Singular perturbations and Lindblad-Kossakowski differential equations

Singular perturbations and Lindblad-Kossakowski differential

equations

Pierre Rouchon

In collaboration withMazyar Mirrahimi

Mines ParisTech Centre Automatique et Systèmes

Mathématiques et Systèmes [email protected] http://cas.ensmp.fr/~rouchon/index.html

IMA, Minneapolis March 2-6, 2009

Outline

The main result onΛ-systems

Optical pumping and coherence population trapping

Extension to V-systems

Proof of the main result

Concluding remarks

Open quantum systems

TheLindbald-Kossakowskiequation d

dtρ=−ı

~[H, ρ] +

N

X

k=1

1 2

2LkρL^†_k−L^†_kLkρ−ρL^†_kLk

is the master equation associated to an ensemble average of quantum trajectories (stochastic jump dynamics of a single quantum system where the "environment is watching"¹).

Contribution:when the Lindblad operatorsL_k are associated to highly unstable excited states, we propose a systematic method to eliminate the resulting fast and asymptotically stable dynamics. The obtained slow dynamics

d

dtρ_s=−ı

~[H_s, ρ_s] +

N

X

k=1

1 2

2L_s,kρ_sL^†_s,k −L^†_s,kL_s,kρ_s−ρ_sL^†_s,kL_s,k

isstill of Lindbald-Kossakowskiform ((ρs,Hs,Ls,k) =fnct(ρ,H,Lk)).

1H.-P. Breuer and F. Petruccione.The Theory of Open Quantum Systems.

Clarendon-Press, Oxford, 2006. S. Haroche and J.M. Raimond.Exploring the Quantum: Atoms, Cavities and Photons.Oxford University Press, 2006.

Prototype of open quantum system: Λ-systems.

1 2

3

N stable states |g_ki, k =1, . . . ,N.

Unstable state|ei

Quasi resonant laser transition,|g_ki|ei with de-tuning δ_k and Rabi pulsationsΩk ∈C. Spontaneous emission rate|ei 7→ |g_ki:Γ_k. Lindbald-Kossakowski master-equation for the density matrixρ

d

dtρ=−ı

~[H, ρ] +

N

X

k=1

1

2(2L_kρL^†_k −L^†_kL_kρ−ρL^†_kL_k)

H

~ =PN

k=1δ_k|g_ki hg_k|+ Ω_k|g_ki he|+ Ω^∗_k|ei hg_k|, L_k =√

Γ_k|g_ki he|. Photon flux (measure):y =PN k=1tr

L^†_kL_kρ . Two time-scales:|δ_k|,|Ω_k| Γk.

Main result: adiabatic elimination of the unstable state|ei² Theslow/fastdynamics

d

dtρ=−ı

~[H, ρ] +

N

X

k=1

1 2

2L_kρL^†_k−L^†_kL_kρ−ρL^†_kL_k

withL_k =√

Γ_k|g_ki he|,Γ = (P

kΓ_k)much larger than ^H

~, y =P

ktr L^†_kL_kρ

, is approximated by theslowdynamics d

dtρs=−ı

~[Hs, ρs]+

N

X

k=1

1 2

2L_s,kρsL^†_s,k−L^†_s,kL_s,kρs−ρsL^†_s,kL_s,k

withρs= (1−P)ρ(1−P)theslow density operator, H_s = (1−P)H(1−P)theslow Hamiltonianand L_s,k =2^L_Γ^kH

~(1−P)theslow jump operators(P=|ei he|). The slow approximation ofy is still given by the standard formula

ys=

n

X

k=1

tr

L^†_s,kL_s,kρs

.

2See, Mirrahimi-R, CDC 2006 and IEEE AC to appear in 2009.

Application to the 3-level system (coherence population trapping³)

Input:Ω₁,Ω₂∈Cand _dt^d∆

Output: photo-detector click times corresponding to jumps from

|eito|g₁ior|g₂i.

Two time-scales:|Ω₁|,|Ω₂|,|∆_e|,|∆| Γ₁,Γ₂

3See, e.g., Arimondo:Progr. Optics, 35:257, 1996.

