters, independently ˆ ˆ ˆ operating on one pair of broadband modes each,√ such that Aj → cos(θj)Aj + ı sin(θj)Cj. The effective coupling constant θj = θ · κj ∝ P takes the role of the beam splitter angle. Its transmission probability – the probability to ﬁnd a photon in the up-converted mode Cˆ j if it 2 initially has been in mode Aˆ j – is ηj = sin (θj).

Figure 7.2: (A1-A3) SFG transfer function f(ωi, ωo) with (A1) and without (A2,A3) frequency correlations. (B1-B3) Coefficients κj for the first four Schmidt mode pairs of the transfer functions. (C1-C3) SFG efficiencies Aˆ j → Cˆ j for the first four Schmidt modes against gating power dependent SFG coupling constant θ

In Fig. 7.2 A1-C1, we illustrate an example for a non-engineered SFG process, as commonly found in pulsed SFG experiments: The transfer function f(ωi, ωo)(Fig. 7.2 A1) exhibits spectral correlations, causing more than one non-zero Schmidt coefﬁcient (Fig. 7.2 B1). This leads to ˆ the simultaneous√ conversion of multiple modes Aj at once with non-zero coupling constants θj ∝ P for any given gating pulse power P (Fig. 7.2 C1). Hence a SFG process in general is not mode-selective.

7.3 Spectral engineering and the Quantum Pulse Gate

However, spectral engineering can make SFG mode-selective by eliminating its spectral correlations so that the frequency of an up-converted photon gives no information about its original frequency. Now, Schmidt decomposition yields one predominant parameter κj ≈ 1 with all others close to zero and a separable transfer function f(ωi, ωo) ≈ κjξj(ωi)ψj(ωo). Also, now the full coupling θj ≈ θ is exploited, allowing for relatively weak gating beams for unit conversion efﬁ- ciency. We achieve this by choosing a SFG process with an already correlation-free phasematching 7.4 Critical group velocity matching and QPG mode-switching 89

function Φ. If the phasematching bandwidth is narrow compared to gating pulse width, spectral correlations are negligible (Fig. 7.2 A2-A3), and we can approximate a separable transfer function (Fig. 7.2 B2-B3). The effective SFG Hamiltonian is now formally a beam splitter Hamiltonian ˆ ˆ ˆ † HQPG = θAjC0 + h. c. (7.20)

meaning that only mode Aˆ j is accepted for conversion. Because of the horizontal phasematching, the target mode is always the Gaussian pulse Cˆ 0. This process implements the QPG, with the bright input pulse used as gate pulse to select a speciﬁc broadband mode. By tuning the central wavelength and spectral distribution of the gating pulse, we can control the selected broadband mode’s shape, width and central wavelength. We compare the effect of different gating pulse forms: Gating with mode u0 (i. e. a Gaussian spectrum, Fig. 7.2 A2-C2) selects input mode Aˆ 0, gating with mode u1 (Fig. 7.2 A3-C3) selects Aˆ 1 from the input pulse for frequency up-conversion. Pure heralded single photons are a crucial resource in many quantum optical applications, but the widely used PDC photon pair sources emit mixed heralded photons in general. We now consider the application of the QPG to “purify” those photons. In type-II PDC, a pump photon decays inside a χ(2) -nonlinear medium into one horizontally polarized signal and one vertically polarized idler photon. For a collinear type-II PDC source pumped by ultrafast pulses the general effective Hamiltonian in terms of broadband modes reads

