Continuous quantum measurement of two coupled quantum dots using a point contact:

Continuous quantum measurement of two coupled quantum dots using a point contact:

A quantum trajectory approach

Hsi-Sheng Goan,^1,*G. J. Milburn,¹H. M. Wiseman,²and He Bi Sun¹

1Center for Quantum Computer Technology and Department of Physics, The University of Queensland, Brisbane Qld 4072, Australia

2School of Science, Griffith University, Nathan, Brisbane Qld 4111, Australia

共Received 21 June 2000; revised manuscript received 11 October 2000; published 13 March 2001兲 We obtain the finite-temperature unconditional master equation of the density matrix for two coupled quantum dots共CQD’s兲 when one dot is subjected to a measurement of its electron occupation number using a point contact共PC兲. To determine how the CQD system state depends on the actual current through the PC device, we use the so-called quantum trajectory method to derive the zero-temperature conditional master equation. We first treat the electron tunneling through the PC barrier as a classical stochastic point process共a quantum-jump model兲. Then we show explicitly that our results can be extended to the quantum-diffusive limit when the average electron tunneling rate is very large compared to the extra change of the tunneling rate due to the presence of the electron in the dot closer to the PC. We find that in both quantum-jump and quantum- diffusive cases, the conditional dynamics of the CQD system can be described by the stochastic Schro¨dinger equations for its conditioned state vector if and only if the information carried away from the CQD system by the PC reservoirs can be recovered by the perfect detection of the measurements.

DOI: 10.1103/PhysRevB.63.125326 PACS number共s兲: 73.63.Kv, 85.35.Be, 03.65.Ta, 03.67.Lx I. INTRODUCTION

The origins and mechanisms of decoherence 共dephasing兲 for quantum systems in condensed-matter physics have at- tracted much attention recently due to a number of studies in nanostructure mesoscopic systems^1–5 and various proposals for quantum computers.^6–9One of the issues is the connec- tion between decoherence and quantum measurements^10,11 for a quantum system. It was reported in a recent experiment³ with a ‘‘which-path’’ interferometer that Aharonov-Bohm interference is suppressed owing to the measurement of which path an electron takes through the double-path interferometer. A biased quantum point contact 共QPC兲 located close to a quantum dot, which is built in one of the interferometer’s arms, acts as a measurement device.

The change of transmission coefficient of the QPC, which depends on the electron charge state of the quantum dot, can be detected. The decoherence rate due to the measurement by the QPC in this experiment has been calculated in Refs. 12–

16.

A quantum-mechanical two-state system, coupled to a dissipative environment, provides a universal model for many physical systems. The indication of quantum coher- ence can be regarded as the oscillation or the interference between the probability amplitudes of finding a particle be- tween the two states. In this paper, we consider the problem of an electron tunneling between two coupled quantum dots 共CQD’s兲 using a low-transparency point contact 共PC兲 or tun- nel junction as a detector共environment兲 measuring the posi- tion of the electron 共see Fig. 1兲. This problem has been ex- tensively studied in Refs. 16–24. The case of measurements by a general QPC detector with arbitrary transparency has also been investigated in Refs. 12–15, 25, and 26. In addi- tion, a similar system measured by a single electron transis- tor rather than a PC has been studied in Refs.

27,21,19,22,24,28,29, and 30. The influence of the detector 共environment兲 on the measured system can be determined by the reduced density matrix obtained by tracing out the envi-

ronmental degrees of the freedom in the total, system plus environment, density matrix. The master equation 共or rate equations兲 for this CQD system have been derived and ana- lyzed in Refs. 16 and 14共here we refer to the rate equations as the first-order differential equations in time for both diag- onal and off-diagonal reduced density matrix elements兲. This 共unconditional兲 master equation is obtained when the results of all measurement records 共electron current records in this case兲 are completely ignored or averaged over, and describes only the ensemble average property for the CQD system.

However, if a measurement is made on the system and the results are available, the state or density matrix is a condi- tional state conditioned on the measurement results. Hence the deterministic, unconditional master equation cannot de- scribe the conditional dynamics of the CQD system in a single realization of continuous measurements that reflects the stochastic nature of an electron tunneling through the PC barrier. Consequently, the conditional master equation

FIG. 1. Schematic representation of two coupled quantum dots 共CQD’s兲 when one dot is subjected to a measurement of its electron occupation number using a low-transparency point contact共PC兲 or tunnel junction. Here␮L and␮Rstand for the chemical potentials in the left and right reservoirs, respectively.

should be employed. In condensed-matter physics usually many identical quantum systems are prepared at the same time and a measurement is made upon the systems. For ex- ample, in nuclear or electron magnetic resonance experi- ments, generally an ensemble of systems of nuclei and elec- trons are probed to obtain the resonance signals. This implies that the measurement result in this case is an average re- sponse of the ensemble systems. On the other hand, for vari- ous proposed condensed-matter quantum computer architectures,^6–9 how to read out physical properties of a single electronic qubit, such as charge or spin at a single electron level, is demanding. This is a nontrivial problem since it involves an individual quantum particle measured by a practical detector in a realistic environment. It is particu- larly important to take account of the decoherence intro- duced by the measurements on the qubit as well as to under- stand how the quantum state of the qubit, conditioned on a particular single realization of measurement, evolves in time for the purpose of quantum computing.

