Sorting photon wave packets using temporal-mode interferometry based on multiple-stage quantum frequency conversion

All classical and quantum technologies that encode in and retrieve information from optical fields rely on the ability to selectively manipulate orthogonal field modes of light. Such manipulation can be achieved with high selectivity for polarization modes and transverse-spatial modes. For the time-frequency degree of freedom, this could efficiently be achieved for a limited choice of approximately orthogonal modes, i.e., nonoverlapping bins in time or frequency. We recently proposed a method that surmounts the selectivity barrier for sorting arbitrary orthogonal temporal modes [Opt. Lett. 39, 2924 (2014).] using cascaded interferometric quantum frequency conversion in nonlinear optical media. We call this method temporal-mode interferometry, as it has a close resemblance to the well-known separated-fields atomic interferometry method introduced by Ramsey. The method has important implications for quantum memories, quantum dense coding, quantum teleportation, and quantum key distribution. Here we explore the inner workings of the method in detail, and extend it to multiple stages with a concurrent asymptotic convergence of temporal-mode selectivity to unity. We also complete our analysis of pump-chirp compensation to counter pump-induced nonlinear phase modulation in four-wave mixing implementations.

Figure 1

Time- and frequency-division multiplexing being contrasted with code-division multiplexing (CDM), with different codes represented by time-frequency Wigner functions for orthogonal temporal modes. The choice of the temporal-mode set is not unique.

Figure 2

Single-stage TWM (a), (b) and FWM (c), (d) dominant Schmidt-mode conversion efficiencies $|{\rho }_n{|}^2$ (a), (c) for various pump powers, with selectivities $S$ listed in the legend. (b), (d) $s$-channel input Schmidt modes, illustrating increased temporal skewness with increasing $\gamma$. The values of $t^{\prime }$ are relative to a $|{\beta }_r^{\prime }-{\beta }_s^{\prime }|l$ of (b) 20 and (d) 10, where $l$ is the medium length. Pump $p$ is Gaussian in shape.

Figure 3

Temporal-mode interferometer using two nonlinear media (QFC 1 and QFC 2) with identical (DC) or opposite-sign (RC) dispersion. Appropriate choices for pump-pulse shapes, pump powers, and the phase shift $\theta$ will selectively frequency convert a specific $s$ (green, ${\omega }_s)$ TM into an $r$ (blue, ${\omega }_r)$ TM at a different central frequency, while not affecting temporally orthogonal $s$-input TMs. The pump $q$ is used only for ${\chi }^{\left(3\right)}$ implementations. WDM stands for wavelength-division multiplexer. PC stands for prechirp modules, which are necessary for ${\chi }^{\left(3\right)}$ implementations. The coupler C contains frequency-dependent delays for the DC case. Reproduced from Ref. [16], with permission.

Figure 4

Temporal-mode interferometry with the two interferometer arms being the frequency channels $s$ (green, lighter shade) and $r$ (blue, darker shade), undergoing two complete collisions in the two fibers and exchanging energy. This visualization plots the color-coded signal-field intensities in the average velocity frame with populated green input and empty blue input at $z=0$, for the RC configuration found by numerical solution of the equations of motion. (a) Both pumps are Gaussian, and the first Schmidt-mode green-to-blue conversion is nearly 100% at phase-shift $\theta =0$. (b) Both pumps are Gaussian, and the second Schmidt-mode conversion is nearly 0%. (c) Both pumps are Gaussian, and the first Schmidt-mode conversion efficiency is suppressed to zero at phase-shift $\theta =\pi$. (d) Pump $q$ is Gaussian, and the shape of pump $p$ is tailored to convert the green input mode from (b) into a Gaussian-like blue output with nearly 100% efficiency at $\theta =0$.

Figure 5

(a), (c), (e), (g) The dominant input and (b), (d), (f), (h) output Schmidt modes for the (a), (b), (c), (d) first and (e), (f), (g), (h) second stages for both (a), (b), (e), (f) $r$ and (c), (d), (g), (h) $s$ channels for both RC (red, lighter shade) and DC (black) configurations of TWM-TMI for a Gaussian-shaped pump, and $\zeta =|{\beta }_r^{\prime }-{\beta }_s^{\prime }|l/{\tau }_p=200$. The values of $t$ are relative to a $|{\beta }_r^{\prime }-{\beta }_s^{\prime }|l$ of 20. Due to the nature of temporal skewing, the inter-stage mode-matching between (b), (d) the first-stage output Schmidt modes and (e), (g) the second-stage input Schmidt modes is larger for the RC than the DC configuration, thus yielding better selectivity. The complete two-stage composite system Schmidt modes for TWM-TMI in the RC configuration are plotted in Fig. 6.

