Control of open quantum systems by reservoir engineering

Control of open quantum systems by reservoir engineering

Pierre Rouchon Mines-ParisTech,

Centre Automatique et Systèmes Mathématiques et Systèmes

pierre.rouchon@mines-paristech.fr

CIRM, November 2012

based on collaborations with H. Amini,M. Brune, C. Le Bris, I. Dotsenko, Serge Haroche, Z. Leghtas, M. Mirrahimi,

JM. Raimond,A.Sarlette,C. Sayrin, R. Sepulchre, R. Somaraju.

Outline

Dynamics of open quantum systems Three quantum features

The LKB photon box: a discrete time open quantum system Discrete time models: Markov chains and Kraus maps Continuous-time models: SDE and Lindblad ODE Two kind of feedback for quantum systems

Measurement-based feedback

Coherent (autonomous) feedback with reservoir engineering Reservoir engineering for discrete-time systems

A general setting

Stabilizing Schrödinger cats in the LKB photon box Reservoir engineering for continuous time systems Appendix: two fundamental quantum systems

A qubit: 2 level system Harmonic oscillator

Three quantum features¹

1. Schrödinger equation: wave function|ψi ∈ H, density operatorρ d

dt|ψi=−iH|ψi, d

dtρ=−i[H, ρ]

2. Origin of dissipation and irreversibility: collapse of the wave packetinduced by the measure of observableOwith spectral decompositionP

µλµPµ:

I measure outcomeλ_µ with proba.pµ=hψ|Pµ|ψi= Tr(ρPµ) depending|ψi,ρjust before the measurement

I measure back-action if outcomeλµ:

|ψi 7→ |ψi⁺= Pµ|ψi

phψ|Pµ|ψi, ρ7→ρ+= PµρPµ

Tr(ρPµ) 3. Tensor product for the description of composite systems(S,M):

I Hilbert spaceH=H^S⊗ H^M

I HamiltonianH=HS⊗IM+Hint+IS⊗HM I observable on sub-systemM only:O=IS⊗ O^M.

1S. Haroche and J.M. Raimond.Exploring the Quantum: Atoms, Cavities and Photons.Oxford Graduate Texts, 2006.

LKB photon Box: HScavity,HM flying atom.

2

The experiment measuring and controlling the photons trapped into the cavity C (Cavity Quantum Electro-Dynamics group at Laboratoire

Kastler-Brossel, ENS de Paris).

2Courtesy of Igor Dotsenko

The LKB Photon-Box: measuring photons with atoms C

B

D R₁

R₂

Atoms get out of boxBone by one, undergo then a first Rabi pulse in Ramsey zoneR₁, become entangled with

electromagnetic field trapped inC, undergo a second Rabi pulse in Ramsey zoneR₂and finally are measured in the detectorD.

The Markov chain model (1)

I SystemS corresponds to a quantized mode inC:

HS = (_∞

X

n=0

ψ_n|ni |(ψ_n)^∞_n=0∈l²(C) )

,

where|nirepresents the Fock state associated to exactlyn photons inside the cavity

I MeterMis associated to atoms : HM =C², each atom admits two-level and is described by a wave function c_g|gi+c_e|eiwith|c_g|²+|c_e|²=1; atoms leavingBare all in state|gi

I When atom comes outB, the state|ΨiB ∈ HS⊗ HM of the composite system atom/field isseparable

|ΨiB =|ψi ⊗ |gi.

The Markov chain model (2)

C

B

D R₁

R₂

I When atom comes outB:|Ψi^B =|ψi ⊗ |gi.

I When atom comes out the first Ramsey zoneR1the state remains separable but has changed to

|ΨiR1 = (I⊗U_R1)|ΨiB=|ψi ⊗(U_R1|gi)

where the unitary transformation performed inR1only affects the atom:

U_R1=e⁻ⁱ

θ₁

2(x1σx+y1σy+z1σz)=cos(^θ₂¹)−isin(^θ₂¹)(x₁σ_x+y₁σ_y+z₁σ_z) corresponds, in the Bloch sphere representation, to a rotation of angleθ₁aroundx1~ı+y1~+z1~k (x₁²+y₁²+z₁²=1)

The Markov chain model (3)

C

B

D R₁

R₂

I When atom comes out the first Ramsey zoneR1:

|Ψi^R1 =|ψi ⊗(UR1|gi).

