Lindblad master equation with multi-photon drive and damping.

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 allowedthe 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.

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Two kinds of quantum feedback

quantum system

classical controller

quantum world

classical world y

u

decoherence

Measurement-based feedback:controller is classical; measurement back-action on the quantum system of Hilbert spaceHis stochas- tic (collapse of the wave-packet); the measured outputyis a classical signal; the control inputu is a classical variable appearing in some con- trolled Schrödinger equation;u(t)depends on the past measurementsy(τ),τ ≤t.

quantum system

quantum controller quantum world

y?

u ?

classical world

decoherence

decoherence Coherent/autonomous feedback and reser-

voir engineering: the system of Hilbert space H is coupled to the controller, an- other quantum system; the composite sys- tem of Hilbert spaceHcontroller⊗H, is an open- quantum system relaxing to some target (sep- arable) state.

4 / 25

Watt regulator: classical analogue of quantum coherent feedback.

From WikiPedia

The first variations of speed δω and governor angleδθobey to

d

dtδω=−aδθ d²

dt²δθ=−Λd

dtδθ−Ω²(δθ−bδω) with (a,b,Λ,Ω) positive parame- ters.

Third order system

d³

dt³δω+ Λd²

dt²δω+ Ω²d

dtδω+abΩ²δω=0.

Characteristic polynomialP(s) =s³+ Λs²+ Ω²s+abΩ²with roots having negative real parts iffΛ>ab:governor damping must be strong enough to ensure asymptotic stability.

Key issues:asymptotic stability and convergence rates.

1J.C. Maxwell: On governors. Proc. of the Royal Society, No.100, 1868.

5 / 25

Reservoir Engineering and coherent feedback

System Reservoir

Engineered interaction dissipation

κ

H

H H

H = H

+ H

+ H

if ρ

→ ρ

⊗ | ψih ¯ ψ| ¯ exponentially on a time scale of τ ≈ 1/κ then . . . .

2See, e.g., the lectures of H. Mabuchi delivered at the "Ecole de physique des Houches", July 2011.

6 / 25

Reservoir Engineering and coherent feedback

System Reservoir

Engineered interaction dissipation

κ

H

H H

γ

H = H

+ H

+ H

. . . . ρ

→ ρ

⊗ | ψih ¯ ψ| ¯ + ∆, if κ γ then k ∆ k 1

2See, e.g., the lectures of H. Mabuchi delivered at the "Ecole de physique des Houches", July 2011.

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Convergence issues of open-quantum systems

Continuous-time models: Lindbald master eq. (quantum Fokker-Planck eq.):

d

dtρ=−A(ρ),−ⁱ

~[H, ρ] +X

ν

LνρL^†ν−(L^†νLνρ+ρL^†νLν)/2

,

of stateρa density operator (Hermitian, non negative, trace-class, trace one) withHHermitian operator andLνarbitrary operators (usually unbounded).

WhenHis of finite dimension,(e^−tA)t≥0is a contraction semi-group for many metrics ( Tr(|ρ−σ|), Tr p√

ρσ√ ρ

, see the work of D. Petz).

Open issuesmotivated byrobustquantum information processing:

1. characterization of theΩ-limit support ofρ: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 realisticHandLνto achieve rapid convergence towards prescribed affine spaces (protection against decoherence).

Goal of this talk:well-posedness and convergence for the infinite dimension system withH=0andLν=a^k−α^kIwithk∈Nandα∈C.

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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

I Hilbert space:

H=n P

n≥0ψ_n|ni, (ψ_n)_n≥0∈l²(C)o

≡L²(R,C)

I Operators and commutations:

a|ni=√

n|n-1i,a^†|ni=√

n+1|n+1i; N=a^†a,N|ni=n|ni;

[a,a^†] =I,af(N) =f(N+I)a;

Dα=e^{αa†−α†a}. a=X +iP= ^√1

2 x+_∂x^∂

, [X,P] =ıI/2.

I Hamiltonian:H/~=ω_ca^†a+u_c(a+a^†).

(associated classical dynamics:

dx

dt =ω_cp, ^dp_dt =−ω_cx−√ 2u_c ).

I Classical pure state≡coherent state|αi α∈C: |αi=P

n≥0

e^{−|α|2/2√αn}

n!

|ni;|αi ≡π^1/41 e^ı

√2x=αe^−(x−

√2<α)2 2

a|αi=α|αi,Dα|0i=|αi.

|0

|1

| 2 ω

| n

ω

u

... ...

