Introduction to Quantum Linear Systems

Transcription

1 The Hong Kong Polytechnic University 29 July 2014 Quantum Control Engineering, Cambridge, 21 July - 15 August 2014

2 Classical linear systems mathematical model (via first-principle or system identification) stability, controllability and observability system architecture (e.g. Kalman canonical decomposition) response of the system to various input signals (impulse, step, sinusoidal,...) control, in particular feedback control (to deliver desired performance)...

3 Outline Quantum linear systems Quantum linear passive systems Some applications Conclusion

4 Quantum linear systems: three examples Beam splitter: where Optical cavity: [ ] [ b 1 b1 = S b 2 b 2 ], [ ] η 1 η S =, (0 < η < 1). 1 η η a static system da(t) = (iω + κ 2 )a(t)dt κdb(t), db out (t) = κa(t)dt +db(t). a passive system

5 Quantum linear systems: three examples Degenerate parametric amplifier (DPA) [ ] da(t) da (t) [ ] dbout (t) dbout (t) = 1 [ ][ κ ǫ a(t) 2 ǫ κ a (t) = [ ] a(t) κ a + (t) [ db(t) db (t) ] [ db(t) κ db (t) ]. ], an active system

6 Quantum linear systems: the mathematical model G: n (interacting) quantum harmonic oscillators driven by m canonical boson fields. Notations a = a 1. a n, a # = ă = [ a a # a 1. a n, a = (a # ) T = [a 1 a n ]. ], ă = (ă # ) T = [a a T ].

7 Quantum linear systems: the mathematical model The (S,L,H) language [Gough & James, IEEE TAC, 54, 2530, 2009]: S C m m is unitary, L = C a+c + a #, C,C + C m n, H = 1 2 ă Ω ă, where Notation [ ] Ω Ω + Ω = = Ω C 2n 2n. Ω # + Ω# (A,B) [ A B B # A # ].

8 Quantum linear systems: the mathematical model Beamsplitter: (S, 0, 0). Optical cavity: (1, κa,ωa a). Degenerate parametric amplifier: (1, κa, 1 2ă Ωă), where Ω = [ 0 iǫ 2 iǫ 2 0 ]. Isolated harmonic oscillator: (,,ωa a). (S, L, H) is able to describe quantum nonlinear systems.

9 Quantum linear systems: the mathematical model Dynamical evolution: du(t) = { ( L ) } L 2 +ih dt +db (t)l L SdB(t)+Tr[(S I m )dλ T (t)] U(t) (t > 0), with U(0) = I. By X(t) U(t) (X I)U(t), B out (t) = U (t)(i B(t))U(t), we have a quantum linear system: where dă(t) = Aă(t)dt +Bd B(t), d B out (t) = Că(t)dt +Dd B(t), A = 1 2 C C ij n Ω, B = C (S,0 m m ), C = (C, C + ) C, D = (S,0 m m ),

10 Quantum linear systems: the mathematical model with Notation Fundamental relations: [ In 0 J n = 0 I n ], C = J n C J m. G (S,C,Ω). A+A +BB = 0 [ă(t), ă (t)] [ă(0), ă (0)], B = C D [ă(t), B out(t)] 0, DD = D D = I 2m D is symplectic. [James, Nurdin, & Petersen, IEEE TAC, 53, 1787, 2008]; [James & GZ, IEEE TAC, 56, 1535, 2011]

11 Quantum linear systems The remaining discussions are heavily biased towards the research or with which I am familiar; in which I am personally involved. A recent survey on the theoretical part: [Petersen, Quantum Linear Systems Theory, Proceedings of the 19th International Symposium on Mathematical Theory of Networks and Systems MTNS 2010, 59 July, 2010, Budapest, Hungary] A more experiment-oriented review on feedback control in quantum optics: [Alessio Serafini, arxiv: , 2012] An excellent book: [Wiseman & Milburn, Quantum Measurement and Control, 2010] (on among others, measurement-based feedback control)

12 Quantum linear systems: stability, controllability, observability Averaging the above linear system gives rise to the following classical linear system: d ă(t) dt d b out (t) dt = A ă(t) +B b(t), = C ă(t) +D b(t ). Definition The quantum linear system G (S,C,Ω) is said to be Hurwitz stable (resp. controllable, observable) if the corresponding classical linear system is Hurwitz stable (resp. controllable, observable). [Zhou, Doyle, & Glover (Sidney Sussex College, Univ. of Cambridge), Robust and Optimal Control, 1996]

13 Quantum linear systems: stability, controllability, observability Proposition Given a quantum linear system G (S,C,Ω), the following statements are equivalent: (i) G is controllable; (ii) G is observable; (iii) rank(o s ) = 2n, where O s C CJ n Ω. C (J n Ω) 2n 1. Stabilizability = detectability, [Wiseman and Doherty, PRL, 94, , 2005] Question: Is it also true in the nonlinear case?

