Direct and Indirect Couplings in Coherent Feedback Control of Linear Quantum Systems

Guofeng Zhang; Matthew R. James

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Direct and Indirect Couplings in Coherent Feedback Control of Linear Quantum Systems

Guofeng Zhang, Matthew R. James

arXiv:1010.4369v2Oct 21, 2010quant-pheess.SYmath.OC

TL;DR

The paper asks how direct and indirect couplings can be modeled and designed for coherent feedback control of linear quantum stochastic systems. It develops general physical models, characterizes key system properties, and extends H∞ and LQG synthesis to direct coupling. Examples demonstrate beneficial performance consequences from designing direct couplings.

Problem
The paper addresses the need to analyze and design both direct and indirect couplings in coherent feedback control of linear quantum stochastic systems.
Method
The paper builds a general physical model and uses complex Lyapunov equations, LMIs, and multistep optimization to extend coherent H∞ and LQG synthesis to direct couplings.
Results
The examples show that direct coupling can have beneficial performance consequences in coherent feedback control.
Takeaways & Limitations
Direct-coupling design can be incorporated into coherent feedback analysis and synthesis for linear quantum stochastic systems.

Abstract

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The purpose of this paper is to study and design direct and indirect couplings for use in coherent feedback control of a class of linear quantum stochastic systems. A general physical model for a nominal linear quantum system coupled directly and indirectly to external systems is presented. Fundamental properties of stability, dissipation, passivity, and gain for this class of linear quantum models are presented and characterized using complex Lyapunov equations and linear matrix inequalities (LMIs). Coherent $H^\infty$ and LQG synthesis methods are extended to accommodate direct couplings using multistep optimization. Examples are given to illustrate the results.

I. INTRODUCTION

The paper studies direct and indirect couplings in coherent feedback control of linear quantum stochastic systems. It develops general models, system characterizations, and synthesis methods that incorporate direct coupling.

I. INTRODUCTION: Direct coupling is bidirectional, whereas indirect coupling carries quantum information directionally through signals or quantum fields.Coherent feedback preserves quantum coherence and can operate at a speed comparable to the plant.
I. INTRODUCTION: Linear quantum systems are studied because they are practically important, particularly in quantum optics, and mathematically tractable.The introduction contrasts coherent feedback with measurement feedback, which necessarily involves measurement and loss of quantum coherence.
I. INTRODUCTION: The paper presents a general physical model for linear quantum systems coupled directly and indirectly to external systems.The model gives explicit dynamical matrices in terms of physical parameters and includes relations needed for physical realizability.
I. INTRODUCTION: The paper characterizes stability, dissipation, passivity, and gain using complex Lyapunov equations and linear matrix inequalities.These results generalize earlier characterizations developed for special cases of linear quantum systems.
I. INTRODUCTION: Coherent H∞ and LQG synthesis methods are extended to accommodate direct couplings through multistep optimization schemes.The paper notes that explicit closed-form solutions are generally unavailable and illustrates direct optimization with simple examples.

II. LINEAR QUANTUM SYSTEMS

This section develops linear quantum-system models from oscillator dynamics and constructs direct interactions through interaction Hamiltonians. It also introduces a general model with direct and indirect external couplings.

II. LINEAR QUANTUM SYSTEMS: A linear quantum system consists of interacting quantum harmonic oscillators whose annihilation operators evolve according to linear differential equations.The operator evolution is defined through unitary dynamics on a Hilbert space that may include external degrees of freedom.
II. LINEAR QUANTUM SYSTEMS: The general model represents an external system as directly coupled to G and another external system as indirectly coupled through a quantum field signal.The construction provides a physical representation of external disturbances and noise acting on the system of interest.
II. LINEAR QUANTUM SYSTEMS: The paper assumes Gaussian initial system states and Gaussian field inputs for the general linear quantum-system analysis.
B. Direct Coupling: Two independent systems can interact directly by exchanging energy through an interaction Hamiltonian containing cross-system operator terms.The resulting dynamics are symmetric: each system’s evolution depends on the other system’s evolution.
B. Direct Coupling: Direct-coupling models use interaction matrices parameterized by K− and K+ within the Hamiltonian representation.The directly coupled system is denoted G1 ⊲⊳G2.

C. Indirect Coupling via Quantum Fields

Indirect coupling connects quantum systems through boson-field channels and is modeled using input-output quantum stochastic dynamics. Cascade connections produce asymmetric information flow between component systems.

C. Indirect Coupling via Quantum Fields: Boson fields interconnect component systems, with field inputs modeled as quantum stochastic processes and as quantum white noise in the vacuum case.The model specifies Itō products and an Itō matrix for the field increments.
C. Indirect Coupling via Quantum Fields: The field-based noise model is idealized and applies when suitable rotating-wave and Markovian conditions hold.
C. Indirect Coupling via Quantum Fields: The system-field interaction is characterized by coupling operators L = C−a + C+a# and yields linear annihilation-operator dynamics.The field also has an output relation, allowing the output to feed another system in a cascade.
C. Indirect Coupling via Quantum Fields: In a cascade connection, the output field of G1 is fed into the input of G2, forming the series product G2 ⊳G1.The resulting Hamiltonian includes Im{L(2)†L(1)}, and the combined field coupling operator is L = L(1) + L(2).

