Modelling, simulation and feedback of open quantum systems

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Modelling, simulation and feedback of open quantum systems

Pierre Rouchon

Centre Automatique et Systèmes Mines ParisTech

PSL Research University

Fall 2018

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Outline of the lectures (Autumn 2018 at Mines ParisTech )

Lect. 1& 2 (Oct. 24: 11:10-13:00 and 14:00–15:50) Feedback for classical and for quantum systems; the first experimental realization of a

quantum-state feedback (LKB photon box); three quantum features (Schrödinger; collapse of the wave packet, tensor product); Quantum Non Demolition (QND) measurement of photons (ideal Markov model and Matlab simulations).

Lect. 3 (Nov. 7: 11:10–13:00) Models of the LKB photon box: entanglement between the probe-qubit and the photons; qubit-measurement back-action on the photons; measurement errors and imperfections; decoherence as unread fictitious measurements; discrete-time Markov chain; Kraus map; quantum trajectories and realistic Matlab simulations of the photon box

Lect. 4 (Nov. 14: 11:10–13:00) Feedback stabilization of photon-number state:

resonant interaction, the Lyapunov feedback scheme, closed-loop simulations / experimental data.

Lect. 5 (Nov. 21: 9:00–10:50) The LPA super-conduction qubit under continuous-time measurements (counting versus homo/hetero-dyne measurements):

continuous-time stochastic master equation (Poisson versus Wiener);

Lindblad master equation; QND measurement of a qubit and Matlab simulations.

Lect. 6 (Nov. 28: 11:10–13:00) Feedback stabilization of the excited state of a qubit;

quantum-state feedback based on QND measurement;closed-loop simulations.

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Main references

I

S. Haroche and J.M. Raimond. Exploring the Quantum:

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

I

H.M. Wiseman and G.J. Milburn. Quantum Measurement and Control. Cambridge University Press, 2009.

I

M.A. Nielsen and I.L. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, 2000.

I

C.W. Gardiner and P. Zoller Quantum Noise. Springer, 2010.

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Outline

Notion of Feedback

Discrete-time systems: The LKB photon-box

Continuous-time systems: qubit in circuit QED

Continuous diffusive-jump SME

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Model of classical systems

control

perturbation

measure system

For theharmonic oscillatorof pulsationωwithmeasured positiony, controlled by the forceuand subject to an additional unknown force w.

x = (x1,x2)∈R², y =x1

d

dtx1=x2, d

dtx2=−ω²x1+u+w

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Feedback for classical systems

feedback observer/controller

perturbation

set point

control

measure system

Proportional Integral Derivative (PID) for _dt^d²2y =−ω²y+u+w with the set pointv =y^c

u=−Kp y −y^c

−Kd

d

dt y−y^c

−Kint

Z

y −y^c with the positivegains(Kp,Kd,Kint)tuned as follows (0<Ω₀∼ω, 0< ξ∼1, 0< 1:

Kp= Ω²₀, Kd =2ξ q

ω²+ Ω²₀, ,Kint=(ω²+ Ω²₀)^3/2.

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Quantum feedback: the back-action of the measurement.

A typical stabilizing feedback-loop for a classical system

system

controller

w

Two kinds of stabilizing feedbacks for quantum systems 1. Measurement-based feedback: controller is classical;

measurement back-action on the systemS is stochastic (collapse of the wave-packet); the measured outputy is a classical signal; the control inputuis a classical variable appearing in some controlled Schrödinger equation;u(t) depends on the past measurementsy(τ),τ≤t.

2. Coherent/autonomous feedback and reservoir engineering:the systemSis coupled tothe controller, another quantum system; the composite system,H^S⊗H_controller, is an

open-quantum system relaxing to some target (separable) state.

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The first experimental realization of a

quantum state feedback

The photon box of the Laboratoire Kastler-Brossel (LKB):

group of S.Haroche (Nobel Prize 2012), J.M.Raimond and M. Brune.

u y

1

Stabilization of a quantum state with exactlyn=0,1,2,3, . . .photon(s).

Experiment:C. Sayrin et. al., Nature 477, 73-77, September 2011.

Theory:I. Dotsenko et al., Physical Review A, 80: 013805-013813, 2009.

