Photon pair states and multiple-photon squeezed states have many applications in quantum information science. In this paper, Green functions are derived for spontaneous four-wave mixing in the low- and high-gain regimes. Nondegenerate four-wave mixing in a strongly-birefringent medium generates signal and idler photons that are associated with only one pair of temporal (Schmidt) modes, for a wide range of pump powers and arbitrary pump shapes. The Schmidt coefficients (expected photon numbers) depend sensitively on the pump powers, and the Schmidt functions (shapes of the photon wavepackets) depend sensitively on the pump powers and shapes, which can be controlled.
© 2017 Optical Society of America
Photons are a key resource for quantum information science [1–4]. Pairs of signal and idler photons can be generated by spontaneous three-wave mixing (TWM) in a second-order nonlinear medium , which is driven by one pump wave , and spontaneous four-wave mixing (FWM) in a third-order nonlinear medium , which can be driven by one or two pump wave(s) . Detecting the idler photons heralds the existence of the associated signal photons, which subsequently can be used in quantum information protocols. Many of these protocols are similar to Hong-Ou-Mandel (HOM) interference , which requires the photons to be in pure indistinguishable states .
For many wave-mixing processes, the frequency bandwidths associated with wavenumber matching are much broader than the pump bandwidths, so the emitted photons occupy many spectral (temporal) modes. Heralding multiple-mode idler photons by direct detection leaves the associated signal photons in mixed states, which are not suitable for interference. One can reduce the numbers of modes and increase the heralded signal purities by using narrow-bandwidth frequency filters, but doing so also reduces the emission probabilities significantly. Alternatively, one can design sources based on wave mixing in such a way that the emitted photons occupy only single pairs of temporal modes [11–14]. Such designs typically involve relations between the inverse group speeds of the pump(s), signal and idler, the pump bandwidth(s) and the medium length. Sources driven by low-power pumps emit photon pairs with low probabilities. One can increase the emission probabilities by increasing the pump power(s), subject to the fundamental constraints that the emitted photons continue to occupy single pairs of modes, and the relative probabilities of quadruplet and pair states remain low.
Sources driven by high-power pumps produce two-continuous-mode squeezed states with multiple-photon wavepackets [15,16]. Such states are entangled in the number-state basis (the numbers of signal and idler photons are strongly correlated [17, 18]) and the quadrature basis (if the signal and idler position quadratures are correlated, then their momentum quadratures are anti-correlated). Squeezed states enable fundamental tests of quantum mechanics [19,20], and can be used to make other interesting states, such as multiple-photon Fock states  and Schrödinger cat states . They also enable many aspects of continuous-variable (CV) quantum information [23,24], including quantum key-distribution [25,26], quantum teleportation [27,28] and quantum-enhanced metrology [29, 30], which utilizes phase squeezing [31, 32], number squeezing [33,34] and number-difference squeezing [35,36] to reduce measurement errors below the shot-noise limit. To work well, these applications also require the signal and idler photons to occupy single pairs of temporal modes (wavepackets with specific shapes). In this paper, we show theoretically that vector FWM in a strongly-birefringent medium (SBM) can satisfy this requirement.
In Sec. 2, we review the mathematics and physics of two-continuous-mode squeezing, in both the Heisenberg picture (HP) and the Schrödinger picture (SP). We also describe a temporal-mode (Schmidt) decomposition [37–40], which allows this two-continuous-mode process to be reformulated as a set of independent, two-discrete-mode processes, the properties of which are known [41,42]. Although it is easy to establish the general properties of Schmidt decompositions, it is often difficult to make specific decompositions. In Sec. 3, we review the generation of photons by nondegenerate FWM in the low-gain regime, and show that processes in which the first pump copropagates with the signal and the second pump copropagates with the idler enable the generation of photons in single pairs of Schmidt modes [43,44]. One such process is vector FWM in a SBM [45–48]. In Sec. 4, we model the generation of multiple-photon wavepackets by this process, in the high-gain regime. We determine the Green functions analytically and decompose them numerically. The results show that single-mode operation is possible for a wide variety of pump powers and arbitrary pump shapes. In Sec. 5, we discuss the effects of nonlinear phase modulation (NPM) briefly. For moderate pump powers, NPM chirps the photon wavepackets, but does not hinder the generation of single Schmidt-mode pairs. Finally, in Sec. 6 we summarize the main results of the paper.