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Wigner function W

for different values of the density operator ρ W

: C 3 ξ →

Tr

D

e

D

ρ

∈ [ − 2/π, 2/π]

Re(ξ)

Im(ξ)

−0.6

−0.4

−0.2 0 0.2 0.4 Fock state |n=0> Fock state |n=3> Coherent state |α=1.8> 0.6

Coherent state |-α> Statistical mixture of

|-α> and |α> Cat state |-α>+|α>

-3-3

0 0

3 3

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Experimental Wigner functions of 2, 3 and 4-leg Schrödinger cat-states³

3Vlastakis, 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, -

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Master equation for a damped and driven (α∈R) harmonic oscillator⁴

d

dtρ=LρL^†−¹₂

L^†Lρ+ρL^†L

with L=a−αI ρcan be represented by itsWigner functionW^ρdefined by

C3ξ=x+ip7→W^ρ(ξ) = _π² Tr

e^{ξa†−ξ∗a}e^iπN e^{−ξa†+ξ∗a}

ρ

With the correspondences

∂

∂ξ =¹₂ ∂

∂x −i ∂

∂p

, ∂

∂ξ^∗ =¹₂ ∂

∂x+i ∂

∂p

W^ρa=

ξ−¹₂ ∂

∂ξ^∗

W^ρ, W^aρ=

ξ+¹₂ ∂

∂ξ^∗

W^ρ W^ρa†=

ξ^∗+¹₂ ∂

∂ξ

W^ρ, W^a†ρ=

ξ^∗−¹₂ ∂

∂ξ

W^ρ

we get the followingPDEforW^ρ:

∂W^ρ

∂t =¹₂ ∂

∂x

(x−α)W^ρ + ∂

∂p

pW^ρ +¹₄∂²

∂x²W^ρ+¹₄ ∂²

∂p²W^ρ

converging toward theGaussianW^ρ∞(x,p) = _π²e^{−2(x−α)2−2p2}.

4See, e.g., S. Haroche and J.M. Raimond: Exploring the Quantum:

Atoms, Cavities and Photons. Oxford University Press, 2006.

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Difficulties to get the semi-groups e^−tA

t≥0from unbounded generatorsA.

Theminimal solution⁵of _dt^dρ=−A(ρ)need not betrace-preserving.

We can see this on this example due to Davies⁶

d

dtρ=−A(ρ) =LρL^†−¹₂

L^†Lρ+ρL^†L

with L= a^†2

Formally withρ≥0,pn=hn|ρ|ni ≥0 and Tr(ρ) =P

npn=1 we get d

dt Tr(ρN) = Tr(ρ2(N+1)(N+2)) =X

n≥0

pn2(n+1)(n+2)

≥2

X

n≥0

pnn 2

+1=2 Tr²(ρN) +1

by convexity ofx7→2(x+1)(x+2)and 2(x+1)(x+2)≥2x²+1 forx≥0.

Withz= Tr(Nρ), we have_dt^dz≥2z²+1 and thus for any initial condition ρ0≥0,z0≥0 andz(t)reaches+∞in finite time.This implies that Tr(ρ)is decreasing and that the above computations have to be re-considered.

5See, 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.

6E. Davies: Quantum dynamical semigroups and the neutron diffusion equation.

Reports on Mathematical Physics, 1977, 11, 169-188

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Quantum information processing with cat-qubits

It is possible withquantum circuitsto design an open quantum system governed by⁷

d

dtρ=LρL^†−¹₂

L^†Lρ+ρL^†L

+aρa^†−^Nρ+ρN₂

withL=a²−α²I.

The supports of all solutionsρ(t)converge to thedecoherence free space spanned by the even and odd cat-state;

|Cα⁺i ∝ |αi+|-αi, |Cα⁻i ∝ |αi − |-αi.

The corresponding PDE forW^ρis oforder 4inxandp.

A similar systemwhereL=a⁴−α⁴Icould be very interesting forquantum information processingwhere the logical qubit is encoded in the planes spanned by even and odd cat-states:

|C⁺_αi,|C_iα⁺i ,

|C_α⁻i,|C_iα⁻i ..

The corresponding PDE forW^ρis oforder 8inxandp.

7Z. Leghtas et al.: Confining the state of light to a quantum manifold by engineered two-photon loss. Science, 2015, 347, 853-857.

8M. Mirrahimi et al: Dynamically protected cat-qubits: a new paradigm for universal quantum computation, New Journal of Physics, 2014, 16, 045014.

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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

15 / 25

A bunch of spaces

I H=

|ψi=P

n∈Nψn|ni

ψn∈C, P

n∈N|ψn|²<+∞

separable Hilbert space.

I H^f =

|ψi=P

n∈Nψn|ni

∃¯n,∀n>n, ψ¯ n=0

dense inH.

I Hk ={|ψi=P

n∈Nψn|ni | P

n∈Nn^k|ψn|²<+∞}}dense inH.