R. Somaraju et al., Rev. Math. Phys., 25, 1350001, 2013.

H. Amini et. al., Automatica, 49 (9): 2683-2692, 2013.

1Courtesy of Igor Dotsenko.Sampling period 80µs. 8 / 39

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Three quantum features emphasized by the LKB photon box

²

1. Schrödinger: wave funct.|ψi ∈ Hor density op. ρ∼ |ψihψ| d

dt|ψi=−_~ⁱH|ψi, d

dtρ=−_~ⁱ[H, ρ], H=H0+uH1

2. Origin of dissipation: collapse of the wave packetinduced by the measurement of observableOwith spectral decomp.P

µλµPµ:

I measurement outcomeµwith proba.

Pµ=hψ|Pµ|ψi= Tr(ρPµ)depending on|ψi,ρjust before the measurement

I measurement back-action if outcomeµ=y:

|ψi 7→ |ψi⁺= P_y|ψi

phψ|P_y|ψi, ρ7→ρ+= P_yρP_y Tr(ρPy) 3. Tensor product for the description of composite systems(S,M):

I Hilbert spaceH=H^S⊗ H^M

I HamiltonianH=H_S⊗I_M+H_int+I_S⊗H_M

I observable on sub-systemM only:O=I_S⊗O_M.

2S. Haroche and J.M. Raimond.Exploring the Quantum: Atoms, Cavities and Photons.Oxford Graduate Texts, 2006.

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Composite system built with an harmonic oscillator and a qubit.

I System

S corresponds to a quantized harmonic oscillator:

HS = ( _∞

X

n=0

ψn|

n

i

(ψn)^∞_n=0∈

l

1; atoms leaving B are all in state

|

g

i

I State of the full system|Ψi ∈ HS⊗ HM

:

|Ψi=

+∞

i ⊗ |

e

i)n∈N

.

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S: quantum harmonic oscillator (spring system)

I Hilbert space:

HS =n P

n≥0ψ_n|ni, (ψ_n)_n≥0∈l²(C)o

≡L²(R,C)

I Quantum state space:

D={ρ∈ L(HS), ρ^† =ρ, Tr(ρ) =1, ρ≥0}.

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_S/~=ω_ca^†a+uc(a+a^†).

(associated classical dynamics:

dx

dt =ω_cp, ^dp_dt =−ω_cx−√ 2uc).

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

_c

| n

ω

_c

u

_c

... ...

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M: 2-level system, i.e. a qubit (half-spin system)

I Hilbert space:

H^M =C²=n

cg|gi+ce|ei, cg,ce∈C o

.

I Quantum state space:

D={ρ∈ L(H^M), ρ^† =ρ, Tr(ρ) =1, ρ≥0}.

I Operators and commutations:

I Hamiltonian:H_M/~=ω_qσ_z/2+u_qσ_x.

I Bloch sphere representation:

D=n

1

2 I+xσx+yσ_y+zσ_z (x,y,z)∈R³, x²+y²+z²≤1o

| g

| e ω

_q

u

_q

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The Markov ideal model (1)

C

B

D R

1

R

₂

B R

₂

I When atom comes outB,|Ψi^B of the full system isseparable

|Ψi^B=|ψi ⊗ |gi.

I Just before the measurement inD, the state is in general entangled(not separable):

|ΨiR2 =U_SM |ψi ⊗ |gi

= M_g|ψi

⊗ |gi+ M_e|ψi

⊗ |ei whereU_SM is a unitary transformation (Schrödinger propagator) defining the linear measurement operatorsMgandMeonHS. SinceU_SMis unitary,M^†_gM_g+M^†_eM_e=I.

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The Markov ideal model (2)

C

B

D R

₁

R

₂

H _{S M}

The unitary propagatorU_SM is derived from Jaynes-Cummings HamiltonianH_SMin the interaction frame.

Two kinds of qubit/cavity Hamiltonians:

resonant,H_SM/~=i Ω(vt)/2

a^†⊗σ_-−a⊗σ₊ , dispersive,H_SM/~= Ω²(vt)/(2δ)

N⊗σ_z, whereΩ(x) = Ω₀e⁻^x

2

w2,x =vt withv atom velocity,Ω₀vacuum Rabi pulsation,wradial mode-width and whereδ=ω_q−ω_c is the detuning between qubit pulsationω_qand cavity pulsationω_c (|δ| Ω₀).

