Enhancing Associative Memory Recall and Storage Capacity Using Confocal Cavity QED

We introduce a near-term experimental platform for realizing an associative memory. It can simultaneously store many memories by using spinful bosons coupled to a degenerate multimode optical cavity. The associative memory is realized by a confocal cavity QED neural network, with the modes serving as the synapses, connecting a network of superradiant atomic spin ensembles,which serve as the neurons. Memories are encoded in the connectivity matrix between the spins and can be accessed through the input and output of patterns of light. Each aspect of the scheme is based on recently demonstrated technology using a confocal cavity and Bose-condensed atoms. Our scheme has two conceptually novel elements. First, it introduces a new form of random spin system that interpolates between a ferromagnetic and a spin glass regime as a physical parameter is tuned—the positions of ensembles within the cavity. Second, and more importantly, the spins relax via deterministic steepest-descent dynamics rather than Glauber dynamics. We show that this nonequilibrium quantum-optical scheme has significant advantages for associative memory over Glauber dynamics: These dynamics can enhance the network’s ability to store and recall memories beyond that of the standard Hopfield model. Surprisingly, the cavity QED dynamics can retrieve memories even when the system is in the spin glass phase. Thus, the experimental platform provides a novel physical instantiation of associative memories and spin glasses as well as provides an unusual form of relaxational dynamics that is conducive to memory recall even in regimes where it was thought to be impossible.

Popular Summary

Principles from biological and quantum systems are driving a revolution in computing. Artificial neural networks, inspired by the architecture of the brain, already outperform the best humans in complex games such as Go. In this work, we present a neural network composed of atoms and light that can learn and recognize arbitrary sets of patterns, an ability known as associative memory. This quantum-optical neural network is found to improve upon the standard Hopfield model of associative memory in terms of both memory recall ability and the number of memories that can be learned.

We describe how this scheme would work, based entirely on already-demonstrated technology. The modes of a degenerate optical cavity serve as the synapses connecting a network of superradiant atomic spin ensembles serving as the neurons. Binary patterns are stored as memories by placing the atoms at specific locations in the cavity. Those memories are accessed through the input and output of patterns of light.

Improvements over the Hopfield model are enabled primarily by the native driven-dissipative dynamics of the open quantum system: The atomic spins recall a stored memory by flipping to the memory state under a discrete form of naturally realized steepest-descent dynamics. Our work provides a sound theoretical foundation for the efficacy of future quantum-optical neuromorphic computing platforms.

Figure 1

Sketch of the confocal cavity QED system with atomic spin ensembles confined by optical tweezers. Shown are the cavity mirrors (gray), superpositions of the cavity modes (blue), Bose-Einstein condensates (BECs) (red), optical tweezer beams (green), transverse pump beam (red), and interference on a charge-coupled device (CCD) camera for holographic imaging of spin states.

Figure 2

Double Raman atomic coupling. The scheme provides a two-level system, corresponding to the two hyperfine states of ${}^{87}\mathrm{Rb}$. Two transversely oriented pump beams are used to generate cavity-assisted two-photon processes coupling the states $|F,m_F⟩=|2,-2⟩\equiv |\uparrow ⟩$ and $|F,m_F⟩=|1,-1⟩\equiv |\downarrow ⟩$.

Figure 3

(a) All-to-all, sign-changing connectivity between spin ensembles is achieved via photons propagating in superpositions of cavity modes. Blue and red indicate ferromagnetic versus antiferromagnetic $J_{ij}$ links. Only four nodes are depicted. Individual spins align within an ensemble due to superradiance, while ensembles organize with respect to each other due to $J_{ij}$ coupling. Cavity emission allows for holographic reconstruction of the spin state: Red and blue fields are $\pi$ out of phase, allowing discrimination between up and down spin ensembles. (b) The spin ensembles realize a Hopfield neural network: a single-layer network of binary neurons $s_i=\pm 1$ that are recurrently fed back and then subjected to a linear transform $J$ and threshold operation at each neuron. (c) The Hopfield model exhibits an energy landscape with many metastable states. Each local minimum encodes a memory spin configuration (pattern) surrounded by a larger basin of attraction of similar spin states. Energy minimizing dynamics drive sufficiently similar spin configurations to the stored local minimum. There is a phase transition from an associative memory to a spin glass once there are too many memories, i.e., when so many minima exist that basins of attraction vanish. (This transition might be a crossover in the CCQED system; see the text.) (d) Schematic of the associative memory problem. Multiple stored patterns (e.g., images of element symbols) may be recalled by pattern completion of distorted input images.

