Temporal modes (TMs) are field-orthogonal broadband wave-packet states of light occupying a common frequency band, and they can encode information in a higher-dimensional alphabet compared to, say, photon polarization. The key—yet still missing—ingredient for the full implementation of a system deploying TMs is a highly selective quantum pulse gate—a multiplexing device that can route photonic packets according to their temporal shape with high temporal-mode discrimination and high efficiency, figures of merit that together we call high selectivity. Here, we present the first implementation of a highly selective quantum pulse gate. The method is a generalization of all-optical Ramsey interferometry, so far demonstrated only for continuous-wave light fields [Phys. Rev. Lett. 117, 223601 (2016) [CrossRef] ]. As applied to temporal modes, we refer to the method as temporal-mode interferometry.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
A quantum Internet will be based on photons carrying information between computing nodes, and there is a pressing need to discover the most efficient ways in which to encode, route, and transmit such quantum information. Recently, a new framework for quantum networks was proposed: the use of high-dimensional temporal-mode encoding onto the temporal shapes of single-photon wave packets . The chief enabling technology necessary for full exploitation of this resource is a high-fidelity quantum pulse gate (QPG) [2,3]. The barrier that had to be overcome to successfully operate a QPG based on sum-frequency generation (the only known modality to date) was a temporal mode (TM)-selectivity limit in processes that rely on spatio-temporally local, nonlinear interactions between pulsed optical modes traveling at independent group velocities . This limit is a consequence of time ordering in the quantum dynamical evolution [3,5], which is predicted to be surmountable by coherently cascading multiple stages of low efficiency, but highly TM-discriminatory QPGs [6–8]. Our experiments show the first to our knowledge demonstration of TM-selective optical Ramsey interferometry and show a significant enhancement in TM selectivity over single-stage schemes, offering the needed capability for quantum information systems based on temporal modes of photonic packets.
All-optical quantum frequency conversion (QFC) by three- or four-wave mixing in nonlinear materials is well known to preserve the quantum state of light [9,10]. These processes are in principle noiseless, and can be used at sub-unity conversion efficiencies (CE) to generate single-photon color-superposition states across disjoint frequency bands. This facet has been posited as a two-level Hilbert space for single-photon qubits . Two QFC stages, each of which is set up for 50% CE, can be cascaded into a two-color interferometer constructed out of frequency-shifting beam splitters. This effect has been shown for single-photon states in dual-pumped third-order nonlinear optical fibers , as well as for weak coherent states in singly pumped second-order nonlinear waveguides . Both of these instances utilized continuous-wave (CW) lasers for pumps and very narrowband signals (single photons and coherent states, respectively) in their experiments.
The CE of QFC devices becomes TM-dependent when using pulsed laser pump(s) [2,13]. To see this, we express the equations of motion for QFC in terms of temporal-mode envelope functions , where the index labels the frequency band [4,13]. For pulsed, three-wave mixing between pump (-band), signal (-band), and register (-band) fields in a single-transverse-mode waveguide, the interaction Hamiltonian in a medium of length is1), the operators can also stand for the classical TM functions of weak-coherent pulses . The pump is assumed to be a strong, nondepleting coherent state, and its pulse-mode envelope is assumed to be square normalized (). The coupling strength is proportional to the transverse-mode overlap integrals, the -nonlinear coefficient, as well as the square-root of the pump pulse energy.
Equation (1) is identical in form to those governing a wide class of physical systems besides all-optical three-wave mixing. For instance, it can represent a pump-mediated interaction between an optical field and a collective Raman transition “spin-wave” in an atomic ensemble quantum memory [15,16], implying that processes analogous to those studied here should also occur in such systems.
The phenomenon can be expressed as a scattering matrix relating input-mode operators to output-mode operators . It has been shown both theoretically and experimentally that for a wide range of system-parameter values (, where is the pump pulse duration) at low effective-interaction strengths (), the frequency conversion is TM-discriminatory, with the target TM being dependent on the complex shape of the pump-pulse envelope . At low interaction strengths, one can easily achieve a large contrast between the CE of the target TM and the CE of all other orthogonal TMs. But increasing the interaction strength to reach higher CE imposes a trade-off between target-TM CE and TM discrimination [4,18], causing said contrast in CEs to decrease at larger . This departure from high TM discrimination at large CEs can be interpreted as an effect of time-ordering corrections to the evolution . The unitary evolution operator may be expressed in a Magnus expansion,6–8] that this limit to TM selectivity can be asymptotically overcome by preferentially weighing the first-order term in the expansion over the higher-order terms. This is achieved by coherently cascading multiple low-efficiency QFC stages in sequence (see Supplement 1). We have previously referred to this technique as temporal-mode interferometry (TMI). The basic schematic for a two-stage TMI is shown in Fig. 1. The pump pulses for both stages are derived from a single pump beam to ensure constant relative phase. The net interferometric phase for the three-field process is .
