Deterministic submanifolds and analytic solution of the stochastic differential equation describing a continuously measured qubit

Deterministic submanifolds and analytic solution of the stochastic differential equation

describing a continuously measured qubit

Joint work with A. Sarlette,P. Campagne-Ibarcq, P. Six, L. Bretheau, M. Mirrahimi, and B. Huard IMA, Quantum and Nano Control, April 11-15, 2016

Pierre Rouchon

CAS, Mines ParisTech, PSL Research University

QUANTIC, Inria-Paris

Outline

Measuring fluorescence of a qubit (slides of Benjamin Huard)

A positivity preserving numerical scheme for quantum filtering

The deterministic surface and integral quantities

When is a qubit confined to a deterministic surface ?

Conclusion

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A positivity preserving numerical scheme: diffusive case

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

ρ_t+dt = Mdy_tρtM^†_dy

t + (1−η)LρtL^†dt Tr

Mdy_tρtM^†_dy

t+ (1−η)LρtL^†dt withM_dyt =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

√

2π e^−s

2 2dt ds

close to a Gaussian law of variancedtand mean√ ηTr

(L+L^†)ρt

dt.

1H. Amini, M. Mirrahimi, P.R. IEEE CDC, 2011. P.R., J. Ralph PRA 2015.

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A positivity preserving numerical scheme: diffusive/jump case

The quantum stateρ_tis usually mixed and obeys to

dρ_t= −i[H, ρ_t] +X ν

L_νρ_tL^†_ν−¹

2(L^†_νL_νρ_t+ρ_tL^†_νL_ν) +V_µρ_tV_µ^†−¹

2(V_µ^†V_µρ_t+ρ_tV_µ^†V_µ)

! dt

+X ν

√ην

L_νρ_t+ρ_tL^†_ν−Tr

(L_ν+L^†_ν)ρ_t ρ_t

dW_ν,t

+X µ







θµρ_t+P

µ0η_µ,µ0Vµρ_tV_µ^† θ_µ+P

µ0η_µ,µ0Tr V_µ0ρ_tV^†

µ0 −ρ_t









dN_µ(t)− θ_µ+X

µ0 η_µ,µ0Tr

V_µ0ρ_tV^† µ0 dt





whereη_ν∈[0,1],θ_µ, η_µ,µ0≥0 withη_µ0=P

µη_µ,µ0≤1 are parameters modelling measurements imperfections.

If, for someµ,N_µ(t+dt)−N_µ(t) =1, we haveρ_t+dt=

θ_µρ_t+P

µ0η_µ,µ0V_µ0ρ_tV^† µ0 θ_µ+P

µ0η_µ,µ0Tr V_µ0ρ_tV^†

µ0 . When∀µ,dN_µ(t) =0, we have

ρ_t+dt=

M_dytρ_tM^†

dyt+P

ν(1−η_ν)L_νρ_tL^†_νdt+P

µ(1−η_µ)V_µρ_tV_µ^†dt Tr

M_dytρ_tM^†

dyt+P

ν(1−η_ν)L_νρ_tL^†_νdt+P

µ(1−η_µ)V_µρ_tV_µ^†dt

withM_dyt =I+

−iH−¹ 2

P

νL^†_νLν+¹₂P µ

η_µTr

Vµρ_tV_µ^† I−V_µ^†Vµ

dt+P

ν

√ηνdyνtLνand

wheredy_ν,t=√ η_νTr

(L_ν+L^†_ν)ρ_t

dt+dW_ν,t.

2H. Amini, C. Pellegrini, P.R.: Russian Journal of Mathematical Physics, 2014.

P.R:. Proceedings of International Congress of Mathematicians, Seoul 2014 (arXiv:1407.7810).