14 Quantum linear systems: stability, controllability, observability Proposition For the quantum linear system G (S,C,Ω), (i) The unobservable subspace is Ker(O s ) (ii) The uncontrollable subspace is Ker(O s J n ) (iii) The uncontrollable and unobservable subspace is Ker(O s ) Ker(O s J n ). {[ v1 Ker(O s ) Ker(O s J n ) = span 0 ] [ 0, v # 1 ] [ vl,, 0 ] [ 0, v # l ]}.

15 Quantum linear systems: stability, controllability, observability Lemma There exists a unitary matrix V = [ V 1 V 2 ] with V 1 C 2n 2l and V 2 C 2n 2(n l) such that Range(V 1 ) = Ker(O s ) Ker(O s J n ), [ ] V Jl 0 J n V =. 0 J n l V 1 = V 2 = [ ] v1 v l v # 1 v # l [ vl+1 v n v # l+1 v n # ]

16 Quantum linear systems: decoherence free subspace (DFS) Theorem Let V be the matrix defined above. If ΩRange(V 1 ) Range(V 1 ), then the transformed system [ ] ădf V ă ă D has the following realization: dă DF (t) = ij l V 1 ΩV 1ă DF (t)dt, ( (CV2 ) ) (CV 2 ) dă D (t) = +ij n l V 2 2 ΩV 2 ă D (t)dt (CV 2 ) Dd ˇB(t), d ˇB out (t) = (CV 2 )ă D (t)dt +Dd ˇB(t). [Ticozzi & Viola, IEEE TAC, 53, 2048, 2008]; [Ticozzi & Viola, Automatica, 45, 2002, 2009]; [Yamamoto, arxiv: v2, 2013]

17 Quantum linear systems: transfer functions A fundamental property G(s) D +C(sI A) 1 B. G(iω) G(iω) = G(iω)G(iω) I 2m, ω R. [Gough, James, & Nurdin, Phys. Rev. A, , 2010]; [GZ & James, IEEE TAC, 58, 1221, 2013] Proposition The transfer function G(s) for the quantum linear system G (S,C,Ω) can be written in the following fractional form G(s) = (I Σ(s))(I +Σ(s)) 1 (S,0), where Σ(s) 1 2 C(sI +ij nω) 1 C, Re[s] > 0.

18 Outline Quantum linear systems Quantum linear passive systems Some applications Conclusion

19 Quantum linear passive systems: model If the matrices C + = 0 and Ω + = 0, the resulting system, parametrized by matrices S, C, Ω, is often called a quantum linear passive system. in which A 1 2 C C iω. Definition da(t) = Aa(t) C SdB(t), (1) db out (t) = C a(t)+sdb(t). (2) (1)-(2) is said to be the realization of the quantum linear passive system G (S,C,Ω ).

20 Quantum linear passive systems: transfer function Clearly, the transfer function of G (S,C,Ω ) is G(s) = S C (si A) 1 C S. Define Σ(s) 1 2 C (si +iω ) 1 C. (3) We have G(s) = (I Σ(s))(I +Σ(s)) 1 S. A fundamental property G(iω) G(iω) = G(iω)G(iω) = I m, ω R. (4) an all-pass filter

21 Quantum linear passive systems: stability, controllability, observability Lemma The following statements for a quantum linear passive system G (S,C,Ω ) are equivalent: (i) G is Hurwitz stable; (ii) G is observable; (iii) G is controllable. (ii) = (iii) and (ii) (i) : [Guta & Yamamoto, arxiv: , 2013] A Hurwitz stable quantum linear passive system cannot have isolated components.