2. Four-wave mixing
In nondegenerate FWM , two strong pump waves (p and q) drive weak signal (s) and idler (r) waves (πp + πq → πr + πs, where πj represents a photon with carrier frequency ωj). In stimulated FWM, the signal and idler (sideband) growth is seeded by an input signal pulse, whereas in spontaneous FWM, the sideband growth is seeded by vacuum fluctuations (which one can think of as virtual pulses). In classical mechanics, the signal and idler waves (modes) evolve according to the coupled-mode equations (CMEs)41, 42]. (The strong pumps are still treated classically.) The simple relation between the classical and quantum CMEs is not accidental: For systems with Hamiltonians that depend quadratically on the mode amplitudes (operators), the Heisenberg equations for the mode operators are identical to the Hamilton equations for the mode amplitudes.
Because the CMEs are linear in the mode operators, their solutions can be written in the input–output (IO) forms
Because quantum evolution is unitary, the output mode operators must satisfy the same commutation relations as the input operators. It follows from two of these relations that(5) and (6) cause the idler Green functions to have the related decompositions 37–40]. The signal Green functions have similar decompositions (which involve the same coefficients). In stimulated FWM, the input functions are the natural shapes of the input pulses, the output functions are the shapes of the associated output pulses and the squared coefficients are the associated energy gains. In spontaneous FWM, the input functions comprise a natural basis with which to describe the vacuum fluctuations, the output functions are the shapes of the associated output wavepackets and the squared coefficients are the photon emission probabilities. The squared output functions describe the relative probilities that photons leave the medium at specific output times. One can interpret the squared input functions as the relative probabilities that photon generation was initiated (seeded) by virtual photons that entered the medium at specific input times. The Schmidt coefficients and functions depend on the physical parameters of the system, such as the pump powers, durations and slownesses, the sideband slownesses and the medium length.
If the sideband operators are decomposed in terms of Schmidt modes , the Schmidt-mode operators satisfy the discrete commutation relations [arm, arn] = 0 and . Each pair of operators undergoes the two-mode stretching (squeezing) transformation 41,42]. In particular, if the input is a vacuum state, then the output variances of the in-phase and out-of-phase quadratures are (µn + νn)2/2 and (µn − νn)2/2, respectively. In the Schrödinger picture (SP), the two-Schmidt-mode output state 42]. For such a state, which is pure and entangled, each photon-number mean is . If one ignores the idler information (by tracing out the idler degrees or freedom), one obtains a thermal signal state, which is mixed, with number mean . The complete state vector is the product of the two-mode state vectors. Thus, the general properties of nondegenerate FWM are known, in both the HP and SP, for arbitrary pump powers. The challenge is to determine the Schmidt coefficients and functions efficiently, and use this knowledge to optimize the designs of experiments.
3. Low-gain regime
Before considering the high-gain regime in detail, it is instructive to consider the low-gain regime, in which only photon pairs are produced with significant probabilities. By using the time-domain perturbation method [50,51], which is based on the approximation that the idler response to a signal impulse is too weak to affect the signal, one obtains the HP Green functions (10) are valid for arbitrary pump shapes and slownesses.Eqs. (7) and (12), one finds that vrn(t) = urn(t − βrl) and vsn(t) = usn(t − βsl). Although the Schmidt functions are determined by the interaction of the pumps and sidebands throughout the whole medium, the output signal (idler) functions are just time-shifted versions of the input signal (idler) functions. There is no particular relation between the signal and idler functions. The argument of the shape function Ap is 43,44]. The idler Schmidt function is the shape function of pump p and the signal Schmidt function is the shape function of pump q. (The signal and idler functions also include the phase factors and , respectively, where ϕr + ϕs = π/2. Henceforth, we will ignore these factors.) These shape functions can be chosen arbitrarily and independently, as required for temporal-mode multiplexing . Similar results were obtained recently for photon frequency conversion by scalar nondegenerate FWM [50,51].
In the SP, the two-photon state vector44]. Integrating the product of µjj (tj,t′) and νij (ti,t′) replaces the input Schmidt function uj (t′) with the output function vj(tj) = uj(tj − βjl). Hence, the TPA has the Schmidt decomposition
To illustrate the low-gain results, it is convenient to choose the pump shapes from the Hermite–Gauss (HG) functionsFig. 1, the cross-function νrs is plotted as a function of the input and output times for two cases. In the first case, the pumps and sidebands copropagate (βr = βp and βs = βq), but the medium is not long enough for a complete pump–pump collision. In the second case, the medium is long enough for a complete collision, but the pumps and sidebands do not copropagate (βr ≠ βp and βs ≠ βq). Both cases exhibit strong correlations between the input and output times.
