Generation and storage of spin squeezing via learning-assisted optimal control

The generation and storage of spin squeezing is an attractive topic in quantum metrology and the foundation of quantum mechanics. The major models to realize the spin squeezing are the one- and two-axis twisting models. Here, we consider a collective spin system coupled to a bosonic field, and show that proper constant-value controls in this model can simulate the dynamical behavior of these two models. More interestingly, a better performance of squeezing can be obtained when the control is time varying, which is generated via a reinforcement learning algorithm. However, this advantage becomes limited if the collective noise is involved. To deal with this, we propose a four-step strategy for the construction of a type of combined controls, which include both constant-value and time-varying controls, but performed at different time intervals. Compared to the full time-varying controls, the combined controls not only give a comparable minimum value of the squeezing parameter over time, but also provide a better lifetime and larger full amount of squeezing. Moreover, the amplitude form of a combined control is simpler and more stable than the full time-varying control. Therefore, our scheme is very promising to be applied in practice to improve the generation and storage performance of squeezing.

Figure 1

The minimum value of ${\xi }^2$ ($min{\xi }^2$) as a function of $\zeta$ in the case of (a) $N=6$ and (b) $N=8$ for both noise and noiseless scenarios. The dash-dotted black and solid blue lines represent $min{\xi }^2$ for the constant-value (cv) controls with and without noise. The dotted cyan and dashed yellow lines represent $min{\xi }^2$ in the noncontrolled scenario with and without noise. The values of $min{\xi }^2$ for the effective OAT- and TAT-type Hamiltonians are shown as the red stars and purple triangles. The decay rates are set to be $\kappa =\gamma =0.01g$ in the plots.

Figure 2

(a) Schematic of control generation for the enhancement of spin squeezing via reinforcement learning in a collective spin-half system coupled to a cavity. (b) Brief flow chart of the DDPG algorithm [30].

Figure 3

The results for unitary dynamics in the case of $N=6$ (a1–a3) and $N=8$ (b1–b3). Panels (a1) and (b1) show the evolution of ${\xi }^2$ (in the unit of dB) with the time-varying (tv) control (dash-dotted green lines), with optimal constant-value (cv) control (dashed red lines), and without control (solid blue lines), respectively. The time-varying controls are generated via the learning with the corresponding total reward given in (a2) and (b2). Panels (a3) and (b3) give the optimal control $\zeta$ with respect to the dash-dotted green lines in (a1) and (b1).

Figure 4

The variety of spin squeezing as a function of particle number $N$ for the unitary dynamics. The dash-dotted green and solid blue lines represent the minimum spin squeezing with the optimal constant-value control and time-varying control. The dashed red line represents the value of spin squeezing with the time-varying control at the time that achieves the minimum squeezing in the optimal constant-value control. The parameters are set to be the same as Fig. 3. The green and blue lines in the inset are the optimal times for the minimum squeezing in the case of optimal constant-value and time-varying controls, respectively.

Figure 5

The results for noisy dynamics in the case of $N=6$ (a1–a3) and $N=8$ (b1–b3). Panels (a1) and (b1) show the evolution of ${\xi }^2$ (in the unit of dB) with the time-varying (tv) control (dash-dotted green lines), with optimal constant-value (cv) control (dashed red lines), and without control (solid blue lines), respectively. The time-varying controls are generated via the learning with the corresponding total rewards given in (a2) and (b2). Panels (a3) and (b3) give the optimal control $\zeta \left(t\right)$ with respect to the dash-dotted green lines in (a1) and (b1). The noisy dynamics in the plots are governed by Eq. (3) with the decay rates $\kappa =\gamma =0.01g$.

Figure 6

The evolution of ${\xi }^2$ with different dephasing rates $\gamma$ with time-varying control or without control in the case of $N=8$. Other parameters are set to be the same as Fig. 5.

Figure 7

The illustration of the combined strategy and the corresponding results. (a) Four steps for the generation of a combined control. Step 1: Choose a reasonable regime $[{\zeta }_a,{\zeta }_b]$ around the optimal point ${\zeta }_{min}$ that gives the minimum value of $min{\xi }^2$ in the scenario of constant-value controls. Step 2: Construct the stitched controls with the constant-value part given in $[{\zeta }_a,{\zeta }_b]$. Step 3: Calculate $S$ for all stitched controls and find the optimal point ${\zeta }_{\mathrm{opt}}$ that gives the maximum $S$. Step 4: Learn the time-varying part of the ${\zeta }_{\mathrm{opt}}$'s stitched control and construct the final combined control. (b) $\mathcal{S}$ as a function of the stitched controls. This plot is used to search ${\zeta }_{\mathrm{opt}}$. (c, d) The comparison of ${\xi }^2\left(t\right)$ given by different controls for $N=6$ and 8. The parameters are set the same as those in previous figures.

Figure 8

The evolution of optimal squeezing angle in the case of the time-varying control under noiseless dynamics (dash-dotted green line) and noisy dynamics (dashed blue line), and in the case of the combined control (solid red line). The dotted purple and black lines represent the angles for $min{\xi }^2$ under noisy and noiseless dynamics. The decay rates for the blue and red lines are set to be $\kappa =\gamma =0.01g$. Other parameters are set to be the same as the previous figures.

Figure 9

The dynamics of the fidelity between the evolved states given by the original Hamiltonian in Eqs. (A1) and (A2) and the effective Hamiltonian in Eq. (A11).