Abstract

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

1. Introduction

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 [5], which is driven by one pump wave [6], and spontaneous four-wave mixing (FWM) in a third-order nonlinear medium [7], which can be driven by one or two pump wave(s) [8]. 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 [9], which requires the photons to be in pure indistinguishable states [10].

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 [21] and Schrödinger cat states [22]. 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 [8], two strong pump waves (p and q) drive weak signal (s) and idler (r) waves (πp + πqπr + 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)

(z+βrt)Ar(t,z)=iγpq(t,z)As*(t,z),
(z+βst)As(t,z)=iγpq(t,z)Ar*(t,z),
where Aj is a mode amplitude and βj is a group slowness (inverse group speed). The pump function γpq (t, z) = γAp (tβpz)Aq (tβq z), where γ = γKϵ(EpEq)1/2 is proportional to the Kerr nonlinearity coefficient and the square roots of the pump energies. For parallel (perpendicular) pumps, ϵ = 2 (2/3). Pump depletion is neglected, so the pump amplitudes Ap and Aq are specified functions, which are normalized in such a way that dt|Aj(t)|2=1. These CMEs describe pulses that interact (unstably) as they convect through the medium. In the Heisenberg picture (HP) of quantum mechanics, one replaces the signal and idler amplitudes by the operators as and ar, which satisfy the continuous commutation relations [aj (t, z), ak (t′, z)] = 0 and [aj(t,z),ak(t,z)]=δjkδ(tt), where δjk is the Kronecker delta and δ(t) is the Dirac delta function [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

br(t)=dt[μrr(t,t)ar(t)+νrs(t,t)as(t)],
bs(t)=dt[μss(t,t)as(t)+νsr(t,t)ar(t)],
where aj and bj denote input (z = 0) and output (z = l) operators, respectively. In these solutions, the transfer (Green) functions µrr (t, t′) and νrs (t, t′) describe the effects on the output idler of impulses applied to the input idler and signal, respectively. These impulses could represent short input pulses or vacuum fluctuations. A similar statement applies to µss (t, t′) and νsr (t, t′). Henceforth, we will state results mainly for the idler, because the corresponding signal results can be deduced by interchanging subscripts (rs).

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

dt[μrr(t1,t)μrr*(t2,t)νrs(t1,t)νrs*(t2,t)]=δ(t1t2),
dt[μrr(t1,t)νsr(t2,t)νrs(t1,t)μss(t2,t)]=0.
Every complex kernel µ(t, t′) has the Schmidt decomposition nνn(t)μnun*(t), where µn is a Schmidt coefficient, and un(t′) and vn(t) are input and output Schmidt functions, respectively. The coefficients are real and non-negative, and the input (output) functions are orthonormal. Constraints (5) and (6) cause the idler Green functions to have the related decompositions
μrr(t,t)=nνrn(t)μnμrn*(t),νrs(t,t)=nνrn(t)νnusn(t),
where n is the Schmidt-mode index and μn2νn2=1 [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 μn2 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 νn2 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 [arn=dturn*(t)ar(t)andbrn=dtνrn*(t)br(t)], the Schmidt-mode operators satisfy the discrete commutation relations [arm, arn] = 0 and [arm,arn]=δmn. Each pair of operators undergoes the two-mode stretching (squeezing) transformation [49]

brn=μnarn+νnasn,bsn=μnasn+νnarn,
the properties of which are known [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 (µnνn)2/2, respectively. In the Schrödinger picture (SP), the two-Schmidt-mode output state
|ψn=k(νn/μn)k|k,k/μn,
where |k, k〉 denotes a substate with k signal photons and k idler photons [42]. For such a state, which is pure and entangled, each photon-number mean is νn2. 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 νn2. 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 [44]

μss(t,t)=δ(ttβsl),νrs(t,t)=iγpq(tc,zc)/βrs,
where l is the medium length, βrs = βrβs > 0 is the relative slowness and the signal–idler collision coordinates
tc=[βrtβs(tβrl)]/βrs,zc=[t(tβrl)]/βrs.
The auxiliary condition t′ + βsl < t < t′ + βrl ensures that the collision occurs within the medium. One can enforce this condition by mutiplying the cross-function νrs by H(tt′ − βst)H(t′ – βrlt), where H(s) is a Heaviside step function. Formulas (10) are valid for arbitrary pump shapes and slownesses.

By combining Eq. (6) with the first of Eqs. (10), one obtains the reciprocity relation

νrs(tr,tsβsl)=νsr(trβrl,ts).
Apart from time shifts, the effect of a signal impulse on the idler equals the effect of an idler impulse on the signal. By combining Eqs. (7) and (12), one finds that vrn(t) = urn(tβrl) and vsn(t) = usn(tβsl). 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
tcβpzc=[βrptβsp(tβrl)]/βrs
and the argument of Aq is similar (pq). In general, the pump function γpq is a nonseparable function of the input and output times, so many Schmidt coefficients are nonzero and the Schmidt functions depend on both pump shapes. However, if the pumps and sidebands copropagate in long medium (βp = βr and βq = βs), the arguments of the shape functions reduce to t − βrl and t′, and the auxiliary conditions can be ignored. In this case, the pump function is separable and there is only one nonzero Schmidt coefficient (ν1=γ/βrs=γ¯), so a heralded signal would have a pure state [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 eiϕr and eiϕs, 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 [52]. Similar results were obtained recently for photon frequency conversion by scalar nondegenerate FWM [50,51].

In the SP, the two-photon state vector

|ψ1=dtrdtsf(tr,ts)ar(tr)as(ts)|vac,
where the two-photon amplitude (TPA)
f(tr,ts)=dtμss(ts,t)νrs(tr,t)=dtμrr(tr,t)νsr(ts,t)
is related to the HP Green functions [44]. 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(tjβjl). Hence, the TPA has the Schmidt decomposition
f(tr,ts)=nνnνrn(tr)νsn(ts),
which involves only the output functions (as it must do). In general, the output is the entangled state nνn|ϕrn|ϕsn where |ϕrn=dtrνrn(tr)ar(tr)|vac, but in the aforementioned special case (copropagating pumps and sidebands), it reduces to the product state ν1|ϕr1〉|ϕs1〉. Once again, notice that the signal and idler Schmidt (basis) functions can be chosen arbitrarily in this case.

To illustrate the low-gain results, it is convenient to choose the pump shapes from the Hermite–Gauss (HG) functions

HGn(s)=Hn(s/τ)exp(s2/2τ2)/π1/4(2nn!τ)1/2,
where Hn(s) is a Hermite polynomial of order n and τ is a duration (temporal width). These functions comprise a potential basis for temporal-mode multiplexing. For example, consider photon generation driven by pumps whose (common) shape function is a Gauss (zeroth-order HG) function (HG0). In Fig. 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 (βrβp and βsβq). Both cases exhibit strong correlations between the input and output times.

 figure: Fig. 1

Fig. 1 Contour plot of the Green function νrs for two cases in which the pump shapes are Gauss (HG0) functions. (a) βpl/τ = βrl/τ = 1 and βql/τ = βsl/τ = −1. (b) βpl/τ = 4 = − βql/τ and βrl/τ = 2 = − βsl/τ. Lighter shading represents higher values, whereas darker shading represents lower values. Time is measured in units of τ. The input and output times are correlated.

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In Fig. 2, the normalized Schmidt coefficient ν/γ¯ 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 [44]. More such results will be illustrated shortly, as low-gain limits of arbitrary-gain results [Figs. 5(a), 7(a)9(a) and 10].

 figure: Fig. 2

Fig. 2 The normalized Schmidt coefficient is plotted as a function of mode number for two HG0 pumps. (a) βpl/τ = βrl/τ = 1 and βql/τ = βsl/τ = −1. (b) βpl/τ = 4 = − βql/τ and βrl/τ = 2 = − βsl/τ. Several coefficients are nonzero.

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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.

 figure: Fig. 3

Fig. 3 Frequency diagrams for vector four-wave mixing in a strongly-birefringent medium [44]. Points on the blue (red) curves denote waves that are aligned with the fast (slow) axes of the medium. The wavenumbers and slownesses (β1) are matched simultaneously for pump frequencies that are (a) degenerate and (b) nondegenerate. Notice that the copropagating waves have different polarizations. The pump and sideband frequencies can be interchanged (pr and qs). The displayed diagrams pertain to the anomalous dispersion regime and similar diagrams pertain to the normal dispersion regime.

