Workshop On Uncertain Dynamical Systems Udine 2011 1 Robust Linear Quantum Systems Robust Linear Quantum Systems Theory Theory Ian R. Petersen † † School of Engineering and Information Technology, University of New South Wales @ the Australian Defence Force Academy
Workshop On Uncertain Dynamical Systems Udine 2011 2 Acknowledgments Acknowledgments � Professor Matthew James, Australian National University � Dr Hendra Nurdin, Australian National University � Dr Aline Maalouf, University of New South Wales � Prof Elanor Huntington, University of New South Wales � Dr. A.J. Shaiju, IIT Madras � Dr. Igor Vladimirov, University of New South Wales � Mr. Shanon Vuglar, University of New South Wales � The Australian Research Council � The Air Force Office of Scientific Research (AFOSR)
Workshop On Uncertain Dynamical Systems Udine 2011 3 Introduction Introduction � This presentation surveys some recent results on the theory of robust control for quantum linear systems. � Quantum linear systems are a class of systems whose dynamics, which are described by the laws of quantum mechanics, take the specific form of a set of linear quantum stochastic differential equations (QSDEs). � Such systems commonly arise in the area of quantum optics and related disciplines. Systems whose dynamics can be described or approximated by linear QSDEs include interconnections of optical cavities, beam-splitters, phase-shifters, optical parametric amplifiers, optical squeezers, and cavity quantum electrodynamic systems.
Workshop On Uncertain Dynamical Systems Udine 2011 4 A linear quantum optics experiment at the University of New South Wales (ADFA). Photo courtesy of Elanor Huntington.
Workshop On Uncertain Dynamical Systems Udine 2011 5 � With advances in quantum technology, the feedback control of such quantum systems is generating new challenges in the field of control theory. � Potential applications of such quantum feedback control systems include quantum computing, quantum error correction, quantum communications, gravity wave detection, metrology, atom lasers, and superconducting quantum circuits. � A recently emerging approach to the feedback control of quantum linear systems involves the use of a controller which itself is a quantum linear system. � This approach to quantum feedback control, referred to as coherent quantum feedback control, has the advantage that it does not destroy quantum information, is fast, and has the potential for efficient implementation.
Workshop On Uncertain Dynamical Systems Udine 2011 6 Quantum System Coherent Quantum Controller Coherent quantum feedback control. � The presentation discusses recent results concerning the synthesis of H ∞ optimal controllers for linear quantum systems in the coherent control case.
Workshop On Uncertain Dynamical Systems Udine 2011 7 � An example of a coherent quantum H ∞ system considered in (Nurdin, James, Petersen, 2008), (Maalouf Petersen 2010) is described by the following diagram: w v y k 1 k 2 a k 3 z u plant 180 ◦ Phase Shift controller ac wc 0 kc 2 kc 1
Workshop On Uncertain Dynamical Systems Udine 2011 8 � The coherent quantum H ∞ control approach of James Nurdin and Petersen (2008) was subsequently implemented experimentally by Hideo Mabuchi of Stanford University:
Workshop On Uncertain Dynamical Systems Udine 2011 9 � In general, quantum linear stochastic systems represented by linear QSDEs with arbitrary constant coefficients need not correspond to physically meaningful systems. � Physical quantum systems must satisfy some additional constraints that restrict the allowable values for the system matrices defining the QSDEs. � In particular, the laws of quantum mechanics dictate that closed quantum systems evolve unitarily , implying that (in the Heisenberg picture) certain canonical observables satisfy the so-called canonical commutation relations (CCR) at all times. � Therefore, to characterize physically meaningful systems, a formal notion of physically realizable quantum linear stochastic systems has been introduced.
Workshop On Uncertain Dynamical Systems Udine 2011 10 Quantum Harmonic Oscillators Quantum Harmonic Oscillators We formulate a class of linear quantum system models. � These linear quantum system models take the form of QSDEs which are derived from the quantum harmonic oscillator. � We begin by considering a collection of n independent quantum harmonic oscillators which are defined on a Hilbert space H = L 2 ( R n , C ) . � Elements of the Hilbert space H , ψ ( x ) are the standard complex valued wave functions arising in quantum mechanics where x is a spatial variable.
