Quantum Systems: Dynamics and Control 1

Transcript

1 Quantum Systems: Dynamics and Control 1 Mazyar Mirrahimi and Pierre Rouchon 3 February 7, See the web page: INRIA Paris, QUANTIC research team 3 Mines ParisTech, QUANTIC research team

2 Outline 1 Photon Box: a key example Open quantum systems in discrete-time

3 Models of open quantum systems are based on three features 4 1 Schrödinger: wave funct. ψ H or density op. ρ ψ ψ d dt ψ = i H ψ, d dt ρ = i [H, ρ], H = H 0 + uh 1 Entanglement and tensor product for composite systems (S, M: Hilbert space H = H S H M Hamiltonian H = H S I M + H int + I S H M observable on sub-system M only: O = I S O M. 3 Randomness and irreversibility induced by the measurement of observable O with spectral decomp. µ λ µp µ : measurement outcome µ with proba. P µ = ψ P µ ψ = Tr (ρp µ depending on ψ, ρ just before the measurement measurement back-action if outcome µ = y: ψ ψ + = P y ψ ψ Py ψ, ρ ρ + = P yρp y Tr (ρp y 4 S. Haroche, J.M. Raimond: Exploring the Quantum: Atoms, Cavities and Photons. Oxford University Press, 006.

4 Composite system built with a harmonic oscillator and a qubit. System S corresponds to a quantized harmonic oscillator: { } H S = H c = c n n (c n n=0 l (C, n=0 where n represents the Fock state associated to exactly n photons inside the cavity Meter M is a qubit, a -level system (idem spin-1/ system : H M = H a = C, each atom admits two energy levels and is described by a wave function c g g + c e e with c g + c e = 1; atoms leaving B are all in state g State of the full system Ψ H S H M = H c H a : Ψ = + n=0 c ng n g + c ne n e, c ne, c ng C. Ortho-normal basis: ( n g, n e n N.

5 Markov model (1 R 1 R C D B B R When atom comes out B, Ψ B of the full system is separable Ψ B = ψ g. Just before the measurement in D, the state is in general entangled (not separable: ( ( Ψ R = U SM ψ g = Mg ψ g + ( M e ψ e where U SM is a unitary transformation (Schrödinger propagator defining the linear measurement operators M g and M e on H S. Since U SM is unitary, M gm g + M em e = I.

6 Markov model ( Just before D, the field/atom state is entangled: M g ψ g + M e ψ e Denote by µ {g, e} the measurement outcome in detector D: with probability P µ = ψ M µm µ ψ we get µ. Just after the measurement outcome µ = y, the state becomes separable: ( Ψ D = 1 (M y ψ y = Py M e ψ k ψk M e M e ψ k M y ψ M y M y ψ ψ y. Markov process: ψ k ψ t=k t, k N, t sampling period, M g ψ k with y k = g, probability P g = ψ k M gm g ψ k ; ψk M ψ k+1 = g M g ψ k with y k = e, probability P e = ψ k M em e ψ k.

7 Dispersive case U R1 = 1 (I + g e e g U R = 1 (I + e iη g e e iη e g U C = g g e iφ(n + e e e iφ(n+i where φ(n = ϑ 0 + ϑn. With η = (ϕ 0 ϑ 0 ϑ π, the measurement operators M g and M e are the following bounded operators: M g = cos(ϕ 0 + Nϑ, M e = sin(ϕ 0 + Nϑ up to irrelevant global phases. Exercise: Show that M gm g + M em e = I.

8 Resonant case: U SM = U R U C U R1 U R1 = e i θ 1 σ y = cos ( θ 1 + sin ( θ1 ( g e e g and ( U C = g g cos Θ ( N + e e cos ( Θ N + g e sin N Θ N + I a e g a and U R = I ( Θ sin N N The measurement operators M g and M e are the following bounded operators: ( M g = cos ( θ 1 ( cos Θ N sin ( Θ θ 1 sin N a M e = sin ( θ 1 ( cos Θ N + 1 cos ( θ 1 N a ( Θ sin N N Exercise: Show that M gm g + M em e = I.