ˆ X ˆ˜ † ˆ † ˆ˜ ˆ HPDC = ζ cj AjBj + AjBj . (7.21) j Using such a photon pair source for the preparation of heralded single photons, one finds that those are usually not in pure, but spectrally mixed states[134], and thus of limited usefulness for most quantum optical applications. We feed the signal photon (containing all broadband modes ˜ ˜ Aˆ j) from the PDC source into the QPG which is mode-matched such that Aˆ 0 = Aˆ 0. We note that for heralding pure single photons or pure Fock states[106], mode-matching is not necessary and an engineered SFG process according to Eq. 7.20 is sufficient. In that case however, the resulting pulse shape is a coherent superposition of all input modes. Here, only the 0th mode is selected, and the higher modes do not interact with the QPG because the according beam splitter transformations yield the identity Aˆ j → Aˆ j for j > 0. We choose the gating pulse power such π that θ0 = 2 for optimal conversion efficiency. Combining the PDC source with a subsequent QPG ˆ ˆ ˆ ˆ 0 −ıHQPG ˆ ıHQPG results transforms the PDC Hamiltonian as HPDC → H = e HPDCe , and we obtain ∞ ˆ 0 ˆ † ˆ † X ˆ˜ † ˆ † H = ıζc0B0C0 + ζ cjAjBj + h. c. (7.22) k=1

Since mode Cˆ 0 is centered at the sum frequency of input and gating pulse, it can be split off easily into a separate beam path with a dichroic mirror. Conditioning on single photon events on the path of Cˆ 0 provides us with pure heralded single photons in mode Bˆ 0. Fig. 7.3 illustrates this scheme: A photon detection event heralds a pure single photon pulse in broadband mode u1. This process can be cascaded to successively pick off several modes Aˆ j from the input beam.

7.4 Critical group velocity matching and QPG mode-switching

Group velocity matching (GVM) for SPDC (see section 3.2) is subtly different from SFG, albeit that difference exists in terminology only. The three involved waves swap roles: What was the pump 90 7 Quantum pulse manipulation

Figure 7.3: A QPG application: Generating pure heralded broadband single photons in different modes from a PDC source of multi-mode photon pairs beam in the former case, is now considered the output beam in the latter, while signal or idler act as pump and input beams. Consequently, to fulﬁll the GVM condition we now are looking for a SFG with the output wave’s group velocity vo between pump and input velocities, vp and vi respectively, so either vp ≤ vo ≤ vi (7.23) or vi ≤ vo ≤ vp (7.24) has to hold. While it would be possible to ﬁnd an uncorrelated case of ultrafast upconversion with regular GVM, just as we did to realize the uncorrelated bi-photon spectrum for our two-mode-squeezer source in section 6, choosing critical GVM comes with a great advantage for the QPM: The ability to switch, during operation, between orthogonal incoming modes. Consider the general form of the SFG mapping function:

f(ωi, ωo) = α(ωo − ωi) × Φ(ωo, ωi) . (7.25)

Critical GVM is distinguished by a locally horizontal or vertical phasematching function. We 1 1 choose input and output group velocities to be equal ( 0 = 0 ), so that in a (ωo, ωo) graph ko(¯ωo) ki(¯ωi) of Φ phasematching is horizontal. After linear expansion of Φ around the central input and output frequencies ω¯i and ω¯o we ﬁnd 2 (¯ωo−ωo) − 2σ Φ(ωo, ωi) = e pm . (7.26) The pump beam we assume to be emitted by a pulse shaper such that has the form of Hermite mode j: 2 (ν −ν ) νo − νi 1 νo − νi − o i α(ωo − ωi) = uj = √ Hj e 2σ2 (7.27) σ p π2jj!σ σ

ν2 − o νo − νi f(ω , ω ) ≈ e 2σpm u (7.28) i o j σ 7.4 Critical group velocity matching and QPG mode-switching 91

Figure 7.4: Critically group velocity matched SFG phase matching pumped with a Gaussian spectrum u0(ωp)

Figure 7.5: Critically group velocity matched SFG phase matching pumped with a Hermite mode spectrum u1(ωp)

σpm For a phasematching width much smaller than pump width σ 1, we can see from Fig. 7.4 that the mapping function’s contour plot is wide and thin; in the regions where f(ωi, ωo) is signiﬁcantly different from zero, the detuning of the output frequency νo is also much smaller νo−νi νi than the input detuning νi, so that the approximation uj σ ≈ uj − σ is justiﬁed. It follows that the mapping function is approximately separable:

ν2 − o −νi f(ω , ω ) ≈ e 2σpm u . (7.29) i o j σ

If one changes the spectral mode of the pump beam from u0 to u1, the resulting two-photon spectrum is still close to separability, owing to the horizontal, narrow phasematching function (c. f. 7.5). The phasematching width is inversely proportional to the SFG interaction length L, so that the above case can in principle be reached in experiment by increasing the length of the nonlinear crystal or waveguide. This can however be impractical, or bring unwanted side effects due to dispersion, absorption or production imperfections. As we have pointed out in section 2.6.7, the SPDC joint spectral amplitude is connected to the 1 effective mode number K of the process via the visibility V = K of a theoretical two-source HOM experiment with low photon pair ﬂux. The relationship between K and f carries over to SFG, even though the experiment does not: 1 Z Z Z Z = dω dω dω dω f ∗(ω , ω ) f ∗(ω , ω ) f(ω , ω ) f(ω , ω ) (7.30) K 1 2 3 4 1 2 3 2 1 4 3 4 92 7 Quantum pulse manipulation

σpm To gain greater insight into how the ratio between phasematching width and pump width r = σ impacts separability, we plot K versus ratio r for the pump in the ﬁrst four Hermite modes in 1 Fig. 7.6(left). As long as r is in the order of 10 or smaller, the effective mode number stays below 1 K = 1.035. For r > 10 however, the effective mode number rises approximately linear: K ∝ r.

1.1 1 K0(r) K1(r)

j K (r) 1.08 2 0,min,j K3(r) 0.995 K3(0.1)=1.035

1.06 0.99 1.04

κ (r) 0.985 0,min,0 κ0,min,1(r)

Eﬀective mode number K Eﬀective mode 1.02 κ0,min,2(r) κ (r)

Minimum Schmidt number κ 0,min,3 κ0,min,3(0.1)=0.991 1 0.98 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 Ratio r Ratio r

Figure 7.6: Left: Effective mode number Kj of a critically group velocity matched SFG process σpm pumped with Hermite mode uj over ratio r = σ . Right: Lower boundary for the ﬁrst Schmidt coefﬁcient of the same process.

We can also make an estimate for the Schmidt coefﬁcient κ0 of upconversion of the strongest Schmidt mode. For a ﬁxed κ0 the worst-case effective mode number is reached when there is 1 P 4 4 4 exactly one other mode, i. e. κ1 > 0 and κj = 0 for j ≥ 2. From K = j |κj| = |κ0| + |κ1| ≥ 1 2 2 1.035 and |κ0| + |κ1| = 1 we calculate 1 |κ |4 + (1 − |κ |2)2 = 0 0 K 1 2 |κ |4 − 2 |κ |2 + 1 − = 0 0 0 K (7.31) 1 r 1 1 |κ |2 = ± − 0 2 2K 4

2 From the two possible solutions for |κ0| we choose the greater one, since we had assumed j = 0 2 2 2 2 1 to be the strongest mode, therefore |κ0| > |κ1| = 1 − |κ0| and consequently |κ0| > 2 . So for K ≥ 1.035 we find as lower boundary for the probability of the first Schmidt mode s 1 r 1 1 κ ≥ + − = 0.991. (7.32) 0 2 2K 4 with the results depending on the ratio r plotted in Fig. 7.6 (right). In Eq. 7.32 we have established that for Hermitian pump modes j ≤ 3 the strongest mode of the Schmidt decomposition of the SFG mapping function f has a Schmidt coefficient κ0 that is still almost at unity, so that we can consider the SFG process almost single-mode. The form of the first Schmidt mode in the limit of a vanishing pump-to-phasematch width ration r → 0 7.5 Experimental feasibility 93

is exactly that of the pump spectrum, only the central frequency is now that of the input beam instead; this is the central idea of the QPG, and also of the closely related, but independently developed pulse shaping method in [71]. For small values of r, we expect small variance from this spectral form as well, but instead of analyzing this in the idealized case of an exactly horizontal, Gaussian-shaped phase-matching function, we present numerical calculations for more realistic experimental parameters in the next section.