In Fig. 2, the normalized Schmidt coefficient is plotted as a function of the mode number n. In both of the aforementioned cases, several Schmidt coefficients are nonzero, so heralded signals have mixed states. The Schmidt functions (not shown) have widths that differ from the (common) pump width τ. Some results for copropagating pumps and sidebands, which confirm the prediction that heralded signals have pure states, were illustrated in . More such results will be illustrated shortly, as low-gain limits of arbitrary-gain results [Figs. 5(a), 7(a)–9(a) and 10].
4. Arbitrary-gain regime
Armed with the insight provided by the low-gain results, it is now appropriate to consider the interaction of copropagating pumps and sidebands (βp = βr and βq = βs) in the arbitrary-gain regime. Simultaneous slowness and wavenumber matching is exhibited by vector FWM in a SBM, which is illustrated in Fig. 3. The pump frequencies can be varied independently.
The CMEs are partial differential equations with space- and time-varying coefficients. Nonetheless, by using standard mathematical methods [53,54], which are sketched in the appendix, one can solve them analytically for arbitrary pump powers and shapes. The final results are
Formulas (18)–(20) are complicated. Nonetheless, it is posssible to make some general observations. According to the Schmidt formalism, which was described above, the Schmidt coefficients and the four sets of Schmidt functions are specified completely by the cross-functions νrs and νsr (or the self-functions µrr and µss). Consider the former pair of Green functions. The input-signal Schmidt functions are eigenfunctions of the signal kernel
For a complete collision, which is illustrated in Fig. 4, both step functions are equal to 1. For the cross-function νrs, in the formula for ξ one can replace the upper limit t′ by ∞ and in the formula for η one can replace the upper limit t − βsl by ∞. With these replacements, the input signal function is determined solely by Aq (t′) and the output idler function is determined solely by Ap (t − βrl). The Green function is proportional to . This reduced function is separable in the low-gain regime and asymptotically separable in the high-gain regime (as explained in the appendix), so the Schmidt decomposition of νrs involves only a single nonzero Schmidt coefficient and a single pair of Schmidt functions in both regimes. For νsr, in the formulas for ξ and η one can replace the lower limits t − βrl and t′ by −∞. With these replacements, the input idler function is determined solely by Ap (t′) and the output signal function is determined solely by Aq (t − βsl). The reduced function is also approximately separable in the low- and high-gain regimes, so the decomposition of νsr also involves only a single nonzero coefficient and a single pair of functions in both regimes. Notice that Ap and Aq are always paired with ξ and η, respectively. The results that follow all pertain to complete collisions.
4.1. Schmidt coefficients
In Fig. 5, the first (largest) Schmidt coefficient ν1 is plotted as a function of the gain parameter , for two different combinations of HG pump shapes. In the low-gain regime , the coefficient is a linear function of gain and does not depend on the pump shapes, as predicted by Eq. (10) and the corresponding limit of Eq. (19). In the high-gain regime , the coefficient is a nearly-exponential function of gain and still does not depend on the pump shapes! The formula stated in the caption was derived by using the high-gain limit of the modified Bessel function. This derivation and an explanation of why the coefficient does not depend on the pump shapes are provided in the appendix. In both regimes, the associated Schmidt state contains photons on average.
In Fig. 6, the ratio of the second Schmidt coefficient to the first (ν2/ν1) is plotted as a function of gain, for two different combinations of pump shapes. For both combinations, this ratio starts at zero, as predicted by perturbation theory, increases to its peak value of 0.1, then decreases monotonically. The relative probability of the second mode pair, which is the square of this ratio, does not exceed 1%. Thus, for all practical purposes, vector FWM produces a single pair of Schmidt modes for arbitrary pump powers and shapes. Similar single-mode behavior is exhibited by transient spontaneous Raman scattering (SRS) . This behavior contrasts with the behavior of frequency conversion, in which the number of active Schmidt modes is an increasing function of the pump powers [50,51].
Based on the preceding results for vector FWM and SRS, one might suspect that every unstable two-sideband process exhibits single-mode operation. There is certainly a tendancy for unstable processes to be dominated by the modes that grow most rapidly in space and time [56–58]. However, we made a preliminary anaysis of another (common) process, in which the signal copropagates with a single pump (or two copropagating pumps), but the idler does not. In this process the pump is always present (or the pumps always overlap) and several temporal modes, with comparable Schmidt coefficients, are produced.
4.2. Schmidt functions
The cross-function νrs [Eq. (19)] is illustrated in Fig. 7, for pump shapes that are Gauss functions, and small and large gain values. In this figure and the ones that follow, time is measured in units of the pump width τ [Eq. (17)]. In the low-gain regime, the cross-function is the symmetic product of two Gauss functions, as predicted by perturbation theory [Eq. (10)]. In the high-gain regime, it is compact, and skewed toward earlier input and output times [Eq. (19)].