The slow/fast master equation

Master equation of theΛ-system d

dtρ=−ı

~[H, ρ] + 1 2

2

X

k=1

(2L_kρL^†_k −L^†_kL_kρ−ρL^†_kL_k),

with jump operatorsL_k =√

Γk|g_ki he|and Hamiltonian H

~ = ∆

2(|g₂i hg₂| − |g₁i hg₁|)−

∆e+∆ 2

|ei he|

+ Ω1|g₁i he|+ Ω^∗₁|ei hg₁|+ Ω2|g₂i he|+ Ω^∗₂|ei hg₂|. Since|Ω₁|,|Ω₂|,|∆_e|,|∆| Γ₁,Γ₂we havetwo time-scales: a fast exponential decay for "|ei"and aslow evolution for

"(|g₁i,|g₂i)".

The slow master equation with bright and dark states.

The above general result leads to areduced master equation that is still of Lindblad type with aslow HamiltonianH_sandslow jump operatorsL_s,k:

d

dtρs=−ı

~[H_s, ρs] +1 2

2

X

k=1

(2L_s,kρsL^†_s,k−L^†_s,kL_s,kρs−ρsL^†_s,kL_s,k),

withH_s = ^∆₂σz = ^{∆(|g2ihg2|−|g}₂ ^1ihg1|),L_s,k =√

γk|g_ki hb_Ω|where γ_k =4^|Ω_(Γ^1|2+|Ω2|2

1+Γ2)² Γ_k and|b_Ωiis thebright state:

|b_Ωi= Ω1

p|Ω₁|²+|Ω₂|²|g₁i+ Ω2

p|Ω₁|²+|Ω₂|²|g₂i For∆ =0,ρ_s converges towards thedark state|d_Ωi:

|d_Ωi=− Ω^∗₂

p|Ω₁|²+|Ω₂|²|g₁i+ Ω^∗₁

p|Ω₁|²+|Ω₂|²|g₂i.

Extension to V-systems: Dehmelt’s electron shelving scheme⁴

A stable state|g₁i. Aquasi-stable state:

|g₂iwith a long life time 1/γ.

Anunstable state:|eiwith a short life time 1/Γ.

Quasi resonant transitions:

I |g₁i|eiwith de-tuning∆and Rabi pulsationΩ∈C.

I |g₁i|g₂iwith de-tuningδand Rabi pulsationω∈C.

Lindbald-Kossakowski master-equation for the density matrixρ d

dtρ=−ı

~[H, ρ] +1

2(2LρL^†−L^†Lρ−ρL^†L) + 1

2(2lρl^†−l^†lρ−ρl^†l) H

~ = ∆|ei he|+ Ω|g1i he|+ Ω^∗|ei hg1|+δ|g2i hg2|+ω|g1i hg2|+ω^∗|g2i hg1| L=√

Γ|g1i he|, l=√

γ|g1i hg2| Photon flux (measure):y =tr L^†Lρ

+tr l^†lρ

. (|δ|,|ω|,|Ω|, γΓ).

4See, e.g., Cohen-Tannoudji-Dalibard: Europhys. Lett., 1986.

The slow master equation

Theslow dynamicsis still of Lindblad type with aslow HamiltonianH_s,slow jump operatorsL_s andl_s =l:

d

dtρs =−ı

~[H_s, ρs] +1

2(2L_sρsL^†_s−L^†_sL_sρs−ρsL^†_sL_s) + 1

2(2l_sρsl_s^†−l_s^†l_sρs−ρsl_s^†l_s) Hs

~ =δ|g₂i hg₂|+ω|g₁i hg₂|+ω^∗|g₂i hg₁| L_s =2

r|Ω|²

Γ |g₁i hg₁|, l_s=l =√

γ|g₁i hg₂| Photon flux (measure):y =tr

L^†_sLsρs

+tr

l_s^†lsρs

.

Slow/fast systems in Tikhonov normal form

(Σ^ε)









 dx

dt = f(x,z, ε) εdz

dt = g(x,z, ε)

withx ∈Rⁿ,z ∈R^p,

0< ε1a small parameter , f andg regular functions.