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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

μss(t,t)=δ(ttβsl)+γ¯Aq(tβsl)(ξ/η)1/2I1[2γ¯(ξη)1/2]Aq(t)×H(ttβsl)H(t+βrlt),
νrs(t,t)=iγ¯Ap(tβrl)I0[2γ¯(ξη)1/2]Aq(t)H(ttβsl)H(t+βrlt),
where Im(s) is a modified Bessel function of order m and the interaction distances
ξ(t,t)=tβrltds|Ap(s)|2,η(t,t)=ttβslds|Aq(s)|2.
The step functions enforce causality: For each input point (t′, 0) they define the region of influence and for each output point (t, l) they define the domain of dependence. The interaction distances are proportional to the pump energies contained within the interaction region, which is the intersection of the aforementioned domain and region. The formulas for the other two Green functions are similar (rs and ξη).

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

ks(ts,ts)=dtrνrs*(tr,ts)νrs(tr,ts),
whereas the output-idler Schmidt functions are eigenfunctions of the idler kernel
kr(tr,tr)=dtsνrs(tr,ts)νrs*(tr,ts).
It is easy to show that if us (ts) is an eigenfunction of the signal kernel, then vr(tr)=dtsvrs(tr,ts)us(ts) is an eigenfunction of the idler kernel and is associated with the same eigenvalue. Conversely, if vr (tr) is an eigenfunction of the idler kernel, then ur(tr)=dtsνrs*(tr,ts)vr(tr) is an eigenfunction of the idler kernel. Hence, the signal and idler kernels have the same eigenvalues, which are non-negative. The (common) Schmidt coefficients are the square roots of these eigenvalues. Similar statements can be made about the input-idler and output-signal Schmidt functions. In general, the Green functions (interaction distances and step functions) are nonseparable functions of the input and output times, so one should expect several Schmidt coefficients to be nonzero and the Schmidt functions to depend on both pump shapes.

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 tβsl by ∞. With these replacements, the input signal function is determined solely by Aq (t′) and the output idler function is determined solely by Ap (tβrl). The Green function is proportional to γ¯I0{2γ¯[ξ(tβrl)η(t)]1/2}. 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 tβrl 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 (tβsl). The reduced function γ¯I0{2γ¯[ξ(t)η(tβsl)]1/2} 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.

 figure: Fig. 4

Fig. 4 Characteristic diagram for (a) idler generation and (b) signal generation. The horizontal axis is distance and the vertical axis is time. The interaction is limited to the region in which the pumps collide, which is the interior of the dashed diamond. The blue and red rays (arrows) represent parts of the signal and idler that interact strongly.

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4.1. Schmidt coefficients

In Fig. 5, the first (largest) Schmidt coefficient ν1 is plotted as a function of the gain parameter γ¯, for two different combinations of HG pump shapes. In the low-gain regime (γ¯0.5), 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 (γ¯2), 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 2ν12 photons on average.

 figure: Fig. 5

Fig. 5 The first Schmidt coefficient is plotted as a function of the gain parameter γ¯. In (a) the solid line is γ¯ and the vertical scale is linear, whereas in (b) the solid curve is exp(2γ¯)/(16πγ¯)1/2 and the scale is logarithmic. Filled circles denote results for two HG0 pumps, whereas empty circles denote results for one HG0 and one HG1 pump. The Schmidt coefficients do not depend on the pump shapes.

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In Fig. 6, the ratio of the second Schmidt coefficient to the first (ν21) 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) [55]. 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].

 figure: Fig. 6

Fig. 6 (a) The ratio of the second Schmidt coefficient to the first is plotted as a function of the gain parameter γ¯. Filled circles denote results for two HG0 pumps, whereas empty circles denote results for one HG0 and one HG1 pump. The relative probability of the second mode, which is the square of the Schmidt ratio, never exceeds 1%. (b) The Schmidt ratio is plotted as a function of mode number for the worst case, in whichγ¯=1.76. The HG0-HG0 and HG0-HG1 results are indistinguishable.

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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)].

 figure: Fig. 7

Fig. 7 Contour plots of the Green function νrs for the gain parameters (a) γ¯=0.3 and (b) γ¯=3.0. Both pump shapes are Gauss (HG0) functions. Lighter shading represents higher values, whereas darker shading represents lower values. High gain localizes the Green function near early input and output times.

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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 [49]. 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.

 figure: Fig. 8

Fig. 8 The (common) signal and idler Schmidt function (dots) is plotted as a function of time for the gain parameters (a) γ¯=0.3 and (b) γ¯=3.0. Both pump shapes (solid curves) are Gauss (HG0) functions. In the low-gain regime the Schmidt function is approximately Gaussian, whereas in the high-gain regime it is a distorted Gaussian.

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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.

 figure: Fig. 9

Fig. 9 Contour plots of the Green function νrs for the gain parameters (a) γ¯=0.3 and (b) γ¯=3.0. The pump shapes are HG0 and HG1 functions. Lighter shading represents higher values, whereas darker shading represents lower values. High gain localizes the Green function near early input and output times.

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 figure: Fig. 10

Fig. 10 The input signal (a) and output idler (b) Schmidt functions (dots) are plotted as functions of time for the gain parameter γ¯=0.3. The pump shapes (solid curves) are HG0 and HG1 functions. In the low-gain regime both Schmidt functions are approximately HG functions.

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 figure: Fig. 11

Fig. 11 The input signal (a) and output idler (b) Schmidt functions (dots) are plotted as functions of time for the gain parameter γ¯=3.0. The pump shapes (solid curves) are HG0 and HG1 functions. In the high-gain regime both Schmidt functions are distorted HG functions.

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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 [62].

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 [62]. 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.

 figure: Fig. 12

Fig. 12 The ratio of the second Schmidt coefficient to the first is plotted as a function of the gain parameter γ¯ for two Gauss (HG0) pumps. The solid curve denotes semi-analytical results that omit nonlinear phase modulation, whereas crosses denote numerical results that include nonlinear phase modulation. The results are qualitatively similar and quantitatively comparable.

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 figure: Fig. 13

Fig. 13 The amplitudes (blue curves) and phases (red curves) of the input signal (a) and output idler (b) Schmidt functions are plotted as functions of time for the gain parameter γ¯=1.55. Both pump shapes are Gauss (HG0) functions. Nonlinear phase modulation imposes opposite chirps and frequency shifts on the signal and idler. The chirps are extremal near the peaks of the copropagating pumps, relative to which the signal and idler are skewed toward early times.

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In Fig. 12, the ratio of the second Schmidt coefficient to the first (ν21) is plotted as a function of the gain parameter γ¯. (For the chosen parameters, γ¯=1 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 [62]. 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 γ¯=4. 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 γ¯=1.55, 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.

6. Summary

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 [52]. 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 [44] 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

(z+βrt)Ar=iγpqAs*+σrδ(z)δ(t),(z+βst)As=iγpqAr*+σsδ(z)δ(t),
where the pump function γpq = γAp (tβpz)Aq(tβqz) and σj = 0 or 1, depending on which mode is seeded by an impulse. The physical CMEs are difficult to solve, because they are partial differential equations with coefficients that depend on distance and time. However, there is at least one important case in which they can be solved.

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 = βrzt and y = tβsz. Then the characteristic CMEs are

yAr=iγ¯Ap(x)Aq(y)As*+σrδ(x)δ(y),xAs=iγ¯Ap(x)Aq(y)Ar*+σsδ(x)δ(y),
where γ¯=γ/βrs, βrs = βr − βs and δ(z)δ(t) = βrsδ(x)δ(y).

Now let Ar/ApAr and As/AqAs, and define the interaction distances ξ=x0|Ap(x)|2dx and η=0y|Aq(y)|2dy (The normalization factors Ap and Aq are evaluated at −x and y, respectively.) Then the canonical CMEs are

ηAr=iγ¯As*+σrAp*δ(ξ)δ(η),ξAs=iγ¯Ar*+σsAq*δ(ξ)δ(η),
where δ(x) = |Ap|2δ(ξ) and δ(y) = |Aq|2δ(η). (The source strengths |Ap|2 and |Aq|2 are evaluated at 0.) Notice that Eqs. (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]
Grs(ξ,η)=iγ¯I0[2γ¯(ξη)1/2]H(ξ)H(η),
Gss(ξ,η)=γ¯(ξ/η)1/2I1[2γ¯(ξη)1/2]H(ξ)H(η)+H(ξ)δ(η),
where Im(s) is a modified Bessel function of order m and H(s) is a Heaviside step function. The Green functions associated with an idler source are similar (rs and ξη). In particular, the cross functions satisfy the reciprocity relation Grs (ξ, η) = Gsr (ξ, η). To account for sources at other locations, one makes the replacements ξξξ′ and ηηη′.