Figure 4

Confocal cavity connectivities. Distribution of normalized $J_{ij}$ couplings shown for widths (a) $w=w_0/2$, (b) $w=w_0$, and (c) $w=4w_0$. Representative $J_{ij}$ connectivity graphs are shown to the right, with red (blue) links indicating ferromagnetic $J_{ij}>0$ (antiferromagnetic $J_{ij}<0$) coupling. The radius $w_0$ is the Gaussian width of the ${\mathrm{TEM}}_{00}$ mode in the transverse plane and is indicated by the dashed circle. (The modes of a multimode cavity extend far beyond the characteristic length scale $w_0$.) (d) Statistical properties of the $J_{ij}$ versus $w$. The correlation function in Eq. (8) between different $J_{ij}$ (black line) shows that they become uncorrelated at large $w$. The probability of generating frustration between three randomly chosen spins (red line) and the probability of generating an antiferromagnetic coupling between two randomly chosen spins (blue line) are also plotted. The dashed lines correspond to the widths used in (a)–(c).

Figure 5

(a) Number of metastable states versus distribution width for various system sizes $N$, as indicated. The average number of metastable states increases with $N$ above the ferromagnetic-to-associative memory transition at $w_{\mathrm{AM}}=0.67(1)w_0$. (b) Scaling collapse of the above in the region $0.5w_0<w<0.8w_0$, with denser sampling. The $x$ axis is rescaled as $N^{1/\nu }(w-w_{\mathrm{AM}})/w_0$, while the $y$ axis is unchanged. The parameters $\nu$ and $w_{\mathrm{AM}}$ are determined by fitting the collapsed data to an exponential form $\sqrt{1+A{e}^{Bx}}$, where $x=N^{1/\nu }(w-w_{\mathrm{AM}})/w_0$. The uncertainties are one standard error.

Figure 6

Eigenvalue spectrum of the CCQED connectivity, demonstrating random matrix statistics. (a) The eigenvalue distribution $f_w$ and (b) level-spacing spectrum for the nonlocal interaction matrix $J_{ij}=\cos (2{\mathbf{r}}_i·{\mathbf{r}}_j/w_0^2)$. $N=1000$ spins are averaged over 50 realizations of $J_{ij}$ matrices. The distributions are shown for $w=2w_0$ (red), $w=3w_0$ (green), and $w=12w_0$ (blue). Each distribution is normalized to have unit variance. The solid black lines show the Wigner semicircle eigenvalue distribution (a) and the Wigner surmise for the level-spacing distribution of Gaussian orthogonal ensemble random matrices (b). (c) shows the Hellinger distance between the averaged CCQED connectivity eigenvalue distribution $f_w$ [shown in (a)] and the Wigner semicircle distribution $f_{\text{Wigner}}$. The distance is plotted as a function of the rescaled width $N^{1/\nu }w/w_0$ for different power laws $\nu =-3,-4,-5$ corresponding to different colors, as indicated. Lines of varying intensity correspond to different $N$ ranging from $N=100$ (faintest) to $N=1000$ (darkest). The traces overlap well for the $\nu =-4$ scaling.

Figure 7

Regimes of CCQED connectivity behavior versus width $w$ and system size $N$. The three shaded regions correspond to ferromagnetic (F), associative memory, and SK-like spin glass (SG) regimes, as discussed in the text. Note that associative memories can still be encoded even in the spin glass regime due to SD dynamics; see Secs. 5 and 7.

