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KFKI-RMKI-28-APR-1993Unique Quantum Paths by Continuous

arXiv:gr-qc/9304046v1 1 May 1993KFKI-RMKI-28-APR-1993Unique Quantum Paths by ContinuousDiagonalization of the Density OperatorLajos Di´osiKFKI Research Institute for Particle and Nuclear PhysicsH-1525 Budapest 114, POB 49, Hungarye-mail: diosi@rmki.kfki.huIn this short note we show that for a Markovian open quantum system it is alwayspossible to construct a unique set of perfectly consistent Schmidt paths, supportingquasi-classicality. Our Schmidt process, elaborated several years ago, is the ∆t →0limit of the Schmidt chain constructed very recently by Paz and Zurek.Typeset Using REVTEX1

I. INTRODUCTIONIn a very recent work, Paz and Zurek [1] discuss Markovian open quantum systems andconstruct Schmidt paths showing exact decoherence. The authors, however, notice theirSchmidt paths are quite unstable under, e.g., varying the number of subsequent projections.They think to eliminate the problem by tuning time intervals between subsequent projectionslarger than the typical decoherence time.In the present short note I propose the opposite.

Let time intervals be infinitely short!It means that the frequency of the subsequent projections is so high that the Schmidt pathwill be defined for any instant t in the period considered.This infinite frequency limitexists and provides unique consistent set of Schmidt paths. The issue was discussed [2]and completely solved [3] several years ago.

Of course, the above limit is only valid up toMarkovian approximation. In fact, the ”infinitesimal” repetition interval is still longer thanthe response time of the reservoir.For the sake of better distinction between ordinary [1] and hereinafter advocated [3]Schmidt paths, let us call them Schmidt chain and Schmidt process, respectively.In the next Sect., Schmidt chains are briefly reviewed.The Sect.

III. will recall the earlierresults available now for Schmidt processes.Subsequently, in Sect.IV., we propose anapplication of Schmidt processes to the quantum Brownian motion where classicality mightbe demonstrated.II.

SCHMIDT PATH—MARKOV CHAINConsider the reduced dynamics of a given subsystem:ρ(t′) = J(t′ −t)ρ(t),(t′ > t)(1)where ρ is the reduced density operator, J is the Markovian evolution superoperator. For agiven sequence t0 < t1 < .

. .

< tn of selection times, let us have the corresponding sequenceof pure state (Hermitian) projectors: {P 0, P 1, . .

. , P n} is a Schmidt chain if2

[P k+1, J(tk+1 −tk)P k] = 0,k = 0, 1, 2, . .

. , n −1(2)cf.

Sect. 4 of Ref.

[1]. For fixed initial state P 0 = ρ(t0), let the probabilitiesp(P 1, P 2, .

. .

, P n) = trhP nJ(tn −tn−1)P n−1 . .

. P 1J(t1 −t0)ρ(t0)i(3)be assigned to Schmidt chains.

Schmidt chains satisfy the following sum rule:XSchmidt pathsp(P 1, P 2, . .

. , P n)P 1 ⊗P 2 ⊗.

. .

⊗P n= ρ(t1) ⊗ρ(t2) ⊗. .

. ⊗ρ(tn)(4)assuring the consistency [4] of the probability assignments (3).Let us construct concrete Schmidt chains.

To satisfy the Eq. (2) for k = 0, let us firstdiagonalize the positive definite operator J(t1 −t0)P 0:J(t1 −t0)P 0 =Xαp1αP 1α.

(5)If our choice is P 1 = P 1α1, whose probability is pα1, consider Eq. (2) for k = 1 and diagonalizeJ(t2 −t1)P 1α1:J(t2 −t1)P 1α1 =Xαp2αP 2α.

(6)Single out P 2 = P 2α2 at random, with probability p2α2, etc.For the Schmidt chain {P 0, P 1α1, P 2α2, . .

. , P nαn} one generates from the fixed initial stateP 0, the probability (3) takes the following factorized form:p(P 1α1, P 2α2, .

. .

, P nαn) ≡p(α1, α2, . .

. , αn) = p1α1p2α2 .

. .

pnαn. (7)Schmidt chain is Markov chain.

Given P 0 = ρ(t0), it will branch at t1 into P 1α1, i.e.,into one of the eigenstate projectors of J(t1 −t0)P 0; the branching probability p1α1 is thecorresponding eigenvalue. In the general case, P kαk will branch into P k+1αk+1, i.e., into a certaineigenstate of J(tk+1 −tk)P kαk, with branching probability pk+1αk+1 given by the correspondingeigenvalue.3

III. SCHMIDT PATH—MARKOV PROCESSIn this Sect., we consider the limiting case of the Schmidt chains when the separationstk+1 −tk go to zero.

The pure state path {P(t); t > t0}, starting from the fixed initial stateP(t0) = ρ(t0), is a Schmidt process if (for t > t0 and ǫ ≡dt > 0) P(t + ǫ) branches into aneigenstate projector Pα(t) of J(ǫ)P(t) while the branching probability is the correspondingeigenvalue pα(t). Branching rates wα(t) are worthwhile to introduce by pα(t) = ǫwα(t).We follow the general results obtained in Ref.

[3]. Let us introduce the Liouville super-operator L generating the Markovian evolution (1):J(ǫ) = 1 + ǫL.