22 Quantum linear passive systems: G and Σ Definition (Lossless Bounded Real ([Maalouf & Petersen, IEEE TAC, 56, 786, 2011].) A quantum linear passive system G = (S,C,Ω ) is said to be lossless bounded real if it is Hurwitz stable and Eq. (4) holds. A Hurwitz stable passive G is naturally lossless bounded real. Definition (Lossless Positive Real.) A function Ξ(s) is said to be positive real if it is analytic in Re[s] > 0 and satisfies Ξ(s)+Ξ(s) 0, Re[s] > 0. Moreover, Ξ(s) is called lossless positive real if is positive real and satisfies Ξ(iω)+Ξ(iω) = 0, where iω is not a pole of Ξ(s).

23 Quantum linear passive systems: G and Σ G(s) = (I Σ(s))(I +Σ(s)) 1 S. Theorem If a quantum linear passive system G (S,C,Ω ) is Hurwitz stable, then (i) G(s) is lossless bounded real. (ii) Σ(s) defined in Eq. (3) is lossless positive real. Quantum linear passive systems are the counterpart of classical linear passive electric networks. [Anderson & Vongpanitlerd, Network Analysis and Synthesis: a Modern Systems Theory Approach, 1973]

24 Quantum linear passive systems: system realization Given a quantum linear passive system G (I,C,Ω ), Figure : r = rank(c ); multi-input-multi-output system realization

25 Quantum linear passive systems: system realization Alternatively, Figure : multi-input-multi-output system realization

26 Quantum linear passive systems: system realization Figure : Cascaded realization: a quantum linear passive system with n system oscillators is realised as a sequence of n components in series, each one having a one-mode oscillator. [Nurdin, IEEE TAC, 55, 2439, 2010]; [Petersen, Automatica, 47, 1757, 2011]

27 Quantum linear passive systems: system realization From now on we focus on the single-input case, that is, there is only one input field channel. Principle Mode... Figure : Bus realization: the principal mode is coupled to n 1 independent auxiliary modes. The principal mode couples to the field, while the auxiliary modes are independent other than that they couple to the principal mode. The transfer function is G (s) = 1 [Gough & GZ, arxiv: , 2013] γ s γ +iω 0 + n 1 k=1. κ k s+iω k

28 Quantum linear passive systems: system realization Principal Mode Figure : Chain-mode realization: the principal mode is coupled to a non-damped mode which in turn is coupled to a finite chain of modes. [Gough & GZ, arxiv: , 2013] The idea is borrowed from the physical chemistry community: [Hughes, Christ, & Burghardt, J. Chem. Phys., 131, , 2009]; [Woods, Groux, Chin, Huelga, & Plenio, J. Math. Phys., 55, , 2014]

29 Quantum linear passive systems: system realization The transfer function is G (s) =I s γ +iω 0 + γ s +i ω κ 1. [Gough & GZ, arxiv: , 2013] κ nmin 2 s +i ω nmin 2 + κ n min 1 s +i ω nmin 1

30 Quantum linear passive systems: system realization Equivalence between the bus and chain-mode realizations where G(s) = 1 γ s + γ 2 +iω 0 + (s), (s) n min 1 k=1 κ k s +iω k = s +i ω 1 + s +i ω κ 1 κ 2. κ nmin 2 s +i ω nmin 2 + κ n min 1 s + ω nmin 1

31 Quantum linear passive systems: system realization If we define then Σ(s) = 1 γ 2s +iω 0 + (s), G(s) = 1 Σ(s) 1+Σ(s), is a physically valid quantum linear passive system.

32 Outline Quantum linear systems Quantum linear passive systems Some applications Conclusion

33 Some applications Figure : Coherent feedback control H control, LQG control, squeezing,...

34 Some applications: H control (a) Schematic (b) Quantum optics experiment Figure : Coherent H quantum feedback control [James, Nurdin, & Petersen, IEEE TAC, 53, 1787, 2008]; [Mabuchi, Phys. Rev. A, 78, , 2008]; [GZ & James, IEEE TAC, 56, 1535, 2011]

35 Some applications: coherent LQG control [Nurdin, James, & Petersen, Automatica, 45, 1837, 2009] [GZ & James, IEEE TAC, 56, 1535, 2011] [GZ, Lee, Huang, & Zhang, SIAM J. Contr. and Optim, 50, 2130, 2012] [Hamerly & Mabuchi, Phys. Rev. A 87, , 2013] As yet, no analytical solution is available to coherent LQG quantum feedback control!