I K¹(H)the Banach space of Hermitian trace-class operators equipped with the trace norm:ρ∈ K¹(H)compact Hermitian operator with spectral decompositionρ=P

µ≥1λµ|ψµihψµ|and such that P

µ≥1|λµ|<+∞. The trace-norm iskρktr = Tr(|ρ|) =P∞ µ=1|λµ|.

We haveρ=ρ⁺−ρ⁻and|ρ|=ρ⁺+ρ⁻with ρ⁺=P

µ≥1max(0, λµ)|ψµihψµ|andρ⁻=P

µ≥1max(0,−λµ)|ψµihψµ|.

I Quantum state-space: the convex set of density operators D=n

ρ∈ K¹(H)

P

µ≥1λµ=1,; λµ≥0 for allµ≥1o . I K^f(H) =n

P¯n

n,n⁰=1fn,n⁰|nihn⁰|

fn,n⁰ =f_n^∗0,n, n¯∈N o

dense inK¹(H).

I For anyρ∈ K¹(H)and any bounded operatorBonHwe have

Tr(Bρ) = Tr(ρB), Tr(Bρ)≤ Tr(|Bρ|) =kBρktr ≤ kBk Tr(|ρ|) =kBk kρktr 16 / 25

Adapted Banach space for _dt^dρ=LρL^†−¹₂(L^†Lρ+ρL^†L)withL=a^k−α^kI.

I The operatorL^†Lwith domainH2k admits a spectral decomposition L^†L=P∞

µ=1dµ|gµihgµ|where |gµi

µ≥1is an Hilbert basis ofHand dµ≥0.

Proof:(I+L^†L)⁻¹is a compact Hermitian operator.

I KL(H),n

ρ∈ K¹(H) Tr

pI+L^†Lρp I+L^†L

<+∞o

equipped with the normkρk_L= Tr

pI+L^†Lρp I+L^†L

is a Banach space.

Moreoverρ∈ KL(H)impliesLρL^†∈ K¹(H).

I We have[L,L^†] =a^k(a^†)^k −(a^†)^ka^k =Mwith

M= (N+I)(N+2I). . .(N+kI) − N(N−I)⁺. . .(N−(k−1)I)⁺≥k!I.

I Tr LρL^†

satisfies_dt^d Tr LρL^†

=−Tr LρL^†M

≤ −k!Tr LρL^† .

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Well posedness in K

( H ) based on Hill-Yosida thm for Banach space

Consider the Cauchy problem_dt^dρ=−A(ρ)associated to the super-operator KL(H)⊃DA3ρ7→A(ρ) = (L^†Lρ+ρL^†L)/2−LρL^†∈ KL(H) withL=a^k−α^k. For any integerk>0, any realα >0 and anyρ0in the domain ofA,

1. there exists a uniqueC¹function[0,+∞[3t7→ρ(t)∈ KL(H), such that ρ(t)belongs to the domain ofAfor allt≥0 and solves the initial value problem withρ(0) =ρ0

2. ∀t≥0, Tr(ρ(t)) = Tr(ρ0),kρ(t)k_L≤ kρ0k_LandkA(ρ(t))k_L≤ kA(ρ0)k_L. 3. Ifρ0is non-negative thenρ(t)remains also non negative.

Proof: for anyλ >0 andf ∈ KL(H), exitsρ∈ KL(H)such thatρ+λA(ρ) =f andkρk_L≤ kfk_L. We prove that(I+λA)⁻¹is a completely positive map, i.e.

a quantum (Kraus) map, fromKL(H)toDA. We combine arguments due to E. Davies⁹with the fact that[L,L^†]>0. See the forthcoming special issue of COCV or the preprinthttp://arxiv.org/abs/1511.03898.

9E. Davies: Quantum dynamical semigroups and the neutron diffusion equation. Reports on Mathematical Physics, 1977, 11, 169-188

10H. Brezis: Analyse fonctionnelle. Masson, Paris, 1987.

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Exponential convergence toward the decoherence-free sub-space of dimensionk

I The set of steady-states characterized byA(ρ) =0 corresponds to Hermitian operatorsρ¯with range included in thek-dimensional complex vector space,the decoherence-free sub-space,

Hα,k =span

|αmi

αm=αe^2iπm/k , m=1,2, ...,k

. I Consider the unique trajectory[0,+∞[3t7→ρ(t)∈ KL(H)solution of

d

dtρ=−A(ρ)with initial conditionρ(0) =ρ0non-negative, of trace one and in the domain ofA. Then there existsρ¯ρ₀∈ KL(H)nonnegative and of trace one, with support inHα,k such thatρconverges toρ¯inKL(H).