I When atom comes out cavityC, the state does not remain separable: atom and field becomes entangled and the state is described by

|Ψi^C=UC|Ψi^R1

where the unitary transformationU_ConHS⊗ H^M is associated to a Jaynes-Cumming Hamiltonian:

H_C= ^∆(t)₂ σ_z+i^Ω(t)₂ (a^†σ₋−aσ+)Parameters:∆(t) =ω_eg−ω_c, Ω(t)depend on timet.

The Markov chain model (4)

C

B

D R₁

R₂

I When atom comes out cavityC:|ΨiC=U_C |ψi ⊗(U_R1|gi) .

I When atom comes out second Ramsey zoneR₂, the state becomes|ΨiR2 = (I⊗U_R2)|ΨiCwithU_R2 =e^{−iθ22(x2σx+y2σy+z2σz)}.

I Just before the measurement inD, the state is given by

|ΨiR2 =U_SM |ψi ⊗ |gi

= M_g|ψi

⊗ |gi+ M_e|ψi

⊗ |ei whereUSM=UR2UCUR1 is the total unitary transformation defining the linear measurement operatorsMg andMeonH^S. SinceU_SMis unitary,M_g^†M_g+M_e^†M_e=I.

The Markov chain model (5)

Just before the measurement inD, the atom/field state is:

|gi ⊗M_g|ψi+|ei ⊗M_e|ψi

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

|Ψi^D= ^√1_p

µ |µi ⊗(Mµ|ψi) = |µi ⊗(Mµ|ψi) q

hψ|Mµ^†Mµ|ψi .

Markov process(density matrix formulation)

ρ+=







M^g(ρ) = ^MgρM

† g

Tr(MgρM_g^†), with probabilitypg = Tr

MgρM_g^†

; Me(ρ) = ^MeρM

† e

Tr(MeρMe^†), with probabilityp_e= Tr

M_eρM_e^† .

Positive Operator Valued Measurement (POVM) (1)

SystemSof interest (aquantized electromagnetic field) interacts with the meterM (a probe atom), and the

experimentermeasures projectively the meterM (theprobe atom). Need for acomposite system:HS⊗ HM whereHS

andHM are the Hilbert space ofSandM.

Measurement process in three successive steps:

1. Initially the quantum state isseparable HS⊗ HM 3 |Ψi=|ψ_Si ⊗ |θMi with a well defined and known state|θ_MiforM.

2. Then aSchrödinger evolutionduring a small time (unitary operatorU_S,M) of the composite system from|ψ_Si ⊗ |θMi and producingU_S,M |ψ_Si ⊗ |θ_Mi

,entangledin general.

3. Finally aprojective measurementof the meterM:

OM =IS⊗ P

µλµPµ

the measured observable for the meter. Projection operatorP_µis a rank-1 projection inHM

over the eigenstate|λµi ∈ HM:P_µ=|λµihλµ|.

Positive Operator Valued Measurement (POVM) (2) Define themeasurement operatorsMµvia

∀|ψ_Si ∈ HS, U_S,M |ψ_Si ⊗ |θ_Mi

=X

µ

Mµ|ψ_Si

⊗ |λµi. ThenP

µM_µ^†M_µ=IS. The set{M_µ}defines aPositive Operator Valued Measurement(POVM).

InHS⊗ HM, projective measurement ofOM =IS⊗ P

µλ_µP_µ with quantum stateU_S,M |ψ_Si ⊗ |θ_Mi

:

1. The probability of obtaining the valueλ_µis given by pµ=hψ_S|Mµ^†Mµ|ψ_Si

2. After the measurement, the conditional (a posteriori) state of the system, given the outcomeλ_µ, is

|ψ_Si+= M_µ|ψ_Si

√p_µ . Formixed state ρ(instead of pure state|ψ_Si):

p_µ= Tr

M_µρM_µ^†

andρ₊ = ^MµρM

† µ

Tr MµρMµ^†

,

Markov chain et Kraus map

I To the POVM onHS is attached a stochastic process of quantum stateρ,ρ^†=ρ≥0, Tr(ρ) =1 P

µM_µ^†M_µ=I ρ₊= M_µρMµ^†

Tr

M_µρMµ^†

with probabilityp_µ= Tr

M_µρM_µ^† .

I For any observableAonHS, itsconditional expectation value after the transition knowing the stateρ

E

where the linear mapρ7→K(ρ) =P

µM_µρMµ^† is aKraus mapdefining a quantum channel.

