Robust Linear Quantum Systems Theory

Transcription

1 Workshop On Uncertain Dynamical Systems Udine Robust Linear Quantum Systems Theory Ian R. Petersen School of Engineering and Information Technology, University of New South the Australian Defence Force Academy

2 Workshop On Uncertain Dynamical Systems Udine 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)

3 Workshop On Uncertain Dynamical Systems Udine 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.

4 Workshop On Uncertain Dynamical Systems Udine A linear quantum optics experiment at the University of New South Wales (ADFA). Photo courtesy of Elanor Huntington.

5 Workshop On Uncertain Dynamical Systems Udine 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.

6 Workshop On Uncertain Dynamical Systems Udine Quantum System Coherent quantum feedback control. Coherent Quantum Controller The presentation discusses recent results concerning the synthesis of H optimal controllers for linear quantum systems in the coherent control case.

7 Workshop On Uncertain Dynamical Systems Udine An example of a coherent quantum H system considered in (Nurdin, James, Petersen, 2008), (Maalouf Petersen 2010) is described by the following diagram: v w y k 1 k a 2 k 3 z u plant 180 Phase Shift controller wc 0 ac kc 2 kc 1

8 Workshop On Uncertain Dynamical Systems Udine The coherent quantum H control approach of James Nurdin and Petersen (2008) was subsequently implemented experimentally by Hideo Mabuchi of Stanford University:

9 Workshop On Uncertain Dynamical Systems Udine 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.

10 Workshop On Uncertain Dynamical Systems Udine 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.

11 Workshop On Uncertain Dynamical Systems Udine Corresponding to this collection of harmonic oscillators is a vector of annihilation operators a = a 1 a 2. a n. Each annihilation operator a i is an unbounded linear operator defined on a suitable domain in H by (a i ψ)(x) = 1 2 x i ψ(x) ψ(x) x i where ψ H is contained in the domain of the operator a i. The adjoint of the operator a i is denoted a i creation operator. and is referred to as a These correspond to the annihilation and creation of a photon.

12 Workshop On Uncertain Dynamical Systems Udine Canonical Commutation Relations are such that the following canonical commutation relations are satisfied The operators a i and a i [a i,a j] := a i a j a ja 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.

13 Workshop On Uncertain Dynamical Systems Udine Notation For a general vector of operators g = g 1 g 2. g n, on H, we use the notation g # = g 1 g 2... g n, to denote the corresponding vector of adjoint operators.

14 Workshop On Uncertain Dynamical Systems Udine Also, g T denotes the corresponding row vector of operators g T = [ g 1 g 2... g n ], and g = ( g #) T. Using this notation, the canonical commutation relations can be written as [ [ ] [ ] ] a a a #, a # = = = J. [ a a # ] [ a a # ] [ I 0 0 I ] ( [ ] # [ ] ) T T a a a # a #

15 Workshop On Uncertain Dynamical Systems Udine 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(t) = A 1 (t) A 2 (t). A m (t).

16 Workshop On Uncertain Dynamical Systems Udine 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.

17 Workshop On Uncertain Dynamical Systems Udine 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] [L,X]L. This defines the dynamics of the open quantum system.

18 Workshop On Uncertain Dynamical Systems Udine 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)

19 Workshop On Uncertain Dynamical Systems Udine Now [V,z ] = V z z V and hence [V,z ] = zv V z = [z,v ] since V is self adjoint. Therefore, and hence 0 ( [V,z ] iw )( [V,z ] iw ) = [V,z ][z,v ] + i[v,z ]w iw [z,v ] + w w i[v,z ]w + iw [z,v ] [V,z ][z,v ] + w w. Substituting this into (1), it follows that G(V ) i[v,h 1 ] + L(V ) + [V,z ][z,v ] + w w i[v,h 1 ] + L(V ) + [V,z ][z,v ] + 1 γ 2 z z using the sector bound condition.

20 Workshop On Uncertain Dynamical Systems Udine Using this inequality, we obtain the following result. Theorem. Suppose that the open quantum system (S, L, H) satisfies the above conditions. Also suppose there exists a non-negative selfadjoint operator V and real constants c > 0, λ 0 such that i[v,h 1 ] + L(V ) + [V,z ][z,v ] + 1 γ 2 z z + cv λ. Then V (t) e ct V + λ c, t 0. Proof. If the conditions of the theorem are satisfied, then it follows from the previous inequality that G(V ) + cv λ. Then the result of the theorem follows from a result of (James, Gough, 2010). Note < > denotes quantum expectation.

21 Workshop On Uncertain Dynamical Systems Udine Linear Uncertain Quantum Systems We now specialize the previous result to the case of linear uncertain quantum systems. In this case, we assume that the nominal Hamiltonian is of the form H 1 = 1 2 [ a a T ] M [ a a # ] where M C 2n 2n is a Hermitian matrix of the form [ ] M1 M 2 M = M # 2 M # 1 and M 1 = M 1, M 2 = M T 2.