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Dispersive and resonant Jaynes-Cummings propagators

The solution of

ı_dt^dU =−_~ⁱH_SM(t)U, withU₀=I

reads

I

for

H_SM(t)/~=

i f

(t) a^†⊗ |

g

ih

e

| −a⊗ |

e

ih

g

|

(resonant)

U_t =

cos

θt

2

√N

⊗ |

g

ih

g

|+

cos

θt

2

| +

sin

θt

2

√N

√N a^†⊗ |

g

ih

e

where

θ(t) =Rt

0

f

(τ)

d

τ

.

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Quantum trajectories

Just beforedetectorDthe quantum state isentangled:

|Ψi_R

2= (Mg|ψi)⊗ |gi+ (Me|ψi)⊗ |ei

Just afteroutcomey, the state becomesseparable³:

|Ψi_D =

My q

hψ|M^†_yM_y|ψi|ψi

⊗ |yi.

Outcomeyobtained with probabilityPy =hψ|M^†yMy|ψi..

Quantum trajectories(Markov chain, stochastic dynamics):

|ψk+1i=







M_g q

hψ_k|M^†_gM_g|ψ_ki|ψki, yk =gwith probabilityhψk|M^†gMg|ψki;

Me

√

hψ_k|M^†_eM_e|ψ_ki|ψki, yk =ewith probabilityhψk|M^†eMe|ψki;

with state|ψkiand measurement outcomeyk ∈ {g,e}at time-stepk:

3Measurement operatorO=IS⊗(|eihe| − |gihg|).

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Quantum Non Demolition (QND) measurement of photons

⁴

|Ψi_R

2 =U_R₂U_CU_R₁ |ψi ⊗ |gi

C

B

D R1

R2

B R2

UR1 =I_S⊗_|gi+|ei

√2

hg|+_−|gi+|ei

√2

he| U_C =e⁻ⁱ

φ₀ 2N

⊗ |gihg|+eⁱ

φ₀ 2N

⊗ |eihe| U_R₂ =U_R₁

UR₁ |ψi ⊗ |gi

=^√¹

2(|ψi ⊗ |gi+|ψi ⊗ |ei) UCUR₁ |ψi ⊗ |gi

=^√¹

2

e⁻ⁱ

φ₀ 2N

|ψi

⊗ |gi+

eⁱ

φ₀ 2N

|ψi

⊗ |ei

|Ψi_R

2 =¹₂

e⁻ⁱ

φ₀ 2N

|ψi

⊗(|gi+|ei) +

eⁱ

φ₀ 2N

|ψi

⊗(−|gi+|ei)

=

−isin(^φ₂⁰N)|ψi

⊗ |gi+

cos(^φ₂⁰N)|ψi

⊗ |ei ThusMg=−isin(^φ₂⁰N)andMe=cos(^φ₂⁰N).

Quantum Monte-Carlo simulations with MATLAB:

IdealModelPhotonBoxWaveFunction.m

4M. Brune, . . . : Manipulation of photons in a cavity by dispersive atom-field coupling: quantum non-demolition measurements and generation of "Schrödinger cat"

states . Physical Review A, 45:5193-5214, 1992.

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The Markov ideal model (3)

Just beforeD, the field/atom state isentangled:

M_g|ψi ⊗ |gi+M_e|ψi ⊗ |ei

Denote byµ∈ {g,e}the measurement outcome in detectorD: with probabilityPµ=hψ|M^†_µMµ|ψiwe getµ. Just after the measurement outcomeµ=y,the state becomes separable:

|Ψi^D=√¹

Py

(M_y|ψi)⊗ |yi= q ^M^y

hψ|M^†_yMy|ψi|ψi

!

⊗ |yi. Markov process(density matrix formulationρ∼ |ψihψ|)

ρ+=







MgρM^†_g

Tr(MgρM^†g), with probabilityPg= Tr

MgρM^†_g

;

MeρM^†_e

Tr(MeρM^†_e), with probabilityPe= Tr

MeρM^†_e .