I IfA¯ is astationary point of the adjoint Kraus mapK^∗, K^∗(¯A) =P

µMµ^†AM¯ µ, then Tr Aρ¯

is amartingale:

E

Models of open quantum systems

Discrete-time modelsareMarkov chains ρ_k+1= 1

pµ(ρ_k)Mµρ_kM_µ^† with proba. pµ(ρ_k) = Tr Mµρ_kM_µ^† with measureµand associated toKraus maps(ensemble average, open quantum channels)

E

µ

Mµρ_kM_µ^† with X

µ

M_µ^†Mµ=I

Continuous-time modelsarestochastic differential systems dρ=

−i[H, ρ] +LρL^†−¹₂(L^†Lρ+ρL^†L) dt +

Lρ+ρL^†− Tr (L+L^†)ρ ρ

dw driven byWiener processes³dw =dy− Tr (L+L^†)ρ

dt with measurey and associated toLindbaldmaster equations:

d

dtρ=−_~ⁱ[H, ρ] +LρL^†−¹₂(L^†Lρ+ρL^†L)

3Another common possibility not considered here: SDE driven by Poisson processes.

From discrete-time to continuous-time: heuristic connection For Monte-Carlo simulations of

dρ=

−i[H, ρ] +LρL^†−¹₂(L^†Lρ+ρL^†L)

dt +

Lρ+ρL^†− Tr

(L+L^†)ρ ρ

dw

take a small sampling timedt, generate a random numberdw_t according to a Gaussian law ofstandard deviation√

dt, and set ρ_t+dt =Mdyt(ρt)where the jump operatorMdyt is labelled by the measurement outcomedy_t = Tr (L+L^†)ρ_t

dt+dw_t: Mdyt(ρt) = ^I+(−iH−

1

2L^†L)dt+Ldyt

ρt I+(iH−¹

2L^†L)dt+L^†dyt

Tr

I+(−iH−¹₂L^†L)dt+Ldy_t

ρt I+(iH−¹₂L^†L)dt+L^†dy_t. Thenρ_t+dt remains always a density operator and using the Ito rules (dw of order√

dt anddw²≡dt) we get the good dρ=ρt+dt −ρt up toO((dt)^3/2)terms.

From discrete-time to continuous-time: heuristic connection (end)

For the Lindblad equation d

dtρ=−_~ⁱ[H, ρ] +LρL^†−¹₂(L^†Lρ+ρL^†L) take a small sampling timedtand set

ρ_t+dt = ^I+(−iH−

1 2L^†L)dt

ρt I+(iH−¹₂L^†L)dt

+dtLρ_tL^† Tr

I+(−iH−¹₂L^†L)dt

ρt I+(iH−¹₂L^†L)dt

+dtLρtL^†. Thenρt+dt remains always a density operator and

d

dtρ= (ρ_t+dt−ρt)/dt up toO(dt)terms.

Measurement-based stabilization of photon-number state⁴

•Control inputu=Ae^ıΦ; measure outputy ∈ {g,e}.

•Sampling time 80µs long enough for numerical computations inK.

4C. Sayrin, I. Dotsenko, X. Zhou, B. Peaudecerf, Th. Rybarczyk, S.

Gleyzes, P. R., M. Mirrahimi, H. Amini, M. Brune, J.M. Raimond, S. Haroche:

Real-time quantum feedback prepares and stabilizes photon number states.

Nature, 477(7362),2011.

Watt regulator: a classical analogue of quantum coherent feedback. ⁵ 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 param- eters.

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 stabilityof the closed-loop system.

5J.C. Maxwell: On governors. Proc. of the Royal Society, No.100, 1868.

Reservoir Engineering⁶and coherent feedback⁷

System Reservoir

Engineered interaction dissipation

κ

H_int

H_sys H_res

H =H_res+H_int+H_syst ifρ →

t→∞|ψ¯ihψ¯| ⊗ρ_res exponentially on a time scale ofτ ≈κthen . . . .

6Introduced by Poyatos, Cirac and Zoller, 1996.

7See, e.g., the lectures of H. Mabuchi delivered at the "école de physique des Houches", July 2011.

Reservoir Engineering⁶and coherent feedback⁷

System Reservoir

Engineered interaction dissipation

κ

H_int

H_sys H_res

γ

H =H_res+H_int+H_syst . . . ρ →

t→∞|ψ¯ihψ¯| ⊗ρ_res+∆, ifκγ thenk∆k 1

6Introduced by Poyatos, Cirac and Zoller, 1996.

7See, e.g., the lectures of H. Mabuchi delivered at the "école de physique des Houches", July 2011.