D. A More General Model

The paper develops a general physical model for linear quantum systems with direct and indirect couplings to external systems, including performance variables and a real quadrature representation. It also states matrix relations that preserve commutation relations and characterize physical realizability.

D. A More General Model: The general model represents a system coupled directly to one external system and indirectly to another through a series connection.The system is specified by physical parameters and performance-variable matrices within the complete interconnection.
D. A More General Model: Performance variables capture selected performance aspects, may include external variables such as reference signals, and need not equal coupling outputs.
D. A More General Model: The model uses annihilation-creation equations with complex matrices and converts them into real quadrature equations through unitary transformations.The real representation may be more convenient for standard matrix-analysis software.
D. A More General Model: The resulting quadrature representation has real matrix entries while retaining the model's input, output, external-variable, and performance-variable structure.
F. Physical Realization: Matrix relations for the general model preserve commutation relations and extend canonical physical-realizability criteria for linear quantum systems.Physical realizability is fundamental when designing quantum systems for coherent feedback.

III. PERFORMANCE CHARACTERISTICS OF SYSTEMS WITH DIRECT AND INDIRECT INTERACTIONS

The paper characterizes stability, dissipation, and passivity for linear quantum systems using energy-like storage functions, Lyapunov criteria, and LMIs. These results distinguish oscillatory, stable, and unstable behavior and provide matrix tests for dissipativity and passivity.

A. Stability: For closed systems with Ω+ = 0, the number of quanta is conserved, so the system is marginally stable but not exponentially stable.The system oscillates rather than decays exponentially.
A. Stability: Stability is assessed through the expected number of quanta and classified as exponential, marginal, or exponential instability according to its time dependence.
A. Stability: Theorem 1 gives a sufficient stability criterion using nonnegative Hermitian matrices P and Q satisfying a matrix inequality, with an exponential bound when P is uniformly positive.The resulting bound controls the expected quadratic storage quantity over time.
B. Dissipativity of Linear Quantum Systems: Dissipativity with respect to a supply rate is characterized if and only if a nonnegative Hermitian matrix P satisfies the corresponding LMI.
C. Positive Real Lemma: Passivity is defined as dissipativity for a supply rate with nonnegative Q, and the Positive Real Lemma gives an equivalent pair of nonnegative Hermitian matrix conditions.The performance variable is formed as z = C_p a in the stated formulation.

D. Bounded Real Lemma

The bounded real lemmas characterize finite L2-gain behavior through dissipativity, stability, and matrix inequalities. Special cases connect lossless bounded realness with physical realizability, while strict bounded realness admits equivalent Lyapunov and Riccati conditions.

Bounded realness is defined by dissipativity with respect to a supply rate for the transfer from u to z.
The bounded real lemma states that finite L2 gain below g is equivalent to the existence of a non-negative Hermitian matrix P satisfying an LMI.
With no direct coupling and the specified special parameters, the gain from w to z is exactly one, yielding the lossless bounded real property.
Physical realizability and lossless bounded realness coincide for the special class but are distinct in general.
Strict bounded realness is equivalent to stability plus a positive-definite Lyapunov inequality, and also to a Riccati equation with a Hurwitz modified system matrix.
When the strict bounded real conditions hold, the associated Hermitian matrices satisfy P1 < P2.

F. Examples

The examples illustrate how direct and indirect couplings affect passivity, stability, gain, and LQG performance in linear quantum systems. Direct coupling can change both channel gain and plant LQG cost, while stability conditions constrain amplifier behavior.

The directly coupled oscillator example is passive with performance variable z = −γa.
For the degenerate parametric amplifier, passivity holds if and only if ϵ ≤ κ, coinciding with marginal or exponential stability.
When ϵ < κ, the L2 gain is (κ + ϵ)/(κ − ϵ), and the system is physically realizable but not lossless bounded real.
5) Effect of Direct Coupling on H∞Performance: Direct coupling changes the L2 gain of an optical-amplifier channel as K− varies with fixed ratio K+ = 3K−.
Direct coupling affects LQG performance both when K+ = 0 and when K+/K− = 1.1, as illustrated by the two plotted examples.

IV. COHERENT FEEDBACK CONTROLLER SYNTHESIS

The synthesis framework models plant-controller feedback with both direct Hamiltonian coupling and indirect field coupling. Controller and coupling parameters are then selected using H∞ and LQG criteria, extending existing coherent synthesis methods.

The paper extends coherent H∞ and LQG controller synthesis methods to include direct couplings.
A. Closed-Loop Plant-Controller System: The plant-controller architecture includes both direct coupling through an interaction Hamiltonian and indirect coupling through field channels.
The closed-loop model combines plant and controller dynamics with direct-coupling terms in its system matrices and quantum-noise inputs.
The formulation assumes compatible dimensions for the plant and controller variables and matrices.
Controller matrices and coupling parameters are synthesized to optimize closed-loop performance criteria.