Kraus map:

E

^(ρ⁺^{/ρ) =}^{K(ρ) =}^M^g^ρM^†^g⁺^M^e^ρM^†^e^.

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LKB photon-box: Markov process with detection efficiency

I With pure stateρ=|ψihψ|, we have ρ₊=|ψ₊ihψ₊|= 1

Tr

MµρM^†_µMµρM^†_µ

when the atom collapses inµ=g,ewith proba. Tr

MµρM^†_µ .

I Detection efficiency:the probability to detect the atom is

η∈[0,1]. Three possible outcomes fory:y =gif detection ing, y =eif detection ineandy =0 if no detection.

The only possible update is based onρ: expectationρ+of|ψ+ihψ+| knowingρand the outcomey ∈ {g,e,0}.

ρ+=











MgρM^†_g

Tr(MgρMg) ify=g, probabilityηTr(MgρMg) MeρM^†_e

Tr(MeρMe) ify=e, probabilityηTr(MeρMe)

MgρM^†_g+MeρM^†_e ify=0, probability 1−η

ρ₊does not remain pure: the quantum stateρ₊becomes a mixed state;|ψ₊ibecomes physically irrelevant.

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LKB photon-box: Markov process with detection errors (1)

I With pure stateρ=|ψihψ|, we have ρ₊=|ψ₊ihψ₊|= 1

Tr

MµρM^†_µMµρM^†_µ

when the atom collapses inµ=g,ewith proba. Tr

MµρM^†_µ .

I Detection error rates: P(y =e/µ=g) =η_g∈[0,1]the probability of erroneous assignation toewhen the atom collapses ing;P(y =g/µ=e) =η_e∈[0,1](given by the contrast of the Ramsey fringes).

Bayes law: expectationρ+of|ψ+ihψ+|knowingρand the imperfect detectiony.

ρ₊=











(1−ηg)MgρM^†_g+η_eMeρM^†_e

Tr(^(1−ηg)MgρM^†g+ηeMeρM^†e)^if^y⁼g, prob. Tr

(1−ηg)MgρM^†_g+ηeMeρM^†_e

; η_gMgρM^†_g+(1−ηe)MeρM^†_e

Tr(^ηgMgρM^†_g+(1−ηe)MeρM^†_e)^if^y⁼e, prob. Tr

ηgMgρM^†_g+ (1−ηe)MeρM^†_e .

ρ+does not remain pure: the quantum stateρ+becomes a mixed

state;|ψ+ibecomes physically irrelevant. _{20 / 39}

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LKB photon-box: Markov process with detection errors (2)

We get

ρ₊=











(1−ηg)MgρM^†_g+η_eMeρM^†_e

Tr(^(1−ηg)MgρM^†g+ηeMeρM^†e), with prob. Tr

(1−ηg)MgρM^†_g+ηeMeρM^†_e

; η_gMgρM^†_g+(1−ηe)MeρM^†_e

Tr(^ηgMgρM^†g+(1−ηe)MeρM^†e) with prob. Tr

ηgMgρM^†_g+ (1−ηe)MeρM^†_e .

Key point:

Tr

(1−η_g)M_gρM^†_g+η_eM_eρM^†_e

and Tr

η_gM_gρM^†_g+ (1−η_e)M_eρM^†_e are the probabilities to detecty =g ande, knowingρ.

Generalizationby merging aKraus mapK(ρ) =P

µMµρM^†_µwhere P

µM^†_µMµ=I with aleft stochastic matrix(ηµ⁰,µ):

ρ₊= P

µη_y,µMµρM^†_µ Tr

P

µη_y,µMµρM^†_µ when we detecty =µ⁰. The probability to detecty =µ⁰knowingρis Tr

P

µη_y,µMµρM^†_µ .

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Cavity decay (decoherence) seen as unread measurements The cavity mirrors play the role of a detector with two possible outcomes:

I zero photon annihilation

during

∆T

: Kraus operator

M₀=I−^∆T₂ L^†₋₁L₋₁

, probability

†

−1

Tr

M−1ρtM^†₋₁

where

L₋₁= q 1

Tcava

is the

Lindbald operator associated to cavity damping

(see bellow the continuous time models) with T

cav

the photon life time and

∆T

T

_cav

the sampling period (T

_cav =

100 ms and

∆T ≈

100

µs

for the LKB photon Box).

