A Photonic Tautochrone

Abstract

We propose to implement an optical analogue of the tautochrone property of the cycloid to allow the focusing of ultrashort pulses inside photonic systems. This allows to enhance nonlinear effects, resulting in orders of magnitude increase of nonlinearity-induced phase shifts, while employing low irradiances. Building upon the optical-mechanical analogy, we show how to produce optical limiters for temporal light pulses, and how to implement temporal bistability and even multistability with large numbers of states. Finally, we move this concept to the quantum realm and predict a tautochrone quantum blockade regime with a stronger antibunching.

Figures (9)

• Figure 1: Photonic Tautochrone. a) A pulsed excitation initializes particles at different positions of a parabolic potential. The particles are then accelerated toward the center corresponding to a focusing of light. b) A space-time plot of intensity at $y=0$ shows that the focusing occurs at $t=\pi/2$ and integer delays of $\pi$ thereafter. The colour scale has been saturated at $1/100$ the peak value. c) The phase of light in the center undergoes a jump of $\pi$ after each focusing event (red); the shift is enhanced in the nonlinear regime (blue). Circles and stars mark the times and phases of the excitation pulses for the optical limiter and bistable regimes, respectively. d) Nonlinearity-induced phase shift for free particles (grey) and in the presence of a parabolic potential (blue). The initial field $\psi(x)$ is normalized to unity. Other parameters: $\sigma_0=5$; $\gamma=0$.
• Figure 2: Pulsed Excitation. (a) Intensity $|\psi(r,t)|^{2}$ (left) and its linecut at $r=0$ (right) under periodic excitation without interactions, computed from Eq. \ref{['eq:Schrodinger']} ($\psi(r,\phi,t)$ is independent of $\phi$, so we do not plot the $\phi$ dependence). The repetition period was $\pi$, $\gamma = 1/(2\pi)$, and all other parameters were the same as in Fig. \ref{['fig:TautochroneScheme']}, with the spatially integrated pulse intensity set to unity. Arrows mark the pulse arrival times, and their orientation encodes the excitation phase; these pulses correspond to the circles shown in Fig. \ref{['fig:TautochroneScheme']}(c). We have chosen the driving term to have a Gaussian form in space. (b) Same analysis as in (a), but with nonlinear interaction included, which leads to a pronounced quenching of the peaks intensity. Intensities in (a) and (b) have been normalised to the maximum peak intensity in (a). For subsequent pulses, the system eventually settles in a state where the intensity is periodic in time.
• Figure 3: Optical Limiter & Bistability. a) Power dependence of a photonic tautochrone under periodic pulsed excitation, compared to the same setup under continuous wave (CW) excitation. $\langle|\psi|^2\rangle$ and $\langle|F|^2\rangle$ represent spatially integrated and time-averaged field and driving intensities, respectively. Parameters were the same as in Fig. \ref{['fig:PeriodicPulsing']}b. As the overall response to the CW excitation is weaker than the pulsed case, $|F|^2$ is multiplied by a factor of $100$ for CW excitation. b) Same as (a) but with the phase delay between subsequent pulses set to $1.25\pi$. The driving field intensity is slowly ramped up and down, revealing a hysteresis region (shaded).
• Figure 4: Multistability. A photonic tautochrone is driven by two pulse trains with a repetition period of $\pi/2$, each encoding a different bit. Under these conditions, a hysteresis curve analogous to that in Fig. \ref{['fig:OpticalLimiter']}b is obtained, and the pulse intensity is chosen to operate within the multistable regime. Depending on the excitation history, four distinct oscillating states can be produced. Panels (a–d) correspond to the four possible bit encodings $\ket{00}$, $\ket{01}$, $\ket{10}$, and $\ket{11}$, respectively. All remaining parameters are the same as in Fig. \ref{['fig:OpticalLimiter']}b. The colour scales have been saturated to $1/10$ of the peak intensity in (d). Unlike in Fig. 2, here we show the intensity after many pulses such that the initial transient dynamics has been completed.
• Figure 5: Photon blockade. For sufficient photon nonlinearity $\alpha$, conventional photon blockade will shift the multi-photon states out of resonance causing antibunching, as is well known under continuous wave (CW) excitation, also in the parabolic trap. The enhancement of interactions through pulses in the tautochrone scheme however, is able to enhance such antibunching. Simulations were done over fixed $\alpha=0.1$$\omega l^2$, where $l,\omega$ are the harmonic oscillator length and frequency, respectively, while varying $F_0$, which affects also the average particle number $N$. Refer to supplementary for details.