Korotkov^18,20 has obtained the Langevin rate equations for the CQD system. These rate equations describe the ran- dom evolution of the density matrix that both conditions and is conditioned by the PC detector output. In his approach, the individual electrons tunneling through the PC barrier were ignored and the tunneling current was treated as a continu- ous, diffusive variable. More precisely, he considered the change of the output current average over some small time␶^, 具^I典, with respect to the average current I_i, as a Gaussian white-noise distribution. He then updated具^I典 in the density- matrix elements using the new values of 具^I典 after each time interval␶. However, treating the tunneling current as a con- tinuous, diffusive variable is valid only when the average electron tunneling rate is very large compared to the extra change of the tunneling rate due to the presence of the elec- tron in the dot closer to the PC. The resulting derivation of the stochastic rate equations is semiphenomenological, based on basic physical reasoning to deduce the properties of the density matrix elements, rather than microscopic.

To make contact with the measurement output, in this paper we present a quantum trajectory 31,35–42,28 measure- ment analysis to the CQD system. We first use the quantum open system approach^31–34to obtain the unconditional Mar- kovian master equation for the CQD system, taking into ac- count the finite-temperature effect of the PC reservoirs. Par- ticularly, we assume that the transparency of the PC detector is small, in the tunnel-junction limit. Subsequently, we de- rive microscopically the zero-temperature conditional master equation by treating the electron tunneling through the PC as a classical stochastic point process 共also called a quantum- jump model兲.^37,42,28 Generally the evolution of the system state undergoing quantum jumps 共or other stochastic pro- cesses兲 is known as a quantum trajectory³¹. Real measure- ments 共for example, the photon number detection兲 that cor- respond approximately to the ideal quantum-jump共or point- process兲 measurement are made regularly in experimental quantum optics. For almost all-infinitesimal time intervals, the measurement result is null共no photon detected兲. The sys- tem in this case changes infinitesimally, but not unitarily.

The nonunitary component reflects the changing probabili-

ties for future events conditioned on past null events. At randomly determined times 共conditionally Poisson distrib- uted兲, there is a detection result. When this occurs, the sys- tem undergoes a finite evolution, called a quantum jump. In reality these point processes are not seen exactly due to a finite frequency response of the circuit that averages each event over some time. Nevertheless, we first take the zero- response time limit and consider the electron tunneling cur- rent consisting of a sequence of random ␦ function pulses, i.e., a series of stochastic point processes. Then we show explicitly that our results can be extended to the quantum- diffusive limit and reproduce the rate equations obtained by Korotkov.^18,20We refer to the case studied by Korotkov^18,20 as quantum diffusion, in contrast to the case of quantum jumps considered here. Hence our quantum trajectory ap- proach may be considered as a formal derivation⁴³of the rate equations in Refs. 18 and 20. We find in both quantum-jump and quantum-diffusive cases that the conditional dynamics of the CQD system can be described by the stochastic Schro¨- dinger equations31,35,37,40,42共SSE’s兲 for the conditioned state vector, provided that the information carried away from the CQD system by the PC reservoirs can be recovered by the perfect detection of the measurements.

This paper is organized as follows. In Sec. II, we sketch the derivation of the finite-temperature unconditional master equation for the QCD system. To determine how the CQD system state depends on the actual current through the PC device, we derive in Sec. III the zero-temperature conditional master equation and the SSE in the quantum-jump model.

Then in Sec. IV we extend the results to the case of quantum diffusion and obtain the corresponding conditional master equation and SSE. The analytical results in terms of Bloch sphere variables for the conditional dynamics are presented in Sec. V. Specifically, we analyze in this section the local- ization rate and mixing rate.^27,21,22Finally, a short conclusion is given in Sec. VI. The Appendix is devoted to the demon- stration of the equivalence between the conditional stochastic rate equations in Refs. 18–20 and those derived microscopi- cally in the present paper.

II. UNCONDITIONAL MASTER EQUATION FOR THE CQD AND PC MODEL

The appropriate way to approach quantum measurement problems is to treat the measured system, the detector共envi- ronment兲, and the coupling between them microscopically.