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 (γ¯1), Grsγ¯. One can rewrite this result in the Schmidt form Grs(ξ,η)v(ξ)vu(η), where v=γ¯ and u(s) = v(s) = H(s)H(1 − s). In the high-gain regime (γ¯1), one can use the asymptotic formula I0(s) ~ exp(s)/(2πs)1/2 to write

Grs(ξ,η)~γ¯exp[2γ¯ξ1/2(1η)1/2](4πγ¯)1/2~γ¯exp[2γ¯γ¯(1ξ)γ¯η](4πγ¯)1/2.
Hence, Grs (ξ, η′) ∼ v(1 − ξ)νu(η′), where ν=exp(2γ¯)/(16πγ¯)1/2 and u(s)=ν(s)=(2γ¯)1/2exp(γ¯s). The signal function is skewed toward smaller values of η′ and the idler function is skewed toward larger values of ξ. In both the low- and high-gain regimes, the decomposition involves only a single pair of Schmidt modes. Second, consider Gsr, for which the effective input (idler) point is (ξ′, 0) and the effective output (signal) point is (1, η). The decomposition of this cross-function is similar to the previous decomposition, the only differences being that the idler function is skewed toward smaller values of ξ′ and the signal function is skewed toward larger values of η. Apart from reflections in characteristic coordinates, the signal and idler Schmidt functions are identical, as required by reciprocity.

To convert the canonical Green functions to physical functions, one multiplies Grr and Gsr by Ap*(0), and Grs and Gss by Aq*(0). 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′, − xtβrz and ytβsz.) The final results are

Grs(t,t)=γ¯Ap(tβrl)I0[2γ¯(ξη)1/2]Aq(t)×H(ttβsl)H(t+βrlt),
Gss(t,t)=δ(ttβsl)+γ¯Ap(tβsl)(ξ/η)1/2I1[2γ¯(ξη)1/2]Aq(t)×H(ttβsl)H(t+βrlt),
where the interaction distances
ξ(t,t)=tβrltds|Ap(s)|2,η(t,t)=ttβslds|Aq(s)|2.
The physical Schmidt coefficients equal the canonical coefficients, and the physical Schmidt functions are the canonical functions multiplied by the appropriate pump-shape functions. (The canonical functions skew the physical functions.) For a system with a Hamiltonian that is quadratic in the mode amplitudes (operators), the Heisenberg equations for the mode operators are identical to the Hamilton equations for the mode amplitudes. Hence, the quantum Green functions are identical to the classical functions derived in this appendix.

Funding

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).

Acknowledgments

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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References

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  1. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).
  2. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
    [Crossref]
  3. P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Downing, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
    [Crossref]
  4. I. A. Walmsley, “Quantum optics: Science and technology in a new light,” Science 348, 525–530 (2015).
    [Crossref] [PubMed]
  5. D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
    [Crossref]
  6. R. W. Boyd, Nonlinear Optics, 3rd Ed. (Academic, 2008).
  7. M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
    [Crossref]
  8. M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge University, 2007).
    [Crossref]
  9. C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
    [Crossref] [PubMed]
  10. I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science 307, 1733–1734 (2005).
    [Crossref] [PubMed]
  11. A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).
  12. K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14870–14886 (2007).
    [Crossref] [PubMed]
  13. P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
    [Crossref] [PubMed]
  14. M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17, 4670–4676 (2009).
    [Crossref] [PubMed]
  15. A. I. Lvovsky, “Squeezed light,” in Photonics: Scientific Foundations, Technology and Applications, Volume 1, edited by D. Andrews, ed. (Wiley, 2015), pp. 121–164.
  16. M. Chekhova, G. Leuchs, and M. Zukowski, “Bright squeezed vacuum: Entanglement of macroscopic light beams,” Opt. Commun. 337, 27–43 (2015).
    [Crossref]
  17. O. Aytur and P. Kumar, “Pulsed twin beams of light,” Phys. Rev. Lett. 65, 1551–1554 (1990).
    [Crossref] [PubMed]
  18. J. E. Sharping, M. Fiorentino, and P. Kumar, “Observation of twin-beam-type quantum correlation in optical fiber,” Opt. Lett. 26, 367–369 (2001).
    [Crossref]
  19. Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
    [Crossref] [PubMed]
  20. M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
    [Crossref]
  21. M. Cooper, L. J. Wright, C. Söller, and B. J. Smith, “Experimental generation of multi-photon Fock states,” Opt. Express 21, 5311–5317 (2013).
    [Crossref]
  22. A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schrödinger cats from photon number states,” Nature 448, 784–786 (2007).
    [Crossref] [PubMed]
  23. S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
    [Crossref]
  24. U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev. 4, 337–354 (2010).
    [Crossref]
  25. D. Gottesman and J. Preskill, “Secure quantum key distribution using squeezed states,” Phys. Rev. A 63, 022309 (2001).
    [Crossref]
  26. A. C. Funk and M. G. Raymer, “Quantum key distribution using non-classical photon number correlations in macroscopic light pulses,” Phys. Rev. A 65, 042307 (2002).
    [Crossref]
  27. A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
    [Crossref] [PubMed]
  28. N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light,” Science 332, 330–332 (2011).
    [Crossref] [PubMed]
  29. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306, 1330–1336 (2004).
    [Crossref] [PubMed]
  30. J. P. Dowling and K. P. Seshadreesan, “Quantum Optical Technologies for Metrology, Sensing, and Imaging,” J. Lightwave Technol. 33, 2359–2370 (2015).
    [Crossref]
  31. M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278–281 (1987).
    [Crossref] [PubMed]
  32. G. Y. Xiang, H. F. Hofmann, and G. J. Pryde, “Optimal Multi-Photon Phase Sensing with a Single Interference Fringe,” Sci. Rep. 32684 (2013).
    [Crossref] [PubMed]
  33. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [Crossref]
  34. R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Commun. 1, 121 (2010).
    [Crossref] [PubMed]
  35. B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
    [Crossref]
  36. F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
    [Crossref] [PubMed]
  37. S. L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A 71, 055801 (2005).
    [Crossref]
  38. W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73, 063819 (2006).
    [Crossref]
  39. C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic theory and simple examples,” Opt. Express 21, 1374–1394 (2013).
    [Crossref] [PubMed]
  40. G. Ferrini, I. Fsaifes, T. Labidi, F. Goldfarb, N. Treps, and F. Bretenaker, “Symplectic approach to the amplification process in a nonlinear fiber: role of signal-idler correlations and application to loss management,” J. Opt. Soc. Am. B 31, 1627–1641 (2014).
    [Crossref]
  41. R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford University, 2000).
  42. C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005). Two-mode squeezed vacuum states are discussed in Sec. 7.7.
  43. B. Fang, O. Cohen, J. B. Moreno, and V. O. Lorenz, “State engineering of photon pairs produced through dual-pump spontaneous four-wave mixing,” Opt. Express 21, 2707–2717 (2013).
    [Crossref] [PubMed]
  44. J. B. Christensen, C. J. McKinstrie, and K. Rottwitt, “Temporally uncorrelated photon-pair generation by dual-pump four-wave mixing,” Phys. Rev. A 94, 013819 (2016).
    [Crossref]
  45. C. J. McKinstrie and G. G. Luther, “The modulational instability of colinear waves,” Phys. Scripta 30, 31–40 (1990).
    [Crossref]
  46. J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
    [Crossref] [PubMed]
  47. P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
    [Crossref]
  48. D. Amans, E. Brainis, M. Haelterman, P. Emplit, and S. Massar, “Vector modulation instability induced by vacuum fluctuations in highly birefringent fibers in the anomalous-dispersion regime,” Opt. Lett. 30, 1051–1053 (2005).
    [Crossref] [PubMed]
  49. M. G. Raymer, “Quantum state entanglement and readout of collective atomic-ensemble modes and optical wave packets by stimulated Raman scattering,” J. Mod. Opt. 51, 1739–1759 (2004).
    [Crossref]
  50. L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime,” Opt. Express 20, 8367–8396 (2012).
    [Crossref] [PubMed]
  51. C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A 85, 053829 (2012).
    [Crossref]
  52. B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: A complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).
  53. M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: Unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A 24, 1980–1993 (1981).
    [Crossref]
  54. R. E. Giacone, C. J. McKinstrie, and R. Betti, “Angular dependence of stimulated Brillouin scattering in homogeneous plasma,” Phys. Plasmas 2, 4596–4605 (1995).
    [Crossref]
  55. M. G. Raymer, K. Rzazewski, and J. Mostowski, “Pulse-energy statistics in stimulated Raman scattering,” Opt. Lett. 7, 71–73 (1982).
    [Crossref] [PubMed]
  56. D. L. Bobroff, “Coupled-modes analysis of the photon-photon parametric backward-wave oscillator,” J. Appl. Phys. 36, 1760–1769 (1965).
    [Crossref]
  57. M. G. Raymer, Z. W. Li, and I. A. Walmsley, “Temporal quantum fluctuations in stimulated Raman scattering: Coherent-modes description,” Phys. Rev. Lett. 63, 1586–1589 (1989).
    [Crossref] [PubMed]
  58. N. Liu, Y. Liu, X. Guo, L. Yang, X. Li, and Z. Y. Ou, “Approaching single temporal mode operation in twin beams generated by pulse pumped high gain spontaneous four wave mixing,” Opt. Express 24, 1096–1108 (2016).
    [Crossref] [PubMed]
  59. L. Mejling, D. S. Cargill, C. J. McKinstrie, K. Rottwitt, and R. O. Moore, “Effects of nonlinear phase modulation on Bragg scattering in the low-conversion regime,” Opt. Express 20, 27454–27475 (2012).
    [Crossref] [PubMed]
  60. B. Bell, A. McMillan, W. McCutcheon, and J. Rarity, “On the effects of self- and cross-phase modulation on photon purity for four-wave mixing photon-pair sources,” Phys. Rev. A. 92, 053849 (2015).
    [Crossref]
  61. G. F. Sinclair and M. G. Thompson, “Effect of self- and cross-phase modulation on photon pairs generated by spontaneous four-wave mixing in integrated optical waveguides,” Phys. Rev. A 94, 063855 (2016).
    [Crossref]
  62. C. J. McKinstrie, J. B. Christensen, K. Rottwitt, and M. G. Raymer, “Single-temporal-mode photon generation beyond the low-power regime,” Quantum Information and Measurement conference, Paris, France, 5–7 April 2017, paper QW3C.6. In the simulations, β1 = 10s/cm, β2 = 0.2 ps2/cm and l = 1.0 cm.