7.5 Experimental feasibility

Finally we give realistic parameters to show the feasibility of an experimental implementation of the QPG, and analyze the mode selection performance. For the SFG process we use a periodically 2 poled LiNbO3 (PPLN) waveguide with an area of 8 × 5µm , a length of L=50 mm, a Λ = 4.2µm periodic poling period and at 175◦C to achieve phasematching for SFG of an input pulse at 1550 nm to 557 nm. It is gated by coherent laser pulses at 870 nm with 2 ps pulse length or a spectrum with 0.635 nm FWHM to ensure a transfer function separability. The uncorrelated, separable transfer functions in Fig. 7.2 (A2-A3) are calculated from these parameters, using gating pulses with u0 and u1 as spectral amplitude, respectively.

Figure 7.7: Overlap between input pulse mode u˜l and QPG Schmidt mode ξj for mode-matched (left) and non-mode-matched (right) case.

In Fig. 7.7 we illustrate the switching capabilities of our QPG, as well as the impact of mode matching. For the given material parameters, we employ gating pulses with pulse form u0 to u10, determine the Schmidt decomposition of the resulting transfer function f(ωi, ωo), and plot the overlap of the predominant Schmidt function ξj (with κj ≈ 1) with an Hermitian input mode u˜l from an incident light pulse. On the left, gating and input pulse have equal frequency FWHM, which is essential for good mode matching. Now, by switching the order j of the gating mode (and without changing the physical parameters of the QPG), we select with high ﬁdelity only the input 2 R ∗ mode j to be converted. For j ≤ 10, the overlap dωu˜j (ω) ξj(ω) exceeds 99%, and the overlap for all other input modes combined therefore is less than 1%: Only a negligible fraction of modes other than the selected input mode are converted. In contrast, Fig. 7.7 (right) has no mode matching, the gating pulse FWHM is twice that of the input pulse. Multiple strong overlaps between SFG Schmidt modes ξj and input modes u˜l appear: 94 7 Quantum pulse manipulation

A wide range of modes is converted for any given input spectrum. The checkerboard pattern reﬂects the fact that only Hermite modes of the same parity overlap, and the SFG Schmidt modes are in good approximation Hermite modes.

7.6 The Quantum Pulse Shaper

The “switchability” of the QPG process hinges on the locally horizontal phasematching function of the underlying SFG; geometrically speaking the spectral shape of the gating beam determines the spectral shape of the QPG input mode, while the output mode follows a perpendicular cut through the phasematching function, leading to a sinc profile, or a Gaussian profile in the approximation. In short, the input function is variable, while the output function is fixed. Any three-wave-mixing process with a vertical phasematching function turns this around, as is illustrated in Fig. 7.8: We find a mode-selective process with a fixed input mode and a variable output mode that follows the spectral shape of the bright pump beam. This is the Quantum Pulse Shaper (QPS)[21], an technique to directly imprint an arbitrary pulse form on a single photon, rather than indirectly as in the scheme in Fig. 7.3. The Hamiltonian again has the structure of a broadband mode beamsplitter: ˆ ˆ ˆ † HQPS = θA0Cj + h. c. (7.33)

The difference to the QPG Hamiltonian is that now the output mode Cˆ j rather than the input mode is switchable, so that the pump spectrum will be repeated by the SFG output mode.

Figure 7.8: Critically group velocity matched SFG with a horizontal phase matching function, pumped with a Hermite mode u1(ωp)

7.7 Time ordering and strongly coupled three-wave-mixing

Our model of three-wave-mixing processes up until now has implicitly assumed that multiple photon pairs in the case of SPDC or multiple conversion events in the case of SFG are independent from each other. However, this is not the case in general, and for processes with a strong coupling constant ζ this self-interaction will inﬂuence its modal structure. We can understand the underlying reasons by considering the time evolution operator of an initially general quantum mechanical process described by the Hamiltonian formalism. The unitary time evolution operator U(ˆ t, t0) associated with a physical process described by Hamiltonian H(ˆ t) propagates an arbitrary state at time t0 to the state at time t:

U(ˆ t, t0) |Ψ(t0)i = |Ψ(t)i . (7.34) 7.7 Time ordering and strongly coupled three-wave-mixing 95

For SPDC, the time difference t − t0 corresponds to the travel time of the pump pulse through the nonlinear material of length L such that t − t = L where v is the pump beam’s group velocity. 0 vp p Up to this point, we have assumed the operator to be deﬁned by

−ı R t dt0H(ˆ t0) ˆ t UTaylor(t, t0) = e 0 (7.35) which can readily be developed into a Taylor series:

Z t Z t 2 ˆ 0 ˆ 0 1 0 ˆ 0 UTaylor(t, t0) = 1 − ı dt H t − dt H t + ... (7.36) t0 2 t0 However, this deﬁnition implicitly assumes that the Hamiltonian operator at different times commutes, but in general we have h i H(ˆ t1) , H(ˆ t2) 6= 0, (7.37)

and for three wave mixing we can easily verify that this is the case by calculating this commutator for H(ˆ t) = âbˆcˆ †e±ı(ωt) + h. c.. We can understand the physical impact of this fact for SPDC: The photons’ bosonic character makes the emission of a photon pair into two already populated mode more probable than into a vacuum mode. Therefore, there is a difference between the first and the second produced photon pair in a PDC process. But the second order term of Eq. 7.36 implies that the creation of both photon pairs is completely independent, as it can be expressed as the non-time-dependent product of two commuting (indeed identical) Hamiltonian operators. We can now see that this is not the case here, and that Eq. 7.36 is a good approximation only if we truncate after the first order term, i. e. in the weakly coupled regime where with high probability only one photon pair is produced. The same argument can be made for SFG, and likewise the simplified Taylor approach is applicable to weak coupling θ 1. The exact form of the time-ordered time evolution operator can be derived by considering the time-dependent Schrödinger equation for an arbitrary quantum state |Ψ(t)i[142]: ∂ H(ˆ t) |Ψ(t)i = ı |Ψ(t)i (7.38) ∂t When we substitute Eq. 7.34 into 7.38, we find ∂ H(ˆ t) U(ˆ t, t ) |Ψ(t )i = ı U(ˆ t, t ) |Ψ(t )i . (7.39) 0 0 ∂t 0 0 We can consider this as a Schrödinger equation for the time evolution operator – rather than the state it is generating – and write ∂ ı U(ˆ t, t ) = H(ˆ t) U(ˆ t, t ) . (7.40) ∂t 0 0 Eq. 7.40 is sometimes referred to as the Tomonaga-Schwinger equation. Transformed into an integral equation through straight-forward integration it reads

Z t U(ˆ t, t0) = 1 − ı dt1H(ˆ t1) U(ˆ t1, t0) (7.41) t0 96 7 Quantum pulse manipulation

Recursive self-substitution results in the Dyson series representation of the time evolution operator: Z t U(ˆ t, t0) = 1 − ı dt1H(ˆ t1) t0 Z t Z t1 − dt1H(ˆ t1) dt2H(ˆ t2) (7.42) t0 t0 Z t Z t1 Z t2 +ı dt1H(ˆ t1) dt2H(ˆ t2) dt3H(ˆ t3) + ... t0 t0 t0

In each summand term, we see that the nested integrals set the constraint t > t1 > t2 > ..., and the time-dependences appear in this descending order; therefore, the series is time-ordered. The commutation relation Eq. 7.37 forbids trivially switching time-dependences out-of-order. If the Hamiltonians at different times do commute, then this series is equal to the Taylor series in Eq. 7.35. In section 2.6.6, we observed that with the help of the Schmidt decomposition of the joint spectral amplitude, one can express the broadband squeezing operator Sˆ as a tensor product of broadband two-mode squeezing operators. The same applies to the non-time-ordered time ˆ evolution operator UTaylor of SFG