Following from Refs. 16,18 and 20, we describe the whole system共see Fig. 1兲 by the following Hamiltonian:

H⫽HCQD⫹HPC⫹Hcou p, 共1兲

where

HCQD⫽ប关␻1c₁^†c₁⫹␻2c₂^†c₂⫹⍀共c1

†c₂⫹c2

†c₁兲兴, 共2兲

HPC⫽ប

兺

_k ^{共␻kLaLk† aLk⫹␻kRaRk† aRk兲}

⫹

兺

_k,q ^{共TkqaLk† aRq⫹Tqk*aRq† aLk兲, 共3兲}

Hcou p⫽

兺

_k,q ^{c1†c1共␹kqaLk† aRq⫹␹qk*aRq† aLk兲. 共4兲}

HCQDrepresents the effective tunneling Hamiltonian for the measured CQD system. For simplicity, we assume strong inner and interdot Coulomb repulsion, so only one electron can occupy this CQD system. We label each dot with an index 1,2 共see Fig. 1兲 and let ci (c_i^†) and ប␻i represent the electron annihilation 共creation兲 operator and energy for a single electron state in each dot, respectively. The coupling between these two dots is given by ប⍀. The tunneling Hamiltonian for the PC detector is represented by HPC

where a_Lk, a_Rkandប␻k L,ប␻k

Rare, respectively, the fermion 共electron兲 field annihilation operators and energies for the left and right reservoir states at wave number k. One should not be confused by the electron in the CQD with the elec- trons in the PC reservoirs. The tunneling matrix element be- tween states k and q in left and right reservoir, respectively, is given by T_kq. Equation共4兲, Hcou p, describes the interac- tion between the detector and the measured system, depend- ing on which dot is occupied. When the electron in the CQD system is close to the PC共i.e., dot 1 is occupied兲, there is a change in the PC tunneling barrier. This barrier change re- sults in a change of the effective tunneling amplitude from T_kq→Tkq⫹␹kq. As a consequence, the current through the PC is also modified. This changed current can be detected, and thus a measurement of the location of the electron in the CQD system is effected.

The total density operator R(t) for the entire system in the interaction picture satisfies

R˙_I共t兲⫽⫺i

ប 关H_I共t兲,RI共0兲兴

⫺ 1

ប²

冕

^{0tdt⬘†HI共t兲,关HI共t⬘兲,RI共t⬘兲兴‡. 共5兲}

The dynamics of the entire system is determined by the time- dependent Hamiltonian:⁴⁴

H_I共t兲⫽

兺

_k,q ^{共Tkq⫹␹kqc1†c1兲aLk† aRqei(␻kL⫺␻kR)t⫹H.c., 共6兲}

where we have treated the sum of the tunneling Hamiltonian parts in HPC andHcou p as the interaction HamiltonianHI, and H.c. stands for Hermitian conjugate of the entire previ- ous term. By tracing both sides of Eq. 共5兲 over the bath 共reservoir兲 variables and then changing from the interacting picture to the Schro¨dinger picture, we obtain^31–33the finite- temperature, Markovian master equation for the CQD system:

␳^˙共t兲⫽⫺ i

ប 关H^CQD,␳共t兲兴⫹D关T⫹⫹X_⫹n₁兴␳共t兲

⫹D关T_⫺*_⫹X⫺*n₁兴␳共t兲, 共7兲 where␳^(t)⫽TrBR(t) and Tr_Bindicates a trace over reservoir variables. In arriving at Eq.共7兲, we have made the following assumption and approximations:共a兲 treating the left and right

fermion reservoirs in the PC as thermal equilibrium free- electron baths, 共b兲 weak system-bath coupling, 共c兲 small transparency of the PC, i.e., in the tunnel-junction limit,共d兲 uncorrelated and factorizable system-bath initial conditions, 共e兲 relaxation time scales of the reservoirs being much shorter than that of the system state, 共f兲 Markovian approxi- mation, 共g兲 兩eV兩,kBTⰆ␮L(R), and 共h兲 energy-independent electron tunneling amplitudes and density of states over the bandwidth of max(兩eV兩,kBT). Here k_B is the Boltzmann con- stant, T represents the temperature, eV⫽␮L⫺␮R is the ex- ternal bias applied across the PC, and ␮L and␮R stand for the chemical potentials in the left and right reservoirs, re- spectively. In Eq. 共7兲, n1⫽c1

†c₁ is the occupation number operator for dot 1. The parametersT_⫾ andX_⫾are given by 兩T_⫾兩²⫽D_⫾⫽2␲^e兩T00兩²g_Lg_RV_⫾/ប, 共8a兲 兩T_⫾⫹X_⫾兩²⫽D_⫾⬘⫽2␲^e兩T00⫹␹00兩²g_Lg_RV_⫾/ប, 共8b兲 where D_⫾ and D_⫾⬘ are the average electron tunneling rates through the PC barrier in positive and negative bias direc- tions at finite temperatures, without and with the presence of the electron in dot 1, respectively. Here the effective finite- temperature external bias potential eV_⫾ is given by the fol- lowing expression:

eV_⫾⬅ ⫾eV

1⫺ exp关⫿eV/共kBT兲兴. 共9兲 T₀₀ and ␹00 are energy-independent tunneling amplitudes near the average chemical potential, and g_L and g_R are the energy-independent density of states for the left and right fermion baths. Note that the average electron currents through the PC barrier is proportional to the difference be- tween the average electron tunneling rate in opposite direc- tions. Hence, the average currents eD⫽e(D_⫹⫺D_⫺) and eD⬘^⫽e(D⫹⬘⫺D_⫺⬘), following from Eqs. 共8兲 and 共9兲, are temperature independent^45,46at least for a range of low tem- peratures k_BTⰆ␮L(R). In addition, the current-voltage char- acteristic in the linear response region兩eV兩Ⰶ␮L(R)is of the same form as that for an Ohmic resistor, though the nature of charge transport is quite different in both cases.

