Time reversal of arbitrary photonic temporal modes via nonlinear optical frequency conversion

Single-photon wave packets can carry quantum information between nodes of a quantum network. An important general operation in photon-based quantum information systems is ‘blind’ reversal of a photon’s temporal wave packet envelope, that is, the ability to reverse an envelope without knowing the temporal state of the photon. We present an all-optical means for doing so, using nonlinear-optical frequency conversion driven by a short pump pulse. The process used may be sum-frequency generation or four-wave Bragg scattering. This scheme allows for quantum operations such as a temporal-mode parity sorter. We also verify that the scheme works for arbitrary states (not only single-photon ones) of an unknown wave packet.


Introduction
Single-photon wave packets can carry quantum information between nodes of a quantum network for distributed quantum-information processing. For efficient transfer, the shapes of the wave packets need to be finely controlled. For example, the temporal shape emitted by a single quantum emitter (atom, quantum dot, etc) in free space is not the shape that leads to efficient absorption by a receiving quantum absorber, even if the emitter and absorber are physically identical. For efficient absorption, the envelope of the emitted wave packet must be time reversed [1][2][3]. Optimal input and output modes of optical cavities have a similar time-reversed relationship [4,5], as has been demonstrated experimentally for weak classical pulses [6] and for single-photon wave packets [7].
Although quantum mechanics is formally time-reversal symmetric, there is no unitary transformation or corresponding physical device operation that can implement full time reversal of a quantum state [2,8,9]. In quantum optics, such an operation would temporally reverse the complex envelope of an optical wave packet and at the same time conjugate its phase, that is E t E t r r , , ) It is known that optical phase conjugation by four-wave mixing can cause such a phase transformation with or without reversal of the envelope [10][11][12], but only at the cost of adding quantum noise (spontaneous photons) to the mode of interest, making the transformation non-unitary [13][14][15][16]. The added noise makes phase conjugation ineffective for implementing quantum-information protocols.
This paper proposes an all-optical operation that reverses only a wave packet's envelope function and not its phase. The method functions without knowing the wave packet; thus we call it 'blind'. Such an operation can in principle be unitary and thus noise free, as needed in quantum information science.
It is known that reflection by an optical cavity designed to have its resonant frequency and cavity decay time matching those of the quantum emitter reverses the envelope function of an incident wave packet generated by that emitter [6]. But this does not represent a universal or 'blind' operation, as it requires that the particular shape of the optical wave packet be known in advance, and fails if the wave packet does not match the cavity. This prevents the technique being used to implement quantum logic operations in a state space of temporal shapes (modes)-an important capability in quantum information science [17].
In addition to noiseless waveform manipulation, in quantum networks there is also a need for noiseless frequency conversion (FC), which together with pulse shaping can enable complete temporal and spectral 'impedance' matching between quantum emitters and absorbers that are not identical (for example, a rubidium vapor and a GaAs quantum dot), having different carrier frequencies and decay times [18][19][20][21]. Such frequency conversion has been demonstrated for single photons using nonlinear three-wave mixing to implement sumfrequency generation (SFG) [22] while maintaining single-photon number statistics [23] and even quantum entanglement [24] with other physical systems. For FC between frequencies separated by smaller differences than can be accommodated by sum-or difference-frequency generation, four-wave mixing has been demonstrated for converting the frequency of single photons [25]. It is known that FC using three-or four-wave mixing can be engineered to be highly waveform selective [26,27]. Ideally, only a single, given TM is frequency converted, while all temporally orthogonal TMs are not. The resulting operation, called a 'quantum pulse gate,' enables a new framework for quantum information science, based on manipulation of single-photon waveforms and carrier frequencies in a higher-dimensional (qudit) state space [17].
In this paper we show theoretically that frequency conversion by three-or four-wave mixing can be engineered to implement noiseless envelope reversal in a manner that is waveform nonselective, which enables quantum information operations such as time reversing a superposition of envelope shapes without knowledge of the state. Within certain bandwidth constraints (to be specified), it operates on the envelope as f t f t ,  -( ) ( ) that is, it reverses the envelope but does not conjugate the phase. In this regard the time-reversal frequency conversion process is similar to a time lens, but with the difference that it also converts the carrier frequency from one band to another. The new scheme provides a single unifying platform that can implement both frequency conversion and noiseless time reversal of arbitrary (that is, unknown) envelopes.