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The deterministic surface

The SME (γ1=1 here) dρt=

σ-ρσ+−σ+σ-ρ+ρσ+σ-

2

dt

+ qη

2 σ-ρ+ρσ+− Tr(σxρ)ρ dW_I+

qη

2 iσ-ρ−iρσ+− Tr(σyρ)ρ dW_Q

reads with the Bloch coordinates(x,y,z)

dxt =−¹₂xtdt+ qη

2

(1+zt−x_t²)dW_I−xtytdW_Q dyt =−¹

2ytdt+ qη

2

−xtytdWI+ (1+zt−y_t²)dWQ

dzt =−(1+zt)dt−q

η 2(1+zt)

xtdW_I+ytdW_Q

For any realization of starting from the same initial point(x0,y0,z0)we have

1

2 x_t²+y_t²

+ct(1+zt)²−(1+zt) =0

wherect = (c₀−^η₂)e^t+^η₂ remains in[¹₂,+∞)andc₀= ^x

2 0+y₀² 2(1+z₀)²+_1+z¹

0.

3Ph. Campagne-Ibarcq et al. PRX 2016.

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The quantum filter is "solvable by elementary quadratures"

The solution of

dxt =−¹

2xtdt+ qη

2

(1+zt−x_t²)dWI−xtytdWQ

dy_t =−¹₂y_tdt+ qη

2

−x_ty_tdW_I+ (1+z_t−y_t²)dW_Q dzt =−(1+zt)dt−

qη

2(1+zt)

xtdWI+ytdW_Q

can be computed from simple integrals of the signalsdIt=q

η

2xtdt+dWIand dQt =

qη

2ytdt+dWQ. This results from d

x 1+z

=¹_{2 x}

t 1+z_t

dt+

qη

2dIt, d

y 1+z

=¹_{2 y}

t 1+z_t

dt+

qη 2dQt

This provides a completion of¹₂ x_t²+y_t²

+ct(1+zt)²−(1+zt) =0 with

x_t 1+z_t =e^t/2

x₀ 1+z₀ +

qη 2

Z t

0

e^{−τ /2}dIτ

, _1+z^yt

t =e^t/2

y₀ 1+z₀+

qη 2

Zt

0

e^{−τ /2}dQτ

.

Related toPicard-Vessiot and Liouvillian extensions of differential fields: the solution of the quantum filter is an algebraic function of some integrals and exponentials of integral of its two inputsIt andQt.

4Ph. Campagne-Ibarcq et al. PRX 2016.

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The second case where the qubit is confined to a deterministic surface

Superconducting qubit dispersively coupled to a cavity traversed by a microwave signal (in- put/output theory). The back-action on the qubit state of a singlemeasure- mentof both output field quadratures It and Qt is described by a simple SME for the qubit density operator. (M. Hatridge et al.: Science, 2013).

dρt= γ(σzρσz−ρt) dt+

qηγ

2 σzρt+ρtσz−2 Tr(σzρt)ρt

dW_I+i qηγ

2 [σz, ρt]dW_Q withdIt=q

ηγ

2 Tr(2σzρt)dt+dWI anddQt =dWQ, whereγ≥0 is related to the measurement strength andη∈[0,1]is the detection efficiency.

The deterministic surface is given here by another ellipsoid of revolution axisz:

x_t²+y_t²+bt(z_t²−1) =0 wherebt =b0e^{−2(1−η)t} andb0=x₀²+y₀² 1−z²₀ .

5A. Sarlette, P.R.: preprint 2016 (arXiv:1603.05402).

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There are only two cases where the qubit is confined on an evolving surface

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

−1 −0.5 0 0.5 1

−1

−0.8

−0.6

−0.4

−0.2 0 0.2 0.4 0.6 0.8 1

Theorem[A. Sarlette, P.R.: arXiv:1603.05402]For any initial stateρ0, the qubit state ρt, solution of the SME (ην∈(0,1))

dρt= X

ν

LνρtL^†_ν−¹₂(L^†_νLνρt+ρtL^†_νLν)

! dt

+X

ν

√ην

Lνρt+ρtL^†_ν− Tr

(Lν+L^†_ν)ρt

ρt

dWν,t,

is restricted to a deterministically evolving 2-dimensional manifold if, and only if, I either existβν, αν∈Cand U∈U(2)such that Lν=βνVσzV^†+αν,∀ν I or existβν∈Cand V ∈U(2)such that Lν=βνVσ-V^†,∀ν.