We have also introduced, in Eq. 共7兲, an elegant superoperator37,28,47–49 D, widely used in measurement theory in quantum optics. Physically the ‘‘irreversible’’ part caused by the influence of the environment in the uncondi- tional master equation is represented by theD superoperator.

Generally superoperators transform one operator into another operator. Mathematically, the expression D关B兴␳ ^{means that} superoperator D takes its operator argument B, acting on␳^. Its precise definition is in terms of another two superopera- tors J and A:

D关B兴␳⫽J关B兴␳⫺A关B兴␳^, 共10兲 where

J关B兴␳⫽B␳^B†, 共11兲 A关B兴␳⫽共B^†B␳⫹␳^B†B兲/2. 共12兲

The form of the master equation 共7兲, defined through the superoperatorD关B兴␳(t), preserves the positivity of the den- sity matrix operator␳(t). Such a Markovian master equation is called a Lindblad⁵⁰form.

To demonstrate the equivalence between the master equa- tion共7兲 and the rate equations derived in Ref. 16, we evalu- ate the density matrix operator in the same basis as in Ref. 16 and obtain

␳^˙aa共t兲⫽i⍀关␳ab共t兲⫺␳ba共t兲兴, 共13a兲

␳^˙ab共t兲⫽i␧␳ab共t兲⫹i⍀关␳aa共t兲⫺␳bb共t兲兴

⫺共兩XT兩²/2兲␳ab共t兲⫹i Im共T_⫹*_X⫹

⫺T_⫺*_X⫺兲␳ab共t兲. 共13b兲 Here ប␧⫽ប(␻2⫺␻1) is the energy mismatch between the two dots, ␳i j(t)⫽具^i兩␳^(t)兩 j典^{, and} ␳aa(t) and ␳bb(t) are the probabilities of finding the electron in dot 1 and dot 2, re- spectively. The rate equations for the other two density ma- trix elements can be easily obtained from the relations:

␳bb(t)⫽1⫺␳aa(t) and␳ba(t)⫽␳ab*(t). Compared to an iso- lated CQD system, the presence of the PC detector intro- duces two effects to the CQD system. First, the imaginary part of the product of T_⫹*_X⫹⫺T⫺*_X⫺ 关the last term in Eq.

共13b兲兴 causes an effective temperature-independent shift in the energy mismatch between the two dots. Here (T_⫹*_X⫹

⫺T_⫺*_X⫹⫽T⫺*_X⫺)⫽T*X is a temperature-independent quan- tity whereT⫽T_⫹(0), i.e.,T_⫹andX_⫹evaluated at zero tem- perature, respectively. Second, it generates a decoherence 共dephasing兲 rate

⌫d⫽兩XT兩²/2 共14兲

for the off-diagonal density matrix elements, where 兩XT兩²

⫽兩X_⫹兩²⫹兩X_⫺兩². We note that the decoherence rate comes entirely from the effect of the measurement revealing where the electron in the CQD’s is located. If the PC detector does not distinguish which of the dots the electron occupies, i.e., X_⫾⫽0, then ⌫d⫽0. The rate equations in Eq. 共13兲 are ex- actly the same as the zero-temperature rate equations in Ref.

16 if we assume that the tunneling amplitudes are real, T₀₀

⫽T00* and␹00⫽␹00*. In that case, the last term in Eq.共13b兲 vanishes and⌫d⫽X²/2⫽(

冑

^D⬘⫺

冑

^D)2/2. Actually, the rela- tive phase between the two complex tunneling amplitudes may produce additional effects on conditional dynamics of the CQD system as well. This will be shown later when we discuss conditional dynamics. Physically, the presence of the electron in dot 1 raises the effective tunneling barrier of the PC due to electrostatic repulsion. As a consequence, the ef- fective tunneling amplitude becomes lower, i.e., D⬘^⫽兩T

⫹X兩²⬍D⫽兩T 兩². This sets a condition on the relative phase

␪ ^betweenX and T: cos␪⬍⫺兩X兩/(2兩T 兩).

The dynamics of the unconditional rate equations at zero temperature was analyzed in Ref. 16. Here, following from Eqs. 共14兲, 共8兲, and 共9兲, we find that the temperature- dependent decoherence rate due to the PC thermal reservoirs has the following expression:

⌫d共T兲

⌫d共0兲⫽e共V_⫹⫹V_⫺兲

eV ⫽coth

冉

^2keVBT

冊

^{. 共15兲}

As expected, ⌫d(T) increases with increasing temperature, although the average tunneling current through the PC is temperature independent^45,46for the same range of low tem- peratures k_BTⰆ␮L(R). This temperature dependence of the decoherence rate is in fact just the temperature dependence of the zero-frequency noise power spectrum of the current fluctuation in a low-transparency PC or tunnel junction.⁵¹ The CQD system weakly coupled to another finite- temperature environment beside the PC detector was dis- cussed in Ref. 20. However, the influence of the finite- temperature PC reservoirs on the CQD system, presented here, was not taken into account. The finite-temperature de- coherence rate of a one-electron state in a quantum dot due to charge fluctuation of a general QPC has been calculated in Ref. 13. In Ref. 26, the temperature-dependent decoherence rate for a two-state system caused by a QPC detector has been discussed specifically in the context of the measure- ment problem.