2016 (3)

J. B. Christensen, C. J. McKinstrie, and K. Rottwitt, “Temporally uncorrelated photon-pair generation by dual-pump four-wave mixing,” Phys. Rev. A 94, 013819 (2016).
[Crossref]

N. Liu, Y. Liu, X. Guo, L. Yang, X. Li, and Z. Y. Ou, “Approaching single temporal mode operation in twin beams generated by pulse pumped high gain spontaneous four wave mixing,” Opt. Express 24, 1096–1108 (2016).
[Crossref] [PubMed]

G. F. Sinclair and M. G. Thompson, “Effect of self- and cross-phase modulation on photon pairs generated by spontaneous four-wave mixing in integrated optical waveguides,” Phys. Rev. A 94, 063855 (2016).
[Crossref]

2015 (5)

B. Bell, A. McMillan, W. McCutcheon, and J. Rarity, “On the effects of self- and cross-phase modulation on photon purity for four-wave mixing photon-pair sources,” Phys. Rev. A. 92, 053849 (2015).
[Crossref]

B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: A complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).

I. A. Walmsley, “Quantum optics: Science and technology in a new light,” Science 348, 525–530 (2015).
[Crossref] [PubMed]

M. Chekhova, G. Leuchs, and M. Zukowski, “Bright squeezed vacuum: Entanglement of macroscopic light beams,” Opt. Commun. 337, 27–43 (2015).
[Crossref]

J. P. Dowling and K. P. Seshadreesan, “Quantum Optical Technologies for Metrology, Sensing, and Imaging,” J. Lightwave Technol. 33, 2359–2370 (2015).
[Crossref]

2014 (2)

2013 (4)

B. Fang, O. Cohen, J. B. Moreno, and V. O. Lorenz, “State engineering of photon pairs produced through dual-pump spontaneous four-wave mixing,” Opt. Express 21, 2707–2717 (2013).
[Crossref] [PubMed]

C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic theory and simple examples,” Opt. Express 21, 1374–1394 (2013).
[Crossref] [PubMed]

G. Y. Xiang, H. F. Hofmann, and G. J. Pryde, “Optimal Multi-Photon Phase Sensing with a Single Interference Fringe,” Sci. Rep. 32684 (2013).
[Crossref] [PubMed]

M. Cooper, L. J. Wright, C. Söller, and B. J. Smith, “Experimental generation of multi-photon Fock states,” Opt. Express 21, 5311–5317 (2013).
[Crossref]

2012 (3)

2011 (1)

N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light,” Science 332, 330–332 (2011).
[Crossref] [PubMed]

2010 (2)

U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev. 4, 337–354 (2010).
[Crossref]

R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Commun. 1, 121 (2010).
[Crossref] [PubMed]

2009 (2)

M. Halder, J. Fulconis, B. Cemlyn, A. Clark, C. Xiong, W. J. Wadsworth, and J. G. Rarity, “Nonclassical 2-photon interference with separate intrinsically narrowband fibre sources,” Opt. Express 17, 4670–4676 (2009).
[Crossref] [PubMed]

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

2008 (1)

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[Crossref] [PubMed]

2007 (3)

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14870–14886 (2007).
[Crossref] [PubMed]

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Downing, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schrödinger cats from photon number states,” Nature 448, 784–786 (2007).
[Crossref] [PubMed]

2006 (1)

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73, 063819 (2006).
[Crossref]

2005 (5)

D. Amans, E. Brainis, M. Haelterman, P. Emplit, and S. Massar, “Vector modulation instability induced by vacuum fluctuations in highly birefringent fibers in the anomalous-dispersion regime,” Opt. Lett. 30, 1051–1053 (2005).
[Crossref] [PubMed]

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

S. L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A 71, 055801 (2005).
[Crossref]

I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science 307, 1733–1734 (2005).
[Crossref] [PubMed]

A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

2004 (2)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref] [PubMed]

M. G. Raymer, “Quantum state entanglement and readout of collective atomic-ensemble modes and optical wave packets by stimulated Raman scattering,” J. Mod. Opt. 51, 1739–1759 (2004).
[Crossref]

2002 (3)

A. C. Funk and M. G. Raymer, “Quantum key distribution using non-classical photon number correlations in macroscopic light pulses,” Phys. Rev. A 65, 042307 (2002).
[Crossref]

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

2001 (2)

J. E. Sharping, M. Fiorentino, and P. Kumar, “Observation of twin-beam-type quantum correlation in optical fiber,” Opt. Lett. 26, 367–369 (2001).
[Crossref]

D. Gottesman and J. Preskill, “Secure quantum key distribution using squeezed states,” Phys. Rev. A 63, 022309 (2001).
[Crossref]

1998 (1)

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref] [PubMed]

1995 (1)

R. E. Giacone, C. J. McKinstrie, and R. Betti, “Angular dependence of stimulated Brillouin scattering in homogeneous plasma,” Phys. Plasmas 2, 4596–4605 (1995).
[Crossref]

1992 (1)

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

1990 (4)

O. Aytur and P. Kumar, “Pulsed twin beams of light,” Phys. Rev. Lett. 65, 1551–1554 (1990).
[Crossref] [PubMed]

C. J. McKinstrie and G. G. Luther, “The modulational instability of colinear waves,” Phys. Scripta 30, 31–40 (1990).
[Crossref]

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
[Crossref] [PubMed]

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[Crossref]

1989 (1)

M. G. Raymer, Z. W. Li, and I. A. Walmsley, “Temporal quantum fluctuations in stimulated Raman scattering: Coherent-modes description,” Phys. Rev. Lett. 63, 1586–1589 (1989).
[Crossref] [PubMed]

1987 (2)

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref] [PubMed]

M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278–281 (1987).
[Crossref] [PubMed]

1986 (1)

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

1982 (1)

1981 (2)

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: Unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A 24, 1980–1993 (1981).
[Crossref]

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[Crossref]

1970 (1)

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[Crossref]

1965 (1)

D. L. Bobroff, “Coupled-modes analysis of the photon-photon parametric backward-wave oscillator,” J. Appl. Phys. 36, 1760–1769 (1965).
[Crossref]

Amans, D.

