Creation, storage, and retrieval of an optomechanical cat state

We analyze a method for the creation, storage, and retrieval of optomechanical Schrödinger cat states, in which there is a quantum superposition of two distinct macroscopic states of a mechanical oscillator. In the quantum memory protocol, an optical cat state is first prepared in an optical cavity, then transferred to the mechanical mode, where it is stored and later retrieved using control fields. We carry out numerical simulations for the quantum memory protocol for optomechanical cat states using the positive-P phase space representation. This has a compact, positive representation for a cat state, thus allowing a probabilistic simulation of this highly nonclassical quantum system. It is essential to use importance sampling to carry out the simulation effectively. To verify the effectiveness of the cat-state quantum memory, we consider several cat-state signatures and show how they can be computed. We also investigate the effects of decoherence on a cat state by solving the standard master equation for a simplified model analytically, allowing us to compare with the numerical results. Focusing on the negativity of the Wigner function as a signature of the cat state, we evaluate analytically an upper bound on the time taken for the negativity to vanish, for a given temperature of the environment of the mechanical oscillator. We show consistency with the numerical methods. These provide exact solutions, allowing a full treatment of decoherence in an experiment that involves creating, storing, and retrieving mechanical cat states using temporally mode-matched input and output pulses. Our analysis treats the internal optical and mechanical modes of an optomechanical oscillator, and the complete set of input and output field modes which become entangled with the internal modes. The model includes decoherence due to thermal effects in the mechanical reservoirs, as well as optical and mechanical losses.

Figure 1

The state transfer protocol. During the writing stage, both the preparation field containing the cat state, and the transfer field that couples the preparation field to the cavity, are turned on for a duration of $t_w$. Both fields are turned off during the storage stage for $t_s$. The stored state is read out by applying a second transfer field for $t_r=t_w$.

Figure 2

Schematic diagram of the optomechanical system.

Figure 3

Probability density distribution $P\left(x\right)$ for the $x$ quadrature of the cat state Eq. (2.1) with ${\alpha }_0=2$ as given in Eq. (3.4).

Figure 4

Probability density distributions $P\left(p\right)$ for the $p$ quadrature of the cat state Eq. (2.1) with ${\alpha }_0=2$ as given in Eq. (3.5).

Figure 5

The Wigner function of a cat state as given in Eq. (3.9) for a coherent amplitude ${\alpha }_0=5$. Here $x$ and $y$ in the plot are the real and imaginary part of $\alpha$ in the Wigner function $W\left(\alpha \right)$, respectively.

Figure 6

The modulus of the density operator for a cat state in the coherent state basis as given in Eq. (3.11) with coherent amplitude ${\alpha }_0=5$ in the coherent state basis based on Eq. (3.11).

Figure 7

The $p$-quadrature probability distribution computed using the positive-P distribution with Eq. (4.18) for ${\alpha }_0=5$ after reading out from the quantum memory. The mean mechanical thermal noise and internal loss rate are chosen to be ${\overline{n}}_{\text{th}}={\mathrm{\Gamma }}_{\text{int}}=0$ throughout the simulation. Here the optomechanical cat state has a low decoherence due to the short storage time compared to the mechanical oscillator lifetime. This figure is for a storage time of $0.02/{\mathrm{\Gamma }}_m$. A total number of four samples are taken.

Figure 8

The $p$-quadrature probability distribution computed using using the positive-P distribution with Eq. (4.18) for ${\alpha }_0=5$. Other parameters as in Fig. 7. Here the optomechanical cat state decoheres for a storage time of $0.3466/{\mathrm{\Gamma }}_m$, which is the time a Wigner function loses its negativity according to Eq. (A12).

Figure 9

The Wigner function computed using the positive-P distribution and Eq. (4.20) for ${\alpha }_0=5$ after reading out from the quantum memory. Here $x$ and $y$ in the plot are the real and imaginary part of $\alpha$ in the Wigner function $W\left(\alpha \right)$ in Eq. (4.20) respectively. Other parameters as in Fig. 7. This figure has a storage time of $0.02/{\mathrm{\Gamma }}_m$, too short for substantial decoherence.

Figure 10

The Wigner function computed using the positive-P distribution and Eq. (4.20) for ${\alpha }_0=5$ after reading out from the quantum memory. As previously, $x$ and $y$ in the plot are the real and imaginary part of $\alpha$ in the Wigner function $W\left(\alpha \right)$ in Eq. (4.20), respectively. Other parameters as in Fig. 8. Here the optomechanical cat state decoheres after a storage time of $0.3466/{\mathrm{\Gamma }}_m$, which is the time a Wigner function loses its negativity according to Eq. (A12).