For this scheme to function, every stage needs to be highly TM-discriminatory, but not necessarily of high efficiency. The best parameter regime for high TM-discrimination at low CE is known to require that one of the weak bands (signal or register) copropagate with the pump pulse with a matching group velocity, and the other band’s group velocity be drastically different [3,4]. In our system, this condition is approximately satisfied through the use of an MgO-doped, periodically poled lithium niobate (PPLN) waveguide quasi-phase-matched for Type-0 second-harmonic generation from 816.6 to 408.3 nm. We center our pump and signal bands at 821 and 812.2 nm, respectively, and choose their bandwidths such that the pump and signal pulse durations () are much larger than the pump-signal inter-pulse walkoff within the length of the medium, thus effectively mimicking the group-velocity matching condition [18,19].
The pump-signal walkoff scales linearly with medium length, implying that the group-velocity matching condition is better approximated in shorter waveguides. However, the single-stage TM discrimination also requires that the register-band pulse walks off from the pump/signal pulse by a large amount . This necessitates that the ratio of the effective interaction time and the pump duration be much larger than unity . We work with a 5 mm long waveguide, yielding , and a pump-signal walkoff of . Our pump and signal fields were derived from a homebuilt, Kerr-lens mode-locked ultrafast Ti:sapphire laser producing pulses centered at 821 nm and with a full width at half-maximum (FWHM) spectral-intensity bandwidth of . The pulse repetition rate was 80 MHz. The laser pulses were then directed into a pulse shaper.
The Fourier-domain, pulse shaper used an 1800 lines/mm holographic grating in near-Littrow configuration and a cylindrical lens of focal length 250 mm. We used a Meadowlark 8-bit, 2D, phase-only, reflective spatial light modulator of pixel resolution and array size of in the Fourier plane of the pulse shaper. This allowed us to perform both amplitude and phase modulation using the Frumker–Silberberg first-order method [18,20]. This system was used to generate three modified Hermite–Gaussian spectral amplitudes for both pump and signal bands. These functions (shown in temporal domain in Fig. 1) are spectrally defined as follows (with bandwidth parameter and normalization constants ):
The width modifications ensure mutual orthogonality while restricting total bandwidth of all three modes to the same neighborhood ().
A Newport ISO-05-800-BB broadband Faraday optical isolator had a sufficiently flat transmission curve over a wide range of wavelengths for it to be deployed without imparting significant dispersion to the pump and signal-in pulses. The dichroic elements labeled DM1 in Fig. 2 are the DMLP650 longpass dichroic mirrors from Thorlabs. The transmission edges of Semrock FF01-810/10 bandpass filters (DM2) and NF03-808E notch filters (DM3) were used to split/combine the -band and -band pulses from/with each other. The signal- and register- mirrors were mounted on -545.140 closed-loop servo linear stages from Physik Instruments, which had a nomimal minimum step size of 6 nm and an encoder-based position read-out precision of 1 nm. Low-profile flexture mounts from Siskiyou Inc. were used in between the two stages to construct a stable interferometer.
The pump and signal-in pulses for the interferometer were generated by carving out their spectral components from the broadband ultrafast laser pulses in the pulse shaper . We chose to work with modified Hermite–Gaussian functions with comparable bandwidths for the two-stage experiment. Specifically, we used three mode shapes shown in Fig. 1 [defined in Eqs. (3)–(5)]. Below, the pump and signal pulse shapes are denoted by and , respectively, where index refers to shape from Fig. 1. The pump and signal-in pulses are then sent towards the interferometer setup illustrated in Fig. 2.
The dispersion of the nonlinear media used in the two stages needs to be identical to ensure phase matching between the same central wavelengths. We therefore reused the same waveguide twice, using a back-reflection-based doublepass “Michelson” scheme detailed in the figure. A broadband Faraday optical isolator enabled us to separate the “forward” and “backward” propagating pulses from the two stages and measure the final outputs at its reject port. Longpass dichroic mirrors (labeled DM1 in Fig. 2) were used to split/combine the -band pulses from the - and -band pulses. Similarly, other dichroic elements (DM2 and DM3 in Fig. 2) were employed to split/combine the different bands.