Workshop On Uncertain Dynamical Systems Udine 2011 11 � Corresponding to this collection of harmonic oscillators is a vector of annihilation operators a 1 a 2 a = . . . . a n � Each annihilation operator a i is an unbounded linear operator defined on a suitable domain in H by 1 2 x i ψ ( x ) + 1 ∂ψ ( x ) √ √ ( a i ψ )( x ) = ∂x i 2 where ψ ∈ H is contained in the domain of the operator a i . � The adjoint of the operator a i is denoted a ∗ i and is referred to as a creation operator. � These correspond to the annihilation and creation of a photon.
Workshop On Uncertain Dynamical Systems Udine 2011 12 Canonical Commutation Relations Canonical Commutation Relations � The operators a i and a ∗ i are such that the following canonical commutation relations are satisfied [ a i , a ∗ j ] := a i a ∗ j − a ∗ j a i = δ ij where δ ij denotes the Kronecker delta multiplied by the identity operator on the Hilbert space H . � We also have the commutation relations [ a i , a j ] = 0 , [ a ∗ i , a ∗ j ] = 0 . � These commutation relations encapsulate Heisenberg’s uncertainty relation.
Workshop On Uncertain Dynamical Systems Udine 2011 13 Notation Notation � For a general vector of operators g 1 g 2 g = , . . . g n on H , we use the notation g ∗ 1 g ∗ g # = 2 , . . . g ∗ n to denote the corresponding vector of adjoint operators.
Workshop On Uncertain Dynamical Systems Udine 2011 14 � Also, g T denotes the corresponding row vector of operators � g 1 g 2 . . . g n g # � T . g T = , and g † = � � � Using this notation, the canonical commutation relations can be written as �� a � a � a � � a �� a � # � a � T � T � † � � † � = − , a # a # a # a # a # a # � � 0 I = 0 − I = J.
Workshop On Uncertain Dynamical Systems Udine 2011 15 Quantum Weiner Processes Quantum Weiner Processes � The quantum harmonic oscillators described above are assumed to be coupled to m external independent quantum fields modelled by bosonic annihilation field operators A 1 ( t ) , A 2 ( t ) , . . . , A m ( t ) which are defined on separate Fock spaces F i defined over L 2 ( R ) for each field operator. � For each annihilation field operator A j ( t ) , there is a corresponding creation field operator A ∗ j ( t ) , which is defined on the same Fock space and is the operator adjoint of A j ( t ) . � The field annihilation operators are also collected into a vector of operators defined as follows: A 1 ( t ) A 2 ( t ) A ( t ) = . . . . A m ( t )
Workshop On Uncertain Dynamical Systems Udine 2011 16 Hamiltonian, Coupling and Scattering Operators Hamiltonian, Coupling and Scattering Operators � In order to describe the joint evolution of the quantum harmonic oscillators and quantum fields, we specify the Hamiltonian operator for the quantum system which is a Hermitian operator on H . This operator describes the internal dynamics of the quantum system. � Also, we specify the coupling operator vector for the quantum system L , which is a vector of operators on H . These operators define the interaction between the quantum system and the light fields which interact with it. � In addition, we define a scattering matrix which is a unitary matrix S ∈ C m × m . This matrix describes the interactions between the light fields.
Workshop On Uncertain Dynamical Systems Udine 2011 17 Robust Stability of Uncertain Quantum Systems Robust Stability of Uncertain Quantum Systems � We consider an open quantum system defined by the parameters ( S, L, H ) where H = H 1 + H 2 . � H 1 corresponds to the nominal (known) part of the Hamiltonian and H 2 corresponds to the uncertain (unknown) part of the Hamiltonian. � The corresponding generator for this quantum system is given by G ( X ) = − i [ X, H ] + L ( X ) where L ( X ) = 1 2 L † [ X, L ] + 1 2 [ L † , X ] L . � This defines the dynamics of the open quantum system.
Workshop On Uncertain Dynamical Systems Udine 2011 18 � We first assume that there exist operator column vectors z and w such that [ V, H 2 ] = [ V, z † ] w − w † [ z, V ] for all non-negative self-adjoint operators V . � Also, we assume the sector bound condition: w † w ≤ 1 γ 2 z † z. � Let V be any non-negative self-adjoint operator and consider G ( V ) . � Then G ( V ) = = − i [ V, H ] + L ( V ) = − i [ V, H 1 ] − i [ V, H 2 ] + L ( V ) = − i [ V, H 1 ] + L ( V ) − i [ V, z † ] w + iw † [ z, V ] . (1)
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