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Cavity decoherence: cavity decay, thermal photon(s)

Three possible outcomes:

I zero photon annihilationduring∆T: Kraus operator M₀=I−^∆T₂ L^†₋₁L₋₁−^∆T₂ L^†₁L₁, probability≈ Tr

M₀ρ_tM^†₀ with back actionρ_t+∆T ≈ _Tr^M(M⁰^ρ0ρ^t^M_tM^†⁰^†₀).

I one photon annihilationduring∆T: Kraus operator M−1=√

∆TL−1, probability≈ Tr

M−1ρ_tM^†₋₁

with back action ρ_t+∆T ≈ ^M⁻¹^ρ^t^M

†

−1

Tr(^M−1ρtM^†₋₁)

I one photon creationduring∆T: Kraus operatorM1=√

∆TL1, probability≈ Tr

M₁ρ_tM^†₁

with back actionρ_t+∆T ≈ _Tr^M(M¹^ρ1^tρ^M_tM^†¹^†₁) where

L−1= q1+nth

Tcav a, L₁=q

nth

Tcava^†

are theLindbald operators associated to cavity decoherence:T_cav the photon life time,∆T T_cav the sampling period andn_this the average of thermal photon(s) (vanishes with the environment

temperature) (n_th≈0.05 for the LKB photon box). ^{23 / 39}

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Experimental results

⁵

Valeur moyenne du nombre de photons le long d’une longue séquence de mesure:

observation d’une trajectoire stochastique

Une trajectoire correspondant au résultat initial n=5

Sauts quantiques vers le vide dus à l’amortissement du champ Des mesures répétées

confirment n=5

Projection de l’état cohérent sur n=5

nNombre moyen de photons A partir de la probabilité Pi(n) inférée après chaque atome, on déduit le nombre moyen de photons:

Première observation des trajectoires stochastiques du champ, en très bon accord avec les prédictions théoriques (simulations

de Monte- Carlo. Voir cours précédents).

n = nPi(n)

n

! ^(6"10)

See the quantum Monte Carlo simulations of the Matlab script:

RealisticModelPhotonBox.m.

5From Serge Haroche, Collège de France, notes de cours 2007/2008.

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Feedback: Kraus operators depend on the control input

u⁶

u=0: dispersiveinteraction with

Mg(0) =cos

φ0N+φR

2

andMe(0) =sin

φ0N+φR

2

,

u=1: resonantinteraction with atom prepared in|ei

Mg(1) = sin_θ

0 2

√ N

√

N a^†andMe(1) =cos θ₀

2

pN+I

u=−1: resonantinteraction with atom prepared in|gi

Mg(−1) =cos_θ

0 2

√ N

andMe(−1) =−a sin_θ

0 2

√ N

^(ρ^k+1|ρ_k) =K(ρ_k) =X

y

K_y(ρ_k)

Continuous-time modelsarestochastic differential systems dρ_t = −_~ⁱ[H, ρ_t] +X

ν

Lνρ_tL^†_ν−¹₂(L^†_νLνρ_t+ρ_tL^†_νLν)

! dt

+X

ν

√ην

Lνρ_t+ρ_tL^†_ν− Tr

(Lν+L^†_ν)ρ_t ρ_t

dW_ν,t driven byWiener process⁷dW_ν,t =dy_ν,t−√ην Tr

(Lν+L^†_ν)ρ_t dt with measuresyν,t, detection efficienciesην ∈[0,1]and

Lindblad-Kossakowskimaster equations (ην ≡0):

d

dtρ=−_~ⁱ[H, ρ] +X

ν

Lνρ_tL^†_ν−¹₂(L^†_νLνρ_t+ρ_tL^†_νLν)

7and/or Poisson processes, see next slides. 29 / 39

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Ito stochastic calculus

Given a SDE

dX_t =F(X_t,t)dt+X

ν

Gν(X_t,t)dW_ν,t,

we have the following chain rule summarized by the heuristic formulae:

dW_ν,t =O(√

dt), dW_ν,tdWν⁰,t =δ_ν,ν⁰dt. Ito’s ruleDefiningft =f(Xt)aC²function ofX, we have

df_t = ∂f

∂X Xt

F(X_t,t) +1 2

X

ν

∂²f

∂X² Xt

(Gν(X_t,t),Gν(X_t,t))

! dt

+X

ν

∂f

∂X Xt

Gν(Xt,t)dWν,t.

