Continuous quantum measurement of two coupled quantum dots ...

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GOAN, MILBURN, WISEMAN, AND SUN PHYSICAL REVIEW B 63 125326The form of the master equation 7, defined through thesuperoperator DB(t), preserves the positivity of the densitymatrix operator (t). Such a Markovian master equationis called a Lindblad 50 form.To demonstrate the equivalence between the master equation7 and the rate equations derived in Ref. 16, we evaluatethe density matrix operator in the same basis as in Ref. 16and obtain˙ aa ti ab t ba t,˙ ab ti ab ti aa t bb tX T 2 /2 ab ti ImT *X T *X ab t.13a13bHere ( 2 1 ) is the energy mismatch between thetwo dots, ij (t)i(t) j, and aa (t) and bb (t) are theprobabilities of finding the electron in dot 1 and dot 2, respectively.The rate equations for the other two density matrixelements can be easily obtained from the relations: bb (t)1 aa (t) and ba (t) ab* (t). Compared to an isolatedCQD system, the presence of the PC detector introducestwo effects to the CQD system. First, the imaginarypart of the product of T *X T *X the last term in Eq.13b causes an effective temperature-independent shift inthe energy mismatch between the two dots. Here (T *X T *X T *X )T*X is a temperature-independent quantitywhere TT (0), i.e., T and X evaluated at zero temperature,respectively. Second, it generates a decoherencedephasing rate d X T 2 /214for the off-diagonal density matrix elements, where X T 2X 2 X 2 . We note that the decoherence rate comesentirely from the effect of the measurement revealing wherethe electron in the CQD’s is located. If the PC detector doesnot distinguish which of the dots the electron occupies, i.e.,X 0, then d 0. The rate equations in Eq. 13 are exactlythe same as the zero-temperature rate equations in Ref.16 if we assume that the tunneling amplitudes are real, T 00T 00* and 00 00* . In that case, the last term in Eq. 13bvanishes and d X 2 /2(DD) 2 /2. Actually, the relativephase between the two complex tunneling amplitudesmay produce additional effects on conditional dynamics ofthe CQD system as well. This will be shown later when wediscuss conditional dynamics. Physically, the presence of theelectron in dot 1 raises the effective tunneling barrier of thePC due to electrostatic repulsion. As a consequence, the effectivetunneling amplitude becomes lower, i.e., DTX 2 DT 2 . This sets a condition on the relative phase between X and T: cos X/(2T ).The dynamics of the unconditional rate equations at zerotemperature was analyzed in Ref. 16. Here, following fromEqs. 14, 8, and 9, we find that the temperaturedependentdecoherence rate due to the PC thermal reservoirshas the following expression: d T d 0 eV V eVcoth eV. 152k B TAs expected, d (T) increases with increasing temperature,although the average tunneling current through the PC istemperature independent 45,46 for the same range of low temperaturesk B T L(R) . This temperature dependence of thedecoherence rate is in fact just the temperature dependenceof the zero-frequency noise power spectrum of the currentfluctuation in a low-transparency PC or tunnel junction. 51The CQD system weakly coupled to another finitetemperatureenvironment beside the PC detector was discussedin Ref. 20. However, the influence of the finitetemperaturePC reservoirs on the CQD system, presentedhere, was not taken into account. The finite-temperature decoherencerate of a one-electron state in a quantum dot dueto charge fluctuation of a general QPC has been calculated inRef. 13. In Ref. 26, the temperature-dependent decoherencerate for a two-state system caused by a QPC detector hasbeen discussed specifically in the context of the measurementproblem.III. QUANTUM-JUMP, CONDITIONALMASTER EQUATIONSo far we have considered the evolution of the reduceddensity matrix when all the measurement results are ignored,or averaged over. To make contact with a single realizationof the measurement records and study the stochastic evolutionof the quantum state, conditioned on a particular measurementrealization, we derive in this section the quantumjump,conditional master equation at zero temperature.The nature of the measurable quantities, such as accumulatednumber of electrons tunneling through the PC barrier,is stochastic. On average, of course, the same current flowsin both reservoirs. However, the current is actually made upof contributions from random pulses in each reservoir, whichdo not necessarily occur at the same time. They are indeedseparated in time by the times at which the electrons tunnelthrough the PC. In this section, we treat the electron tunnelingcurrent consisting of a sequence of random -functionpulses. In other words, the measured current is regarded as aseries of point processes a quantum-jump model. 37,42,28 Thecase of quantum diffusion will be analyzed in Sec. IV.Before going directly to the derivation, we discuss somegeneral ideas concerning quantum measurements. If the systemunder observation is in a pure quantum state at the beginningof the measurement, then it will still be in a pureconditional state after the measurement, conditioned on theresult, provided no information is lost. For example, if theinitial normalized state is (t), the unnormalized final stategiven the result at the end of the time interval t,tdt)ofthe measurement becomes˜ tdtM dtt, 16where M (t) represents a set of operators that define themeasurements and satisfies the completeness condition M † tM t1. 17125326-4