Slow approximation (zero order inε)

As soon asg(x,z,0) =0admits a solution,z =ρ(x), withρ smooth function ofx and ∂g

∂z(x, ρ(x),0)is a stable matrix, the approximation of

(Σ^ε)









 dx

dt = f(x,z, ε) εdz

dt = g(x,z, ε)

by (Σ⁰)





 dx

dt = f(x,z,0) 0 = g(x,z,0) isvalid for time intervals of length 1.

Forlonger intervals of length 1/ε, correction terms of order 1 in εshould be included inΣ⁰. They can be computed viacenter manifold techniques and Carr’s approximation lemma⁵.

5See, e.g., J. Carr: Application of Center Manifold Theory. Springer, 1981.

Proof based on matrix computations only⁶

WithQ_k =|g_ki he|,Γ_k = ^Γ_ε^k and 0< ε1 the slow/fast master equation reads

d

dtρ=−i

~[H, ρ] +

N

X

k=1

Γ_k

2ε(2Q_kρQ^†_k−Q^†_kQ_kρ−ρQ^†_kQ_k).

Change of variablesρ7→(ρ_f, ρ_s)to put the system in Tikhonov normal form (P=|ei he|): ρ_f =Pρ+ρP−PρP,

ρ_s = (1−P)ρ(1−P) + ¹

PN k=1Γk

PN

k=1Γ_k QkρQ_k^†, with inverse ρ=ρ_s+ρ_f − ¹

PN k=1Γ_k

PN

k=1Γ_k Q_kρ_fQ_k^†. The dynamics in(ρ_s, ρ_f)"Tikhonov coordinates":

d

dtρ_s = (1−P) −ıH

~ , ρ

(1−P) + 1 PN

k=1Γ_k

N

X

k=1

Γ_kQ_k −ıH

~ , ρ

Q_k^†

εd

dtρ_f =− PN

k=1Γ_k

2 (ρ_f+Pρ_fP)−εı

~(P[H, ρ] + [H, ρ]P−P[H, ρ]P).

6See, Mirrahimi-R, CDC 2006 and IEEE AC to appear in 2009.

Order zero approximation inε

I Settingεto 0 in d

dtρ_s = (1−P) −ıH

~ , ρ

(1−P) + 1 PN

k=1Γ_k

N

X

k=1

Γ_kQk

−ıH

~ , ρ

Q_k^†

εd dtρ_f =−

PN k=1Γ_k

2 (ρ_f +PρfP)−εı

~(P[H, ρ] + [H, ρ]P−P[H, ρ]P).

yields to thecoherentdynamics

ı~d

dtρs = [(1−P)H(1−P), ρ_s] ρ_f =0

withy =0.

I Need for higher order corrections termsinε

High order approximation via center manifold techniques⁷ Consider the slow/fast system (f andg are regular functions)

d

dtx =f(x,z), εd

dtz =−Az+εg(x,z)

where all the eigenvalues of the matrixAhave strictly positive real parts, and 0< ε1. Theslow invariant attractive manifold admits for equation (boundary layer)

z=εA⁻¹g(x,0) +O(ε²)

and therestriction of the dynamicson this slow invariant manifold reads

d

dtx =f(x, εA⁻¹g(x,0))+O(ε²) =f(x,0)+ε∂f

∂z|_(x,0)A⁻¹g(x,0)+O(ε²).

Center-manifold approximationsyield to second order terms in the expansion ofz:

z=εA⁻¹g(x,0)+ε²A⁻¹ ∂g

∂z|_(x,0)A⁻¹g(x,0)−A⁻¹∂g

∂x|_(x,0)f(x,0)

+O(ε³).

7See, e.g., Fenichel J. Diff. Eq. 1979 or Duchêne-R Chem. Eng. Sci.

1996.