Furthermore

E

d dtf_t

X_t

=

E

_∂X^∂f

Xt

F(X_t,t) +1 2

X

ν

∂²f

∂X² Xt

(Gν(X_t,t),Gν(X_t,t))

! .

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Continuous/discrete-time diffusive SME

With a single imperfect measuredyt=√ ηTr

(L+L^†)ρt

dt+dWtand detection efficiencyη∈[0,1], the quantum stateρtis usually mixed and obeys to

dρt=

−ⁱ

~[H, ρt] +LρtL^†−1 2

L^†Lρt+ρtL^†L dt +√

η

Lρt+ρtL^†− Tr

(L+L^†)ρt

ρt

dWt

driven by the Wiener processdWt(Gaussian law of mean 0 and variancedt).

WithIt¯o rules, it can be written as the following "discrete-time" Markov model

ρt+dt =

Mdy_tρtM^†_dy

t + (1−η)LρtL^†dt

Tr

Mdy_tρtM^†_dy

t+ (1−η)LρtL^†dt

withMdy_t =I+

−ⁱ

~H−¹₂ L^†L

dt+√

ηdytL. The probability to detectdytis given by the following density

P

dyt∈[s,s+ds]

= Tr

MsρtM^†_s+ (1−η)LρtL^†dt

√

2πdt e⁻^s

2 2dt ds

close to a Gaussian law of variancedtand mean√ ηTr

(L+L^†)ρt

dt.

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A key physical example in circuit QED

⁸

Superconducting qubit dispersively coupled to a cavity traversed by a microwave signal (input/output theory).

The back-action on the qubit state of a single measurement of both output fieldquadraturesIt

andQt is described by a simple SME for the qubit density operator.

dρ_t = −2ⁱ[uσx+vσ_y, ρ_t] +γ(σ_zρσ_z−ρ_t) dt +p

ηγ/2 σ_zρ_t+ρ_tσ_z−2 Tr(σ_zρ_t)ρ_t

dW_t^I+ıp

ηγ/2[σz, ρ_t]dW_t^Q withIt andQt given bydI_t =p

ηγ/2 Tr(2σzρ_t)dt+dW_t^I and

dQ_t =dW_t^Q, whereγ≥0 is related to the measurementstrengthand η∈[0,1]is the detection efficiency.uandvare the two control inputs.

8M. Hatridge et al. Quantum Back-Action of an Individual Variable-Strength Measurement. Science, 2013, 339, 178-181.

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Qubit, density matrix and Bloch sphere

With

|ψi=ψg|

g

i+ψe|

e

i

satisfying

ı~_dt^d|ψi=H|ψi

, the density operator corresponds to

ρ=|ψihψ|

. Then

ρ

is non negative Hermitian operator such that Tr

(ρ) =

1 and obeying to

d

dt

ρ=−ı

~[H, ρ].

For mixed states,

ρ= ^I^+xσ^x^+yσ₂ ^y^+zσ^z

ρ²

=

x

²+

y

²+

z

²≤

1):

d dt

M

~ = (u~ı+

v~

)×

M.

~

is another formulation of

_dt^dρ=−2ⁱ[uσx+

v

σy, ρ]. Here

u stands for the

rotation speed

around x -axis and v the rotation speed around y -axis.

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(34)

QND measurement: asymptotic behavior in open-loop

Almost such convergence:Consider the SME

dρ_t = −2ⁱ[uσ_x+vσ_y, ρ_t] +γ(σ_zρσ_z−ρ_t) dt +p

dW_t^I+ıp

ηγ/2[σz, ρ_t]dW_t^Q

withu=v =0 andη >0.

I For any initial stateρ₀, the solutionρ_t converges almost surely ast → ∞to one of the states|gihg|or|eihe|.

