Fast amplification and rephasing of entangled cat states in a qubit-oscillator system

We study a qubit-oscillator system, with a time-dependent coupling coefficient, and present a fast scheme for generating entangled Schrödinger-cat states with large mean photon numbers and also a scheme that protects the cat states against dephasing caused by the nonlinearity in the system. We focus on the case where the qubit frequency is small compared to the oscillator frequency. We first present the exact quantum state evolution in the limit of infinitesimal qubit frequency. We then analyze the first-order effect of the nonzero qubit frequency. Our scheme works for a wide range of coupling strength values, including the recently achieved deep-strong-coupling regime.

Figure 1

Evolution of $\tilde{g}\left(t\right)$ under time-dependent $g\left(t\right)$ with infinitesimal $\mathrm{\Delta }$. (a) Three different time dependencies of $g\left(t\right)$ as a function of time $t$. (b) The trajectories of $\tilde{g}\left(t\right)$ corresponding to (a), which, as seen in Eq. (4), determine the evolution of the dynamical evolution eigenstates. We set the initial condition as $\tilde{g}\left(0\right)/g\left(0\right)=1$. (c) The time dependencies of $g\left(t\right)$ and $|\tilde{g}\left(t\right)|$ as functions of time $t$, in the case of sinusoidal driving force $g\left(t\right)=g\left(0\right)cos\omega t$. (d) The trajectories of $g\left(t\right)$ and $\tilde{g}\left(t\right)$ corresponding to (c). The amplitude of $\tilde{g}\left(t\right)$ keeps increasing, showing cat-state amplification. The blue and red dots in (c) and (d) indicate where the modulation stops ($t=4\pi /\omega$) in the scheme shown in Fig. 2.

Figure 2

Cat-state amplification and rephasing in the Wigner representation of the oscillator state projected onto the qubit state ${|+\rangle }_x+{|-\rangle }_x$. The $x$ and $y$ axes are the oscillator's dimensionless field quadratures $p$ and $q$. The parameters are modulated as explained in the text with $\mathrm{\Delta }=0.1\omega$ and $g\left(0\right)=0.833\omega$. The insets show the central parts of the figures. (a) The initial state, which is taken to be the ground state, reasonably resembles the cat state $|-\frac{g\left(0\right)}{\omega }\rangle +|+\frac{g\left(0\right)}{\omega }\rangle$, with a fidelity of 0.99986. After the initial state goes through two oscillator periods of the sinusoidally driven cat-state-amplification process, the resulting state is shown in (b), which resembles the target cat state $|-\frac{\tilde{g}(4\pi /\omega )}{\omega }\rangle +|+\frac{\tilde{g}(4\pi /\omega )}{\omega }\rangle$ with $|\tilde{g}/\omega |=5.3$ and a fidelity of 0.989. We now let the state in (b) freely evolve for ten oscillator periods, and the resulting state of the oscillator is shown in (c). The fidelity between the state of (c) and the target cat state is only 0.933, since the state is dephased by the nonzero $\mathrm{\Delta }$. However, if we insert one ${\widehat{\sigma }}_x-\pi$ pulse in the middle of the ten oscillator periods, the evolved state, shown in (d), has a 0.989 fidelity with the target cat state, analogous to a Hahn spin-echo rephasing effect.

Figure 3

Cat-state amplification and rephasing in the Wigner representation of the oscillator state projected onto the qubit state ${|+\rangle }_x+{|-\rangle }_x$. The parameters are $\mathrm{\Delta }=0.1\omega$; $g\left(0\right)=0.833\omega$, and the coupling coefficient is modulated as $g\left(t\right)=g\left(0\right)cos\omega t$. (a) The initial state, which is taken to be the ground state, reasonably resembles the cat state $|-\frac{g\left(0\right)}{\omega }\rangle +|+\frac{g\left(0\right)}{\omega }\rangle$, with a fidelity of 0.99986. After the initial state goes through four oscillator periods of the sinusoidally driven cat-state-amplification process, the resulting state is shown in (b), which is the cat state $|-\frac{\tilde{g}(8\pi /\omega )}{\omega }\rangle +|+\frac{\tilde{g}(8\pi /\omega )}{\omega }\rangle$, with $|\tilde{g}/\omega |=10.5$ and a fidelity of 0.988. We now let the state in (b) freely evolve for ten oscillator periods, and the resulting state of the oscillator is shown in (c). The fidelity between the state of (c) and the state $|-\frac{\tilde{g}(8\pi /\omega )}{\omega }\rangle +|+\frac{\tilde{g}(8\pi /\omega )}{\omega }\rangle$ is only 0.946, since the cat state is dephased by the nonzero $\mathrm{\Delta }$. However, if we insert one ${\widehat{\sigma }}_x-\pi$ pulse in the middle of the ten oscillator periods, the evolved state, shown in (d), has a 0.980 fidelity with the state in (b), analogous to a Hahn spin-echo rephasing effect. Zoom-in of the central part in (a) to (d) is shown in (a′) to (d′), respectively.

Figure 4

Cat-state amplification and rephasing in the Wigner representation of the oscillator state projected onto the qubit state ${|+\rangle }_x+{|-\rangle }_x$. The parameters are $\mathrm{\Delta }=0.1\omega$; $g\left(0\right)=0.1\omega$. (a) The initial state, which is taken to be the ground state, reasonably resembles the cat state $|-\frac{g\left(0\right)}{\omega }\rangle +|+\frac{g\left(0\right)}{\omega }\rangle$, with a fidelity of 1.000. (b) After the initial state goes through five oscillator periods of cat-state-amplification process using ${\sigma }_z-\pi$ pulses at every half of the oscillator period, the resulting state of the oscillator is shown, which is far from the cat state, due to the effect of the finite $\mathrm{\Delta }$. (c) After the initial state goes through five oscillator periods of cat-state-amplification process using ${\sigma }_y-\pi$ pulses at every half of the oscillator period, the resulting state of the oscillator is shown, which is the cat state, $|-\frac{\tilde{g}(10\pi /\omega )}{\omega }\rangle +|+\frac{\tilde{g}(10\pi /\omega )}{\omega }\rangle$, with $|\tilde{g}/\omega |=2.1$ and a fidelity of 0.999953. In (c), by using ${\sigma }_y-\pi$ pulses instead of ${\sigma }_z-\pi$ pulses, cat state amplification and rephasing are simultaneously realized.