ˆ −i P θκ Aˆ Cˆ †+θ∗κ∗Aˆ †Cˆ −i θκ Aˆ Cˆ †+θ∗c∗Aˆ †Cˆ ˆ −ıH j j j j j j j O j j j j j j UTaylor(t, t0) = e = e = e (7.43) j where Hˆ is once again the effective SFG Hamiltonian. But while this is a very good approximation for weakly coupled SFG with θ 1, in general it cannot be applied to the exact evolution operator U(ˆ t, t0), since it implicitly assumes that photon sum frequency conversion events are independent from each other. We can understand this from the definition of Uˆ in Eq. 7.42: Starting point for the Schmidt decomposition is the mapping function f(ωi, ωo) from the effective SFG Hamiltonian Hˆ, which is the integral of the interaction Hamiltonian over the interaction time. The operator Uˆ features Hˆ only in the first non-constant term, all higher terms are time-ordered ˆ nested integrals over the interaction Hamiltonian Hint(t) that represent the interaction between multiple conversion events. Therefore in the general, non-perturbative case the operator Uˆ cannot be decomposed to single-mode operators according to Eq. 7.43, and consequently its broadband mode basis must differ from the Schmidt modes of the perturbative case. To gain insight into the mode structure of SFG (or any three wave mixing process) with a classical pump beam for arbitrary coupling strengths, we consider the general input-output relations or Bogoliubov transformations for the mode operators â and ˆc. Since it is known that every time evolution U(ˆ t, t0) of a bi-linear Hamiltonian generates linear Bogoliubov transformations, we write as ansatz[76, 87, 31]: Z Z out † 0 0 0 0 0 0 â (ω) = U(ˆ t, t0) â(ω) Uˆ (t, t0) = dω Ca ω, ω â ω + dω Sa ω, ω ˆc ω Z Z (7.44) out † 0 0 0 0 0 0 ˆc (ω) = U(ˆ t, t0) ˆc(ω) Uˆ (t, t0) = dω Cc ω, ω ˆc ω + dω Sc ω, ω â ω

The operator transformations are governed by the bivariate integral kernels Ca,Sa,Cc and Sc. According to the Bloch-Messiah theorem for bosons[20, 141, 87], there exist four ultrafast spectral 7.7 Time ordering and strongly coupled three-wave-mixing 97

ˆ 0 ˆ 0 ˆ out ˆ out mode bases Aj, Cj, Aj and Cj that decompose the integral kernels such that 7.44 can be re-written as ˆ out ˆ 0 ˆ 0 Aj = cos(κj) Aj + ısin(κj) Cj (7.45) ˆ out ˆ 0 ˆ 0 Cj = cos(κj) Cj − ısin(κj) Aj.

ˆ 0 ˆ out We note that the input mode sets {Aj} and {Aj } in general differ from each other as well as from the input Schmidt mode set {Aˆ j} that arises from the decomposition of the perturbative ˆ approximation for the time evolution UTaylor. The form of the Bloch-Messiah decomposition and thus the spectral shape of the modes now depends on the coupling constant θ. We performed a π decomposition of the QPG’s SFG process for the experimental parameters in section 7.5 and θ0 = 2 , by solving numerically for the integral kernels and applying a singular value decomposition to them.

Figure 7.9: QPG input mode (left) and output mode (right) solution for the perturbative (blue) and rigorous, time-ordered (red) case.

Figure 7.10: Mode occupation coefﬁcients κj for the perturbative (red) and time-ordered (blue) case.

In Fig. 7.9, we see the impact of time ordering on the mode shapes of the QPG: The input mode is slightly wider than the perturbative solution predicts, but still of a Gaussian shape. In 98 7 Quantum pulse manipulation experiment this can be easily compensated for by using a spectrally narrower pump beam. The output mode’s sinc shape is still visible but slightly “washed out” in the rigorous solution. The comparison of mode occupation coefﬁcients κj in Fig. 7.10 reveals that for strong coupling, the 2 occupation probability of the ground mode drops to |κ9| ≈ 90% while the probabilities of the higher modes increase. We conclude that taking into account time ordering for strongly coupled, approximately single mode SFG decreases the “single-modedness” and increases the effective mode number K, lowering the working ﬁdelity of the QPG.

