Quantum annealing with ultracold atoms in a multimode optical resonator

A dilutely filled $N$-site optical lattice near zero temperature within a high-$Q$ multimode cavity can be mapped to a spin ensemble with tailorable interactions at all length scales. The effective full site to site interaction matrix can be dynamically controlled by the application of up to $N(N+1)/2$ laser beams of suitable geometry, frequency, and power, which allows for the implementation of quantum annealing dynamics relying on the all-to-all effective spin coupling controllable in real time. Via an adiabatic sweep starting from a superfluid initial state one can find the lowest-energy stationary state of this system. As the cavity modes are lossy, errors can be amended and the ground state can still be reached even from a finite temperature state via ground-state cavity cooling. The physical properties of the final atomic state can be directly and almost nondestructively read off from the cavity output fields. As an example we simulate a quantum Hopfield associative memory scheme.

Figure 1

A partially filled optical lattice with $N$ sites inside a multimode optical resonator is pumped from the side by several lasers with frequencies close to cavity resonances.

Figure 2

Top: Input parameters for the two input states ${\mathbf{\chi }}_1$ (dark/orange) and ${\mathbf{\chi }}_2$ (light/green) given in the main text for a chosen set of modes $\mathcal{B}$, where the second transverse mode index $m=0$. Bottom: These input parameters can be realized by the pump strengths ${\eta }_m=\sqrt{-f_m({\mathrm{\Delta }}_{c,m}^2+{\kappa }_m^2)/\left(\hslash {\mathrm{\Delta }}_{c,m}\right)}$, where ${\kappa }_m=1000J/\hslash$ and ${\mathrm{\Delta }}_{c,m}=\mathrm{sgn}(-f_m){\kappa }_m$. Due to weak coupling, e.g., the first mode needs to be pumped strong in both cases.

Figure 3

Spectrum of the lowest few eigenvalues ${\epsilon }_n$ of $H_{\mathrm{sp}}$, Eq. (9), as a function of $\zeta$ for a recall of the input pattern ${\mathbf{\chi }}_1$, leading to the recovered interaction matrix ${\tilde{A}}_{{\chi }_1}$. The dotted line at $\zeta /J=0.28$ shows the position of the smallest gap between ground state and first excited state, while the brightness of the lines encodes the overlap of the target memory state with the eigenstates $|\langle {\varphi }_n\left(\zeta \right)|{\mathit{w}}_1\rangle {|}^2$ from light ($=0$) to dark ($=1$). Already at $\zeta /J=2$, the ground state is very close to the target state: $|\langle {\varphi }_0(\zeta =2J)|{\mathit{w}}_1\rangle {|}^2=0.976$.

Figure 4

Time evolution of the overlap between the instantaneous ground state $|{\varphi }_0\left(\zeta \right)\rangle$ and the solution of the time-dependent Schrödinger equation $\left|\psi \right(t)\rangle$ for the linear ramp $\zeta \left(t\right)=2Jt/\tau$ and different annealing times $\tau$. The vertical dotted line shows the location of the smallest gap, as in Fig. 3.

Figure 5

Time evolution of the expectation values $\langle {\sigma }_i^z\rangle$ for each lattice site $i$ for an annealing time of $J\tau =50$ and a linear ramp $\zeta \propto t$ (see Fig. 4). The overlap with the target state in the end is $|\langle \psi \left(\tau \right)|{\mathit{w}}_1\rangle {|}^2=0.959$. Due to the finite annealing time there is a fraction in the excited states and thus the curves do not converge to 1 and $-1$ exactly.

Figure 6

Different atomic states give rise to distinct intensity patterns, which can be measured. Here they are shown for the states ${\mathit{w}}_1$ (dark/orange) and ${\mathit{w}}_2$ (light/green) at $\zeta /J=2$. The parameters are as given in Fig. 2.

Figure 7

The lattice location within the cavity used in the example in Sec. 5. The (blue) dots indicate positions of individual lattice sites. The depicted cavity mode is $(n,l)=(100,2)$. The (black) arcs depict the cavity mirrors.

Figure 8

The energies depending on the choice of $\nu$ when recalling ${\mathbf{\chi }}_1$. The recall bias $E_{\mathrm{AM}}\left({\mathbf{\chi }}_1\right)$ (solid blue) has to be smaller than $E_{\mathrm{AM}}\left({\mathit{w}}_1\right)$ (dashed green), hence we need to choose $0<\nu <4$. $E_{\mathrm{AM}}\left({\mathit{w}}_2\right)$ (dotted red) is not affected by $\nu$ due to $\langle {\mathbf{\chi }}_1,{\mathit{w}}_2\rangle =0$. Here $P=2$ such that the memory patterns are degenerate.