Figure 8

(a) The spin-flip rate function $K(\delta \epsilon )$ in the confocal limit (solid line) versus spin-flip energy $\delta \epsilon$. Units are in terms of the pump-cavity detuning $|{\mathrm{\Delta }}_C|$. Rates for Metropolis-Hastings (dashed line) and Glauber (dotted line) dynamics are also shown. (b) Ratio of energy-decreasing to energy-increasing spin-flip rates $K(\pm \delta \epsilon )$ versus spin-flip energy (solid line). Comparing finite-temperature Metropolis Hastings dynamics (dashed line) to the actual cavity dynamics allows one to identify a low-energy effective temperature $T_{\mathrm{eff}}=({\mathrm{\Delta }}_C^2+{\kappa }^2)/4|{\mathrm{\Delta }}_C|$ [59, 60, 95].

Figure 9

(a) Simulation of a system of $N=100$ ensembles. All are initialized in a local minimum state except five ensembles that are flipped in the wrong orientation. Each ensemble is composed of $M=2S={10}^5$ spins. For this (realistic) number of spins, the statistical fluctuations, which are present, are not resolvable. A comparison is made between dynamics derived from the equations of motion in the thermodynamic limit (dashed lines) and a full stochastic unraveling of the master equation (solid lines). The five traces show relaxation of those five spin ensembles to the local minimum state. The other $N-3$ spin ensembles remain magnetized at $\pm 1$ and are plotted in black. Data from the stochastic master equation simulation lie close to that of the equation of motion. The timescale associated with spontaneous decay is marked by a dotted vertical line; the system reaches steady state beforehand. Spontaneous emission can induce motional heating and demagnetization of the ensembles if mitigating actions are not taken, such as repumping and sideband cooling; see the text. All parameters are as listed in Sec. 2 and ${\omega }_z=1\mathrm{MHz}$. (b) The total energy versus time, in units of $S|{\mathrm{\Delta }}_C|$. As expected of discrete steepest descent dynamics, steps are clearly visible, and they occur in order of the energy removed per spin flip.

Figure 10

(a) Recall probability as a function of the input error applied, quantified by the Hamming distance between the input and pattern state. This probability is plotted for three forms of dynamics: 0TMH (red), SD (blue), and CCQED (orange). The latter is given by Eq. (26). This plot illustrates how the dynamics affects recall probability and, thus, basin size (defined by the point where the recall curve drops below 95%). We use a connectivity matrix $J_{ij}$ corresponding to a pseudoinverse Hopfield model with $N=100$ spins and $P=0.6N$ patterns stored. We choose this learning rule because it illustrates a large difference between the two dynamics. (b) Average basin size of attractors for connectivity of a Hebbian learning rule. The intensity of the trace increases with system size, $N=200$, 400, 800, while color indicates the form of dynamics. The critical ratio ${\alpha }_c\approx 0.138$ marks the known pattern loading threshold where Hebbian learning fails in the thermodynamic limit [16]. Inset: overlap between the attractor of the dynamics ${\mathbf{s}}^{\mu }$ and the bare memory ${\mathbf{\xi }}^{\mu }$, as discussed in the text. (c) Average basin size for the pseudoinverse learning rule versus the ratio of patterns stored to number of nodes, $P/N$. Steepest descent dynamics outperforms 0TMH in all cases, yielding larger basin sizes for $P/N\gtrsim 0.1$ and, thus, greater memory capacity.

Figure 11

(a) Average basin size of randomly found local minima for the SK model using 0TMH (red) or SD (blue). The shaded region corresponds to $\pm 1$ standard deviation in the basin sizes. For each $N$, the basin size is averaged over 50 realizations of $J_{ij}$ matrices, with 500 random initial states per $J_{ij}$ realization. (b) Histogram of basin sizes using 0TMH (red) and SD (blue) for the $N=1000$ data plotted in (a). The mean basin size for SD is marked by the vertical dashed line.