Reservoir engineering (coherent feedback) v.s. measurement-based feedback

Advantages over measurement-based feedback

I Does not require knowing the measurement result.

I No external intervention on small time scale.

Difficulty

I For each target state|ψ¯i, engineer a coupling to the reservoir which drivesρtoρres⊗ |ψ¯ihψ¯|, compatible with lab constraints.

Reservoir engineering for discrete-time systems

Data:HSwith HamiltonianH_S, a pure goal stateρ¯_S=|ψ¯_Sihψ¯_S|. Find a "realistic" meter systemof Hilbert spaceHM with initial state

|θ_Mi, with HamiltonianH_Mand interaction HamiltonianH_int such that 1. the propagatorUS,M =U(T)between 0 and timeT

(_dt^dU =−i(HS+HM+Hint)U,U(0) =I) reads:

∀|ψ_Si ∈ HS, U_S,M |ψ_Si ⊗ |θ_Mi

=X

µ

Mµ|ψ_Si

⊗ |λµi

where|λ_µiis a ortho-normal basis ofH^M.

2. the resulting measurement operatorsMµadmit|ψ¯_Sias common eigen-vector, i.e.,ρ¯_S is a fixed point of the Kraus map

K(ρ) =P

µMµρM_µ^†: K( ¯ρ_S) = ¯ρ_S.

3. iterates ofKconverge toρ¯_Sfor any initial conditionρ₀:

k7→+∞lim ρ_k = ¯ρ_Swhereρ_k =K(ρk−1).

Here the reservoir is made of the infinite set of identical meter systems with initial state|θ_Miatt = (k −1)T and interacting withHS

during[(k−1)T,kT],k =1,2, . . ..

Reservoir stabilizing "Schrödinger cats"⁸

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

HereH^S=H^c ={P

n≥0ψ_n|ni, (ψ_n)_n≥0∈l²(C)}and H^M =H^q={cg|gi+ce|ei, cg,ce∈C}.

HS+HM+Hint is the Jaynes-Cumming Hamiltionian H(t) =ω_ca^†a+^δ(t)₂ σ_z+i^Ω(vt)₂ (a^†|gihe| −a|eihg|)

which is time varying with controlδ(t) =ω_q(t)−ω_cand Gaussian radial profileΩ(x) = Ω₀e^−x

2

w2,x =vtwithv atom velocity.

8A. Sarlette, Z. Leghtas, M. Brune, J.M. Raimond, P.R.: Stabilization of nonclassical states of one and two-mode radiation fields by reservoir engineering. Phys. Rev. A 86, 012114 (2012)

Composite interaction

ge

time t t_r/2

-t_r/2

-T/2 T/2

δ(t)δ₀

-δ₀ 0

0

top cavity mirror

bottom cavity mirror Ω(vt)

U =Uoff-resonant

Composite interaction

ge

time t t_r/2

-t_r/2

-T/2 T/2

δ(t)δ₀

-δ₀ 0

0

top cavity mirror

bottom cavity mirror Ω(vt)

U =UresonantUoff-resonant

Composite interaction

ge

time t t_r/2

-t_r/2

-T/2 T/2

δ(t)δ₀

-δ₀ 0

0

top cavity mirror

bottom cavity mirror Ω(vt)

U =Uoff-resonant^† UresonantUoff-resonant

Composite interaction

ge

time t 0 t_r/2

-t_r/2

-T/2 T/2

δ(t)δ₀

-δ₀ 0

top cavity mirror

bottom cavity mirror Ω(vt)

U ≈e^−iφKerrn2Uresonante^iφKerrn2

Simulation: convergence toward the cat from vacuumρ₀=|0ih0| Simulation: convergence after a cat jump

Reservoir engineering for continuous time systems

Data:HSwith HamiltonianH_S, a pure goal stateρ¯_S=|ψ¯_Sihψ¯_S|. Find a "realistic" controllerof Hilbert spaceHM, with Hamiltonian H_M and interaction HamiltonianH_int and Lindblad operatorsLµ acting only onHM such that

1. the Lindblad master equation of the composite systemHS⊗ H^M governing the density operatorρevolution

d dtρ=−i

HS+HM+Hint, ρ

+X

µ

LµρL^†_µ−¹2L^†_µLµρ−¹2ρL^†_µLµ

admits a separable steady stateρ¯= ¯ρ_S⊗ρ¯_Mfor some density operatorρ¯_MonHM.