III. QUANTUM-JUMP, CONDITIONAL MASTER EQUATION

So far we have considered the evolution of the reduced density matrix when all the measurement results are ignored, or averaged over. To make contact with a single realization of the measurement records and study the stochastic evolu- tion of the quantum state, conditioned on a particular mea- surement realization, we derive in this section the quantum- jump, conditional master equation at zero temperature.

The nature of the measurable quantities, such as accumu- lated number of electrons tunneling through the PC barrier, is stochastic. On average, of course, the same current flows in both reservoirs. However, the current is actually made up of contributions from random pulses in each reservoir, which do not necessarily occur at the same time. They are indeed separated in time by the times at which the electrons tunnel through the PC. In this section, we treat the electron tunnel- ing current consisting of a sequence of random ␦^-function pulses. In other words, the measured current is regarded as a series of point processes共a quantum-jump model兲.^37,42,28The case of quantum diffusion will be analyzed in Sec. IV.

Before going directly to the derivation, we discuss some general ideas concerning quantum measurements. If the sys- tem under observation is in a pure quantum state at the be- ginning of the measurement, then it will still be in a pure conditional state after the measurement, conditioned on the result, provided no information is lost. For example, if the initial normalized state is兩␺^(t)典, the unnormalized final state given the result␣ at the end of the time interval关t,t⫹dt) of the measurement becomes

兩␺^˜␣共t⫹dt兲典^⫽M␣共dt兲兩␺共t兲典^{, 共16兲} where 兵^M␣(t)其 represents a set of operators that define the measurements and satisfies the completeness condition

兺

_␣ ^{M␣†共t兲M␣共t兲⫽1. 共17兲}

Equation共17兲 is simply a statement of conservation of prob- ability. The corresponding unnormalized density matrix, fol- lowing from Eq.共16兲, is given by

␳

˜␣共t⫹dt兲⫽兩␺^˜_␣共t⫹dt兲典具␺^˜_␣共t⫹dt兲兩⫽J 关M_␣共dt兲兴␳共t兲, 共18兲 where␳^(t)⫽兩␺^(t)典具␺^(t)兩 and the superoperatorJ is defined in Eq. 共11兲. Of course, if the measurement is made but the result is ignored, the final state will not be pure but a mixture of the possible outcome weighted by their probabilities. Con- sequently, the unconditional density matrix can be written as

␳共t⫹dt兲⫽

兺

_␣ ^{˜␳␣共t⫹dt兲⫽}

兺

_␣ ^{Pr关␣兴␳␣}共t⫹dt兲, 共19兲 where Pr关␣兴⫽Tr关^˜␳␣(t⫹dt)兴 stands for the probability for the system to be observed in the state ␣^{, and} ␳␣(t⫹dt)

⫽^˜␳␣(t⫹dt)/Pr关␣兴 is the normalized density matrix at time t⫹dt.

Now we proceed to derive the quantum-jump, conditional master equation in the following. Only two measurement op- erators M_␣(dt) for ␣⫽0,1 are needed for a measurement record that is a point process. For most of the infinitesimal time intervals, the measurement result is␣⫽0, regarded as a null result. On the other hand, at randomly determined times, there is a result␣⫽1, referred as a detection of an electron tunneling through the PC barrier. Formally, we can write the current through the PC as

i共t兲⫽edN共t兲/dt, 共20兲

where e is the electronic charge and dN(t) is a classical point process that represents the number共either zero or one兲 of tunneling events seen in an infinitesimal time dt. We can think of dN(t) as the increment in the number of electrons N(t) in the drain in time dt. It is this variable, the accumu- lated electron number transmitted through the PC, which is used in Refs. 16, 27, and 22. The point process is formally defined by the conditions on the classical random variable dN_c(t):

关dNc共t兲兴²⫽dNc共t兲, 共21兲

E关dNc共t兲兴⫽Tr关^˜␳1c共t⫹dt兲兴

⫽Tr兵J关M1共dt兲兴␳c共t兲其^⫽P1c共t兲dt. 共22兲 Here we explicitly use the subscript c to indicate that the quantity to which it is attached is conditioned on previous measurement results, the occurrences 共detection records兲 of the electrons tunneling through the PC barrier in the past.

E关Y 兴 denotes an ensemble average of a classical stochastic process Y. Equation 共21兲 simply states that dNc(t) equals either zero or one, which is why it is called a point process.

Equation共22兲 indicates that the ensemble average of dNc(t) equals the probability 共quantum average兲 of detecting elec- trons tunneling through the PC barrier in time dt. In addi- tion, dN_c(t) is of order dt and obviously all moments共pow- ers兲 of dNc(t) are of the same order as dt. Note here that the

density matrix ␳c(t) is not the solution of the unconditional reduced master equation, Eq.共25a兲. It is actually conditioned by dN_c(t⬘^{) for t}⬘^⬍t.

