Lindblad master equation with multi-photon drive and damping. Nonlinear Partial Differential Equations and Applications A conference in the honor of Jean-Michel Coron Institut Henri Poincaré 2016, June 20–24 Pierre Rouchon Centre Automatique et Systèmes, Mines ParisTech, PSL Research University QUANTIC, Inria-Paris / ENS Paris / Mines ParisTech Joint work with Rémi Azouit and Alain Sarlette 1 / 25
Outline Motivation: coherent feedback and reservoir engineering Harmonic oscillator with single-photon drive and damping Well posedness and convergence for multi-photon drive and damping Conclusion: many other examples of physical interest 2 / 25
Nobel Prize in Physics 2012 Serge Haroche David J. Wineland " This year’s Nobel Prize in Physics honours the experimental inventions and discoveries that have allowed the measurement and control of individual quantum systems. They belong to two separate but related technologies: ions in a harmonic trap and photons in a cavity . . . " From the Scientific Background on the Nobel Prize in Physics 2012 compiled by the Class for Physics of the Royal Swedish Academy of Sciences, 9 october 2012. 3 / 25
Two kinds of quantum feedback decoherence Measurement-based feedback: controller is classical ; measurement back-action on the u quantum quantum system of Hilbert space H is stochas- system tic (collapse of the wave-packet); the measured quantum world output y is a classical signal; the control input u y is a classical variable appearing in some con- classical world trolled Schrödinger equation; u ( t ) depends on classical the past measurements y ( τ ) , τ ≤ t . controller classical world Coherent/autonomous feedback and reser- decoherence u ? voir engineering: the system of Hilbert quantum system space H is coupled to the controller, an- other quantum system ; the composite sys- quantum world y? tem of Hilbert space H controller ⊗ H , is an open- quantum system relaxing to some target (sep- quantum controller arable) state. decoherence 4 / 25
Watt regulator: classical analogue of quantum coherent feedback. 1 From WikiPedia The first variations of speed δω and governor angle δθ obey to d dt δω = − a δθ d 2 dt 2 δθ = − Λ d dt δθ − Ω 2 ( δθ − b δω ) with ( a , b , Λ , Ω) positive parame- ters. Third order system dt 3 δω + Λ d 2 d 3 dt 2 δω + Ω 2 d dt δω + ab Ω 2 δω = 0 . Characteristic polynomial P ( s ) = s 3 + Λ s 2 + Ω 2 s + ab Ω 2 with roots having negative real parts iff Λ > ab : governor damping must be strong enough to ensure asymptotic stability. Key issues: asymptotic stability and convergence rates. 1 J.C. Maxwell: On governors. Proc. of the Royal Society, No.100, 1868. 5 / 25
Reservoir Engineering and coherent feedback 2 dissipation κ H int Reservoir System Engineered H res H interaction H = H res + H int + H t →∞ ρ res ⊗ | ¯ ψ �� ¯ if ρ → ψ | exponentially on a time scale of τ ≈ 1 /κ then . . . . . . 2 See, e.g., the lectures of H. Mabuchi delivered at the "Ecole de physique des Houches", July 2011. 6 / 25
Reservoir Engineering and coherent feedback 2 dissipation κ γ H int Reservoir System Engineered H res H interaction H = H res + H int + H t →∞ ρ res ⊗ | ¯ ψ �� ¯ . . . . . . ρ → ψ | + ∆ , if κ ≫ γ then � ∆ � ≪ 1 2 See, e.g., the lectures of H. Mabuchi delivered at the "Ecole de physique des Houches", July 2011. 6 / 25
Convergence issues of open-quantum systems Continuous-time models: Lindbald master eq. (quantum Fokker-Planck eq.): � � d � dt ρ = − A ( ρ ) � − i L ν ρ L † ν − ( L † ν L ν ρ + ρ L † � [ H , ρ ] + ν L ν ) / 2 , ν of state ρ a density operator (Hermitian, non negative, trace-class, trace one) with H Hermitian operator and L ν arbitrary operators (usually unbounded). When H is of finite dimension, ( e − t A ) t ≥ 0 is a contraction semi-group for many �� √ ρσ √ ρ � metrics ( Tr ( | ρ − σ | ) , Tr , see the work of D. Petz). Open issues motivated by robust quantum information processing: 1. characterization of the Ω -limit support of ρ : decoherence free spaces are affine spaces where the dynamics are of Schrödinger types; they can be reduced to a point (pointer-state); 2. Estimation of convergence rate and robustness. 3. Reservoir engineering: design of realistic H and L ν to achieve rapid convergence towards prescribed affine spaces (protection against decoherence). Goal of this talk: well-posedness and convergence for the infinite dimension system with H = 0 and L ν = a k − α k I with k ∈ N and α ∈ C . 7 / 25