Andersen, U. L.

U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev. 4, 337–354 (2010).
[Crossref]

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Aytur, O.

O. Aytur and P. Kumar, “Pulsed twin beams of light,” Phys. Rev. Lett. 65, 1551–1554 (1990).
[Crossref] [PubMed]

Bachor, H. A.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Banaszek, K.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73, 063819 (2006).
[Crossref]

A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

Bell, B.

B. Bell, A. McMillan, W. McCutcheon, and J. Rarity, “On the effects of self- and cross-phase modulation on photon purity for four-wave mixing photon-pair sources,” Phys. Rev. A. 92, 053849 (2015).
[Crossref]

Benichi, H.

N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light,” Science 332, 330–332 (2011).
[Crossref] [PubMed]

Betti, R.

R. E. Giacone, C. J. McKinstrie, and R. Betti, “Angular dependence of stimulated Brillouin scattering in homogeneous plasma,” Phys. Plasmas 2, 4596–4605 (1995).
[Crossref]

Bobroff, D. L.

D. L. Bobroff, “Coupled-modes analysis of the photon-photon parametric backward-wave oscillator,” J. Appl. Phys. 36, 1760–1769 (1965).
[Crossref]

Bowen, W. P.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics, 3rd Ed. (Academic, 2008).

Brainis, E.

Braunstein, S. L.

S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77, 513–577 (2005).
[Crossref]

S. L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A 71, 055801 (2005).
[Crossref]

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref] [PubMed]

Brecht, B.

B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: A complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).

Bretenaker, F.

Burnham, D. C.

D. C. Burnham and D. L. Weinberg, “Observation of simultaneity in parametric production of optical photon pairs,” Phys. Rev. Lett. 25, 84–87 (1970).
[Crossref]

Cargill, D. S.

Cavalcanti, E. G.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
[Crossref]

Cemlyn, B.

Chekhova, M.

M. Chekhova, G. Leuchs, and M. Zukowski, “Bright squeezed vacuum: Entanglement of macroscopic light beams,” Opt. Commun. 337, 27–43 (2015).
[Crossref]

Christensen, J. B.

J. B. Christensen, C. J. McKinstrie, and K. Rottwitt, “Temporally uncorrelated photon-pair generation by dual-pump four-wave mixing,” Phys. Rev. A 94, 013819 (2016).
[Crossref]

C. J. McKinstrie, J. B. Christensen, K. Rottwitt, and M. G. Raymer, “Single-temporal-mode photon generation beyond the low-power regime,” Quantum Information and Measurement conference, Paris, France, 5–7 April 2017, paper QW3C.6. In the simulations, β1 = 10s/cm, β2 = 0.2 ps2/cm and l = 1.0 cm.

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Clark, A.

Cohen, O.

Cooper, M.

M. Cooper, L. J. Wright, C. Söller, and B. J. Smith, “Experimental generation of multi-photon Fock states,” Opt. Express 21, 5311–5317 (2013).
[Crossref]

Dowling, J. P.

Downing, J. P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Downing, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Drummond, P. D.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[Crossref]

Dudley, J. M.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[Crossref]

Emplit, P.

Erdmann, R.

A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

Fang, B.

Ferrini, G.

Fiorentino, M.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

J. E. Sharping, M. Fiorentino, and P. Kumar, “Observation of twin-beam-type quantum correlation in optical fiber,” Opt. Lett. 26, 367–369 (2001).
[Crossref]

Fsaifes, I.

Fuchs, C. A.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref] [PubMed]

Fulconis, J.

Funk, A. C.

A. C. Funk and M. G. Raymer, “Quantum key distribution using non-classical photon number correlations in macroscopic light pulses,” Phys. Rev. A 65, 042307 (2002).
[Crossref]

Furusawa, A.

N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light,” Science 332, 330–332 (2011).
[Crossref] [PubMed]

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref] [PubMed]

Garay-Palmett, K.

Gerry, C. C.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005). Two-mode squeezed vacuum states are discussed in Sec. 7.7.

Giacone, R. E.

R. E. Giacone, C. J. McKinstrie, and R. Betti, “Angular dependence of stimulated Brillouin scattering in homogeneous plasma,” Phys. Plasmas 2, 4596–4605 (1995).
[Crossref]

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref] [PubMed]

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Goldfarb, F.

Gottesman, D.

D. Gottesman and J. Preskill, “Secure quantum key distribution using squeezed states,” Phys. Rev. A 63, 022309 (2001).
[Crossref]

Grangier, P.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schrödinger cats from photon number states,” Nature 448, 784–786 (2007).
[Crossref] [PubMed]

Grice, W. P.

A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

Guo, X.

Haelterman, M.

Halder, M.

Harvey, J. D.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[Crossref]

Hofmann, H. F.

G. Y. Xiang, H. F. Hofmann, and G. J. Pryde, “Optimal Multi-Photon Phase Sensing with a Single Interference Fringe,” Sci. Rep. 32684 (2013).
[Crossref] [PubMed]

Hong, C. K.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref] [PubMed]

Hudelist, F.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Huntington, E.

N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light,” Science 332, 330–332 (2011).
[Crossref] [PubMed]

Jeong, H.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schrödinger cats from photon number states,” Nature 448, 784–786 (2007).
[Crossref] [PubMed]

Jing, J.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Karlsson, M.

Kennedy, T. A. B.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[Crossref]

Kimble, H. J.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref] [PubMed]

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
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M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278–281 (1987).
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Klauder, J. R.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

Knight, P. L.

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005). Two-mode squeezed vacuum states are discussed in Sec. 7.7.

Kok, P.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Downing, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Kong, J.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Kumar, P.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

J. E. Sharping, M. Fiorentino, and P. Kumar, “Observation of twin-beam-type quantum correlation in optical fiber,” Opt. Lett. 26, 367–369 (2001).
[Crossref]

O. Aytur and P. Kumar, “Pulsed twin beams of light,” Phys. Rev. Lett. 65, 1551–1554 (1990).
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Labidi, T.

Lam, P. K.

R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Commun. 1, 121 (2010).
[Crossref] [PubMed]

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Lee, N.

N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light,” Science 332, 330–332 (2011).
[Crossref] [PubMed]

Leonhardt, R.

P. D. Drummond, T. A. B. Kennedy, J. M. Dudley, R. Leonhardt, and J. D. Harvey, “Cross-phase modulational instability in high-birefringence fibers,” Opt. Commun. 78, 137–142 (1990).
[Crossref]

Leuchs, G.

M. Chekhova, G. Leuchs, and M. Zukowski, “Bright squeezed vacuum: Entanglement of macroscopic light beams,” Opt. Commun. 337, 27–43 (2015).
[Crossref]

U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev. 4, 337–354 (2010).
[Crossref]

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Li, X.

Li, Z. W.

M. G. Raymer, Z. W. Li, and I. A. Walmsley, “Temporal quantum fluctuations in stimulated Raman scattering: Coherent-modes description,” Phys. Rev. Lett. 63, 1586–1589 (1989).
[Crossref] [PubMed]

Liu, C.

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Liu, N.

Liu, Y.

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306, 1330–1336 (2004).
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Lorenz, V. O.

Loudon, R.

R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford University, 2000).

Lundeen, J. S.

Luther, G. G.

C. J. McKinstrie and G. G. Luther, “The modulational instability of colinear waves,” Phys. Scripta 30, 31–40 (1990).
[Crossref]

Lvovsky, A. I.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73, 063819 (2006).
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A. I. Lvovsky, “Squeezed light,” in Photonics: Scientific Foundations, Technology and Applications, Volume 1, edited by D. Andrews, ed. (Wiley, 2015), pp. 121–164.

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: Beating the standard quantum limit,” Science 306, 1330–1336 (2004).
[Crossref] [PubMed]

Mandel, L.

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
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Marhic, M. E.

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge University, 2007).
[Crossref]

Massar, S.

Mavalvala, N.

R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Commun. 1, 121 (2010).
[Crossref] [PubMed]

McCall, S. L.

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033–4054 (1986).
[Crossref]

McClelland, D. E.