Because the pump shapes are identical, so also are the signal and idler Schmidt functions. The (common) Schmidt function is illustrated in Fig. 8. In the low-gain regime, the Schmidt function is approximately Gaussian. In the high-gain regime it is skewed toward earlier input and output times, like the Green function with which it is associated. This phenomenon can be explained. Idler generation is illustrated in Fig. 4(a). In physical terms, as the input signal ray (shown) propagates to the lower right, it seeds idler rays that propagate to the upper right (not shown). In turn, these idler rays seed signal rays (not shown) that propagate to the lower left and interact with the idler rays. Thus, the sidebands grow in the part of the collision region that is above and to the right of the input signal ray. Earlier idler rays (one of which is shown) represent parts of the idler that interact with stronger parts of the signal and grow more. In mathematical terms, the argument of the modified Bessel function is proportional to the square root of the area of the growth region, which is larger for signal and idler rays that correspond to earlier input and output times. Similar skewing is exhibited by transient SRS . This phenomenon is characteristic of instabilities, in which the output is dominated by the source points (times) associated with higher gains. For signal generation [Fig. 4(b)], the input idler and output signal functions (not shown) are advanced relative to the previous ones, and are skewed toward later times, for a reason similar to that stated above.
Similar behavior occurs for other pump shapes. The cross-function νrs is illustrated in Fig. 9, for pumps that are zeroth- and first-order HG functions, and small and large gain values. The associated input and output Schmidt functions are illustrated in Figs. 10 and 11, respectively. In the low-gain regime, the Green and Schmidt functions are approximately HG functions, as predicted by perturbation theory. In the high-gain regime, all three functions are skewed toward earlier input and output times.
5. Nonlinear phase modulation
Equations (1) and (2) do not include the effects of NPM, which can chirp the pumps, signal and idler, and dephase the interaction [44,59–61]. To obtain a Green matrix, we solved the CMEs for FWM and NPM numerically, for many HGn input signals, and decomposed the output idlers in terms of the same basis functions. Subsequently, we decomposed this matrix numerically .
Some results of such simulations are shown in Figs. 12 and 13. In all the simulations, the medium length l = 80 m, the relative slowness β1 = βrs = 0.5 ps/m, the second-order dispersion coefficient β2 = 15 ps2/Km and the Kerr nonlinearity coefficient γK = 10/Km-W. The input pumps have Gauss (HG0) shape functions with (common) width τ = 5 ps. The fast pump (q) is delayed by 10 ps at the input and the slow pump (p) is advanced by 10 ps, so the pumps overlap completely in the middle of the medium. These parameters are typical of silica fibers. Experiments are also feasible in silicon chips, in which higher birefringence and nonlinearity compensate for shorter length . The nonlinear effect of each pump on itself (self-phase modulation) is a chirp, whereas the effect of each pump on the other (cross-phase modulation) is a wavenumber shift. We chose to prechirp the input pumps so that they are approximately unchirped in the middle of the medium, where their interaction with the signal and idler is strongest.
In Fig. 12, the ratio of the second Schmidt coefficient to the first (ν2/ν1) is plotted as a function of the gain parameter . (For the chosen parameters, corresponds to a peak pump power of 5.6 W.) The numerical Schmidt ratios are higher than the semi-analytical ratios. However, the numerical and semi-analytical curves are qualitatively similar and for no value of the gain parameter (in the chosen range) does the numerical result exceed the semi-analytical result by more than a factor of 2. In particular, the maximal value of the numerical ratio is 0.13, which corresponds to a relative Schmidt-mode probability of only 1.7%. For the parameters chosen, NPM does not increase the Schmidt ratio significantly, because most of the induced phase shifts are chirps (separable functions of t and t′). We also simulated photon generation by shorter (1-ps) pulses, for which the effects of dispersion are stronger . In these simulations, the numerical and semi-analytical results were also comparable for small and medium values of the gain parameter, but the numerical results started to diverge from the semi-analytical ones at . This value corresponds to a gain of more than 40 dB (expected photon number larger than 104), so NPM does not prevent single-mode operation for a useful range of pump powers.
In Fig. 13, the amplitudes and phases of the input-signal and output-idler Schmidt functions are plotted as functions of time for the gain parameter , which corresponds to expected signal and idler photon numbers of 10. The amplitude profiles of the signal and idler are similar, but not identical. Both pulses are skewed toward early times, as predicted by theory, but the idler is skewed slightly more than the signal. The signal and idler are chirped in opposite ways and the chirps are centered on the peaks of the copropagating pumps. The signal phase has a net negative slope, which corresponds to a positive frequency shift, whereas the idler phase has a net positive slope, which corresponds to a negative frequency shift. These shifts are measured relative to the frequencies that correspond to wavenumber matching at zero pump power. The signal and idler frequencies are shifted toward the pump frequencies, as required to achieve wavenumber matching for nonzero power [44,48]. For reference, the input idler and output signal functions (not shown) are time-reversed images of the displayed input and output functions.
