Continuous quantum measurement of two coupled quantum dots ...

GOAN, MILBURN, WISEMAN, AND SUN PHYSICAL REVIEW B 63 125326where mixThe mean and variance of this distribution are equal and (1/t mix ) represents the mixing rate. For finite , the rate atgiven by ENVar(N)Rt. In the quantum-jump case which the variables x(t) and y(t) relax can be found fromfrom Eq. 30, if the electron is in dot 1 then the rate of the real part of the eigenvalues of the matrix in the first termelectrons passing through the PC is D. If the electron is in on the right hand side of Eq. 45a. This gives the decaycase, the relation d loc t ms mix , DiVincenzo, J. R. Friedman, and V. Sverdlov.dot 2, then the rate is just D. Requiring the difference in decoherence rate d for the off-diagonal variables, x(t) andmeans of the two probability distributions, p(N,), being of y(t). The variable z(t)0 represents an equal probabilitythe order of the square root of the sum of twice the variances for the electron in the CQD’s to be in each dot. Hence theat time yieldsrate at which the variable z(t) relaxes to zero corresponds tothe mixing rate, 22 mix . Under the assumption of d mixDD2D2D. 53 for effective measurements, the variables x(t) and y(t)Solving this for 1 yields a characteristic rate which is thesame as loc defined in Eq. 50.A similar conclusion is reached in Refs. 27,21, and 22.The measurement time, t ms , in Refs. 27,21, and 22 istherefore relax at a rate much faster than that of the variablez(t). As a result, it is valid to substitute the steady-statevalue of y(t) obtained from Eq. 45a into ż(t), Eq. 45b, tofind the mixing rate. Consequently, we obtainroughly the inverse of the localization rate given here. However,there the condition for being able to distinguish the twoprobability distributions is slightly different from the conditiondiscussed here. The measurement time 27,21,22 42 ddztis denoteddt 2 d zt mixzt.2as the time at which the separation in the means of the two57distributions is larger than the sum of the widths, i.e., thesum of the square roots of twice the individual varianceIt is easy to see that the mixing rate Eq. 57 vanishes asrather than the square root of the sum of twice the variance.→0. Finally, the self-consistent requirement for the assumption d mix yields, from Eq. 57, the condition If this condition is applied here, instead of Eqs. 51 and53, we have( 2 d 2 /2). The mixing rate Eq. 57 is in agreementwith the result found in Ref. 22 under a similar requiredT 2 2TXt ms T 2 t ms 2T 2 t ms 2T 2 t ms 54 condition. 54for the quantum-diffusive case andDt ms Dt ms 2Dt ms 2Dt ms 55VI. CONCLUSIONWe have obtained the unconditional master equation forfor the quantum-jump case. We find from Eqs. 54 and 55the CQD system, taking into account the effect of finite temperatureof the PC reservoirs under the weak system-that the inverse of the measurement time t ms is the same forboth quantum-diffusive and quantum-jump cases, and isenvironment coupling and Markovian approximations. Weequal to the decoherence rate:have also presented a quantum trajectory approach to derive,t ms X 22 DD 2for both quantum-jump and quantum-diffusive cases, the2d 1 d , 56 zero-temperature conditional master equations. These conditionalmaster equations describe the evolution of the measuredCQD system, conditioned on a particular realization ofwhere d (1/ d ) is the decoherence time. This is in agreementwith the result in Refs. 21 and 22. Our condition the measured current. We have found in both cases that theshows, on the other hand, the different localization rates for dynamics of the CQD system can be described by the SSE’sthe quantum-jump and quantum-diffusive cases. This is consistentwith the initial rates obtained from the ensemble av-carried away from the system by the PC reservoirs can befor its conditional state vector provided that the informationerage of Bloch variable, Ez 2 c (t).recovered by perfect measurement detection. Furthermore,we have analyzed for both cases the localization rates atThere is another time scale denoted as mixing time, t mix ,which the electron becomes localized in one of the two dotsdiscussed in Refs. 27,21, and 22. It is the time after whichthe information about the initial quantum state of the CQD’swhen 0. We have shown that the localization time discussedhere is slightly different from the measurement timeis lost due to the measurement-induced transition. This transitionarises because of the nonzero coupling term in thedefined in Refs. 27, 21, and 22. The mixing rate at which theCQD Hamiltonian, which does not commute with the occupationnumber operator of dot 1 the measured quantity andtwo possible states of the CQD’s become mixed when 0 has been calculated as well and found in agreement withthus mixes the two possible states of the CQD system. Belowthe result in Ref. 22.we estimate the mixing time using the differential equationsfor the Bloch variables. It is expected that effective and successfulquantum measurements require t mix t ms t loc d .In other words, the readout should be achieved long beforethe information about the measured initial quantum state islost. In terms of different characteristic rates, we have, in thisACKNOWLEDGMENTSWe thank M. Büttiker and A. M. Martin for bringing ourattention to Ref. 15. H.S.G. is grateful for useful discussionswith A. N. Korotkov, D. V. Averin, K. K. Likharev, D. P.125326-10