(8)Assume the Lindblad form [5]:Lρ = −i[H, ρ] −12XλF †λFλρ + ρF †λFλ −2FλρF †λ(9)where H is the Hamiltonian and {Fλ} are the Lindblad generators. Following the methodof Ref.

[3], introduce the frictional (i.e. nonlinear-nonhermitian) Hamiltonian:HP = H −12iXλ< F †λ > Fλ −H.C.−i2XλF †λ−< F †λ >(Fλ−< Fλ >) + i2w(10)and the nonlinear positive definite transition rate operator:WP = (Fλ−< Fλ >) PF †λ−< F †λ >(11)where, e.g., < Fλ >≡tr(FλP).

We need the unit expansion of the transition rate operator:WP =∞Xα=1wαPα. (12)Observe that, due to the identity WPP ≡0, each Pα is orthogonal to P. The wα’s are calledtransition (branching) rates.

The total transition (branching) rate then follows from Eqs. (11) and (12):4

w ≡Xαwα =< F †λFλ > −< F †λ >< Fλ > . (13)How to generate Schmidt processes?Given the initial pure state ρ(t0) = P(t0), thepure state P(t) evolves according to the deterministic frictional Schr¨odinger-von Neumannequation:ddtP = −i(HPP −PH†P)(14)except for discrete orthogonal jumps (branches)P(t + 0) = Pα(t)(15)occurring from time to time at random with P(t)-dependent partial transition rates wα(t).It is worthwhile to note that neither HP nor WP depend on the concrete Lindblad represen-tation (9) of L, as it is clear in Ref.

[3].Mathematically, the above Schmidt path is pure-state-valued Markov process of gener-alized Poissonian type. During a given infinitesimal period (t, t + dt), the probability of thebranch-free (i.e., jump-free, continuous) evolution is 1−w(t)dt.

Consequently, one obtains [6]the a priori probability of continuous evolution for an arbitrarily given period (t1, t2) asexp−Z t2t1w(t)dt.(16)IV. CLASSICALITYSchmidt processes assure maximum classicality in ”measurement situations”.It hasbeen shown in Ref.

[6] that for large enough t, Schmidt process converges to one of thepointer states while the overall probability of further branches tends to zero. Convergence isthen dominated by the asymptotic solutions of the deterministic frictional Schr¨odinger-vonNeumann Eq.

(14).To test classicality of Schmidt processes in less artificial situations, let us start with the(modified [7]) Caldeira-Leggett [8] master equation:5

ddtρ = Lρ = −i 12M [p2, ρ] −iγ[q, {p, ρ}]−12γλ−2dB[q, [q, ρ]] −12κγλ2dB[p, [p, ρ]](17)where γ is (two times) the friction constant, λdB stands for the thermal deBroglie lengthof the Brownian particle of mass M. In Ref. [7] the value κ = 4/3 has been suggested.

Forsimplicity’s, we have omitted the usual renormalized potential term in the Hamiltonian,assuming it is zero or small enough. Hence we can model quantum counterpart of purefrictional motion.Applying mechanically the Eqs.

(10) and (11), we calculate both the frictional Hamilto-nian and the transition rate operator:HP =12M p2 + 12γ{q−< q >, p−< p >}−i2γλ−2dB((q−< q >)2 −σ2qq) + κλ2dB((p−< p >)2 −σ2pp),(18)WP = γλ−2db (q−< q >)P(q−< q >) + κγλ2db(p−< p >)P(p−< p >)−iγ(q−< q >)P(p−< p >) −(p−< p >)P(q−< q >)(19)where σ2qq =< q2 > −< q >2 and σ2pp =< p2 > −< p >2. The total transition (branching)rate (13) obtains the simple form:w = γ(λ−2dBσ2qq + κλ2dBσ2pp −1)(20)as can be easily verified by observing w = trWP.For most of the time the Schmidt process is governed by the frictional Hamiltonian(18), via the nonlinear Eq.

(14). This equation itself possesses a stationary solution P(∞)with simple Gaussian wave function representing a standing particle.

Furthermore, one canheuristically guess that the nonhermitian terms establish quasi-classicality for arbitrarilygiven initial states.Obviously, the random jumps (15) will interrupt the deterministicevolution of the Schmidt process. If, however, in the quasi-classical regime the rate (20)of jumps were much smaller than the speed of relaxation due to the frictional Hamiltonian(18) then the infrequent jumps would only cause slight random walk and breathing to theotherwise quasi-classical wave function.6

V. CONCLUSIONSchmidt processes offer a certain solution to the preferred basis problem of quantummechanics, at least when the subsystem’s reduced dynamics can be considered Markovian.It will be interesting to carry on with analytic calculations for the Schmidt process of theBrownian motion, not at all exhausted in Sec. IV.This work was supported by the Hungarian Scientific Research Fund under Grant OTKA1822/1991.7

REFERENCES[1] J. P. Paz and W. Zurek, preprint (1993). [2] L. Di´osi, Phys.

Lett. 112A, 288 (1985).

[3] L. Di´osi, Phys. Lett.

114A, 451 (1986). [4] R. B. Griffiths, J. Stat.

Phys. 36, 219 (1984).

[5] G. Lindblad, Commun. Math.

Phys. 48, 119 (1976).

[6] L. Di´osi, J. Phys. A21, 2885 (1988).

[7] L. Di´osi, Europhys. Lett.

22, 1 (1993). [8] A. O. Caldeira and A. J. Leggett, Physica A12, 587 (1983).8

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