Feedback-stabilized dynamical steady states in the Bose-Hubbard model

The implementation of a combination of continuous weak measurement and classical feedback provides a powerful tool for controlling the evolution of quantum systems. In this paper, we investigate the potential of this approach from three perspectives. First, we consider a double-well system in the classical large-atom-number limit, deriving the exact equations of motion in the presence of feedback. Second, we consider the same system in the limit of small atom number, revealing the effect that quantum fluctuations have on the feedback scheme. Finally, we explore the behavior of modest-sized Hubbard chains using exact numerics, demonstrating the near-deterministic preparation of number states, a tradeoff between local and nonlocal feedback for state preparation, and evidence of a feedback-driven symmetry-breaking phase transition.

Figure 1

System and measurement schematic. An ensemble of locally interacting (with strength $U$) neutral atoms described by the bosonic field operators ${\widehat{a}}_j$ are confined in a lattice potential with tunneling strength $J_j$ between sites $j$ and $j+1$. These atoms interact with independent modes of an optical field described by bosonic field operators ${\widehat{b}}_j$. After interacting, the optical modes are imaged via phase contrast imaging, implementing a spatially resolved optical homodyne detection providing a measurement sensitive to the quadrature field operator ${\widehat{X}}_j=({\widehat{b}}_j+{\widehat{b}}_j^{†})/2$. This idealized measurement assumes that the incident laser field is only forward scattered, essentially making the paraxial approximation.

Figure 2

Classical spin model computed for $f=32$, $Uf=2$, $J_0/\left(Uf\right)=0.35$, and $\mathrm{\Delta }=0$. (a) Trajectories as a function of $\theta ,\varphi$ with respect to the ${\mathbf{e}}_x$ axis. These orbits are separated by the red curve into regions of Josephson oscillations (e.g., orange curve) and oscillations around the macroscopically self trapping point (e.g., blue and green curves). The color indicates energy, with black denoting the lowest energy and white the largest. (b), (c) Orbits around both JO and MST fixed points. Red curves denote the result of our numerical simulation while the black dashed curves plot the result of linear response theory. Panel (b) plots the case with no measurement or feedback and panel (c) plots the case with feedback with $\tau =1/2$ and $t_m=0.01$. Feedback in $V$ [bottom two plots in (c)] damps to the JO and MST fixed points for $f{G}_V=-1$ and $1/4$, respectively. Feedback in $J$ [bottom plot in (c)] damps to the JO fixed point with $G_J=2$. Because the feedback is nonlinear, no linear response theory is displayed in the bottom plot.

Figure 3

Comparison of classical (red) and quantum (blue) simulations with identical parameters. (a) Free evolution around the JO fixed point for the same parameters as in Fig. 2, including $\tau =0.5$. (b) Evolution including measurement noise, backaction, and feedback for the same parameters as in Fig. 2, as well as a measurement strength $f^{1/2}\phi =t_m^{1/2}$ with $t_m=0.01$. These trajectories were averaged over 1024 realizations to show their average behavior. (c) Evolution for $N=2$, with parameters suitably scaled by $f$ to yield same classical dynamics as (b).

Figure 4

Average overlap $|\langle {\psi }_2|\psi \left(T\right)\rangle {|}^2$ for different values of $G$ and $\kappa$ for the nearest-neighbor approach using 256 trajectories.

Figure 5

Optimal behavior of the two feedback approaches using 2048 trajectories for each data point except for the nearest-neighbor approach for ${\psi }_3$, which uses 4096 trajectories. (a) Optimal gain $G_{\mathrm{opt}}$ as a function of $\kappa$. (b) Optimal target state overlap as a function of $\kappa$. The solid (dotted) lines correspond to the nearest-neighbor (imbalance) approach. Blue, red, and tan lines correspond to ${\psi }_T={\psi }_1,{\psi }_2,{\psi }_3$, respectively.

Figure 6

Average overlap $|\langle {\psi }_{\mathrm{T}}|\psi \left(T\right)\rangle {|}^2$ for different values of $G$ when the initial states are close to the target state using 4096 trajectories. The solid (dotted) lines correspond to the nearest-neighbor (imbalance) approach. Blue, red, and tan lines correspond to ${\psi }_T={\psi }_1,{\psi }_2,{\psi }_3$, respectively.

Figure 7

Solid blue line shows tunneling feedback used for symmetry-breaking transition as a function of $\delta {\epsilon }_j$. Red dotted lines correspond to the feedback used in the state preparation section, with the left (right) curve corresponding to the target state $|020202\rangle$ ($|202020\rangle$).

Figure 8

Twenty smallest effective energies of steady-state $H_{ss}$ as a function of $G$. Eigenvalues are calculated after symmetrizing ${\rho }_{ss}$ with respect to translations and reflections, leading to degeneracy for states related through these symmetries; quantitatively similar behavior emerges without symmetrization, although the eigenvalues are no longer as degenerate. The orange dots are doubly degenerate and correspond approximately to $|202020\rangle ,|020202\rangle$, with the strongest overlap occurring for the larger gaps. For $G<.5U$, ${\rho }_{ss}$ is averaged over 128 trajectories from $t=100U^{-1}$ to $t=400U^{-1}$. For $G\ge .5U$, ${\rho }_{ss}$ is averaged over 32 trajectories from $t=20U^{-1}$ to $t=320U^{-1}$. The initial time delay for averaging ensures that the system has relaxed to the steady state.