R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Commun. 1, 121 (2010).
[Crossref] [PubMed]

McCutcheon, W.

B. Bell, A. McMillan, W. McCutcheon, and J. Rarity, “On the effects of self- and cross-phase modulation on photon purity for four-wave mixing photon-pair sources,” Phys. Rev. A. 92, 053849 (2015).
[Crossref]

McGuinness, H. J.

McKinstrie, C. J.

J. B. Christensen, C. J. McKinstrie, and K. Rottwitt, “Temporally uncorrelated photon-pair generation by dual-pump four-wave mixing,” Phys. Rev. A 94, 013819 (2016).
[Crossref]

C. J. McKinstrie and M. Karlsson, “Schmidt decompositions of parametric processes I: Basic theory and simple examples,” Opt. Express 21, 1374–1394 (2013).
[Crossref] [PubMed]

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime,” Opt. Express 20, 8367–8396 (2012).
[Crossref] [PubMed]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A 85, 053829 (2012).
[Crossref]

L. Mejling, D. S. Cargill, C. J. McKinstrie, K. Rottwitt, and R. O. Moore, “Effects of nonlinear phase modulation on Bragg scattering in the low-conversion regime,” Opt. Express 20, 27454–27475 (2012).
[Crossref] [PubMed]

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14870–14886 (2007).
[Crossref] [PubMed]

R. E. Giacone, C. J. McKinstrie, and R. Betti, “Angular dependence of stimulated Brillouin scattering in homogeneous plasma,” Phys. Plasmas 2, 4596–4605 (1995).
[Crossref]

C. J. McKinstrie and G. G. Luther, “The modulational instability of colinear waves,” Phys. Scripta 30, 31–40 (1990).
[Crossref]

C. J. McKinstrie, J. B. Christensen, K. Rottwitt, and M. G. Raymer, “Single-temporal-mode photon generation beyond the low-power regime,” Quantum Information and Measurement conference, Paris, France, 5–7 April 2017, paper QW3C.6. In the simulations, β1 = 10s/cm, β2 = 0.2 ps2/cm and l = 1.0 cm.

McMillan, A.

B. Bell, A. McMillan, W. McCutcheon, and J. Rarity, “On the effects of self- and cross-phase modulation on photon purity for four-wave mixing photon-pair sources,” Phys. Rev. A. 92, 053849 (2015).
[Crossref]

Mejling, L.

Milburn, G. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Downing, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Moore, R. O.

Moreno, J. B.

Mosley, P. J.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[Crossref] [PubMed]

Mostowski, J.

M. G. Raymer, K. Rzazewski, and J. Mostowski, “Pulse-energy statistics in stimulated Raman scattering,” Opt. Lett. 7, 71–73 (1982).
[Crossref] [PubMed]

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: Unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A 24, 1980–1993 (1981).
[Crossref]

Munro, W. J.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Downing, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Nemoto, K.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Downing, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
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Nielsen, M. A.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

Ou, Z. Y.

N. Liu, Y. Liu, X. Guo, L. Yang, X. Li, and Z. Y. Ou, “Approaching single temporal mode operation in twin beams generated by pulse pumped high gain spontaneous four wave mixing,” Opt. Express 24, 1096–1108 (2016).
[Crossref] [PubMed]

F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

C. K. Hong, Z. Y. Ou, and L. Mandel, “Measurement of subpicosecond time intervals between two photons by interference,” Phys. Rev. Lett. 59, 2044–2046 (1987).
[Crossref] [PubMed]

Ourjoumtsev, A.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schrödinger cats from photon number states,” Nature 448, 784–786 (2007).
[Crossref] [PubMed]

Peng, K. C.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

Pereira, S. F.

Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, “Realization of the Einstein-Podolsky-Rosen paradox for continuous variables,” Phys. Rev. Lett. 68, 3663–3666 (1992).
[Crossref] [PubMed]

Polzik, E. S.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref] [PubMed]

Preskill, J.

D. Gottesman and J. Preskill, “Secure quantum key distribution using squeezed states,” Phys. Rev. A 63, 022309 (2001).
[Crossref]

Pryde, G. J.

G. Y. Xiang, H. F. Hofmann, and G. J. Pryde, “Optimal Multi-Photon Phase Sensing with a Single Interference Fringe,” Sci. Rep. 32684 (2013).
[Crossref] [PubMed]

Radic, S.

Radzewicz, C.

W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73, 063819 (2006).
[Crossref]

Ralph, T. C.

P. Kok, W. J. Munro, K. Nemoto, T. C. Ralph, J. P. Downing, and G. J. Milburn, “Linear optical quantum computing with photonic qubits,” Rev. Mod. Phys. 79, 135–174 (2007).
[Crossref]

Rangel-Rojo, R.

Rarity, J.

B. Bell, A. McMillan, W. McCutcheon, and J. Rarity, “On the effects of self- and cross-phase modulation on photon purity for four-wave mixing photon-pair sources,” Phys. Rev. A. 92, 053849 (2015).
[Crossref]

Rarity, J. G.

Raymer, M. G.

B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: A complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A 85, 053829 (2012).
[Crossref]

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime,” Opt. Express 20, 8367–8396 (2012).
[Crossref] [PubMed]

K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14870–14886 (2007).
[Crossref] [PubMed]

A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science 307, 1733–1734 (2005).
[Crossref] [PubMed]

M. G. Raymer, “Quantum state entanglement and readout of collective atomic-ensemble modes and optical wave packets by stimulated Raman scattering,” J. Mod. Opt. 51, 1739–1759 (2004).
[Crossref]

A. C. Funk and M. G. Raymer, “Quantum key distribution using non-classical photon number correlations in macroscopic light pulses,” Phys. Rev. A 65, 042307 (2002).
[Crossref]

M. G. Raymer, Z. W. Li, and I. A. Walmsley, “Temporal quantum fluctuations in stimulated Raman scattering: Coherent-modes description,” Phys. Rev. Lett. 63, 1586–1589 (1989).
[Crossref] [PubMed]

M. G. Raymer, K. Rzazewski, and J. Mostowski, “Pulse-energy statistics in stimulated Raman scattering,” Opt. Lett. 7, 71–73 (1982).
[Crossref] [PubMed]

M. G. Raymer and J. Mostowski, “Stimulated Raman scattering: Unified treatment of spontaneous initiation and spatial propagation,” Phys. Rev. A 24, 1980–1993 (1981).
[Crossref]

C. J. McKinstrie, J. B. Christensen, K. Rottwitt, and M. G. Raymer, “Single-temporal-mode photon generation beyond the low-power regime,” Quantum Information and Measurement conference, Paris, France, 5–7 April 2017, paper QW3C.6. In the simulations, β1 = 10s/cm, β2 = 0.2 ps2/cm and l = 1.0 cm.

Reddy, D. V.

B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: A complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).

Reid, M. D.

M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81, 1727–1751 (2009).
[Crossref]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Rothenberg, J. E.

J. E. Rothenberg, “Modulational instability for normal dispersion,” Phys. Rev. A 42, 682–685 (1990).
[Crossref] [PubMed]

Rottwitt, K.

J. B. Christensen, C. J. McKinstrie, and K. Rottwitt, “Temporally uncorrelated photon-pair generation by dual-pump four-wave mixing,” Phys. Rev. A 94, 013819 (2016).
[Crossref]

L. Mejling, C. J. McKinstrie, M. G. Raymer, and K. Rottwitt, “Quantum frequency translation by four-wave mixing in a fiber: low-conversion regime,” Opt. Express 20, 8367–8396 (2012).
[Crossref] [PubMed]

C. J. McKinstrie, L. Mejling, M. G. Raymer, and K. Rottwitt, “Quantum-state preserving optical frequency conversion and pulse reshaping by four-wave mixing,” Phys. Rev. A 85, 053829 (2012).
[Crossref]

L. Mejling, D. S. Cargill, C. J. McKinstrie, K. Rottwitt, and R. O. Moore, “Effects of nonlinear phase modulation on Bragg scattering in the low-conversion regime,” Opt. Express 20, 27454–27475 (2012).
[Crossref] [PubMed]

C. J. McKinstrie, J. B. Christensen, K. Rottwitt, and M. G. Raymer, “Single-temporal-mode photon generation beyond the low-power regime,” Quantum Information and Measurement conference, Paris, France, 5–7 April 2017, paper QW3C.6. In the simulations, β1 = 10s/cm, β2 = 0.2 ps2/cm and l = 1.0 cm.