Order one approximation inε

Addition offirst order correction termsinεare related to decoherenceand thus to Lindblad terms:

d

dtρ_s=−ı

~[Hs, ρ_s]+2ε

N

X

k=1

Γ_k

2Qs,kρ_sQ_s,k^† −Q_s,k^† Qs,kρ_s−ρ_sQ_s,k^† Qs,k

where

H_s = (1−P)H(1−P) and Q_s,k = 1

~ PN

l=1Γ_l(1−P)Q_kH(1−P).

Theboundary layerreads ρ_f = −2ı ε

~ PN

k=1Γ_k

(PHρs−ρsHP) +O(ε²).

and the output (measure)

y(t) =4ε X^N

k=1

Γ_k

tr Pρs

+O(ε²),

Concluding remarks

I The proposed adiabatic reduction mixingnon commutative computationswith operators anddynamical systems theory(geometric singular perturbations theory, invariant manifold) preserves the"physics"(CPT slow dynamics).

I In the slow master equation, the decoherence terms depend on the control inputΩ_k:influence on controllability and optimal control?⁸

I Straightforward extensions to: several unstable states|e_ri with fast relaxation to stable states|g_ki; slow decoherence between the "stable" states|g_ki.

I A method to approximateslow/fast quantum trajectoriesby slow quantum trajectorieswhere the jumps from|eito|g_ki are replaced by jumps inside the "slow space"⁹

8See, e.g., Altafini and Bonnard-Chyba-Sugny for the recent results on controllability and optimal control of such dissipative systems.

9For mathematical justifications see: Bouten-Silberfarb: Commun. Math.

Phys., 2008; Bouten-vanHandel-Silberfarb: Journal of Functional Analysis, 2008; Gough-vanHandel: J. Stat. Phys., 2007.

Quantum trajectories¹⁰associated to the slow master equation Setγ_k =4^|Ω_(Γ^1|2+|Ω2|2

1+Γ2)² Γ_k fork =1,2.

At each infinitesimal time stepdt,

I ρsjumps

I towards the state|g₁i hg₁|with ajump probabilitygiven by:

hb_Ω|ρ_s|b_Ωi γ₁dt.

I or towards the state|g₂i hg₂|with ajump probabilitygiven by:hbΩ|ρ_s|bΩi γ₂dt.

I orρsdoes not jump with probability

1− hb_Ω|ρs|b_Ωi (γ1+γ2)dt

and then evolves on theBloch sphereaccording to 1

γ d

dtρ_s=−ı∆

2γ[σ_z, ρ_s]−|b_Ωi hb_Ω|ρ_s+ρ_s|b_Ωi hb_Ω|

2 +hbΩ|ρ_s|bΩiρ_s. withγ =γ1+γ2.

10See, e.g., Haroche-Raimond book, 2006.

Quantum trajectories in Bloch-sphere coordinates Setβ=2 arg(Ω₁+ıΩ2)and

ρ_s =1+X(|bi hd|+|di hb|) +Y(ı|bi hd| −ı|di hb|) +Z(|di hd| − |bi hb|) 2

At each infinitesimal time stepdt, the point(X,Y,Z)∈S²,

I jumps

I towards the state(sinβ,0,cosβ)with ajump probability given by: ^(1−Z)₂ γ₁dt.

I or towards the state(−sinβ,0,−cosβ)with ajump probabilitygiven by: ^(1−Z₂ ⁾ γ₂dt.

I or does not jump with probability 1−^(1−Z₂ ⁾(γ₁+γ₂)dt and then evolves according to

d

dtX =−∆cosβY −γXZ 2 d

dtY = ∆cosβX −∆sinβZ −γYZ 2 d

dtZ = ∆sinβY+ γ(1−Z²) 2

Convergence of the no-jump dynamics

For|∆|< ^γ₂, the nonlinear system inS² d

dtX =−∆cosβY −γXZ 2 d

dtY = ∆cosβX−∆sinβZ −γYZ 2 d

dtZ = ∆sinβY +γ(1−Z²) 2

admits a two equilibirum points, one is unstable and the other one isquasi-global asymptotically stable.

Proof: based on Poincaré-Bendixon on the sphere.¹¹

11See, Mirrahimi-R, 2008, arxiv:0806.1392v1