Both the signal-mid and register-mid pulses were backreflected off of flat mirrors mounted on high-precision linear translation stages. These mirrors, referred to as “-mirror” (for signal) and “-mirror” (for register), were used to ensure proper pulse overlap in the second stage as well as to impart interferometric phase via subwavelength displacements. For example, a fine displacement of in the signal-mid arm would impart a phase of to the interferometer, with the factor of 2 resulting from the doublepass path-length change.
All pulses were coupled into (and out of) the 5 μm wide PPLN waveguide using aspheric lenses. The coupling efficiency into the waveguide was around 30%. Since the conversion efficiencies in the two stages need to match each other to ensure sufficient pump energy in the second stage, we derived the pump-2 pulse afresh from the output of the shaper. A half-wave plate and polarizing beam splitter combination, along with the pulse shaper, allowed us independent control over the power in various beams. This freedom also enabled us to verify the identical operation of the waveguide in both directions by reproducing the single-stage results from  for both directions of propagation. A coupled average pump power of 0.47 mW at a pulse rate of 80 MHz yielded a CE of 0.5 in both directions.
The fringe visibility of the two-stage interferometer is sensitive to any imbalance in effective losses between the signal and register arms. Kobayashi et al.  have modeled all the loss channels in the two-color interferometer as disjoint unitary beam splitters placed at various beam locations in the setup. We matched the losses (due to absorption in various elements, as well as inefficient coupling into waveguide) between the signal-mid and register-mid beams by inserting a spatially varying neutral density filter (not shown) into the beam path of the better coupled (signal-mid) arm.
All-optical Ramsey-interference fringes were observed when recording the “internal” CE (defined as fraction of signal power depleted in the presence of pumps, which is independent of coupling losses) versus mirror displacements. Figure 3 plots the CE versus -mirror displacement for various combinations of pump and signal TM shapes. The spatial peak-to-peak period was found to be roughly half the signal-band wavelength, consistent with the backreflection setup. Correspondingly, -mirror displacements resulted in a spatial period of half the register-band wavelength [Fig. 4(a)] (The relative phase shifts among the fringe patterns, both across and within the subplots, are due to system drifts between data runs).
Figure 4(a) plots the CE fringe patterns for displacements of the -mirror alone (10 nm steps, legend item “”), both -mirror and -mirror in opposite directions (10 nm and 20 nm steps, respectively, legend item “”), as well as both in the same direction (legend item “”). Here, the direction is “positive” towards the waveguide, i.e., shortening path length. Combined mirror displacements in opposite directions halved the fringe period, while the CE was nearly unchanging for matching directional moves, confirming the relative signs in the net interferometric phase formula (). Note that the register frequency is not an exact harmonic of the signal frequency.
The peak CE for the cases where the pump and signal TMs matched in shape far exceeded the shape-mismatched ones, as seen in Fig. 4(b). Also plotted with dashed lines are the simulated single-stage CE for the theoretically predicted exact, optimal Schmidt modes at the pump powers required to match the maximum CE at each pump-pulse shape. The three-mode TM separability (the ratio of the CE of the target TM to the sum of the CEs of the three TMs considered) are printed in the insets, as are the corresponding quantity for the single-stage case at a matching target-TM CE (in parentheses). This parameter is enhanced for most shape combinations, even for the signal TMs we used, which simply match the pump shapes (i.e., no attempt was made to optimize the signal pulses to match the exact Schmidt modes of the two-stage process). The single-stage results in Ref.  for the same waveguide and shaper slightly underperformed the numerical estimates.
The numerics assumed a parameter value . This must roughly equal the ratio of the temporal width of the register-mid pulse to that of the pump/signal-in/signal-mid pulse . We can estimate these widths by scanning the delay between pump-2 and the signal (register-mid) using ()-mirror displacement and measuring the stage-2 CE while blocking the register (signal)-mid beam, as shown in Fig. 4(c). The widths of pump-2 and signal-mid pulses will add in quadrature, since their group velocities are nearly equal. The pump-2 and idler-mid pulses, however, walk off relative to each other in stage-2, resulting in a near-triangular CE curve. The data in Fig. 4(c) indicates a . It also demonstrates the expected temporal stretching of the register-mid pulse , related to the short second harmonic generation (SHG) bandwidth of such waveguides [21,22].