Figure 12

(a) The average basin size in the CCQED system versus width of the spin ensemble distribution $w$. Metastable states are found by relaxing random initial seeds. Basin sizes for both 0TMH dynamics (red) and the (native) SD dynamics (blue) are shown. System sizes ranging from $N=3$ to $N=800$ are considered, with darker traces corresponding to larger sizes. We extrapolate to the limit $N\to \infty$ by fitting a linear function of $1/N$ to the finite $N$ data. The intercept at $1/N=0$ yields the thermodynamic limit (dashed lines). The ferromagnetic-to-associative memory phase transition, indicating the onset of multiple metastable states, is marked by a vertical black dashed line at $w_{\mathrm{AM}}$. The green line shows the typical basin size under SD dynamics for the SK connectivity, $0.0135(1)N$, as extracted from Fig. 11. (b) Histogram of the basin sizes found for widths $w=1.5w_0$ and $w=4w_0$.

Figure 13

Average basin size of patterns stored in a CCQED system as a function of $P/N$ using the linear transformation encoding scheme (yellow line). Results are shown for $N=800$; however, similar results are found for system sizes of $N=50–800$. For comparison, the average basin sizes are plotted for the Hebbian (red line) and pseudoinverse (blue line) learning methods. In each case, we show the results of both SD (more intense lines) and 0TMH (less intense lines) dynamics; $w=1.5w_0$.

Figure 14

The effect of weight chaos in experimental parameter regimes, using an ensemble placement uncertainty of $1\mu \mathrm{m}$ and $1/\sqrt{M}$ fluctuations in atom number. The drift of metastable states between different realizations of noise weights is plotted in (a) as a function of the width $w$ of the ensemble distribution and for different system sizes $N$ [see the legend in (b)]. The states $\mathbf{s}$ and ${\mathbf{s}}^{\prime }$ refer to metastable states of two realizations $\mathbf{J}$ and ${\mathbf{J}}^{\prime }$, respectively, as described in the text. (b) The direct change in the $\mathbf{J}$ connectivity matrix is plotted as a function of the width.

Figure 15

The spin-flip rate function for an Ohmic noise source. The Lorentzian peak centered at ${\mathrm{\Delta }}_C=-2\mathrm{MHz}$ has a width set by $\kappa =150\mathrm{kHz}$ and provides cooling, while the peak at zero causes heating if the noise source broadens it too much. Plotted is the spin-flip rate function with no noise source (solid line), in which the central peak is a zero-width delta function, and with a classical Ohmic noise sufficient to completely suppress superpositions of $S^x$ states of up to $M={10}^4$ spins per ensemble (dashed line). The rate functions overlap away from the point of zero spin-flip energy. The central peak is broadened but still remains small compared to the typical spin-flip energy $\lesssim {\mathrm{\Delta }}_C$; as such, it does not interfere with the cooling dynamics.

Figure 16

The spin-flip rate versus energy for an Ohmic quantum bath for various bath coupling strengths ${\alpha }_Q$. Three regimes of dynamics are apparent. The spin-flip rate diverges for ${\alpha }_Q<1$ near $\delta \epsilon =0$, giving rise to dynamics that perform the smallest energy-lowering spin flips possible. For ${\alpha }_Q\simeq 1$, the spin-flip rate is roughly flat for negative $\delta \epsilon$, matching 0TMH dynamics. Finally, for ${\alpha }_Q>1$, the spin-flip rate rises sharply as $\delta \epsilon$ approaches zero. This spin-flip rate describes dynamics akin to SD, in which the most energy-lowering spin flip has the highest spin-flip rate.

Figure 17

Comparison between the mean-field equations of motion (dashed lines) in Eq. (f3) and a full stochastic unraveling of the cavity master equation (solid lines) for a system with $N=100$ spin ensembles each with $M={10}^3$ atoms. Four ensembles are initially misaligned from a local minimum spin configuration of the Hopfield energy landscape and are corrected by the cavity dynamics.