36 Some applications: squeezing enhancement Figure : Lossy DPA In the (S,L,H) language, assume the lossy DPA has parameters S = I, Ω = 0, Ω + = iǫ [ ] κ 2, C =, C γ + = 0.

37 Some applications: squeezing enhancement Figure : Lossy DPA in feedback loop [Yanagisawa & Kimura, IEEE TAC, 48, 2121, 2003]; [Gough & Wildfeuer, Phys. Rev. A, 80, 42107, 2009] Experimental realization: [Iida, Yukawa, Yonezawa, Yamamoto, & Furusawa, IEEE TAC, 57, 2045, 2012]; with two OPOs as controller: [Crisafulli, Tezak, Soh, Armen, & Mabuchi, Optics Express, 21, 18371, 2013]

38 Some applications: squeezing enhancement Figure : Schematic of plant-controller system K = (S K,L K,H K ).

39 Some applications: squeezing enhancement S K = L K = H K [ ] 1 0, 0 1 [ ] LK1 = L K2 [ = 1 2ă K [ ] [ a K ] ak, i i ] ă K. Table : Numerical comparison µ p κ = 0.6;γ = 0.2;ǫ = 0.2 κ = 0.5;γ = 0.4;ǫ = 0.2 Lossy DPA GW The proposed scheme [Bian, GZ, & Lee, Int. J. Control, 85, 1865, 2012]

40 Some applications: squeezing enhancement u i e y P 1 M 1 L k1 D 1 S 1 L k2 S 2 w in P 2 i e w out M 2 D 2 D 0 H k Figure : Possible implementation of the coherent feedback controller [Nurdin, James, & Doherty, SIAM J. Contr. and Optim, 48, 2686, 2009] [Bian, GZ, & Lee, I. J. Contr., 85, 1865, 2012]

41 Some applications: photon processing Define B(ξ) := ξ (r)b(r)dr, B (ξ) := ξ(r)b (r)dr. single-photon light pulses A continuous-mode single-photon state ξ can be defined as ξ := B (ξ) 0, with Mean: ξ b(t) ξ = 0. Covariance function: [ ] R(t,r) = E ξ [ b(t) b 1 0 (r)] = δ(t r) 0 0 }{{} vacuum ξ(t) 2 dt = 1. [ ξ(t)ξ + (r) 0 ]. } 0 ξ (t)ξ(r) {{} pulse shape

42 Some applications: photon processing b G bout Question: ξ G??? How does a linear quantum system process an input field containing a definite number of photons? For the single-photon case ξ out [s] = G[s]ξ in [s]. The multi-photon case has to use tensor computation. the single-photo case: [GZ & James, IEEE TAC, 58, 1221, 2013]; the multi-photon case: [GZ, arxiv: , 2013]

43 Some applications: photon processing [ η 1 η b out (t) = ]b(t). 1 η η If the input state is ν 1 ν 2, the output state is Ψ out = η(1 η)b 1 (ν 1)B 1 (ν 2) ηb 1 (ν 1) 0 1 B 2 (ν 2) 0 2 (1 η)b 1(ν 2 ) 0 1 B 2(ν 1 ) 0 2 η(1 η) 0 1 B 2 (ν 1)B 2 (ν 2) 0 2.

44 Some applications: photon processing For a balanced beamsplitter, η = 1/2, and indistinguishable single-photon state (ν 1 ν 2 ), the steady-state output state is Ψ out = The HOM interference effect. [Hong, Ou, & Mandel, PRL. 59, 2044, 1987]

45 Some applications: photon processing Given a beam splitter S = [ R T T R ], R,T C and an input 2-photon state of the form Ψ in = the output state is Ψ out = R T + R 2 + T 2 dt 1 dt 2 ψ(t 1,t 2 ) b 1 (t 1)b 2 (t 2) 0 0, +T R dt 1 dt 2 ψ(t 1,t 2 )b 1(t 1 )b 1(t 2 ) 0 0 dt 1 dt 2 ψ(t 1,t 2 )b 1 (t 1)b 2 (t 2) 0 0 dt 1 dt 2 ψ(t 1,t 2 )b 1 (t 2)b 2 (t 1) 0 0 dt 1 dt 2 ψ(t 1,t 2 )b 2(t 1 )b 2(t 2 ) 0 0 [Eq. (6.8.7), Loudon,The Quantum Theory of Light, 2000]