Outline Motivation: coherent feedback and reservoir engineering Harmonic oscillator with single-photon drive and damping Well posedness and convergence for multi-photon drive and damping Conclusion: many other examples of physical interest 8 / 25
Quantum harmonic oscillator ◮ Hilbert space: � � � n ≥ 0 ψ n | n � , ( ψ n ) n ≥ 0 ∈ l 2 ( C ) ≡ L 2 ( R , C ) H = | n ◮ Operators and commutations: √ a | n � = √ n | n - 1 � , a † | n � = n + 1 | n + 1 � ; N = a † a , N | n � = n | n � ; ... ... [ a , a † ] = I , a f ( N ) = f ( N + I ) a ; D α = e α a † − α † a . | 2 u c � � ω c 1 x + ∂ a = X + i P = , [ X , P ] = ı I / 2. √ ∂ x 2 | 1 ◮ Hamiltonian: H / � = ω c a † a + u c ( a + a † ) . ω c (associated classical dynamics: √ dx dp dt = ω c p , dt = − ω c x − 2 u c ). | 0 ◮ Classical pure state ≡ coherent state | α � √ √ � � 2 ℜ α ) 2 α ∈ C : | α � = � e −| α | 2 / 2 α n 2 x ℑ α e − ( x − 1 π 1 / 4 e ı | n � ; | α � ≡ √ 2 n ≥ 0 n ! a | α � = α | α � , D α | 0 � = | α � . 9 / 25
Wigner function W ρ for different values of the density operator ρ �� � � W ρ : C ∋ ξ → 2 D ξ e i π N D † π Tr ρ ∈ [ − 2 /π, 2 /π ] ξ Fock state |n=0> Fock state |n=3> Coherent state |α=1.8> 0.6 0.4 0.2 0 3 Statistical mixture of Coherent state |-α> Cat state |-α>+|α> Im(ξ) |-α> and |α> −0.2 0 −0.4 −0.6 -3 -3 0 3 Re(ξ) 10 / 25
Experimental Wigner functions of 2, 3 and 4-leg Schrödinger cat-states 3 3 Vlastakis, B.; Kirchmair, G.; Leghtas, Z.; Nigg, S. E.; Frunzio, L.; Girvin, S. M.; Mirrahimi, M.; Devoret, M. H., Schoelkopf, R. J. "Deterministically Encoding Quantum Information Using 100-Photon Schrödinger Cat States". Science, 2013, - 11 / 25
Master equation for a damped and driven ( α ∈ R ) harmonic oscillator 4 � � dt ρ = L ρ L † − 1 d L † L ρ + ρ L † L with L = a − α I 2 ρ can be represented by its Wigner function W ρ defined by �� � � e ξ a † − ξ ∗ a e i π N e − ξ a † + ξ ∗ a C ∋ ξ = x + ip �→ W ρ ( ξ ) = 2 π Tr ρ With the correspondences � ∂ � ∂ � � ∂ ∂ x − i ∂ ∂ ∂ x + i ∂ ∂ξ = 1 ∂ξ ∗ = 1 , 2 2 ∂ p ∂ p � ∂ � � ∂ � W ρ a = W a ρ = W ρ , W ρ ξ − 1 ξ + 1 2 ∂ξ ∗ 2 ∂ξ ∗ W ρ a † = � � � � ∂ ∂ ξ ∗ + 1 W a † ρ = ξ ∗ − 1 W ρ , W ρ 2 2 ∂ξ ∂ξ we get the following PDE for W ρ : � ∂ ∂ W ρ ∂ 2 ∂ 2 � + ∂ ∂ x 2 W ρ + 1 � ( x − α ) W ρ � � pW ρ � = 1 + 1 ∂ p 2 W ρ 2 4 4 ∂ t ∂ x ∂ p π e − 2 ( x − α ) 2 − 2 p 2 . converging toward the Gaussian W ρ ∞ ( x , p ) = 2 4 See, e.g., S. Haroche and J.M. Raimond: Exploring the Quantum: Atoms, Cavities and Photons. Oxford University Press, 2006. 12 / 25
e − t A � � Difficulties to get the semi-groups t ≥ 0 from unbounded generators A . The minimal solution 5 of d dt ρ = − A ( ρ ) need not be trace-preserving. We can see this on this example due to Davies 6 � � dt ρ = − A ( ρ ) = L ρ L † − 1 a † � 2 d L † L ρ + ρ L † L � with L = 2 Formally with ρ ≥ 0, p n = � n | ρ | n � ≥ 0 and Tr ( ρ ) = � n p n = 1 we get d � dt Tr ( ρ N ) = Tr ( ρ 2 ( N + 1 )( N + 2 )) = p n 2 ( n + 1 )( n + 2 ) n ≥ 0 � 2 � � + 1 = 2 Tr 2 ( ρ N ) + 1 ≥ 2 p n n n ≥ 0 by convexity of x �→ 2 ( x + 1 )( x + 2 ) and 2 ( x + 1 )( x + 2 ) ≥ 2 x 2 + 1 for x ≥ 0. dt z ≥ 2 z 2 + 1 and thus for any initial condition With z = Tr ( N ρ ) , we have d ρ 0 ≥ 0, z 0 ≥ 0 and z ( t ) reaches + ∞ in finite time. This implies that Tr ( ρ ) is decreasing and that the above computations have to be re-considered. 5 See, e.g., chapter 4 written by F. Fagnola and R. Rebolledo in the book edited by Attal, S.; Joye, A.; Pillet, C.-A. (Eds.) Open Quantum Systems III: Recent Developments Springer, Lecture notes in Mathematics 1882, 2006. 6 E. Davies: Quantum dynamical semigroups and the neutron diffusion equation. Reports on Mathematical Physics, 1977, 11, 169-188 13 / 25
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