Moreover, we haveexponential convergence towardsHα,k in the sense:

Tr

L(ρ(t)−ρ¯ρ₀)L^†

≤ Tr

L|ρ0−ρ¯ρ₀|L^† e^−k!t .

Proof:the Lyapunov functionV(ρ) = Tr LρL^†

and _dt^dV ≤ −k!V.

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Invariant and conserved quantities

There existk²linearly independentHermitian bounded operators Qm,m⁰,m,m⁰ =1,2, ...,k, which areinvariantunder _dt^dρ=−A(ρ), i.e. for which

Tr(Q_m,m⁰ρ_t) = Tr(Q_m,m⁰ρ₀) for any trajectory[0,+∞)3t 7→ρ_t ∈ K^L(H).

Moreover, the linear space of invariant Hermitian operators spanned by{Q_m,m⁰}m,m⁰=1...k contains in particular thek operators

Q^cos_m =X

n∈N

cos

2πmn k

|nihn| for m=0,1, ...,d^k−12 e;

Q^sin_m =X

n∈N

sin

2πmn k

|nihn| for m=1, ...,b^k−12 c.

Proof:use the fact thatρ7→limt7→+∞e^−tA(ρ)is a complete positive map, i.e. a quantum channel and the fact that the dual ofK¹(H)is the set of bounded operators.

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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

21 / 25

Reservoir with the cavity deoherence (1/κ photon life-time)

Cavity mode (system) atom (reservoir)

R1

R2

Box of atoms

Aim:

engineer atom-mode interaction, to stabilize |-α +|α

DC field:

(controls atom frequency) ENS experiment

d dt

ρ =

reservoir relaxation

z }| {

a − α)ρ(a − α)

−

(a − α)

(a − α)ρ + ρ(a − α)

(a − α)

+ κ ae

ρe

a

−

(a

aρ + ρa

a)

| {z }

decoherence

.

11A. Sarlette, ; Brune, M.; Raimond, J.M.; P.R. "Stabilization of nonclassical states of the radiation field in a cavity by reservoir engineering", Phys. Rev.

Lett., 2011, 107, 010402.

A. Sarlette ; Leghtas, Z.; Brune, M.; Raimond, J.; P.R. " Stabilization of nonclassical states of one and two-mode radiation fields by reservoir engineering." Phys. Rev. A, 2012, 86, 012114

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Robustness of the reservoir stabilizing the two-leg cat.

SinceW^{eiπNρe−iπN}(ξ) =W^ρ(−ξ) the master Lindblad equation

d dtρ=

reservoir relaxation

z }| {

a−α)ρ(a−α)^†−¹₂ (a−α)^†(a−α)ρ+ρ(a−α)^†(a−α) +κ ae^iπNρe^−iπNa^†−¹₂(a^†aρ+ρa^†a)

| {z }

decoherence

.

yields to the followingnon localdiffusion PDE (quantum Fokker-Planck equation):

∂W^ρ

∂t (x,p)

=^1+κ₂ ∂

∂x

(x−α)W^ρ + ∂

∂p pW^ρ

+¹₄∆W^ρ

(x,p)

+κ

(x²+p²+¹₂)

W^ρ|_(−x,−p)−W^ρ|_(x,p) +₁₆¹

∆W^ρ|_(−x,−p)−∆W^ρ|_(x,p)

−κ ^x₂ ∂W^ρ

∂x _(−x,−p)

+ ∂W^ρ

∂x (x,p)

!

+^p₂ ∂W^ρ

∂p _(−x,−p)

+ ∂W^ρ

∂p (x,p)

!!

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Spin-spring systems

Lindblad master equation

d

dtρ=−_~ⁱ[H, ρ] +X

ν

LνρL^†_ν−(L^†_νLνρ+ρL^†_νLν)/2

,

forcomposite systemsmade of qubit(s) (Pauli operatorσ_x,σy andσ_z) and harmonic oscillator(s) (annihilation operatora, number operator N) with e.g. (Hamiltonian coupling)

H =ω_cN⊗I_q+χN²⊗I_q+u_c(a+a^†)⊗I_q +^ωq

2I_c⊗σ_z+u_qI_c⊗σ_x+g(a+a^†)⊗σ_x andlocal decoherence

L₁=p

κ(1+n_th)a⊗I_q, L₂=√

κn_tha^†⊗I_q, L₃=

q1

T1I_c⊗(σ_x−iσ_y), L₄=q

1

TφI_c⊗σ_z.

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Conclusion pour Jean-Michel

Bon anniversaire

Jean-Michel

grand merci

1. la communauté du contrôle,

2. la théorie mathématique des systèmes,

3. et plus largement les mathématiques en France et dans le Monde,

4. . . .

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