CONTINUOUS QUANTUM MEASUREMENT OF TWO . . . PHYSICAL REVIEW B 63 125326Equation 17 is simply a statement of conservation of probability.The corresponding unnormalized density matrix, followingfrom Eq. 16, is given by˜ tdt˜ tdt˜ tdtJ M dtt,18where (t)(t)(t) and the superoperator J is definedin Eq. 11. Of course, if the measurement is made but theresult is ignored, the final state will not be pure but a mixtureof the possible outcome weighted by their probabilities. Consequently,the unconditional density matrix can be written astdt ˜ tdt Pr tdt, 19where PrTr˜ (tdt) stands for the probability forthe system to be observed in the state , and (tdt)˜ (tdt)/Pr is the normalized density matrix at timetdt.Now we proceed to derive the quantum-jump, conditionalmaster equation in the following. Only two measurement operatorsM (dt) for 0,1 are needed for a measurementrecord that is a point process. For most of the infinitesimaltime intervals, the measurement result is 0, regarded as anull result. On the other hand, at randomly determined times,there is a result 1, referred as a detection of an electrontunneling through the PC barrier. Formally, we can write thecurrent through the PC asitedNt/dt,20where e is the electronic charge and dN(t) is a classicalpoint process that represents the number either zero or oneof tunneling events seen in an infinitesimal time dt. We canthink of dN(t) as the increment in the number of electronsN(t) in the drain in time dt. It is this variable, the accumulatedelectron number transmitted through the PC, which isused in Refs. 16, 27, and 22. The point process is formallydefined by the conditions on the classical random variabledN c (t):dN c t 2 dN c t,EdN c tTr˜ 1c tdtTrJM 1 dt c tP 1c tdt.2122Here we explicitly use the subscript c to indicate that thequantity to which it is attached is conditioned on previousmeasurement results, the occurrences detection records ofthe electrons tunneling through the PC barrier in the past.EY denotes an ensemble average of a classical stochasticprocess Y. Equation 21 simply states that dN c (t) equalseither zero or one, which is why it is called a point process.Equation 22 indicates that the ensemble average of dN c (t)equals the probability quantum average of detecting electronstunneling through the PC barrier in time dt. In addition,dN c (t) is of order dt and obviously all moments powersof dN c (t) are of the same order as dt. Note here that thedensity matrix c (t) is not the solution of the unconditionalreduced master equation, Eq. 25a. It is actually conditionedby dN c (t) for tt.The stochastic conditional density matrix at a later timetdt can be written as˜ 1c tdt c tdtdN c tTr˜ 1c tdt˜ 0c tdt1dN c tTr˜ 0c tdt . 23Equation 23 states that when dN c (t)0 a null result, thesystem changes infinitesimally via the operator M 0 (dt)and hence c (tdt) 0c (tdt). Conversely, if dN c (t)1 a detection, the system goes through a finite evolutioninduced by the operator M 1 (dt), called a quantum jump. Thecorresponding normalized conditional density matrix thenbecomes 1c (tdt). One can see, with the help of Eq. 20,that in this approach the instantaneous system state conditionsthe measured current see Eq. 22, while the measuredcurrent itself conditions the future evolution of the measuredsystem see Eq. 23 in a self-consistent manner. It isstraightforward to show that the ensemble average of theconditional density matrix equals the unconditional one,E c (t)(t). Tracing over both sides of Eq. 19 for 0,1, we obtainTr˜ 0c tdt1Tr˜ 1c tdt.24Then taking the ensemble average over the stochastic variablesdN c (t) on both sides of Eq. 23, replacing EdN c (t)by using Eq. 22, and comparing the resultant equation withEq. 19 completes the verification.Next we find the specific expression of ˜ 1c (tdt) and˜ 0c (tdt) and derive the conditional master equation for theCQD system measured by the PC. If a perfect PC detectoror efficient measurement is assumed, then whenever anelectron tunnels through the barrier, there is a measurementrecord corresponding to the occurrence of that event; thereare no ‘‘misses’’ in the count of the electron number. As aresult, the information lost from the system to the reservoirscan be recovered using a perfect detector. Here we assume azero-temperature case for the efficient measurement. At finitetemperatures, the electrons can, in principle, tunnel throughthe PC barrier in both directions. But experimentally the detectormight not be able to detect these electron tunnelingprocesses on both sides of the PC barrier. This may result ininformation loss at finite temperatures. Hence, at zero temperaturethe unconditional master equation 7 reduces to˙ ti H CQD ,tDTXn 1 t i H CQDiF*XFX*n 1 /2,tDXn 1 TFt,25a25b125326-5
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