46 Some applications: photon processing Two-channel two-photon input state: Ψ in = 2 η(1 η) Ψ out = B N 1 (ξ11 )B 1 (ξ12 )B 1 (ξ21 )B 1 (ξ22 ) N 22 j=1 1 N2j 2 Bj (ξ jk ) 0 2 k=1 + η η(1 η) B N 1 (ξ11 )B ( 1 (ξ12 ) B 1 (ξ22 )B 2 (ξ21 ) + B 1 (ξ21 )B ) 0 2 (ξ22 2 ) 21 N 22 η(1 η)(1 η) B1 N (ξ21 )B ( 1 (ξ22 ) B1 (ξ12 )B2 (ξ11 ) + B1 (ξ11 )B ) 0 2 (ξ12 2 ) 21 N 22 η 2 + B N 21 1 (ξ11 )B 1 (ξ12 )B 2 (ξ21 )B 2 (ξ22 ) 0 2 N 22 (1 η)2 + B N 1 (ξ21 )B 1 (ξ22 )B 2 (ξ11 )B 2 (ξ12 ) N 22 η(1 η) ( B1 (ξ11 )B2 (ξ12 ) + B1 (ξ12 )B2 (ξ11 ) )(B 1 (ξ21 )B2 (ξ22 ) + B1 (ξ22 )B2 (ξ21) 2 0 N 21 N 22 η η(1 η) ( B1 (ξ11 )B2 (ξ12 ) + B1 (ξ12 )B ) 2 (ξ11 ) B2 (ξ21 )B 2 (ξ22 ) 0 2 N 21 N 22 η(1 η)(1 η) ( + B 1 (ξ21 )B 2 (ξ22 ) + B 1 (ξ22 )B ) 2 (ξ21 ) B 2 (ξ11 )B 2 (ξ12 ) 0 2 N 21 N 22 η(1 η) + B N 2 (ξ11 )B 2 (ξ12 )B 2 (ξ21 )B 2 (ξ22 ) N 22

47 Some applications: photon processing For a balanced beamsplitter, η = 1/2, and all the pulse shapes are identical, the steady-state output state is 3 Ψ out = 8 4, ,0 0, ,4, which is the same as (15) in [Ou, Int. J. Mod. Phys. B, 21, 5033, 2007]. Question: When photons are not distinguishable, the continuous-variable version reveals the same phenomena as its discrete-variable counterpart does?

48 Some applications: photon processing Let us see how an empty optical cavity processes a 2-photon state Ψ in = 1 N2 ψ(t 1,t 2 )b (t 1 )b (t 2 )dt 1 dt 2 0, where ψ(t 1,t 2 ) = with ( 1 exp 1 1/2 2π Σ 2 Σ = [ 1 ρ ρ 1 [ t1 1 t 2 1 ] Σ 1 [ t1 1 t 2 1 ], 1 < ρ < 1. ]), A pure state of two photons sharing a Gaussian temporal profile

49 Some applications: photon processing!!!!!!!!!"#$ %&'!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!(#$$ %&)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!*#$$ %&+'!!!!!!!!!!!!!!!!!!!!!!,#$$ %&'!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!-#$$ )!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!.#$$ /!! Multi-photon output pulse shaping via an optical cavity [GZ, arxiv: , 2013]!

50 Some applications: quantum memory A quantum linear model for gradient echo memory (GEM) was recently proposed in [Hush, Carvalho, Hedges, & James, New J. Physics, 15, , 2013]

51 Outline Quantum linear systems Quantum linear passive systems Some applications Conclusion

52 Conclusion Linear quantum systems appear simple; are often computationally tractable; are nonetheless able to tell us interesting things are thus the first, and perhaps essential, step in the study of general theory.

53 Conclusion Linear quantum systems appear simple; are often computationally tractable; are nonetheless able to tell us interesting things are thus the first, and perhaps essential, step in the study of general theory.

54 Conclusion Linear quantum systems appear simple; are often computationally tractable; are nonetheless able to tell us interesting things are thus the first, and perhaps essential, step in the study of general theory.

55 Conclusion Linear quantum systems appear simple; are often computationally tractable; are nonetheless able to tell us interesting things are thus the first, and perhaps essential, step in the study of general theory.

56 Thank you for your attention!