A comprehensive set of simulation results will be described in a future publication. However, the representative results described herein show that NPM is not a fundamental obstacle to the generation of signal and idler photons in single pairs of Schmidt modes. If necessary, the signal and idler chirps can be removed by propagation in a dispersive medium.
In summary, photon generation by vector four-wave mixing (FWM) in a strongly birefringent medium was studied in detail. In this process, the signal copropagates with one pump, the idler copropagates with the other, and the medium is long enough for a complete pump–pump collision. This configuration enables the generation of signal and idler photons in a single pair of temporal (Schmidt) modes. The shapes of the signal and idler wavepackets (Schmidt functions) can be specified arbitrarily and indepedently by controlling the pump shapes, as required for temporal-mode multiplexing . For low pump powers, the Schmidt coefficients (square roots of the photon probabilities) are less than 1 and increase linearly with the power parameter (square root of the product of the pump energies). The Schmidt functions match the pump shapes closely and the output functions (which describe the detection probabilities of emitted photons) are just time-shifted versions of the input functions (which describe the seeding probabilities of virtual input photons). For high pump powers, the Schmidt coefficients are greater than 1 and increase nearly exponentially with power. The Schmidt functions are distorted versions of the pump shapes. The input signal and output idler functions are skewed toward earlier times, whereas the input idler and output signal functions are skewed toward later times. Such skewing maximizes the gain experienced by the signal and idler during their spatial–temporal interaction. As the pump powers increase, the relative probability of the second pair of Schmidt modes to the first pair increases from 0 to 1%, then decreases. Nonlinear phase modulation does not increase the modal diversity significantly for gains less than 40 dB. Thus, for all practical purposes, vector FWM produces a single pair of temporal modes for a wide range of pump powers and arbitrary pump shapes. We believe that these properties and frequency tunability  make vector FWM a versatile tool for quantum information science.
Appendix: Derivation of the Green functions
In the undepleted-pump regime, the CMEs for the signal and idler amplitudes are
For vector FWM in a SBM [45–48], there exist pump, signal and idler frequencies for which βp = βr and βq = βs. (Similar conditions facilitated the analysis of frequency conversion in an isotropic fiber [50, 51].) Without loss of generality, suppose that βr > βs, and define the characteristic variables x = βrz − t and y = t − βsz. Then the characteristic CMEs are
Now let Ar/Ap → Ar and As/Aq → As, and define the interaction distances and (The normalization factors Ap and Aq are evaluated at −x and y, respectively.) Then the canonical CMEs areEqs. (25) do not depend explicitly on the pump shapes. They are mathematically equivalent to the CMEs for SBS and SRS, which involve only a single pump and its associated interaction distance. Their solutions can be written in the IO forms of Eqs. (4). The Green functions associated with a signal source of unit strength at the origin are [53,54]
For any parametric process, the Green functions all have Schmidt decompositions, which involve Schmidt coefficients and functions. Because these decompositions are related, it is sufficient to decompose the cross-functions Grs (which involves the input signal and output idler functions) and Gsr (which involves the input idler and output signal functions). First, consider Grs. For a complete collision, the effective (signal) input point is (0, η′) and the effective (idler) output point is (ξ, 1). (For incomplete collisions, the formulas for the input and output points can be complicated.) In the low-gain regime , . One can rewrite this result in the Schmidt form , where and u(s) = v(s) = H(s)H(1 − s). In the high-gain regime , one can use the asymptotic formula I0(s) ~ exp(s)/(2πs)1/2 to write
To convert the canonical Green functions to physical functions, one multiplies Grr and Gsr by , and Grs and Gss by . One also multiplies Grr and Grs by Ap (−x) and Gsr and Gss by Aq (y). Then one rewrites the characteristic variables in terms of the physical variables (0 → t′, − x → t – βrz and y → t − βsz.) The final results areappendix.
Danish National Research Foundation (DNRF-123); Danish Council for Independent Research (DFF-4184-00433); DFF Sapere Aude (NANO-SPECs); Huawei Technologies; National Science Foundation (NSF-1521466).
This work was initiated during a visit by CM to the Technical University of Denmark, which was supported by the Silicon Photonics for Optical Communications Centre. CM would like to thank Leif Oxenløwe for his hospitality.
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