In order to capitalize on their promise of high-bandwidth, ultra-low power operation, future nanophotonic devices will have to contend with quantum and coherent effects inherent to the small-volume, few-photon limit. The theoretical development of general, systematic frameworks for describing quantum feedback networks enables us to leverage established optical technologies—from single- and multi-atom cavity quantum electrodynamics to ultrafast multimode optics—as an essential testbed for establishing direct contact between coherent-feedback control theory and first generation nanophotonic circuits.

## Cavity QED

In atomic cavity QED experiments, an atom or cloud of atoms (in our case, cesium) is placed in a resonant optical cavity, entangling the quantum state of the atom(s) with the state of the resonant optical mode. We use these systems as a testbed to explore the ideas of coherent feedback control theory.

### Coherent Self-Feedback Control

For interesting regimes for photonic logic, we want to work where switching energies become on the order of attojoules. Within atomic cavity QED systems, this regime occurs where the number of atoms within the topical cavity is on the order of 1-10. In this regime, each atom is a quantum emitter and must be treated with a full quantum model. As the atom number is increased, a mean field theory can be applied and the dynamics can be modelled using the optical Bloch equations. Full quantum modelling of systems with the atom number > 10 are not tractable computationally and a simpler semi-classical model does not fully capture the dynamics correctly. Thus we must investigate an intermediate regime where it may not be necessary to keep track of all quantum degrees of freedom and treat them semi-classically. Studying experimental data to help formulate new models will greatly help in the push towards practical optical non-linear systems for ultra low power photonic logic.

With a cavity QED system like the one above, we have an ideal setup to study the effect of coherent feedback control on a quantum system. We implement self feedback to this multi-atom system to study the effects on the system dynamics. The feedback allows us to dynamically control the cavity QED parameters of the overall system in real time by varying the phase of the self feedback in the reflected path as shown below.

### Optical Switching via Nonlinear Bistability

When driven by a laser, a cavity QED system can become bistable. Semiclassically, it has two stable equilibria and the system settles into one of the two, functioning as a “switch” or “memory”.

We have observed optical bistability in a regime of ~500 atoms and 1000 photons. This bistability, described by the semiclassical Maxwell-Bloch equations, can then be used to demonstrate an all-optical memory (SR latch) with switching energies of 1 fJ.

Single atoms can also exhibit bistability, but will undergo spontaneous switching due to quantum noise, which becomes large relative to the signal at low photon numbers.

Coherent feedback can be used to suppress spontaneous switching in such a bistable optical system. The simplest imaginable controller is a linear static controller that routes the output from one port back into another port, adding a fixed phase shift. Tuning this phase shift changes the effective decay rate of the cavity. This type of controller is only effective at stabilizing either the low state or the high state, not both.

A nonlinear dynamic controller, however, can stabilize both states. It must be designed to impart a dynamic phase shift that stabilizes the low state when the state is low and stabilizes the high state when the state is high.

Feedback can also be used to create new dynamics, e.g. creating an optical latch from two (non-bistable) Kerr cavities.

### Coherent Feedback with Linear Amplification

We are developing an experiment that will use a multi-atom cavity as a highly nonlinear controller for a below-threshold non-degenerate optical parametric amplifier (NOPA) pumped at 532 nm. Coherent feedback using the NOPA’s signal mode at 852 nm (resonant to a Cs transition) will be used to induce nonlinear input-output behavior for the idler port at 1416 nm, thus mapping the nonlinearity from the narrow-band atom transition frequency into an arbitrary wavelength.

## OPO Networks for Coherent Computing

[Overview of sub-topic]

### 1D Ising Chains

The “Ising machine” is a network of coupled optical parametric oscillators (OPOs), driven slowly through threshold, which starts from squeezed vacuum, bifurcates and relaxes into a final state that solves for the ground state of the Ising problem. This concept has generated a lot of interest because general Ising problem is NP-hard, meaning that no one knows how to solve it efficiently on a computer. Even quantum computers can’t solve NP-hard problems.

In a collaboration with Hiroki Takesue’s group at NTT Basic Research Labs, I modeled the dynamics of a prototype 10000-bit Ising machine with 1D nearest-neighbor couplings. Instead of relaxing into the ground state, this machine tended to form discrete ferromagnetic domains separated by defects, and that the defect density depended on the pump power. By modeling their system with semiclassical Wigner methods (see section above) one can correctly predict the domain-wall density in 1D systems and make predictions for 2D and frustrated lattices. In the process, one develops theory on how Ising machines work: a {\it growth stage} where linear dynamics selects out the dominant eigenvectors of the coupling matrix, and a {\it saturation stage} where the system relaxes into a valid Ising state with amplitudes 1 (Refs. [6, 7]).

Future work is focusing on Ising machines with more general connectivity, and other devices such as “XY machines” based on nondegenerate OPOs and “Heisenberg machines” based on injection-locked lasers, and ways to use the existing OPO networks to solve other, non-Ising problems.

### All-optical and hybrid Ising machines

We are designing both all-optical and hybrid optical-electronic computing systems. Our focus is on constructing programmable physical machines that can find ground states and low-energy excited states of arbitrary classical Ising spin models. Our work is inspired by recent advances in experimental quantum annealing, in which Ising spin models are realized by networks of superconducting qubits, but our systems work with a somewhat different physical mechanism. We instead realize spin models using pulses inside degenerate optical parametric oscillators (DOPOs). DOPOs produce pulses that have binary phase, which is a natural degree of freedom with which we can represent spins. Interactions between spins are realized using interference between pulses by adding delay lines or a measurement-feedback apparatus.

This work is performed in collaboration with the groups of Prof. Yoshihisa Yamamoto, Prof. Martin Fejer, and Prof. Robert Byer.

Initial experimental results: http://dx.doi.org/10.1038/nphoton.2014.249

### Multimode Squeezing via Linear Coherent Feedback

In the ultrafast regime, the coherent multimode nature of pulsed light provides yet another potential resource for information processing. Following recent work in optical Ising machines for solving combinatorial problems, we are investigating the nature of multimode squeezing in time-multiplexed networks of synchronously-pumped degenerate parametric oscillators (SPOPOs) with coherent linear couplings, which is conjectured to yield speedups in the Ising machine relative to an electronic implementation. Experimentally, we are working towards a system of 16 time-multiplexed SPOPOs pumped with ~20 fs pulses at a 1 GHz repetition rate.

## Coherent LQG Control

LQG stands for Linear Quadratic Gaussian: control of a linear system ({\it plant}) subject to Gaussian noise, where the cost function is quadratic. This is a well-studied classical control problem, and the answer can be obtained by solving a Ricatti equation. The optimal control involves a state estimator ({\it Kalman filter}) and a feedback element based on the estimated state of the plant. Translating this to quantum systems, one can define an optimal {\it measurement-based} controller, where the outputs of the plant are sent into a homodyne detector and we perform LQG-optimal control on the measurement signal.

In Refs. [1-2], we showed that {\it coherent} LQG control, where a quantum system coherently processes the plant output rather than measuring it, does better than measurement-based control for two systems: an optical cavity and an optomechanical oscillator. The intuition is that the coherent controller, being a quantum system, can process both quadratures simultaneously without adding extra noise, whereas the measurement-based controller must measure one quadrature and throw the other away (homodyne) or measure both with a noise penalty (heterodyne).