Adiabatic elimination for bipartite open quantum systems 4th Workshop on Quantum Non-Equilibrium Dynamics 24-26 April 2017, University of Nottingham (UK)

Adiabatic elimination for bipartite open quantum systems 4th Workshop on Quantum Non-Equilibrium Dynamics

24-26 April 2017, University of Nottingham (UK)

Pierre Rouchon

Centre Automatique et Systèmes, Mines ParisTech, PSL Research University Quantic Research Team, Inria

Joint work with R. Azouit, F. Chittaro and A. Sarlette (arXiv.1704.00785)

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Adiabatic elimination for bipartite open quantum systems

Slow/fast bipartite master equations

Model reduction and geometric singular perturbations

Geometric singular perturbations for bipartite quantum systems

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Several contributions on adiabatic elimination (partial list)

I Lambda systems :

E. Brion, L.H. Pedersen, K. Mølmer : Adiabatic elimination in a lambda system Journal of Physics A : Mathematical and Theoretical, 2007, 40, 1033.

M. Mirrahimi, PR :. Singular perturbations and Lindblad-Kossakowski differential equations IEEE Trans.

Automatic Control , 2009, 54, 1325-1329

F. Reiter, A. Sørensen : Effective operator formalism for open quantum systems Phys. Rev. A, 2012, 85, 032111-

I Slow/fast Lindblad dynamics :

E.M. Kessler : Generalized Schrieffer-Wolff formalism for dissipative systems. Phys. Rev. A, 2012, 86, 012126-

D. Burgarth et al. : Non-Abelian Phases from a Quantum Zeno Dynamics. Phys. Rev. A 88, 042107 (2013) P. Zanardi, L. Campos Venuti : Coherent quantum dynamics in steady-state manifolds of strongly dissipative systems. Phys. Rev. Lett. 113, 240406 (2014)

K. Macieszczak, M. Guta, I. Lesanovsky,J.P. Garrahan : Towards a Theory of Metastability in Open Quantum Dynamics. Phys. Rev. Lett. 116, 240404 (2016)

L. Campos Venuti, P. Zanardi : Dynamical Response Theory for Driven-Dissipative Quantum Systems.

Phys. Rev. A 93, 032101 (2016)

I Quantum stochastic models :

J. Gough, R. van Handel : Singular perturbation of quantum stochastic differential equations with coupling through an oscillator model. J. Stat. Phys. 2007, 127 pp :575.

L. Bouten, A. Silberfarb : Adiabatic elimination in quantum stochastic model, Commun. Math. Phys., 283, 491-505 (2008)

L. Bouten, R. van Handel, A. Silberfarb : Approximation and limit theorems for quantum stochastic models with unbounded coefficients. Journal of Functional Analysis 254 (2008) 3123-3147.

O. Cernotik, D. Vasilyev, K. Hammerer : Adiabatic elimination of Gaussian subsystems from quantum dynamics under continuous measurement Phys. Rev. A, , 92, 012124 (2015)

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Bipartite slow/fast open quantum systems

I Sub-systemAwith Hilbert spaceHArelaxing rapidly towards a unique equilibrium density operatorρ_Avia the Lindbladian evolution :

d

dtρA=LA(ρA)with lim

t7→+∞ρA(t) =ρ_A;

I Sub-systemBwith Hilbert spaceH^B having a slow Lindbladian evolution

d

dtρB=LB(ρB)with 0≤1

I Weak(A,B)couplingvia the HamiltonianPm

k=1A_k ⊗B^†_k Bipartite Hilbert space(A,B)with Hilbert spaceH=H^A⊗ H^Band density operatorρgoverned by

d

dtρ=LA(ρ)−i

" _m X

k=1

A_k⊗B^†_k , ρ

#

+LB(ρ)

