Many quantum devices are networks: cavities, waveguides, beam splitters, amplifiers and atoms connected by propagating fields. Linear quantum systems theory describes such networks with the tools of systems engineering — state-space models, transfer functions, interconnections — while respecting quantum mechanics. Its most distinctive idea is coherent feedback. The output field of a system is processed by another quantum system and fed back, with no measurement in the loop.

Input–output and quantum stochastic models

A system coupled to a field is modelled by quantum stochastic differential equations: the field acts as quantum noise driving the system, and the system modifies the outgoing field [1, 2]. In the SLH framework, each component is described by a triple \((S, L, H)\): a scattering matrix for the fields, coupling operators to the fields, and an internal Hamiltonian. Networks are assembled with the series and concatenation products [3, 4].

When the dynamics is linear in the quadratures \(x\) — cavities, beam splitters, parametric amplifiers, squeezers — the model takes the familiar engineering form

$$dx = A\,x\,dt + B\,dw, \qquad dy = C\,x\,dt + D\,dw,$$

with the field noise \(dw\) as input and the output field \(dy\). Quantum mechanics enters through physical realizability conditions that constrain \((A, B, C, D)\) so that commutation relations are preserved [5, 6].

Control design

  • H-infinity control. Robust controllers that bound the effect of disturbances can be designed for linear quantum systems, including controllers that are themselves quantum [7].
  • Coherent LQG. The linear-quadratic-Gaussian problem can be formulated with a quantum controller and fully quantum plant outputs [8].
  • Experiments. An all-optical coherent-feedback controller with a dynamic compensator achieved about 7 dB of broadband disturbance rejection on an optical ring resonator [9]. Coherent feedback enhanced optical squeezing, with time delays and losses limiting the benefit [10].

Coherent versus measurement-based feedback

Which is better depends on the task. In some settings all-optical and electro-optical feedback give identical effects on the controlled cavity [11]. Coherent feedback, in principle, avoids measurement back-action [12]. For cooling optical and optomechanical resonators, coherent schemes can outperform the best measurement-based LQG schemes in the quantum regime, because the coherent controller can process non-commuting output quadratures simultaneously [13]. A systematic comparison within linear systems theory maps out where each approach is advantageous [14]. See Measurement-based feedback.

Why it matters