The stochastic conditional density matrix at a later time t⫹dt can be written as

␳c共t⫹dt兲⫽dNc共t兲 ␳^˜1c共t⫹dt兲 Tr关␳^˜1c共t⫹dt兲兴

⫹关1⫺dNc共t兲兴 ^˜␳0c共t⫹dt兲

Tr关^˜␳0c共t⫹dt兲兴. 共23兲 Equation共23兲 states that when dNc(t)⫽0 共a null result兲, the system changes infinitesimally via the operator M₀(dt) and hence ␳c(t⫹dt)⫽␳0c(t⫹dt). Conversely, if dNc(t)

⫽1 共a detection兲, the system goes through a finite evolution induced by the operator M₁(dt), called a quantum jump. The corresponding normalized conditional density matrix then becomes␳1c(t⫹dt). One can see, with the help of Eq. 共20兲, that in this approach the instantaneous system state condi- tions the measured current关see Eq. 共22兲兴, while the measured current itself conditions the future evolution of the measured system 关see Eq. 共23兲兴 in a self-consistent manner. It is straightforward to show that the ensemble average of the conditional density matrix equals the unconditional one, E关␳c(t)兴⫽␳(t). Tracing over both sides of Eq. 共19兲 for ␣

⫽0,1, we obtain

Tr关^˜␳0c共t⫹dt兲兴⫽1⫺Tr关␳^˜1c共t⫹dt兲兴. 共24兲 Then taking the ensemble average over the stochastic vari- ables dN_c(t) on both sides of Eq.共23兲, replacing E关dNc(t)兴 by using Eq.共22兲, and comparing the resultant equation with Eq. 共19兲 completes the verification.

Next we find the specific expression of ^˜␳1c(t⫹dt) and

␳

˜_0c(t⫹dt) and derive the conditional master equation for the CQD system measured by the PC. If a perfect PC detector 共or efficient measurement兲 is assumed, then whenever an electron tunnels through the barrier, there is a measurement record corresponding to the occurrence of that event; there are no ‘‘misses’’ in the count of the electron number. As a result, the information lost from the system to the reservoirs can be recovered using a perfect detector. Here we assume a zero-temperature case for the efficient measurement. At finite temperatures, the electrons can, in principle, tunnel through the PC barrier in both directions. But experimentally the de- tector might not be able to detect these electron tunneling processes on both sides of the PC barrier. This may result in information loss at finite temperatures. Hence, at zero tem- perature the unconditional master equation共7兲 reduces to

␳^˙共t兲⫽⫺ i

ប 关H^CQD,␳共t兲兴⫹D关T⫹Xn1兴␳共t兲 共25a兲

⫽⫺i

ប 关H^CQD⫺iប共F*_X⫺FX*_兲n1/2,␳共t兲兴

⫹D关Xn1⫹T⫹F兴␳共t兲, 共25b兲

⬅L␳共t兲, 共25c兲 whereD is defined in Eq. 共10兲. Here F is an arbitrary com- plex number,^48,49 while we are usingT and X to represent, respectively, the quantitiesT_⫹andX_⫹in Eq.共8兲 evaluated at zero temperature.

Requiring that the ensemble average of the conditioned density matrix E关␳c(t⫹dt)兴⫽␳^(t⫹dt) satisfies the uncon- ditional master equation共25兲 leads to

␳

˜_0c共t⫹dt兲⫹␳^˜1c共t⫹dt兲⫽共1⫹dtL兲␳c共t兲. 共26兲 Here we have explicitly used the stochastic Itoˆ calculus^52,53 for the definition of time derivatives as ␳^{˙ (t)}⫽lim_dt→0关␳^(t

⫹dt)⫺␳^(t)兴/dt. This is in contrast to the definition ␳^{˙ (t)}

⫽lim_dt→0关␳^(t⫹dt/2)⫺␳^(t⫺dt/2)兴/dt, used in another sto- chastic calculus, the Stratonovich calculus.^52,53 Recall that Eq.共22兲 indicates that E关dNc(t)兴/dt equals the average elec- tron tunneling rate through the PC barrier. From Eq.共8兲, the electron tunneling rates are D⫽兩T 兩² when n₁⫽0 and D⬘

⫽兩T⫹X兩² when n₁⫽1. From Eq. 共22兲 we thus have the correspondence

Tr关M1共dt兲␳c共t兲M1

†共dt兲兴

⫽Tr兵␳c共t兲关T*_⫹n1␹*兴关T⫹n1␹兴其^dt. 共27兲 Also, for Eq.共26兲 to reproduce the master equation 共25b兲 we must have^48,49

M₁共dt兲⫽

冑

^dt共Xn1⫹T⫹F兲 共28兲 for some arbitrary complex numberF. By inspection of Eq.