An alternative method that can reverse an unknown, arbitrary-shaped envelope (within a given spectral band) without adding quantum noise is the so-called time lens [28][29][30]. The time lens uses fast modulation of the refractive index to temporally vary the phase of a light pulse, analogous to the spatial variation of phase imposed by a lens. As a single lens spatially inverts an image in the image plane, the temporal 'image' of the incident pulse is time reversed. The same operation can also stretch (magnify) or compress (demagnify) the waveform, leaving the carrier frequency unchanged [31]. Relatively small shifts of carrier frequencies (up to 100s of GHz) can be implemented using the same class of phase modulators used in the time lens [32]. The time lens reshapes a wave packet, but is limited in its ability for frequency conversion compared to the nonlinear-optical method. The presently proposed scheme can implement envelope reversal and large-reach frequency conversion in a single device, creating a useful tool for quantum information networks.
Optical processes that use inhomogeneously broadened atomic media to implement time reversal are photon echoes and quantum memories based on controlled reversible inhomogeneous broadening, also known as gradient echo memory [33][34][35][36]. They are in principle noiseless [37], but extraneous quantum noise associated with atomic or molecular fluctuations and spontaneous emission, as well as the bandwidth and wavelength constraints imposed by the use of specific resonant materials, may render these approaches less flexible for use in quantum information systems than the all-optical schemes such as the time lens and the presently proposed method.

Time reversal considerations
First, consider the general quantum theoretical problem of time reversing an optical pulse. In a one-dimensional model, we restrict our interest to a frequency band denoted W centered at 0 w with bandwidth B. (We do not need this restriction here, but it will prove useful later.) The forward-traveling 'photon field' operator (scaled so that A A -+( ) ( ) is a photon flux) in this band has positive-frequency part: | |/ is the propagation constant (wavenumber) in free space and the single-frequency creation operators satisfy the commutator a a , 2 .
A pure state of a given temporal mode (wave packet), denoted by the index j within the spectral band , W can be expressed in terms of the photon creation operator corresponding to that temporal mode [38,39]: If the set of spectral amplitudes j w Y { ( )}form an orthonormal set under inner product 2 d , where F is some normalizable function. The corresponding temporal modes are given by the spatial-temporal amplitudes: which form a discrete, orthogonal set under an inner product involving integration over time t or longitudinal distance z. Full time reversal (better named motion reversal) corresponds to replacing the temporal amplitude by [8,9] (where * means complex conjugation) which is equivalent to making the replacements: Full time reversal is seen to be antilinear, defined for an operator P in quantum theory as c c* P Y = P Ŷ ( )ˆ( ) [8,9]. Therefore, no physical process can implement it. To illustrate this point, the time reversal operator applied to one half of an entangled state can produce a state described by a density matrix that is unphysical, evidenced by having at least one negative eigenvalue.
Phase conjugation by four-wave mixing can perform an operation that looks like full time reversal of a signal mode [10,12], but with added spontaneous noise arising from coupling to an extra degree of freedom-the idler mode [15].
In contrast to full time reversal, time reversing only the envelope of an optical pulse can in general be carried out by a physical process described by a unitary transformation without any added noise. Consider the temporal mode as a product of an optical carrier at 0 w and a slowly varying envelope z t , : The temporal envelope function is related to the spectral amplitude by (defining where 0 A indicates a 'base band' centered at zero. Envelope reversal (and reversing the direction of propagation) corresponds to z t , , This transformation is unitary and linear (does not involve complex conjugation of the state).

Envelope reversal-perturbative theory
Consider an initial temporal mode in the spectral band centered at frequency . s w The state of this TM (singlephoton, squeezed, coherent, etc) is arbitrary and unknown. The goal is to time reverse the TM while converting it to a band with a new center frequency , r w otherwise preserving the state. The process proposed below, driven by a pump field centered at frequency , p w reverses the TM's envelope but not its direction of propagation, which can be accomplished trivially by reflection from an ordinary mirror.
We assume the two signal bands are connected via a nonlinear optical three-wave-mixing process driven by the strong, nondepleting pump field, such that the band centers are related by .
The process is known as sum-frequency generation (SFG). We assume that competing processes, such as dissipative loss, crossphase modulation, and higher-order dispersion, are all negligible. Denote group slowness (inverse group velocity ( ) accounts for the linear dispersion of the medium. (Four-wave mixing could also be used, as discussed later, but we focus on three-wave mixing for clarity. ) We find that to achieve envelope reversal it is sufficient to have the group slownesses ordered as , that is, one of the signals travels faster than the pump, while the other signal travels slower than the pump, and to have the medium long enough so that pulses with unequal slownesses pass completely through one another. We show in the appendix A that this condition can be satisfied for realistic media. Then the medium acts like an infinitely long one, leading to exact conservation of momentum among the fields, as explained below.