22 Workshop On Uncertain Dynamical Systems Udine In addition, we assume the coupling operator vector L is of the form L = [ N 1 N 2 ] [ a a # ] where N 1 C m n and N 2 C m n. Also, we write [ L L # ] = N [ a a # ] = [ N1 N 2 N # 2 N # 1 ] [ a a # ].

23 Workshop On Uncertain Dynamical Systems Udine In addition we assume that V is of the form V = [ a a T ] P [ a a # ] where P C 2n 2n is a positive-definite Hermitian matrix of the form [ ] P1 P 2 P = P # 2 P # 1 and P 1 = P 1, P 2 = P T 2.

24 Workshop On Uncertain Dynamical Systems Udine Also, we assume H 2 is of the form H 2 = 1 2 [ ζ ζ T ] (t)[ ζ ζ # ] where (t) C 2m 2m is a Hermitian matrix of the form [ ] 1 (t) (t) = 2 (t) 2 (t) # 1 (t) # and 1 (t) = 1 (t), 2 (t) = 2 (t) T. Also, ζ = E 1 a + E 2 a #.

25 Workshop On Uncertain Dynamical Systems Udine We let w = 1 2 [ 1 (t) 2 (t) 2 (t) # 1 (t) # ][ ζ ζ # ] = 1 2 [ 1 (t)ζ + 2 (t)ζ # 2 (t) # ζ + 1 (t) # ζ # ] and z = [ ζ ζ # ] = Hence, H 2 = w z = 1 2 [ E1 E 2 E # 2 E# 1 ][ a a # ] = E [ a a # ]. [ a a T ] E (t)e [ a a # ]. From this it follows that for any self-adjoint operator V [V,H 2 ] = [V,z ]w w [z,v ] The sector bound condition is equivalent to (t) 2 γ.

26 Workshop On Uncertain Dynamical Systems Udine Now we calculate i[v,h 1 ]: i[v,h 1 ] = i 1 [ [ a a ] T P 2 = [ a a # ], [ a a T ] M [ ] [ ] a a a # [PJM MJP] a #. [ a a # ]] Also, we calculate L(V ) = 1 2 L [V,L] [L,V ]L ( [ ] ) = tr PJN I 0 NJ [ ] a ( N 2 a # JNJP + PJN JN )[ ] a a #.

27 Workshop On Uncertain Dynamical Systems Udine In addition, we calculate [z,v ] = 2EJP [ a a # ]. and therefore, [V,z ][z,v ] = 4 [ a a # ] PJE EJP [ a a # ]. Also, z z = [ a a # ] E E [ a a # ].

28 Workshop On Uncertain Dynamical Systems Udine Hence, we obtain i[v,h 1 ] + L(V ) + [V,z ][z,v ] + z z γ 2 + cv [ ] ( a = a # F P + PF + 4PJE EJP + E E +λ γ 2 + cp )[ a a # ] where F = ijm 1 2 JN JN and ( [ ] ) I 0 λ = tr PJN NJ. 0 0 Then, the condition in the above theorem is satisfied if and only if the following LMI is satisfied: F P + PF + 4PJE EJP + E E γ 2 < 0.

29 Workshop On Uncertain Dynamical Systems Udine It follows from the bounded real lemma that this LMI will have a solution P > 0 if and only if the following H norm bound condition is satisfied: where D = ije. E (si F) 1 D < γ 2 In this case, it follows from the theorem and P > 0 that [ a(t) a # (t) ] [ a(t) a # (t) ] e ct [ a(0) a # (0) + λ cλ min [P] ] [ a(0) a # (0) t 0. ] λ max [P] λ min [P]

30 Workshop On Uncertain Dynamical Systems Udine Quantum Stochastic Differential Equations Given (S,L,H 1 ) as above, we can construct the corresponding QSDE model as follows: [ ] [ ] [ ] da(t) a(t) da(t) da(t) # = F a(t) # dt + G da(t) # ; [ ] [ ] [ ] da out (t) a(t) da(t) da out (t) # = H a(t) # dt + K da(t) #, where F = ijm 1 [ ] S 0 2 JN JN; G = JN 0 S # [ ] S 0 H = N; K = 0 S # ; ;

31 Workshop On Uncertain Dynamical Systems Udine Also, we can introduce a change of variables to convert these into real QSDEs in terms of position and momentum operators: [ q p ] = Φ [ a a # ] ; [ Q P ] [ ] [ ] A Q out = Φ A # ; P out [ ] A out = Φ A out# [ where the unitary matrices Φ have the form Φ = I I 1 2 ii ii ]. Then [ ] dq dp [ ] dq out dp out [ ] [ ] p dq = A dt + B q 2 ; dp [ ] [ ] q dp = C 2 dt + D p 22 dq, where A = ΦFΦ 1, B 2 = ΦGΦ 1, C 2 = ΦHΦ 1, D 22 = ΦKΦ 1 are real matrices.