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Adiabatic elim. of fast qubit

dispersively coupled to slow qubit

|e

|e

|g

|g

u

zA B z -A

fast qubit A slow qubit B

The slow/fast dynamics dρ

dt =uh

σ₊^A−σ_-^A, ρi +κ

σ_-^Aρσ₊^A−^{σ+Aσ-Aρ+ρσ}2 ^+Aσ-A

−iχh

σ_z^A⊗σ_z^B, ρi

Slow dynamics(second order versus=χ/κ) : dρs

dt =i_κ2^χκ+8u²²[σ_z, ρs] + ^(64κχ_(κ²₂^u_+8u^2)(κ₂²₎^+2u₃ ²⁾ σ_zρsσ_z−ρs Kraus (CPTP) map:ρ=(I−iQ⊗σ_z)(ρ_A⊗ρs)(I+iQ^†⊗σ_z)with ρ_A= _κ2^4κu

+8u²σ_x−κ²+8u^{κ2 2}σ_z+¹₂I andQ=•σ_x+•σ_y+•σ_z+•I 1. R. Azouit, F. Chittaro, A. Sarlette, P.R., IFAC world congress 2017.

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Two-photon pumping in super-conducting circuits

B A

d

dtρ=LA(ρ)−i[Hint, ρ]+LB(ρ)where L^A(ρ)= [ua^†−u^∗a, ρ] +κD^a(ρ)

Hint=g[a(b^†)²+a^†b², ρ]

+χ(a^†a)(b^†b) +^χ₂^a(a^†a)² LB(ρ)=−i^χ₂^b[(b^†b)², ρ]

withκmax

|g|,|χ|,|χa|,|χb| . Theslow dynamics(second order approximation,α=2u/κ) :

d

dtρs=−ih

α²χb^†b+^χ₂^b(b^†b)², ρs

i−iαg[b²+(b^†)², ρs]+

4g² κ

D^Ls(ρ) withL_s =b²+^α_g

χb^†b+^χa(1+2α₂ ²⁾I . Kraus (CPTP) map : ρ= I−iM

|αihα| ⊗ρs

I+iM^† with M= (a^†−α^∗)⊗_2g

κb²+^2αχ_κ b^†b+^2α(1+2α_κ ^2)χaI

+^α2_κ^χa(a^†−α^∗)²⊗I. 2. M. Mirrahimi, Z. Leghtas, V.V. Albert, S. Touzard, R.J. Schoelkopf, L. Jiang, and M.H. Devoret. Dynamically protected cat-qubits : a new paradigm for universal quantum computation. New J. of Physics, 16 :045014, 2014.

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Outline

Slow/fast bipartite master equations

Model reduction and geometric singular perturbations

Geometric singular perturbations for bipartite quantum systems

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What is model reduction ?

A possible answer for _dt^dx =v(x):

restriction to an attractive invariant manifoldΣ.

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Slow/fast systems (coordinate free setting)

Geometric definition independent of coordinates due to Fenichel³:

I x 7→v(x)close tox 7→v(x).

I v(x) =0 define a manifoldΣof dimensionns<n=dim(x)of steady-states forv(x).

I n_f =n−n_seigenvalues of ^∂v_∂x

_Σare stable.

3. N. Fenichel : Geometric singular perturbation theory for ordinary differential equations. J. Diff. Equations, 1979, 31, 53-98.

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Tikhonov normal form and model reduction

Any slow/fast system, can be put, after a suitable change of coordinates, in to aquasi-vertical vector fieldv :

d

dtxs =vs(xs,xf) =ε˜vs(xs,xf, ε), d

dtxf =vf(xs,xf) with 0< ε1.

The reduced system _dt^dxs=vs(xs,x_f)with 0=v_f(xs,x_f)is correct if

d

dtξ_f =v_f(xs, ξ_f)stable for any fixedxs.

In general, modeling variablesx arenotTikhonov variables.

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Model reduction with modeling variables

vf xs xf 0

The reduced model of _dt^dxs=vs(xs,xf, ), _dt^dxf =vf(xs,xf, )is⁴

d

dtx_s = 1+∂vs

∂xf

∂vf

∂xf

−2

∂vf

∂xs

!−1

v_s(x_s,x_f, )+O(²), v_f(x_s,x_f, ) =0.

4. J. Carr : Application of Center Manifold Theory. Springer, 1981.

P. Duchêne, P.R. : Kinetic scheme reduction via geometric singular perturbation techniques. Chem. Eng. Science, 1996, 51, 4661-4672.