We denote the (positive-frequency) photon annihilation operators for propagating fields in the three bands by , , , 10 where , s W r W and p W denote the relevant spectral bands, and a z a z , , The dominant term in the interaction Hamiltonian is: proportional to the second-order nonlinear susceptibility and ha is the Hermitian adjoint. To first order, the evolution operator is where L is the medium's length. The phase mismatch is approximated, neglecting higher-order dispersion, as and we assumed the zeroth-order terms sum to zero due to either birefringent phase matching or periodic poling to achieve quasi-phase matching. Then the phase matching function evaluates to LD , This is equivalent to requiring the medium to be long enough so the three pulses can begin being spatially nonoverlapping and pass completely through one another while in the medium. In this limit, the evolution operator is equal to in agreement with temporal-mode envelope reversal as described by equation (9). This supports our claim that to achieve envelope reversal it is necessary to have one of the signals travel faster than the pump, while the other signal travels slower than the pump, and to have the medium long enough so that the pulses pass completely through one another.
In the more general case that m is negative but not necessarily equal to −1, temporal reversal occurs with either a stretching or compressing effect, by a factor m , | | which we call the spectral magnification. Spectral compression by frequency conversion has been observed in experiments [40].
To prove the result explicitly for a single-photon state, and to see what requirements are placed on the state of the pump field, we consider an initial state: Now we see explicitly that the spectral amplitude of the generated r signal, , j n -( ) is the spectrally mirrored version of the spectral amplitude of the s signal, . j ñ ( ) In order to have perfect envelope reversal of the signal, the pump spectrum p j w ( )must be constant across the signal's spectrum (shifted to the pump's band for comparison), satisfied if the pump is sufficiently broad band, B B B , .
p s r  That is, the pump pulse must be much shorter in duration than the signal pulses.
Equation (19) also shows that in first-order perturbation theory a pump field in a coherent state does not get entangled with the signal fields (because the coherent state is an eigenstate of a p ). In first-order perturbation theory, a pump field in any other type of pure state will become entangled (because it is not an eigenstate of a p ). In higher-order perturbation theory even a pump field in a coherent state gets entangled with the signals (because it is not an eigenstate of a p † ). However, for a large-amplitude pump field that entanglement is small (because the coherent state is 'almost' an eigenstate of a . p † More precisely, a p p añ | † has a large overlap with p añ | ). This section has shown from general considerations the conditions required on the pump field and dispersion properties to achieve envelope reversal by FC. We also have seen that, in first-order perturbation theory, the signal fields do not become entangled with the pump field if the pump starts in a coherent state. We also argued it is justified to treat a coherent-state pump as a classical field even in the nonperturbative regime, as in usually done in the FC literature. We shall do so in the following sections.

Envelope reversal-nonperturbative theory
The above analysis leaves open two questions: does envelope reversal by frequency conversion work for high conversion efficiencies, and does it work for arbitrary (but low-photon-number) states of the input signal? The answers to both questions are yes, as we now show.
Above we showed that sufficient conditions for envelope reversal are: (i) the bandwidth relation B B B , , It is convenient to define a new effective 'local time' variable defined by the pump's velocity, as t t z.
(Local time is like a collection of clocks strewn along the path, which are progressively delayed according to their location, like time zones on earth.) In terms of this variable, the pump A t p LT ( )is stationary, that is, we are representing the propagation in the 'frame' moving along with the pump, and in this frame one signal appears to move to the right (increasing z) and the other to the left (decreasing z). We assume the three pulses are timed so they intersect maximally somewhere within the medium. Now, for notational convenience, we drop the subscript on t , LT and write the equations of motion using local time: We next show that, for a short pump pulse, these equations predict that both the amplitude and phase time dependence of the signal's envelope are reversed upon frequency conversion, while the phase of the carrier wave is not conjugated, consistent with noise-free operation.

Analytical solution
Because the equations of motion (22) are linear in the signal operators, they may be solved as if they were nonoperator equations. In fact, the input-output relations derived thereby can be viewed as 'classical' mode transformations, in this case for temporal modes. This will allow predicting output states for arbitrary (but weak) input states.