Rzazewski, K.

Schnabel, R.

R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Commun. 1, 121 (2010).
[Crossref] [PubMed]

Seshadreesan, K. P.

Sharping, J. E.

M. Fiorentino, P. L. Voss, J. E. Sharping, and P. Kumar, “All-fiber photon-pair source for quantum communications,” IEEE Photon. Technol. Lett. 14, 983–985 (2002).
[Crossref]

J. E. Sharping, M. Fiorentino, and P. Kumar, “Observation of twin-beam-type quantum correlation in optical fiber,” Opt. Lett. 26, 367–369 (2001).
[Crossref]

Silberhorn, C.

B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: A complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).

U. L. Andersen, G. Leuchs, and C. Silberhorn, “Continuous-variable quantum information processing,” Laser Photon. Rev. 4, 337–354 (2010).
[Crossref]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[Crossref] [PubMed]

A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

Sinclair, G. F.

G. F. Sinclair and M. G. Thompson, “Effect of self- and cross-phase modulation on photon pairs generated by spontaneous four-wave mixing in integrated optical waveguides,” Phys. Rev. A 94, 063855 (2016).
[Crossref]

Smith, B. J.

M. Cooper, L. J. Wright, C. Söller, and B. J. Smith, “Experimental generation of multi-photon Fock states,” Opt. Express 21, 5311–5317 (2013).
[Crossref]

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
[Crossref] [PubMed]

Söller, C.

M. Cooper, L. J. Wright, C. Söller, and B. J. Smith, “Experimental generation of multi-photon Fock states,” Opt. Express 21, 5311–5317 (2013).
[Crossref]

Sorensen, J. L.

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref] [PubMed]

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N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light,” Science 332, 330–332 (2011).
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Takeno, Y.

N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light,” Science 332, 330–332 (2011).
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Thompson, M. G.

G. F. Sinclair and M. G. Thompson, “Effect of self- and cross-phase modulation on photon pairs generated by spontaneous four-wave mixing in integrated optical waveguides,” Phys. Rev. A 94, 063855 (2016).
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Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
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A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical Schrödinger cats from photon number states,” Nature 448, 784–786 (2007).
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U’Ren, A. B.

P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14870–14886 (2007).
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A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

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I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science 307, 1733–1734 (2005).
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A. B. U’Ren, C. Silberhorn, K. Banaszek, I. A. Walmsley, R. Erdmann, W. P. Grice, and M. G. Raymer, “Generation of pure-state single-photon wavepackets by conditional preparation based on spontaneous parametric downconversion,” Laser Phys. 15, 146–161 (2005).

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P. J. Mosley, J. S. Lundeen, B. J. Smith, P. Wasylczyk, A. B. U’Ren, C. Silberhorn, and I. A. Walmsley, “Heralded generation of ultrafast single photons in pure quantum states,” Phys. Rev. Lett. 100, 133601 (2008).
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N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light,” Science 332, 330–332 (2011).
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Nature (1)

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M. Chekhova, G. Leuchs, and M. Zukowski, “Bright squeezed vacuum: Entanglement of macroscopic light beams,” Opt. Commun. 337, 27–43 (2015).
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N. Liu, Y. Liu, X. Guo, L. Yang, X. Li, and Z. Y. Ou, “Approaching single temporal mode operation in twin beams generated by pulse pumped high gain spontaneous four wave mixing,” Opt. Express 24, 1096–1108 (2016).
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K. Garay-Palmett, H. J. McGuinness, O. Cohen, J. S. Lundeen, R. Rangel-Rojo, A. B. U’Ren, M. G. Raymer, C. J. McKinstrie, S. Radic, and I. A. Walmsley, “Photon pair-state preparation with tailored spectral properties by spontaneous four-wave mixing in photonic-crystal fiber,” Opt. Express 15, 14870–14886 (2007).
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J. B. Christensen, C. J. McKinstrie, and K. Rottwitt, “Temporally uncorrelated photon-pair generation by dual-pump four-wave mixing,” Phys. Rev. A 94, 013819 (2016).
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G. F. Sinclair and M. G. Thompson, “Effect of self- and cross-phase modulation on photon pairs generated by spontaneous four-wave mixing in integrated optical waveguides,” Phys. Rev. A 94, 063855 (2016).
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W. Wasilewski, A. I. Lvovsky, K. Banaszek, and C. Radzewicz, “Pulsed squeezed light: Simultaneous squeezing of multiple modes,” Phys. Rev. A 73, 063819 (2006).
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M. G. Raymer, Z. W. Li, and I. A. Walmsley, “Temporal quantum fluctuations in stimulated Raman scattering: Coherent-modes description,” Phys. Rev. Lett. 63, 1586–1589 (1989).
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B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon temporal modes: A complete framework for quantum information science,” Phys. Rev. X 5, 041017 (2015).

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G. Y. Xiang, H. F. Hofmann, and G. J. Pryde, “Optimal Multi-Photon Phase Sensing with a Single Interference Fringe,” Sci. Rep. 32684 (2013).
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Science (5)

A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706–709 (1998).
[Crossref] [PubMed]

N. Lee, H. Benichi, Y. Takeno, S. Takeda, J. Webb, E. Huntington, and A. Furusawa, “Teleportation of nonclassical wave packets of light,” Science 332, 330–332 (2011).
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I. A. Walmsley, “Quantum optics: Science and technology in a new light,” Science 348, 525–530 (2015).
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I. A. Walmsley and M. G. Raymer, “Toward quantum-information processing with photons,” Science 307, 1733–1734 (2005).
[Crossref] [PubMed]

Other (7)

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge University, 2007).
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R. W. Boyd, Nonlinear Optics, 3rd Ed. (Academic, 2008).

A. I. Lvovsky, “Squeezed light,” in Photonics: Scientific Foundations, Technology and Applications, Volume 1, edited by D. Andrews, ed. (Wiley, 2015), pp. 121–164.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2000).

R. Loudon, The Quantum Theory of Light, 3rd Ed. (Oxford University, 2000).

C. C. Gerry and P. L. Knight, Introductory Quantum Optics (Cambridge University, 2005). Two-mode squeezed vacuum states are discussed in Sec. 7.7.

C. J. McKinstrie, J. B. Christensen, K. Rottwitt, and M. G. Raymer, “Single-temporal-mode photon generation beyond the low-power regime,” Quantum Information and Measurement conference, Paris, France, 5–7 April 2017, paper QW3C.6. In the simulations, β1 = 10s/cm, β2 = 0.2 ps2/cm and l = 1.0 cm.

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Figures (13)