Stage-2 can separately be employed as a measurement device to demonstrate the effect of time ordering in the single-stage process. Figure 5 plots the CE of the signal-mid amplitude in stage-2 versus -mirror displacement for signal shapes s0 and s1 with the register-mid beam blocked. Both pump-1 and pump-2 have shape p0, and the pump-2 beam is weak (averaging 0.1 mW at 80 MHz pulse rate). Figure 5(a) shows that the signal-mid pulse temporally skews to earlier times, as well as compresses in width relative to the signal-in pulse, as pump-1 power is increased. This illustrates the departure from the perturbative regime, and the distortion mimics the shape of the theoretically predicted single-stage output Schmidt modes  (see Supplement 1). The same is true for signal shape s1 [Fig. 5(b)], as the second Schmidt mode for a Gaussian pulse skews to later times .
The direction of the temporal skewing in Fig. 5(a) is consistent with the sign of the signal-register relative group slownesses (). The register amplitude generated inside the waveguide lags behind the copropagating pump-signal pulses, causing an enhanced depletion of the latter half of the signal pulse. The skewing direction is not due to the small difference in the group velocities of the pump and signal bands. It would remain invariant under an exchange of labels between the - and -bands (and a corresponding exchange of powers/amplitudes, as the “pump”-band is defined as that which does not deplete).
The two-stage all-optical Ramsey interferometer was not only able to enhance TM selectivity over the single-stage variants (even for unoptimized, pump-shape-matched signal TMs), but was able to achieve large CE with a more efficient pump-energy budget. The results in Figs. 3 and 4(b) required beam powers of about 0.47 mW coupled in for both pump-1 and pump-2 (at 80 MHz pulse rates). Single-stage setups with the same waveguide could not reach such large CE for these net pump energies , even in theory [4,8]. This is due to the double passage through the nonlinear medium. The gain in interaction strength due to extension of medium length is superior to that from increased pump power, as the latter has a lower time-ordering penalty. The likeness of the exact Schmidt modes to the pump-pulse shape at low CE [2–4,6,7,18] enabled us to demonstrate enhancement in TM selectivtiy without having to optimize the signal-pulse shapes to exactly match the said Schmidt modes.
The device performance can be further improved if the signal-pulse shapes can be made to better match the process’s native input Schmidt modes, or vice versa. Computerized search algorithms such as the one employed in Ref.  over an appropriate parameter space can bridge the gap between any model approximations and the actual physical device properties. The non-zero pump-signal walkoff is another limiting factor. Custom dispersion engineering of nonlinear waveguides could provide us perfect group-velocity matching between accessible pump and signal frequency bands . This would enable one to work with longer waveguides, thus increasing the ratio of the effective interaction time and the pump duration without having to simultaneously increase the pump-signal walkoff.
Lastly, the overall input-to-output transfer efficiency is proportional to the product of the in-coupling efficiencies of the two stages. This can be mitigated via superior engineering of coupling mechanisms. Out-of-plane grating-assisted coupling of free-space beams into chip-based waveguides has been shown to reach coupling efficiencies of 70% in practice , and up to 98% in theory  over significantly wide bandwidths. We envision a fully integrated implementation of the QPG, where the light fields would not travel in free space in between the two (or multiple) frequency-conversion stages, but would instead remain confined to guided modes within on-chip photonic waveguides, thereby avoiding coupling losses.
We have demonstrated temporal-mode-selective quantum frequency conversion in a two-stage all-optical Ramsey interferometer by utilizing shape-tailored strong laser pump pulses and weak coherent signal pulses, and double passing through a single nonlinear optical waveguide in a Michelson configuration. We showed that there is a significant selectivity enhancement over single-stage implementations in agreement with theoretical predictions. We note that by changing the phase of one of the fields inside the Ramsey interferometer, our two-stage device also acts as a frequency-conversion switch that operates on one selected temporal mode and (approximately) not on the others from the orthogonal subspace. We verified the temporal distortion of the signal pulse during nonlinear interaction predicted by theory, and related that to time-ordering effects in general multi-field interaction systems that are governed by similar dynamical equations. This technique provides fruitful insight into the nature of time ordering in pulsed scattering processes and paves the way towards designing a fully selective quantum pulse gate, thus opening up the temporal mode basis space for quantum information processing.
National Science Foundation (NSF) (1521466).
We owe much to the theoretical analysis of Dr. Colin McKinstrie. We also thank Prof. Steven van Enk and Prof. Brian J. Smith for discussions and suggestions regarding the experiment. We thank Phil Battle and David Walsh of AdvR for providing us with the waveguide.
See Supplement 1 for supporting content.
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