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Outline

Slow/fast bipartite master equations

Model reduction and geometric singular perturbations

Geometric singular perturbations for bipartite quantum systems

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Geometric singular perturbations for bipartite open quantum systems⁵

slow invariant attractive sub-manifold 0

s(t)

(t)

s 0

fast relaxation

slow evolution

d

dt ^A i Hint B

A B

s B

0 1

d

dt ^{s s 1 s}

2 s 2 s

s 0 s t

Lindbladian slow dynamicsin a copyHsofHB

d

dtρs =Ls(ρs) =L_s,1(ρ_s) +²L_s,2(ρ_s) +. . .

withKraus mapto recover the physical density operatorρfromρs: ρ=K(ρs) =K₀(ρ_s) +K₁(ρ_s) +. . .

5. R. Azouit et al. IEEE CDC 2016.

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An iterative procedure based on center manifold approximation

d

dtρs =L^s(ρs) =L^s,1(ρs) +²L^s,2(ρs) +. . .into the invariance condition

LA(K(ρs))−i

Hint,K(ρs)

+LB(K(ρs)) = d

dtρ=K(Ls(ρs)) and identify terms of same order :

order 1 :LA(K1(ρs)) +L^int(K0(ρs)) +LB(K0(ρs)) =K0(Ls,1(ρs))

order 2 :L^A(K²(ρs)) +L^int(K¹(ρs)) +L^B(K¹(ρs)) =K⁰(L^s,2(ρs)) +K¹(L^s,1(ρs)) . . .

At each order

1. take the trace versusAto get the correction toLs

2. compute the correction toKvia−L⁻¹A , a super operator for zero-trace operatorsW onHA

−L⁻¹A (W) = Z +∞

0

e^tLA(W)dt

that coincides with the restriction to zero-trace operators of a

completely positive (CP) map. 14 / 18

First order expansion for a bipartite system : Zeno dynamics

The full dynamics

d

dtρ=LA(ρ)−i

" _m X

k=1

A_k ⊗B^†_k , ρ

#

+LB(ρ)

can be approximated by

d

dtρs=−i

" _m X

k=1

tr(Akρ_A)B^†_k, ρs

#

+L^B(ρs) +O(²) ρ= I−iM

(ρ_A⊗ρ_s) I+iM^†

+O(²) whereM =Pm

k=1Fk⊗B^†_k withFk given by

Fkρ_A=−L⁻¹A (Ak ρ_A −tr(Ak ρ_A)ρ_A).

6. A. Azouit et al. arXiv.1704.00785

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Second order dynamics

d

dtρ=LA(ρ)−i

" _m X

k=1

A_k ⊗B^†_k , ρ

#

+LB(ρ)

can be approximated by

d

dtρs =−i



X

k

tr(A_kρ_A)B_k +²X

k,j

y_k,jBkB^†_j , ρs





+LB(ρs) +²

m

X

k=1

DLk(ρ_s)+O(³) ρ= I−iM

(ρ_A⊗ρs) I+iM^†

+O(²) whereyk,j =_2i¹ tr

Fjρ_AA^†_k −Ajρ_AF^†_k

andLk =Pm

j=1λj,kBj with matrixλgiven byλλ^†=x andx_k,j =tr

F_jρ_AA^†_k +A_jρ_AF^†_k 7. A. Azouit et al. arXiv.1704.00785

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Conclusion

Interest of such geometric adiabatic elimination preserving the quantum structure (Lindblad master equation, CPTP maps) :

Some non Markovian dynamics might be modeled via a Lindbladian dynamics on a small Hilbert space and via a CPTP map towards the physical Hilbert space of large dimension.

Quantum feedback where the quantum controller is designed faster than the quantum system to be controlled ((S,L,H) theory of Gough/James).

Extension whenH=L

kHAk⊗ HBk and the slow manifold is parameterized via

ρs=X

k

ρ_Ak⊗ρs,k withρs,k ≥0 andtr(ρs,k)∈[0,1]

(talk of Katarzyna Macieszczak this Monday).

Conjecture : at any order it is always possible to obtain, up-to higher order terms, Lindbladian dynamics forρsand CPTP maps relatingρtoρs.

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