B. Stabilization

The stabilization and synthesis examples show that direct coupling can stabilize an otherwise unstable coherent feedback loop and optimize input-output performance. The general design formulation uses multistep optimization because direct-coupling terms make the LMI conditions nonlinear.

B. Stabilization: Both plant and controller can be individually Hurwitz while their indirectly coupled closed loop remains unstable.
B. Stabilization: Direct coupling with K− = −0.6 and K+ = −0.42 makes the previously unstable closed-loop A-matrix Hurwitz.
C. H∞Synthesis: The H∞ example minimizes the L2 gain from w1 to z with respect to the direct coupling parameter K−, alongside the indirect-coupling parameter ω.
C. H∞Synthesis: The resulting local minimum is 0.6284 at ω = 0 and K− = 17.7135, compared with 0.9035 without direct coupling.
C. H∞Synthesis: As κ3 approaches infinity, the L2 gain approaches zero because strong controller-field coupling provides an additional path for incoming energy.
2) LMI Formulation:: The direct-coupling synthesis inequalities remain nonlinear because of products such as N B21 X and Y B12 M^T, motivating a multistep optimization procedure.

3) Multi-step Optimization:

The paper develops a multi-step optimization procedure for designing indirect and direct couplings while preserving physical realizability. The procedure can improve disturbance attenuation, but the choices of intermediate matrices may require care because they can produce ill-conditioned controllers.

Multi-step procedure: The multi-step procedure alternates LMI-based design of indirect and direct coupling parameters, then repeats optimization with selected parameters fixed.Step 1 computes indirect-coupling parameters and matrices M,N; Step 2 computes direct-coupling parameters; Step 3 restarts optimization with selected values fixed.
Numerical conditioning: Choosing M and N from earlier steps can produce ill-conditioned controller parameters, so Step 3 may require carefully selected matrices for physical meaning.The paper identifies this issue as delicate and illustrates it with an example.
Physical realizability: The resulting controller can be made fully quantum because indirect and direct couplings are constructed to satisfy physical realizability.The procedure first obtains a physically realizable indirect coupling, constructs the direct coupling, and uses the fact that direct coupling does not affect indirect coupling.
Example: The optimization framework is applied to an optical-cavity control problem whose objective is to minimize the influence of one field on another.The example reuses the optical-cavity formulation and compares controllers under parameter uncertainty.
Example: 0.0618 disturbance attenuation with direct coupling approaches the original 0.0487 and improves substantially over the uncertain indirect-only value 0.1702.The uncertain optical-cavity parameter changes the original controller’s attenuation to 0.1702; adding direct coupling reduces it to 0.0618, close to 0.0487.

6) Example 2:

The examples examine H∞ and LQG synthesis with direct and indirect couplings, including stability constraints and numerical optimization difficulties. Direct coupling can stabilize otherwise unstable configurations and improve performance beyond indirect coupling alone.

H∞ example: The second H∞ example reduces the norm from 1.7252 with indirect coupling to 1.6889 after adding direct coupling.The result is obtained by first designing indirect coupling, then adding direct coupling through Step 2.
H∞ example: The first example shows that selecting M and N carelessly can yield an ill-conditioned controller, whereas identity choices produce a usable combined design.The authors explicitly identify the choice of M and N as a delicate issue requiring caution.
H∞ example: The combined controller achieves an H∞ norm of 1.6056 after direct and indirect coupling are jointly designed.The construction follows the multi-step procedure and yields a fully quantum controller.
LQG example: General-purpose optimization is unreliable when many coupling choices violate the Hurwitz constraint, although bounded searches can find a feasible LQG solution.The paper reports that fminsearch does not work well when many candidate closed-loop matrices are not Hurwitz.
LQG example: Direct coupling can stabilize a plant that is unstable without it, while real coupling parameters alone may be insufficient for stability.The example permits complex K− and K+ because no real values of that form yield a stable composite system.

2) The General Case:

The paper extends coherent quantum LQG synthesis to controllers combining direct and indirect couplings. In the general example, this combination lowers the LQG cost relative to indirect coupling alone and substantially outperforms a classical controller.

General case: The LQG design minimizes an infinite-horizon cost subject to the closed-loop Lyapunov equation, stability, and physical-realizability constraints.The closed-loop A-matrix must be Hurwitz, and the positive-definite cost matrix is defined through a Lyapunov equation.
General case: LQG synthesis is more challenging than H∞ synthesis because separation of control and physical realizability no longer holds.The paper motivates semidefinite-programming and multi-step procedures for designing indirect and direct couplings.
General case: The indirect-only quantum controller obtains J∞ = 4.1793, outperforming the classical controller’s J∞ = 5.4.The comparison is made for the same fully quantum plant and LQG objective.
General case: The combined direct-and-indirect controller achieves LQG cost J∞ = 4.000049633093338, improving on indirect coupling alone.The paper states that the combination improves performance relative to the indirect-only design.
Conclusion: The paper concludes that direct couplings have beneficial performance consequences and can be designed systematically through optimization-based methods.Further practical applications of direct-coupling synthesis in quantum optics are left for future work.