共27兲 we must have F⫽0, so that

␳

˜1c共t⫹dt兲⫽J 关T⫹Xn1兴␳c共t兲dt. 共29兲 Substituting Eq.共29兲 into 共22兲 yields

E关dNc共t兲兴⫽Tr关^˜␳1c共t⫹dt兲兴⫽关D⫹共D⬘⫺D兲具ⁿ1典c共t兲兴dt, 共30兲 where 具ⁿ1典c(t)⫽Tr关n1␳c(t)兴. The remaining part, except the jump of Eq. 共29兲, on the right hand side of Eq. 共26兲 in time dt, corresponds to the effect of a measurement giving a null result on ␳c(t):

␳

˜_0c共t⫹dt兲⫽␳c共t兲⫺dt

⫻

再

^{A关T⫹Xn1兴␳c共t兲⫺ប 关Hi CQD,␳c共t兲兴}

冎

^,

共31兲 whereA is defined in Eq. 共12兲. The corresponding measure- ment operator is

M₀共dt兲⫽1⫺dt关共i/ប兲HCQD

⫹共1/2兲共T*_⫹X*n₁兲共T⫹Xn1兲兴. 共32兲 Finally, substituting Eqs. 共29兲, 共31兲, 共24兲, and 共30兲 into Eq. 共23兲, expanding, and keeping the terms of first order in

dt, we obtain the stochastic master equation, conditioned on the observed event in time dt:

d␳c共t兲⫽dNc共t兲

冋

^{J关T⫹XnP1c共t兲1兴⫺1}

册

^␳c共t兲

⫹dt

再

^{⫺A关T⫹Xn1兴␳c共t兲⫹P1c共t兲␳c共t兲}

⫹ i

ប 关H^CQD,␳c共t兲兴

冎

^{, 共33兲}

where

P1c共t兲⫽D⫹共D⬘⫺D兲具ⁿ1典c共t兲. 共34兲 Note that dN_c(t), from Eq.共30兲, is of order dt. Hence terms proportional to dN_c(t)dt are ignored in Eq.共33兲. Again av- eraging this equation over the observed stochastic process by setting E关dNc(t)兴 equal to its expected value, Eq. 共30兲, gives the unconditional, deterministic master equation共25a兲. Equa- tion 共33兲 is one of the main results in this paper.

So far we have assumed perfect detection or efficient measurement. In this case, the stochastic master equation for the conditioned density-matrix operator 共33兲 is equivalent to the following stochastic Scho¨dinger equation 共SSE兲 for the conditioned state vector:

d兩␺c共t兲典^⫽

冋

^dNc共t兲

冉 ^冑

^{T⫹XnP1c共t兲1⫺1}

冊

^⫺dt

^冉

^{បHi CQD}

⫹共T*_⫹X*n₁兲共T⫹Xn1兲

2 ⫺P1c共t兲

2

冊 册

^{兩␺c共t兲典.}

共35兲 This equivalence can be easily verified using the stochastic Itoˆ calculus^52,53

d␳c共t兲⫽d共兩␺c共t兲典具␺c共t兲兩兲

⫽关d兩␺c共t兲典^]具␺c共t兲兩⫹兩␺c共t兲典^d具␺c共t兲兩

⫹关d兩␺c共t兲典兴关d具␺c共t兲兩兴, 共36兲 and keeping terms up to order dt. Since the evolution of the system can be described by a ket state vector, it is obvious that an efficient measurement or perfect detection preserves state purity if the initial state is a pure state. In this descrip- tion of the SSE, the quantum average is now defined, for example, as 具ⁿ1典c(t)⫽具␺c(t)兩n1兩␺c(t)典. The unconditional density-matrix operator is equivalent to the ensemble aver- age of quantum trajectories generated by the SSE, ␳^(t)

⫽E关兩␺c(t)典具␺c(t)兩兴, provided that the initial density opera- tor can be written as ␳⁽⁰⁾⫽兩␺c(0)典具␺c(0)兩.

The interpretation³⁷ for the measured system state condi- tioned on the measurement, in terms of gain and loss of information, can be summarized and understood as follows.

In order for the system to be continuously described by a state vector 共rather than a general density matrix兲, it is nec-

essary 共and sufficient兲 to have maximal knowledge of its change of state. This requires perfect detection or efficient measurement, which recovers and contains all the informa- tion lost from the system to the reservoirs. If the detection is not perfect and some information about the system is un- traceable, the evolution of the system can no longer be de- scribed by a pure state vector. For the extreme case of zero efficiency detection, the information共measurement results at the detector兲 carried away from the system to the reservoirs is 共are兲 completely ignored, so that the stochastic master equation共33兲 after being averaged over all possible measure- ment records reduces to the unconditional, deterministic master equation 共25a兲, leading to decoherence for the sys- tem. This interpretation highlights the fact that a density- matrix operator description of a quantum state is only neces- sary when information is lost irretrievably. The purity- preserving, conditional state evolution for a pure initial state and gradual purification for a nonpure initial state have been discussed in Refs. 18–20 and 23 in the quantum-diffusive limit.