9 Markov process with detection inefficiency With pure state ρ = ψ ψ, we have 1 ρ + = ψ + ψ + = ( M µ ρm Tr M µ ρm µ µ ( when the atom collapses in µ = g, e with proba. Tr M µ ρm µ. Detection efficiency: the probability to detect the atom is η [0, 1]. Three possible outcomes for y: y = g if detection in g, y = e if detection in e and y = 0 if no detection. The only possible update is based on ρ: expectation ρ + of ψ + ψ + knowing ρ and the outcome y {g, e, 0}. M gρm g Tr(M if y = g, probability η Tr (M gρm g gρm g ρ + = M eρm e Tr(M if y = e, probability η Tr (M eρm e eρm e M g ρm g + M e ρm e if y = 0, probability 1 η For η = 0: ρ + = M g ρm g + M e ρm e = K(ρ = E ( ρ + ρ defines a Kraus map.

10 LKB photon-box: Markov process with detection errors (1 With pure state ρ = ψ ψ, we have 1 ρ + = ψ + ψ + = ( M µ ρm Tr M µ ρm µ µ ( when the atom collapses in µ = g, e with proba. Tr M µ ρm µ. Detection error rates: P(y = e/µ = g = η g [0, 1] the probability of erroneous assignation to e when the atom collapses in g; P(y = g/µ = e = η e [0, 1] (given by the contrast of the Ramsey fringes. Bayesian law: expectation ρ + of ψ + ψ + knowing ρ and the imperfect detection y. (1 η gm gρm g +ηemeρm e ((1 Tr((1 η gm gρm g +η em eρm e if y = g, prob. Tr η gm gρm g + η em eρm e ; ρ + = η gm gρm g +(1 ηemeρm e (η Tr(η gm gρm g +(1 η em eρm e if y = e, prob. Tr gm gρm g + (1 η em eρm e. ρ + does not remain pure: the quantum state ρ + becomes a mixed state; ψ + becomes physically irrelevant.

11 LKB photon-box: Markov process with detection errors ( We get ρ + = (1 η gm gρm g +ηemeρm e Tr((1 η gm gρm g +η em eρm e, ((1 with prob. Tr η gm gρm g + η em eρm e ; η gm gρm g +(1 ηemeρm e Tr(η gm gρm g +(1 η em eρm e ( with prob. Tr η gm gρm g + (1 η em eρm e. Key point: ( Tr (1 η g M g ρm g + η e M e ρm e ( and Tr η g M g ρm g + (1 η e M e ρm e are the probabilities to detect y = g and e, knowing ρ. Reformulation with quantum maps : set K g (ρ = (1 η g M g ρm g+η e M e ρm e, K e (ρ = η g M g ρm g+(1 η e M e ρm e. ρ + = K y(ρ Tr (K y (ρ when we detect y The probability to detect y knowing ρ is Tr (K y (ρ. We have the following Kraus map: E ( ρ + ρ = K g (ρ + K e (ρ = K(ρ = M g ρm g + M e ρm e.

12 Outline 1 Photon Box: a key example Open quantum systems in discrete-time

13 General structure of Markov model in discrete time Any open model of quantum system in discrete time is governed by a Markov chain of the form ρ k+1 = K y k (ρ k Tr (K yk (ρ k, with the probability Tr (K yk (ρ k to have the measurement outcome y k knowing ρ k 1. The structure of the super-operators K y is as follows. Each K y is a linear completely positive map (a quantum operation, a partial Kraus map 5 and y K y(ρ = K(ρ is a Kraus map, i.e. K(ρ = µ K µρk µ with µ K µk µ = I. 5 Each K y admits the expression K y(ρ = µ K y,µρk y,µ where (K y,µ are bounded operators on H.

14 Schrödinger view point of ensemble average dynamics Without measurement record, the quantum state ρ k obeys to the master equation ρ k+1 = K(ρ k. since E (ρ k+1 ρ k = K(ρ k (ensemble average. K is always a contraction (not strict in general for the following two such metrics. For any density operators ρ and ρ we have K(ρ K(ρ 1 ρ ρ 1 and F (K(ρ, K(ρ F (ρ, ρ where the trace norm 1 and fidelity F are given by ( ρρ ρ ρ 1 Tr ( ρ ρ and F (ρ, ρ Tr ρ.