On-demand generation of traveling cat states using a parametric oscillator

We theoretically propose a method for on-demand generation of traveling Schrödinger cat states, namely, quantum superpositions of distinct coherent states of traveling fields. This method is based on deterministic generation of intracavity cat states using a Kerr-nonlinear parametric oscillator (KPO) via quantum adiabatic evolution. We show that the cat states generated inside a KPO can be released into an output mode by dynamically controlling the parametric pump amplitude. We further show that the quality of the traveling cat states can be improved by using a shortcut-to-adiabaticity technique.

Figure 1

On-demand generation of traveling cat states using an KPO. Superconducting-circuit implementation of the KPO is assumed [15, 17, 18, 19, 20], where the loop with two Josephson junctions depicted by crosses is a dc superconducting quantum interference device (SQUID). The KPO is capacitively coupled to a transmission line for the output mode. Generated traveling cat states can be directly observed by Wigner tomography.

Figure 2

Simulation results of traveling cat-state generation using a KPO. Left: Time evolutions of the real part of the pump amplitude, $p\left(t\right)$, and the average photon number in the KPO, $\langle a^{†}a\rangle$, which is proportional to the square of the pulse envelope $f_p\left(z\right)$ (see Appendix pp1). In (c) and (d), the imaginary part of the pump amplitude, $p^{\prime }\left(t\right)$, for shortcut to adiabaticity is also plotted. Right: Wigner function, $W\left(\beta \right)$, of the output pulse. See Table 1 for the settings of (a)–(d).

Figure 3

$J$ dependence of the final photon number, $n_{\mathrm{out}}={\sum }_{j=1}^J\langle {\tilde{b}}_j^{†}{\tilde{b}}_j\rangle$, in the output mode. (a)–(d) correspond to Figs. 2–2, respectively. Circles represent simulation results. Solid lines are fitted curves with $n_{\mathrm{out}}\left(J\right)=n_0-b/J$, where $n_0$ and $b$ are the fitting parameters. Horizontal dashed lines represent $n_{\mathrm{out}}\left(J\right)=n_0$. Arrows indicate the discrepancies between the data and $n_0$.

Figure 4

Simulation results for shortcut to adiabaticity. (a) Pump amplitudes $p\left(t\right)$ (black long-dashed line), $p^{\prime }\left(t\right)$ in our method (cyan solid line), and $p^{\prime }\left(t\right)$ in Ref. [15] (red short-dashed line). (b) Average photon number in KPO, $\langle a^{†}a\rangle$. (c) Fidelity between the final state and the ideal even cat state with amplitude of $\sqrt{2}$. (d) Magnification of (c) around the final time. In (b)–(d), cyan solid, red short-dashed, and black long-dashed lines correspond to our method, the method proposed in Ref. [15], and the case of no $p^{\prime }\left(t\right)$ (without shortcut-to-adiabaticity technique), respectively.