2. For any initial conditionρ(0), limt7→+∞ρ(t) = ¯ρ.

Example:cavity cooling towardsρ¯_S =|0ih0|with a qubit-controller H_S=ω_ca^†a, H_M =^δ₂(|eihe| − |gihg|), H_int =^Ω₂(a^†|gihe|+a|eihg|) when|eiis unstable of life-timeTq: L=q

1

Tq|gihe|,ρ¯=|0ih0| ⊗ |gihg| and_dt^d Tr( ¯ρρ) = Tr(|0ih0|⊗|eihe|ρ)

Tq ≥0 as Lyapunov function .

A qubit: 2 level system

I State space: Hq={c_g|gi+c_e|ei, c_g,c_e ∈C}.

I Operators: σz =|eihe| − |gihg|,σx =|eihg|+|gihe|, σ_y =−i|eihg|+i|gihe|.

I Hamiltonian: H_q=ω_qσ_z/2+u_qσ_x.

| g

| e ω _q

u _q

A cavity: quantum harmonic oscillator

I State space:H^c={P

n≥0ψ_n|ni, (ψ_n)_n≥0∈l²(C)}.

I D={ρ∈ L(H^c), ρ^†=ρ, Tr(ρ) =1, ρ≥0}.

I Operators:

a|ni=√

n|n−1i, a^†|ni=√

n+1|n+1i, n|ni=a^†a|ni=n|ni, Dα=e^{αa†−α†a}.

I Hamiltonian:H_c=ω_ca^†a+uc(a+a^†).

I Coherent state of amplitudeα∈C:

|αi=P

n≥0

e^{−|α|2/2√αn}

n!

|ni.

I a|αi=α|αi.

I Dα|0i=|αi.

|0

|1

|2 ω_c

|n ω_c u_c

... ...

Cavity state representation: the Wigner function

W_ρ:C3ξ→ 2 π Tr

e^iπa†aD_−ξρD_ξ

∈R

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

Harmonic oscillator⁹(1): quantization and correspondence principle Classical Hamiltonian formulation of _dt^d2₂x =−ω²x

d

dtx =ωp= ∂H

∂p, d

dtp=−ωx =−∂H

∂x, H= ω

2(p²+x²).

Quantization: probability wave function|ψit ∼(ψ(x,t))_x∈Rwith

|ψi^t ∼ψ(,t)∈L²(R,C)obeys to the Schrödinger equation (~=1 in all the lectures)

id

dt|ψi=H|ψi, H =ω(P²+X²) =−ω 2

∂²

∂x² +ω 2x² whereHresults fromHby replacingx by position operator

√2X andpby impulsion operator√

2P=−i_∂x^∂ .

PDE model:i^∂ψ_∂t(x,t) =−^ω₂^∂_∂x^2ψ2(x,t) +^ω₂x²ψ(x,t), x ∈R.

9Two references: C. Cohen-Tannoudji, B. Diu, and F. Laloë.Mécanique Quantique, volume I& II. Hermann, Paris, 1977.

M. Barnett and P. M. Radmore.Methods in Theoretical Quantum Optics.

Oxford University Press, 2003.

Harmonic oscillator (2): annihilation and creation operators Averaged position_hXi^t =hψ|X|ψiand impulsion_hPi^t=hψ|P|ψi

10:

hXit = ^√1

2

Z +∞

−∞

x|ψ|²dx, , hPit =−^√i₂ Z +∞

−∞

ψ^∗∂ψ

∂xdx. Annihilationaandcreationoperatorsa^†:

a=X +iP= ^√1

2

x+ ∂

∂x

, a^†=X −iP = ^√1

2

x − ∂

∂x

Commutation relationships:

[X,P] = ₂ⁱ, [a,a^†] =1, H =ω(P²+X²) =ω

a^†a+¹

2

. SetX_λ= ¹₂ e^−iλa+e^iλa^†

for any angleλ:

h

X_λ,X_λ+π 2

i

= ₂ⁱ.

10We assume everywhere that for eacht,x7→ψ(x,t)is of the Schwartz class (fast decay at infinity + smooth).

Harmonic oscillator (3): spectral decomposition and Fock states [a,a^†] =1 and Ker(a)of dimension one imply that thespectrum ofN=a^†aisnon-degenerateand isN. More we have the useful commutations for any entire functionf:

a f(N) =f(N+I)a, f(N)a^†=a^†f(N+I).