Proof based on the martingalesVg(ρ) = Tr(|gihg|ρ) = (1−z)/2 and Ve(ρ) = Tr(|eihe|ρ) = (1+z)/2, and on the sub-martingale

V(ρ) = Tr²(σ_zρ) =z²:

E

^dV^g|ρ_t

=

E

^(dV^e|ρ_t) =0,

E

^(dV|ρ_t) =2ηγ 1−z²2

dt ≥0.

Confirmed by the quantum Monte Carlo simulations:

IdealModelQubit.m

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Adding decoherence due to finite life-time of

|ei

d

ρt = −2ⁱ[uσx+

v

σy, ρt] +γ(σzρσz −ρt)

dt

+p

ηγ/2 σzρt+ρtσz−

2 Tr

(σzρt)ρt

dW_t^I+ıp

ηγ/2[σz, ρt]dW_t^Q +

LeρtL^†_e− ¹₂

L^†_eLeρt +ρtL^†_eLe

.

Confirmed by the quantum Monte Carlo simulations:

RealisticModelQubit.m

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Feedback stabilization of the excited state

dρ_t = −2ⁱ[uσx+vσ_y, ρ_t] +γ(σ_zρσ_z−ρ_t) dt +p

dW_t^I+ıp

ηγ/2[σz, ρ_t]dW_t^Q

Withuandv arbitrary, we have forV(ρ) = Tr(σ_zρ) =z,

E

^(dV^t^|^ρ^t^{) =}^u ^Tr^(σ^y^ρ^t⁾⁻^v ^Tr^(σ^x^ρ^t^{) =}^uy⁻^vx.

With the quantum state feedback

u= sign(y)

T (1−V_t), v =−sign(x)

T (1−V_t)

we get in closed loop

E

^(dV^t^|^ρ^t^{) =} ^|x|+|y|T (1−V_t)and thusV tends to converge towards 1, i.e.,z tends to converge towards 1. Confirmed by the closed-loop simulations:IdealFeedbackQubit.m

Robustness of such feedback illustrated by the more realistic simulations:

RealisticFeedbackQubit.m

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(37)

Continuous/discrete-time jump SME

With Poisson processN(t),hdN(t)i=

θ+ηTr Vρ_tV^† dt,and detection imperfections modeled byθ≥0 andη∈[0,1], the quantum stateρ_t is usually mixed and obeys to

dρ_t =

−i[H, ρ_t] +Vρ_tV^†−¹₂(V^†Vρ_t+ρ_tV^†V) dt +

θρ_t+ηVρ_tV^† θ+ηTr(Vρ_tV^†)−ρ_t

dN(t)−

θ+ηTr Vρ_tV^† dt

ForN(t+dt)−N(t) =1we haveρ_t+dt= θρ_t+ηVρ_tV^† θ+ηTr(Vρ_tV^†). FordN(t) =0we have

ρ_t+dt= M₀ρ_tM₀^†+ (1−η)Vρ_tV^†dt Tr

M0ρ_tM₀^†+ (1−η)Vρ_tV^†dt withM₀=I+ −iH+¹₂ η Tr Vρ_tV^†

I−V^†V dt.

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Continuous/discrete-time diffusive-jump SME

The quantum stateρtis usually mixed and obeys to dρt =

−i[H, ρt] +LρtL^†−¹

2(L^†Lρt +ρtL^†L) +VρtV^†−¹

2(V^†Vρt+ρtV^†V)

dt +√

η

Lρt+ρtL^†− Tr

(L+L^†)ρt

ρt

dWt

+

θρt+ηVρtV^† θ+ηTr(VρtV^†)−ρt

dN(t)−

θ+ηTr

VρtV^† dt

ForN(t+dt)−N(t) =1we haveρt+dt= θρt+ηVρtV^† θ+ηTr(VρtV^†). FordN(t) =0we have

ρt+dt = Mdy_tρtM_dy^†

t+ (1−η)LρtL^†dt+ (1−η)VρtV^†dt

Tr

Mdy_tρtM_dy^†

t + (1−η)LρtL^†dt+ (1−η)VρtV^†dt

withMdy_t =I+ −iH−¹₂L^†L+¹₂ ηTr VρtV^†

I−V^†V dt+√

ηdytL.

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Modelling, simulation and feedback of open quantum systems