Figure 11

The reconstructed density operator computed using the positive-P distribution and Eq. (5.1) for ${\alpha }_0=5$ after reading out from the quantum memory. The mean mechanical thermal noise and internal loss rate are chosen to be ${\overline{n}}_{\text{th}}={\mathrm{\Gamma }}_{\text{int}}=0$, and the storage time is $0.02/{\mathrm{\Gamma }}_m$. A total number of four samples are taken.

Figure 12

The reconstructed density operator computed using the positive-P distribution and Eq. (5.1) for ${\alpha }_0=5$ after reading out from the quantum memory. The mean mechanical thermal noise and internal loss rate are chosen to be ${\overline{n}}_{\text{th}}={\mathrm{\Gamma }}_{\text{int}}=0$. Here the optomechanical cat state decoheres due to the finite mechanical lifetime after a storage time of $0.3466/{\mathrm{\Gamma }}_m$, which is the time a Wigner function loses its negativity according to Eq. (A12). A total number of four samples are taken.

Figure 13

The Wigner negativity of the read-out state as a function of the dimensionless storage time (in multiples of $1/{\mathrm{\Gamma }}_m$) for cat amplitudes ${\alpha }_0=2$, 3, 4, and 5. The mean mechanical thermal occupation number and internal loss rate are chosen to be ${\overline{n}}_{\text{th}}={\mathrm{\Gamma }}_{\text{int}}=0$. The corresponding data points in circles are analytical values based on Eq. (5.3). The dashed vertical line is the upper bound of the time for a Wigner function to lose its negativity, as given in Eq. (A12). For ${\overline{n}}_{\text{th}}=0$, the upper bound, in multiples of $1/{\mathrm{\Gamma }}_m$, is 0.3466. A total number of four samples are taken. The error bars denote the time-step error in the phase-space simulations.

Figure 14

The Wigner negativity of the read-out state as a function of the dimensionless storage time (in multiples of $1/{\mathrm{\Gamma }}_m$) for cat amplitudes ${\alpha }_0=2$, 3, 4, and 5. The mean mechanical thermal occupation number ${\overline{n}}_{\text{th}}=2$, and the internal loss is ${\mathrm{\Gamma }}_{\text{int}}=0$. The dashed vertical line is the upper bound of the time for a Wigner function to lose its negativity, as given in Eq. (A12). For ${\overline{n}}_{\text{th}}=2$, the upper bound, in multiples of $1/{\mathrm{\Gamma }}_m$, is 0.0912. A total number of $2\times {10}^5$ samples are taken. The error bars include both the sampling error and time-step error.

Figure 15

The Wigner negativity of the read-out state as a function of the dimensionless storage time $T_s$ (in multiples of $1/{\mathrm{\Gamma }}_m$) and the mean mechanical thermal number ${\overline{n}}_{\text{th}}$ for a cat amplitude ${\alpha }_0=2$. The internal loss is ${\mathrm{\Gamma }}_{\text{int}}=0$. A total number of $2\times {10}^5$ samples are taken, except when ${\overline{n}}_{\text{th}}=0$, where four samples are taken instead.

Figure 16

The Wigner negativity of the read-out state as a function of the dimensionless storage time (in multiples of $1/{\mathrm{\Gamma }}_m$) for cat amplitudes ${\alpha }_0=2$, 3, 4, and 5. The mean mechanical thermal occupation number ${\overline{n}}_{\text{th}}=0$. The internal cavity decay rate is nonzero and contributes to further decoherence of the cat state. Here the internal cavity decay rate is set to be ${\mathrm{\Gamma }}_{\text{int}}=0.05$. A total number of four samples are taken. The error bars denote the time-step error in the phase-space simulations.

Figure 17

The Wigner negativity of the read-out state as a function of the dimensionless storage time (in multiples of $1/{\mathrm{\Gamma }}_m$) for cat amplitudes ${\alpha }_0=2$, 3, 4, and 5. The mean mechanical thermal occupation number ${\overline{n}}_{\text{th}}=2$. The internal cavity decay rate is nonzero and contributes to further decoherence of the cat state. Here the internal cavity decay rate is set to be ${\mathrm{\Gamma }}_{\text{int}}=0.05$ and the initial mechanical mode has an occupation number of 0.5. A total number of $2\times {10}^5$ samples are taken. The error bars include both the sampling error and time-step error.