Fock statewithnphoton(s): the eigen-state ofNassociated to the eigen-valuen:

N|ni=n|ni, a|ni=√

n|n−1i, a^†|ni=√

n+1|n+1i. Theground state|0i(0 photon state or vacuum state) satisfies a|0i=0 and corresponds to theGaussian function:

|0i ∼ψ₀(x) = 1

π^1/4exp(−x²/2).

The operatora(resp. a^†) is the annihilation (resp. creation) operator since it transfers|nito|n−1i(resp. |n+1i) and thus decreases (resp. increases) the quantum numbernby one unit.

Harmonic oscillator (4): displacement operator Quantization of _dt^d22x =−ω²x−ω√

2u H=ω

a^†a+¹

2

+u(a+a^†).

The associated controlled PDE i∂ψ

∂t (x,t) =−ω 2

∂²ψ

∂x²(x,t) +

ω

2x²+√ 2ux

ψ(x,t).

Glauberdisplacement operatorD_α (unitary) withα ∈C: Dα=e^{αa†−α∗a}=e^{2i=αX−2ı<αP}

FromBaker-Campbell Hausdorf formulavalid for any operators AandB,

e^ABe^−A=B+ [A,B] +_2!¹[A,[A,B]] +_3!¹[A,[A,[A,B]]] +. . . we get theGlauber formulawhen[A,[A,B]] = [B,[A,B]] =0:

e^A+B =e^Ae^B e^−12[A,B].

(show thatCt =e^t(A+B)−e^tAe^tBe⁻

t² 2[A,B]

satisfies_dt^dC= (A+B)C)

Harmonic oscillator (5): identities resulting from Glauber formula WithA=αa^† andB=−α^∗a, Glauber formula gives:

Dα=e⁻

|α|²

2 e^αa†e^−α∗a=e⁺

|α|²

2 e^−α∗ae^αa† D−αaDα=a+α and D−αa^†Dα=a^†+α^∗. WithA=2i=αX ∼i√

2=αx andB=−2ı<αP∼ −√

2<α_∂x^∂ , Glauber formula gives¹¹:

Dα=e^−i<α=αeⁱ

√2=αx

e⁻

√2<α^∂_∂x

(Dα|ψi)_x,t =e^−i<α=αeⁱ

√2=αxψ(x−√ 2<α,t)

Exercice

For anyα, β, ∈C, prove that Dα+β =e

α^∗β−αβ^∗ 2 DαDβ

Dα+D−α=

1+^α∗−α₂ ^∗

I+a^†−^∗a+O(||²)

d

dtDα

D−α=

α_dt^dα^∗−α^∗_dt^dα 2

I+

d

dtα

a^†−

d

dtα^∗

a.

11Remember that a time-delay ofrcorresponds to the operatore^−rdtd.

Harmonic oscillator (6): lack of controllability

Take|ψisolution of thecontrolled Schrödinger equation i_dt^d|ψi= ω a^†a+¹₂

+u(a+a^†)

|ψi. Sethai=hψ|aψi. Then d

dthai=−iωhai −iu.

Froma=X+iP, we havehai=hXi+ihPiwherehXi=hψ|X|ψi ∈R andhPi=hψ|P|ψi ∈R. Consequently:

d

dthXi=ωhPi, d

dthPi=−ωhXi −u.

Consider thechange of frame|ψi=e^−iθtD_hait |χiwith θ_t =

Z t

0 |hai|²+u<(hai)

, D_hait =e^{haita†−hai∗ta},

Then|χiobeys toautonomous Schrödinger equation i d

dt|χi=ωa^†a|χi.

The dynamics of|ψican be decomposed into two parts:

I acontrollable part of dimension twoforhai

I an uncontrollable part of infinite dimension for|χi.

Harmonic oscillator (7): coherent states as reachable ones from|0i Coherent states

|αi=D_α|0i=e⁻

|α|² 2

+∞

X

n=0 αⁿ

√

n!|ni, α∈C

are the states reachable from vacuum set. They are also the eigen-stateofa: a|αi=α|αi.

A widely known result in quantum optics¹²: classical currents and sources (generalizing the role played byu) only generate classical light (quasi-classical statesof the quantized field generalizing the coherent state introduced here)

We just propose here a control theoretic interpretation in terms of reachable set from vacuum¹³

12See complementBIII, page 217 of C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg.Photons and Atoms: Introduction to Quantum

Electrodynamics.Wiley, 1989.

13see also: MM-PR, IEEE Trans. Automatic Control, 2004 and MM-PR, CDC-ECC, 2005.