Figure 6

Schmidt modes for TWM-TMI in the RC configuration with Gaussian pump, and $\zeta =|{\beta }_r^{\prime }-{\beta }_s^{\prime }|l/{\tau }_p=200$, yielding a selectivity of $0.9846$. The magnitudes of the first two Schmidt modes are shown for (a) $r$-input, (b) $r$-output, (c) $s$-input, and (d) $s$-output. Values of $t$ are relative to a $|{\beta }_r^{\prime }-{\beta }_s^{\prime }|l$ of 20. Fig. 7 shows the conversion efficiencies of the first four Schmidt modes for two different pump shapes.

Figure 7

(a) Conversion efficiencies for Gaussian-pumped-TWM-TMI Schmidt modes in the RC configuration. The CE for a Gaussian pump are shown in green (darker shade), and those for a custom pump shape tailored to drop the second Schmidt mode are shown in magenta (lighter shade). (b) The corresponding pump shapes shown with matching colors. Values of $t$ are relative to a $|{\beta }_r^{\prime }-{\beta }_s^{\prime }|l$ of 20.

Figure 8

Dominant Schmidt mode conversion efficiencies for TWM-TMI in the RC configuration, illustrating asymptotic improvement in selectivity with decreasing pump-pulse width ${\tau }_p$ relative to interaction time $({\beta }_r^{\prime }-{\beta }_s^{\prime })l$.

Figure 9

The first Schmidt modes of (a) $r$-input, (b) $r$-output, (c) $s$-input, and (d) $s$-output channels for both DC (black) and RC (red, lighter shade) configurations of a two-stage TMI implemented using TWM with a single Gaussian pump, and $|{\beta }_r^{\prime }-{\beta }_s^{\prime }|l/{\tau }_p=10$. Values of $t$ are relative to a $|{\beta }_r^{\prime }-{\beta }_s^{\prime }|l$ of 10.

Figure 10

Multistage TWM implementation of TMI, showing the ratio of converted to total energy of the first Schmidt mode versus $z^{\prime }$ (propagation distance). $N$ is the number of stages and $l$ is the medium length for each stage.

Figure 11

(a), (c), (e) The first-stage and (b), (d), (f) second-stage pump chirps used to derive the three interstage mode-matched Schmidt-mode phase profiles shown in Fig. 12. The chirp parameters {${\epsilon }_p,{\epsilon }_q$} are {1, 0} in (a), (b), {2, 0} in (c), (d), and {0.5, 0.5} in (e), (f). Values of $t$ are relative to a $|{\beta }_r^{\prime }-{\beta }_s^{\prime }|l$ of 10.

Figure 12

(a), (c), (e), (g) The dominant input and (b), (d), (f), (h) output Schmidt modes for (a), (b), (c), (d) the first and (e), (f), (g), (h) second stages for both (a), (b), (e), (f) $r$ and (c), (d), (g), (h) $s$ channels for RC configuration of FWM-TMI for a Gaussian-shaped pumps, and complete pump collisions $\left[\right|{\beta }_r^{\prime }-{\beta }_s^{\prime }|l/({\tau }_p+{\tau }_q)=5]$. Also shown are (i)–(p) the corresponding Schmidt-mode phase profiles for three separate choices of pump chirps for the two stages. Due to pump-induced self- and cross-phase modulation, the second-stage pump chirps have to be related to the first-stage pump chirps in the manner specified in Eq. (16) for good interstage mode-matching between (b), (d), (j), (l) and (e), (g), (m), (o) respectively. The complete two-stage composite system Schmidt modes for FWM-TMI in the RC configuration are plotted in Fig. 13. Values of $t$ are relative to a $|{\beta }_r^{\prime }-{\beta }_s^{\prime }|l$ of 10.

Figure 13

(a), (b), (c), (d) The first two Schmidt modes, and (e), (f), (g), (h) first Schmidt-mode phase profiles for FWM-TMI in the RC configuration with Gaussian pumps. The first two Schmidt modes are shown for (a) $r$-input, (b) $r$-output, (c) $s$-input, and (d) $s$-output. Pump prechirp parameters used were ${\epsilon }_p=2,{\epsilon }_q=0$. The selectivity was $0.9873$. Values of $t$ are relative to a $|{\beta }_r^{\prime }-{\beta }_s^{\prime }|l$ of 10.