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Sketch of the proof (1)

Stroock-Varadhan theorem⁶

Consider a stochastic differential equation

dxt=F(xt)dt+

m

X

j=1

Gj(x)◦ dW_t^j,

with xt ∈R^N the state, dW_t¹,dW_t², ...,dW_t^mindependent Wiener processes, x0fixed and the dynamics to be understood in the Stratonovitch sense (we therefore put the◦ symbol).

The support of the distribution of xtcan be described as the closure, for the natural Banach topology on C([0,1],R^N), of the set of solutions of the following controlled system:

d˜xt =F(˜xt)dt+

m

X

j=1

G_j(˜x)du_t^j,

with˜x₀=x₀, for all possible control signals u¹_t,u_t², ...,u^m_t in H¹([0,1],R^m).

6D.W. Stroock and S.R.S. Varadhan, "On the support of diffusion processes with applications to the strong maximum principle", Proc. 6th Berkeley Symp. Mathematical Statistics and Probability vol.3, pp.333-359, 1972.

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Sketch of the proof (2)

I Strong accessibility theorem⁷The control system

d

dtx=F(x) +Pm

j=1Gj(x)ujwith analytic vector fields F,G1, ...,Gmis strongly accessible at x0if, and only if, thedrift-preserved Lie algebraGF 8has full dimension N at x0.

Moreover, ifGFhas dimension at most N−n<N for all x0, then the system stays on a (time-dependent) manifold of dimension N−n, independently of the control inputs.

I Stratonovitch form of the SME:

dρ_t=X ν

(1−ην)

Lνρ_tL^†_ν−¹

2(L^†_νLνρ_t+ρ_tL^†_νLν) dt

− X ν

ην 2

L^†_νL_νρ+ρL^†_νL_ν+ (L_ν)²ρ+ρ(L^†_ν)²

! dt

+ X

ν ην

2 Tr

L^†_νL_νρ+ρL^†_νL_ν+ (L_ν)²ρ+ρ(L^†_ν)²

ρ

! dt

+ X

ν ηνTr

Lνρ+ρL^†_ν

(Lνρ+ρL^†_ν)−ην

Tr

Lνρ+ρL^†_ν2 ρ

! dt

+X ν

√ην

L_νρt+ρtL^†_ν−Tr

(L_ν+L^†_ν)ρt

ρt

◦dW_ν,t.

7A. Isidori,Nonlinear Control Systems: An Introduction, Springer, Berlin, 1985

8GFis the smallest Lie algebra containingG(the Lie algebra generated by vector fieldsG1,G2, ...,Gm) and closed under Lie brackets withF, i.e. for anyG∈GF we have[F,G]∈G_F.

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Concluding remarks

1. Stochastic master equations govern the dynamics of open quantum systems by taking into accountmeasurement back-actionanddecoherence(unread measurement).

2. Future work could investigate how general confinement of the density operator to submanifolds is in higher-dimensional Hilbert spaces.

3. Interest of continuous fluorescence signals(dI_t,dQ_t)for the characterization of a dephasing noiseξt:

dxt = −^γ₂²xt+ξtyt−v(t)zt dt+

qηγ₁ 2

(1+zt−x_t²)dW_I−xtytdW_Q dyt = −ξtxt−^γ₂²yt+u(t)zt

dt+ qηγ₁

2

−xtytdW_I+ (1+zt−y_t²)dW_Q dzt = v(t)xt−u(t)yt−γ1(1+zt)

dt− qηγ₁

2 (1+zt)

xtdWI+ytdW_Q dIt =

qηγ₁

2 xtdt+dWI

dQt ==

qηγ₁

2 ytdt+dWQ

whereu(t)andv(t)are well chosen open-loop controls (see Lorenza Viola presentation and work).

11 / 11