7.8 Conclusion

In conclusion, we have introduced the concept of the QPG, a flexible device to split well-defined broadband modes from a light pulse based on spectrally engineered SFG. The selected mode can be switched by shaping the gating pulse spectrum and converted with high efficiency. Further, we have given a realistic set of experimental parameters for a QPG realized in a PPLN waveguide and demonstrated the high flexibility of the QPG achieved through shaping the gating pulse form. We proposed as an initial application the preparation of pure heralded single photons from an arbitrary type II PDC source. We investigated the effects of time-ordering in the strong coupling regime and found that for a QPG approaching unit conversion efficiency we have to take into account coupling-strength dependent changes in spectral shape of the input and output modes as well as working fidelity. 8 Conclusion and outlook

In the course of this thesis, we have investigated the means to create and manipulate ultrafast quantum pulses with χ(2) -nonlinear optical processes. We have designed and implemented the first waveguide source of ultrafast two-mode squeezed vacuum states in the telecom wavelength regime based on type II SPDC[42]. To minimize the spectral correlations between signal and idler beams, we took advantage of the dispersion properties of the nonlinear KTP crystal used in the experiments. After setting up the basic experiment and detecting photon pairs, we built a single photon pair fiber spectrometer to map the joint spectrum of our source, demonstrated our control over the form of the spectral correlations, and found the optimal spectral pump width for spectrally uncorrelated beams at 1.95 nm. We then characterized the squeezed vacuum output state with the help of a measurement of the second order correlation function g(2) to ensure state separability on the single photon pair level. The measured value is g(2) = 1.80, and after background analysis we find a corrected value of g(2) = 1.95, corresponding to an effective Schmidt mode number K = 1 = 1.05, meaning we generate a state close to 1−g(2) a two-mode squeezed vacuum with g(2) = 2 and K = 1. For moderate pump pulse energies of 75 pJ, our source produces on average 2.8 photon pairs per pump pulse with a conversion efficiency of 6.0 × 10−8. This is equivalent to a two-mode squeezing of 11.2 dB. In terms of mode number and efficiency, our source constitutes a considerable improvement over previous separable photon pair/two-mode squeezed vacuum experiments both in χ(2) -nonlinear bulk crystal materials and χ(3)-nonlinear optical fibers. The quantum light pulses at telecom wavelengths are well suited for transmission in wide area fiber communication networks. Thanks to its implementation in a waveguide chip of 10 mm length, this compact source is ideal for the generation of high-photon number single mode quantum pulses or alternatively high-purity single photon pulses in integrated optics experiments. For an even higher level of integration one could implement be the separation of the pump beam from the output state with a waveguide-integrated Bragg grating, and pig-tail the waveguide with polarization-maintaining fibers aligned to the SPDC input and output polarization directions. The first priority in improving the waveguide source itself in the future should be the suppression of uncorrelated background photons created through coupling to radiation modes. This can be achieved with a steeper refractive index step between waveguide and surrounding, by using 100 8 Conclusion and outlook different dopants or different concentration of dopants to define the waveguides, using ridge waveguide structures, or using a different waveguide material than KTP altogether. The next logical step towards a CV quantum repeater to counter the decoherence threatening the security of long distance quantum communication is a photon subtraction experiment along the lines of [98] to improve, by increasing the available two-mode squeezing, the teleportation fidelity of a CV light state. With the fiber spectrometer[8] we created a useful tool to characterize the spectra of single- or few-photon signals. With no moving parts involved, it is easy to set up and very robust, and already has seen adoption in the field of quantum optics[23, 53]. We also have introduced the quantum pulse gate[41], a flexible filter device sensitive for broadband modes or, equivalently, pulse forms based on spectrally engineered SFG in a PPLN waveguide. Given an arbitrary input state of light and a bright, coherent gating pulse, it converts one broadband mode with high fidelity to one well-defined broadband mode in another wavelength regime and lets transmit all broadband modes orthogonal to the selected one. The selected mode is determined by the gating pulse shape, and the converted mode is constant. A broadband-mode selection cannot be achieved with standard spectral filters since they transmit part of every broadband mode. Subsequently, the converted mode can be conveniently split off by a dichroic mirror. While this new concept still awaits experimental demonstration, and the modal distortions through self-interaction at high powers need to be fully investigated, already several quantum optical applications are conceivable: The implementation of a source of pure heralded photons with the ability to control the output broadband mode of the photon, the de-multiplexing of multiple quantum information channels in orthogonal broadband modes, or the extraction of a two mode squeezed vacuum state from a multimode squeezer. Also it has spawned the quantum pulse shaper[21]: By reversing the quantum pulse gate’s working principle, it is possible to create a three-wave-mixing process that allows to directly shape a quantum light pulse into an arbitrary pulse form at another wavelength regime. Acknowledgments