Under the stated conditions, an excellent approximation to the exact solution can be found, which is verified numerically below. Choose a local time interval t t , is faster than the pump and enters this window from the 'right' (i.e., m 0 < ). In the figure, local time is on the horizontal axis and distance increases in the downward vertical direction; therefore the envelope of a slower pulse will move progressively 'rightward' as z increases, while the envelope of a faster pulse will appear to move progressively 'leftward' as z increases. (Think of the amplitudes as those that would be recorded by a detector placed at a fixed z value.) Using a 'ray tracing' description, a field value A t 0, s -| | In the pump region the rays interact; some portion of the signal becomes the idler and vice versa. Equating these two expressions for z 0 yields the relation between t and t , 2 that is, t t Mt Mt, / is the temporal magnification, expressing stretching or compressing of the output pulses relative to their input counterparts. In the limit of a short pump pulse, we can approximate t t t ,  With a very short pump pulse, the local interaction between spatial segments of the three pulses occurs in a sufficiently short time interval that spatial propagation has little effect within each segment. Therefore, we can neglect the spatial z derivatives in equation (22). Then, defining an effective time variable in terms of the pump function (assumed real): we arrive at: r r s s s s = = | | | | These can also be expressed as: This shows that the transmitted components of the signal and idler are not time reversed, whereas the 'reflected' (frequency converted) components, which are generated by the other mode, are time reversed and dilated.
A fruitful way to represent these input and output fields in terms of a complete set of TMs, t j j { ( )}is:   Figure 2(b) shows an asymmetric, nonGaussian input pulse A t 0, , s ( ) which changes sign (phase) at the point where the amplitude touches zero. The frequency-converted envelope A z t , r ( )is seen to be time reversed relative to the input pulse, as expected. Figure 2(c) shows the same asymmetric input pulse A t 0, , s ( ) with the coupling coefficient reduced to a value 1.74, g = chosen such that 1 2 , t r » » / so the expected frequency conversion efficiency is 0.5. Indeed, the converted and unconverted pulses are seen to have equal amplitudes at the output, as predicted. and this is precisely what is seen in the figure: the converted pulse has half the duration as the input pulse.
The above examples considered input pulses with constant phase. The analytical theory predicts that for pulses with time-varying phases, the phase evolution will also be time reversed along with the (positive) envelope (although, again, the carrier wave is not phase conjugated). Consider an input pulse having the same envelope function as that in figure 2, but with a phase evolution arbitrarily chosen to have a quadratic time dependence, which corresponds to the linear spectral chirp that develops during propagation in a typical transparent medium such as an optical fiber: Using the same parameters as in figure 2 yields the results in figure 3, where the envelopes and phases of the wave packets at the input and output of the medium are shown. If the output wave packet is flipped in time and delayed to match up with the input packet, both phase profile and envelope are found to match essentially (c) With lower pump power the input is converted 50% to a time-reversed replica. (d) With unequal signal velocities, the converted replica is temporally compressed relative to the input. Throughout, the time span is 20 units, the length of the medium is 20 units, and the input location is taken to be at z 0. = Figure 3. Envelopes and phases of input 'red' signal field and output 'blue' field versus time. Both envelope and phase are seen to be time reversed. The phase values are normalized by a factor . p (The unit of the envelopes is arbitrary.) 2.5 cm, with the limit growing linearly with length. Achieving a broader range of magnifications requires different media or wavelength choices. A means of overcoming this limitation is to use novel techniques like photonic crystals to introduce waveguide dispersion. For example, we consider the photonic crystal fiber used for frequency conversion using four-wave-mixing Bragg scattering in [25], which has been engineered to have zero-group-velocity-dispersion points at two wavelengths, as shown in figure A3. Such waveguides do not have monotonic relations between group-velocity and wavelength.
The particular fiber from figure A3 is phase-matched for frequency conversion through Bragg scattering [25] between the 1200 nm band and the 1692.4 nm band when pumped with strong, coherent lasers at 632.8 and 747.5 nm. Thus, when pumped asymmetrically [47] with continuous wave (or temporally long) pulses of helium-neon laser light at 632.8 nm and ∼10 ps pulses of, say, Ti:sapphire laser at 747.5 nm, a fiber of length 100 m can time reverse a 3.7 ns long pulse centered at 1692.4 nm to create a 1.5 ns pulse in the 1200 nm band. These temporal durations are more conducive to coupling into known quantum optical memory systems than are the ps-duration pulses considered earlier. Careful dispersion engineering can be employed to modify the signal and idler wavelengths into more suitable bands such as telecom (1550 nm).