32 Workshop On Uncertain Dynamical Systems Udine If we define B 1 = ΦDΦ 1 and C 1 = ΦEΦ 1, then our previous robust stability condition becomes C 1 (si A) 1 B 1 < γ 2. We can then set up a corresponding quantum H control problem to achieve this defined by the controlled quantum uncertain system [ dq dp ] [ ] p = A dt + B q 1 dw + B 2 du; [ ] [ ] q q dz = C 1 dt; dy = C p 2 dt + D p 22 du; A recent result (James, Nurdin, Petersen, 2008) shows that this problem can be solved via the two Riccati equation method (with the addition of some small perturbations to make the problem non-singular and some loop shifting to deal with the D 22 term).

33 Workshop On Uncertain Dynamical Systems Udine Physical Realizability Not all QSDEs satisfy the laws of quantum mechanics. This motivates a notion of physical realizability. Definition. QSDEs of the form considered above are physically realizable if there exist suitably structured complex matrices Θ = Θ, M = M, N, S such that S S = I, and F = iθm 1 [ ] S 0 2 ΘN JN;G = ΘN 0 S # ; [ ] S 0 H = N;K = 0 S # ; Physical realizability means that the QSDEs correspond to a quantum harmonic oscillator.

34 Workshop On Uncertain Dynamical Systems Udine Theorem. The above QSDEs are physically realizable if and only if there exist complex matrices Θ = Θ and S such that S S = I, Θ is of the form above, and FΘ + ΘF + GJG = 0; [ ] S 0 G = ΘH 0 S # ; [ ] S 0 K = 0 S #. Note that the first of these conditions is equivalent to the preservation of the commutation relations for all times.

35 Workshop On Uncertain Dynamical Systems Udine (J, J)-unitary Transfer Function Matrices We now relate the physical realizability of the above QSDEs to the (J, J)-unitary property of the corresponding transfer function matrix Γ(s) = [ Γ11 (s) Γ 12 (s) Γ 21 (s) Γ 22 (s) ] = H (si F) 1 G + K. Definition. A transfer function matrix Γ(s) of the above form is (J, J)- unitary if Γ(s) JΓ(s) = J for all s C. Here, Γ (s) = Γ( s ).

36 Workshop On Uncertain Dynamical Systems Udine Theorem. (Shaiju and Petersen) Suppose the linear quantum system defined by the above QSDEs is minimal and that λ i (F)+λ j (F) 0 for all eigenvalues λ i (F), λ j (F) of the matrix F. Then this linear quantum system is physically realizable if and only if the following conditions hold: (i) The system transfer function matrix Γ(s) is (J, J)-unitary; [ ] S 0 (ii) The matrix K is of the form K = 0 S # where S S = I. In solving the quantum H control problem, if the controller is to be a coherent controller, implemented as a quantum system itself, then it must be physically realizable. A result of (James, Nurdin, Petersen, 2008) showed that any LTI controller (such as obtained using the two Riccati solution to the H control problem) can be made physically realizable by suitably adding quantum noises.

37 Workshop On Uncertain Dynamical Systems Udine Robust Stabilization of an Optical Parametric Oscillator An optical parametric oscillator (OPO) can be used to produce squeezed light in which the quantum noise in one quadrature is squeezed relative to the noise in the other quadrature and yet Heisenberg s uncertainty relation still holds. The following is a schematic diagram of an OPO. Optical Cavity optical isolator MgO:LiNbO 3 Laser and second harmonic generator nonlinear optical material output beam partially reflective mirror fully reflective mirror

38 Workshop On Uncertain Dynamical Systems Udine An approximate linearized QSDE model of an OPO is as follows: da = ( κ 2 + iδ(t))adt + χa dt κda; da out = κadt + da where δ(t) represents the mismatch between the OPO resonant frequency and the frequency of the driving laser, which will be treated as an uncertain parameter. The uncertainty Hamiltonian is H 2 = 1 2 δ(t)a a. Also, H 1 = 1 2 iχ( (a ) 2 a 2), L = κa, S = I. This corresponds to M 1 = 0, M 2 = iχ, N 1 = (κ), N 2 = 0; E 1 = 1, E 2 = 0, 1 = δ, 2 = 0. Furthermore, we choose the parameters values κ = 2000, χ = 2000, γ = 20 and set up a corresponding H control problem.

39 Workshop On Uncertain Dynamical Systems Udine Converting to real coordinates, solving the H control problem, converting back to the original coordinates, we obtain the controller dx c = F c x c dt + G c dy;du = H c x c dt, [ ] F c = ; G c = [ ] H c = ; [ ] ; We now consider whether this controller can be implemented as a coherent controller (with minimum number of additional noises). Do there exist matrices G co, H co, such that the system dx c = F c x c dt + G c dy + G co d w; du = H c x c dt + d w; dũ = H co x c dt + dy is physically realizable; i.e., (J, J)-unitary?

40 Workshop On Uncertain Dynamical Systems Udine In a recent result (Vuglar, Petersen, 2011), it is shown that this holds if and only if the Riccati equation F c X + XF c + H cjh c + XG c JG c = 0 has a solution X = TJT where T is non-singular and then, G co = X 1 H c J. [ ] In this example, we find X = and then [ ] G co =