First and foremost I would like to thank my supervisor Prof. Christine Silberhorn for providing me with the opportunity to work within the IQO group as both a diploma and a PhD student at the MPL Erlangen on such a fascinating topic, while always having the freedom and support to pursue my own ideas. I would also like to thank my collaborators within the group: Andreas Christ, who first as a diploma student supported me in the lab, and from whose theoretical work as a PhD student this thesis has greatly benefited. Benjamin Brecht, who worked out the experimental parameters for the quantum pulse gate. Malte Avenhaus, who together with me implemented the fiber spectrometer, especially (but far from only) for taking care of the computer hardware side of things. Peter J. Mosley, whose experience and practical help in the lab was invaluable in realizing the SPDC source. And finally Thomas Lauckner, whose thesis on modal dispersion in waveguides helped to launch several new research projects within the group. My thanks go also to our intern Patrick Bronner and student helpers Peter Vogt, Philip Weber and Thomas Dirmeier, who decided to stay on and further develop my project within the QIV group at the MPL. I want to thank my other colleagues, present and past, for making our group a place I enjoyed working at. Hendrik Coldenstrodt-Ronge showed me the ropes when I had never seen a quantum optics lab from the inside. I thank Christoph Söller, Kaisa Laiho, and Andreas Schreiber for being always ready to help, discuss physics or play badminton, and also Peter Rohde, Katiúsca Cassemiro, Wolfgang Mauerer and Felix Just. Many people outside the group were also involved in making this thesis a success. Prof. Berhard Schmauss from engineering department at Erlangen University and Prof. Georgy Onishchukov from the MPL generously provided us with the DFC coils for our spectrometer. My special thanks go to the QIV group at MPL, to name just a few, to Christoph Marquardt, Christoffer Wittmann and Josef Fürst, for loaning us any lab equipment we happened to need in a hurry, and also to Philip Hölzer, Sebastian Stark, and the Russel Division in general, for lending us their fiber cleavers and optical spectrum analyzers on a regular basis. A thank you goes also to Chris Poulton, on whose mode solver code the work of Thomas Lauckner was originally based. Prof. Jan-Peter Meyn was a great source of knowledge for all things phase-matching, and supported our first steps towards lithium niobate technology by providing PPLN-samples, which were fabricated by Birgit Stiller. I would like to thank Marga Schwender, Tina Schwender, Sabine König, Ulrike Bauer-Buzzoni, Margit Dollinger, Carolin Haßler, Nadine Danders, Manfred Eberler, Michael Zeller, Benjamin Klier, Bernard Thoman and Robert Gall for support on many occasions, be it an urgent order, a guest who needed rooming, a networking problem, impossibly heavy equipment to install, or a flooded lab, and generally for keeping the institute running. I also thank the IQO group Mk. II at the University of Paderborn for welcoming me as a guest during the last stages of my work, and I especially want to thank Irmgard Zimmermann for her help in finding a flat. Finally, I would like to thank my parents for their ongoing love and support that made this thesis possible.

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