Fig. 1
Fig. 1 Contour plot of the Green function νrs for two cases in which the pump shapes are Gauss (HG0) functions. (a) βpl/τ = βrl/τ = 1 and βql/τ = βsl/τ = −1. (b) βpl/τ = 4 = − βql/τ and βrl/τ = 2 = − βsl/τ. Lighter shading represents higher values, whereas darker shading represents lower values. Time is measured in units of τ. The input and output times are correlated.
Fig. 2
Fig. 2 The normalized Schmidt coefficient is plotted as a function of mode number for two HG0 pumps. (a) βpl/τ = βrl/τ = 1 and βql/τ = βsl/τ = −1. (b) βpl/τ = 4 = − βql/τ and βrl/τ = 2 = − βsl/τ. Several coefficients are nonzero.
Fig. 3
Fig. 3 Frequency diagrams for vector four-wave mixing in a strongly-birefringent medium [44]. Points on the blue (red) curves denote waves that are aligned with the fast (slow) axes of the medium. The wavenumbers and slownesses (β1) are matched simultaneously for pump frequencies that are (a) degenerate and (b) nondegenerate. Notice that the copropagating waves have different polarizations. The pump and sideband frequencies can be interchanged (pr and qs). The displayed diagrams pertain to the anomalous dispersion regime and similar diagrams pertain to the normal dispersion regime.
Fig. 4
Fig. 4 Characteristic diagram for (a) idler generation and (b) signal generation. The horizontal axis is distance and the vertical axis is time. The interaction is limited to the region in which the pumps collide, which is the interior of the dashed diamond. The blue and red rays (arrows) represent parts of the signal and idler that interact strongly.
Fig. 5
Fig. 5 The first Schmidt coefficient is plotted as a function of the gain parameter γ ¯. In (a) the solid line is γ ¯ and the vertical scale is linear, whereas in (b) the solid curve is exp ( 2 γ ¯ ) / ( 16 π γ ¯ ) 1 / 2 and the scale is logarithmic. Filled circles denote results for two HG0 pumps, whereas empty circles denote results for one HG0 and one HG1 pump. The Schmidt coefficients do not depend on the pump shapes.
Fig. 6
Fig. 6 (a) The ratio of the second Schmidt coefficient to the first is plotted as a function of the gain parameter γ ¯. Filled circles denote results for two HG0 pumps, whereas empty circles denote results for one HG0 and one HG1 pump. The relative probability of the second mode, which is the square of the Schmidt ratio, never exceeds 1%. (b) The Schmidt ratio is plotted as a function of mode number for the worst case, in which γ ¯ = 1.76. The HG0-HG0 and HG0-HG1 results are indistinguishable.
Fig. 7
Fig. 7 Contour plots of the Green function νrs for the gain parameters (a) γ ¯ = 0.3 and (b) γ ¯ = 3.0. Both pump shapes are Gauss (HG0) functions. Lighter shading represents higher values, whereas darker shading represents lower values. High gain localizes the Green function near early input and output times.
Fig. 8
Fig. 8 The (common) signal and idler Schmidt function (dots) is plotted as a function of time for the gain parameters (a) γ ¯ = 0.3 and (b) γ ¯ = 3.0. Both pump shapes (solid curves) are Gauss (HG0) functions. In the low-gain regime the Schmidt function is approximately Gaussian, whereas in the high-gain regime it is a distorted Gaussian.
Fig. 9
Fig. 9 Contour plots of the Green function νrs for the gain parameters (a) γ ¯ = 0.3 and (b) γ ¯ = 3.0. The pump shapes are HG0 and HG1 functions. Lighter shading represents higher values, whereas darker shading represents lower values. High gain localizes the Green function near early input and output times.
Fig. 10
Fig. 10 The input signal (a) and output idler (b) Schmidt functions (dots) are plotted as functions of time for the gain parameter γ ¯ = 0.3. The pump shapes (solid curves) are HG0 and HG1 functions. In the low-gain regime both Schmidt functions are approximately HG functions.
Fig. 11
Fig. 11 The input signal (a) and output idler (b) Schmidt functions (dots) are plotted as functions of time for the gain parameter γ ¯ = 3.0. The pump shapes (solid curves) are HG0 and HG1 functions. In the high-gain regime both Schmidt functions are distorted HG functions.
Fig. 12
Fig. 12 The ratio of the second Schmidt coefficient to the first is plotted as a function of the gain parameter γ ¯ for two Gauss (HG0) pumps. The solid curve denotes semi-analytical results that omit nonlinear phase modulation, whereas crosses denote numerical results that include nonlinear phase modulation. The results are qualitatively similar and quantitatively comparable.
Fig. 13
Fig. 13 The amplitudes (blue curves) and phases (red curves) of the input signal (a) and output idler (b) Schmidt functions are plotted as functions of time for the gain parameter γ ¯ = 1.55. Both pump shapes are Gauss (HG0) functions. Nonlinear phase modulation imposes opposite chirps and frequency shifts on the signal and idler. The chirps are extremal near the peaks of the copropagating pumps, relative to which the signal and idler are skewed toward early times.

Equations (31)

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( z + β r t ) A r ( t , z ) = i γ p q ( t , z ) A s * ( t , z ) ,
( z + β s t ) A s ( t , z ) = i γ p q ( t , z ) A r * ( t , z ) ,
b r ( t ) = d t [ μ r r ( t , t ) a r ( t ) + ν r s ( t , t ) a s ( t ) ] ,
b s ( t ) = d t [ μ s s ( t , t ) a s ( t ) + ν s r ( t , t ) a r ( t ) ] ,
d t [ μ r r ( t 1 , t ) μ r r * ( t 2 , t ) ν r s ( t 1 , t ) ν r s * ( t 2 , t ) ] = δ ( t 1 t 2 ) ,
d t [ μ r r ( t 1 , t ) ν s r ( t 2 , t ) ν r s ( t 1 , t ) μ s s ( t 2 , t ) ] = 0 .
μ r r ( t , t ) = n ν r n ( t ) μ n μ r n * ( t ) , ν r s ( t , t ) = n ν r n ( t ) ν n u s n ( t ) ,
b r n = μ n a r n + ν n a s n , b s n = μ n a s n + ν n a r n ,
| ψ n = k ( ν n / μ n ) k | k , k / μ n ,
μ s s ( t , t ) = δ ( t t β s l ) , ν r s ( t , t ) = i γ p q ( t c , z c ) / β r s ,
t c = [ β r t β s ( t β r l ) ] / β r s , z c = [ t ( t β r l ) ] / β r s .
ν r s ( t r , t s β s l ) = ν s r ( t r β r l , t s ) .
t c β p z c = [ β r p t β s p ( t β r l ) ] / β r s
| ψ 1 = d t r d t s f ( t r , t s ) a r ( t r ) a s ( t s ) | vac ,
f ( t r , t s ) = d t μ s s ( t s , t ) ν r s ( t r , t ) = d t μ r r ( t r , t ) ν s r ( t s , t )
f ( t r , t s ) = n ν n ν r n ( t r ) ν s n ( t s ) ,
HG n ( s ) = H n ( s / τ ) exp ( s 2 / 2 τ 2 ) / π 1 / 4 ( 2 n n ! τ ) 1 / 2 ,
μ s s ( t , t ) = δ ( t t β s l ) + γ ¯ A q ( t β s l ) ( ξ / η ) 1 / 2 I 1 [ 2 γ ¯ ( ξ η ) 1 / 2 ] A q ( t ) × H ( t t β s l ) H ( t + β r l t ) ,
ν r s ( t , t ) = i γ ¯ A p ( t β r l ) I 0 [ 2 γ ¯ ( ξ η ) 1 / 2 ] A q ( t ) H ( t t β s l ) H ( t + β r l t ) ,
ξ ( t , t ) = t β r l t d s | A p ( s ) | 2 , η ( t , t ) = t t β s l d s | A q ( s ) | 2 .
k s ( t s , t s ) = d t r ν r s * ( t r , t s ) ν r s ( t r , t s ) ,
k r ( t r , t r ) = d t s ν r s ( t r , t s ) ν r s * ( t r , t s ) .
( z + β r t ) A r = i γ p q A s * + σ r δ ( z ) δ ( t ) , ( z + β s t ) A s = i γ p q A r * + σ s δ ( z ) δ ( t ) ,
y A r = i γ ¯ A p ( x ) A q ( y ) A s * + σ r δ ( x ) δ ( y ) , x A s = i γ ¯ A p ( x ) A q ( y ) A r * + σ s δ ( x ) δ ( y ) ,
η A r = i γ ¯ A s * + σ r A p * δ ( ξ ) δ ( η ) , ξ A s = i γ ¯ A r * + σ s A q * δ ( ξ ) δ ( η ) ,
G r s ( ξ , η ) = i γ ¯ I 0 [ 2 γ ¯ ( ξ η ) 1 / 2 ] H ( ξ ) H ( η ) ,
G s s ( ξ , η ) = γ ¯ ( ξ / η ) 1 / 2 I 1 [ 2 γ ¯ ( ξ η ) 1 / 2 ] H ( ξ ) H ( η ) + H ( ξ ) δ ( η ) ,
G r s ( ξ , η ) ~ γ ¯ exp [ 2 γ ¯ ξ 1 / 2 ( 1 η ) 1 / 2 ] ( 4 π γ ¯ ) 1 / 2 ~ γ ¯ exp [ 2 γ ¯ γ ¯ ( 1 ξ ) γ ¯ η ] ( 4 π γ ¯ ) 1 / 2 .
G r s ( t , t ) = γ ¯ A p ( t β r l ) I 0 [ 2 γ ¯ ( ξ η ) 1 / 2 ] A q ( t ) × H ( t t β s l ) H ( t + β r l t ) ,
G s s ( t , t ) = δ ( t t β s l ) + γ ¯ A p ( t β s l ) ( ξ / η ) 1 / 2 I 1 [ 2 γ ¯ ( ξ η ) 1 / 2 ] A q ( t ) × H ( t t β s l ) H ( t + β r l t ) ,
ξ ( t , t ) = t β r l t d s | A p ( s ) | 2 , η ( t , t ) = t t β s l d s | A q ( s ) | 2 .

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