IV. QUANTUM-DIFFUSIVE, CONDITIONAL MASTER EQUATION

In this section, we extend the results obtained in the pre- vious section and derive the conditional master equation when the average electron tunneling current is very large compared to the extra change of the tunneling current due to the presence of the electron in the dot closer to the PC. This limit is studied and called a ‘‘weakly coupling or responding detector’’ limit in Refs. 18 and 20. Here, on the other hand, we will refer to this case as quantum diffusion in contrast to the case of quantum jumps. In the quantum-diffusive limit, many electrons, (N⬎关(D⬘⫹D)/(D⬘⫺D)兴²Ⰷ1), pass through the PC before one can distinguish which dot is oc- cupied. In addition, individual electrons tunneling through the PC are ignored and time averaging of the currents is performed. This allows electron counts, or the accumulated electron number, to be considered as a continuous variable satisfying a Gaussian white-noise distribution. In Refs. 18 and 20 a set of Langevin equations for the random evolution of the CQD system density-matrix elements conditioned on the detector output was presented, based only on basic physi- cal reasoning. In this section, we show explicitly, under the quantum-diffusive limit, that our microscopic approach reproduces⁴³the rate equations in Refs. 18 and 20.

In quantum optics, a measurement scheme known as ho- modyne detection^31,47,48 is closely related to the measure- ment of the CQD system by a weakly responding PC detec- tor. In both cases, there is a large parameter to allow the photocurrent or electron current to be approximated by a continuous function of time. We will follow closely the deri- vation of a smooth master equation for homodyne detection given in Ref. 48 共sketched first by Carmichael³¹兲 for the CQD system.

There are two ideal parameters T and X for the CQD system. In the quantum-diffusive limit, we assume 兩T兩 Ⰷ兩X兩, which is consistent with the assumption, (D⫹D⬘⁾ Ⰷ(D⬘⫺D), made in Refs. 18 and 20 for the weakly cou-

pling or weakly responding PC detector. Consider the evolu- tion of the system over the short-time interval关t,t⫹␦^{t). We} relate the three parameters, X, T, and␦t in our problem as 兩X兩²␦^t⬃⑀^{3/2, where} ⑀⫽(兩X 兩/兩T 兩)Ⰶ1. This scaling is cho- sen so that in time ␦t, the number of detections 共electron counts兲 with dot 1 being unoccupied scales as␦^N⬃兩T 兩²␦^t

⬃⑀^⫺1/2Ⰷ1. However, the extra change in electron number detections due to the presence of the electron in dot 1 scales as 兩X 兩²␦^t⬃⑀^3/2Ⰶ1. To be more specific, the average num- ber of detections, following Eq.共30兲, up to order of⑀^1/2is

E关␦^N共t兲兴⫽兩T 兩²␦^t关1⫹2⑀^cos␪具ⁿ1典c共t兲兴, 共37兲 where␪ is the relative phase betweenX and T. The variance in ␦N will be dominated by the Poisson statistics of the current eD⫽e兩T 兩²in time␦t. Since the number of counts in time ␦t is very large, the statistics will be approximately Gaussian. Indeed, it has been shown⁴⁷ that the statistics of

␦N are consistent with that of a Gaussian random variable of mean given by Eq.共37兲 and the variance up to order of⑀^⫺1/2 is␴N

2⫽兩T 兩²␦t. The fluctuation␴N

2 is necessarily as large as expressed here in order for the statistics of␦N to be consis- tent with Gaussian statistics. Thus,␦N can be approximately written as a continuous Gaussian random variable:^52,53

␦^N共t兲⫽兵兩T 兩²关1⫹2⑀^cos␪具ⁿ1典c共t兲兴⫹兩T 兩␰共t兲其␦^t, 共38兲 where␰(t) is a Gaussian white noise characterized by

E关␰共t兲兴⫽0, E关␰共t兲␰共t⬘兲兴⫽␦共t⫺t⬘兲. 共39兲 Here E denotes an ensemble average and␦^(t⫺t⬘) is a delta function. In stochastic calculus,^52,53␰^(t)dt⫽dW(t) is known as the infinitesimal Wiener increment. In Eq.共38兲, the accu- racy in each term is only as great as the highest-order ex- pression in⑀^1/2. But it is sufficient for the discussions below.

Although the conditional master equation 共33兲 requires dN_c(t) to be a point process, it is possible, in the quantum- diffusive limit, to simply replace dN_c(t) by the continuous random variable␦^Nc(t), Eq.共38兲. This is because each jump is infinitesimal, so the effect of many jumps is approximately equal to the effect of one jump scaled by the number of jumps. This can be justified more rigorously as in Ref. 47.

Finally, expanding Eq. 共33兲 in power of ⑀, substituting dN_c(t)→␦^Nc(t), keeping only the terms up to the order

⑀^3/2, and letting ␦^t→dt, we obtain the conditional master equation

␳^˙c共t兲⫽⫺ i

ប 关H^CQD,␳c共t兲兴⫹D关T⫹Xn1兴␳c共t兲

⫹␰共t兲 1

兩T 兩 关T*_Xn1␳c共t兲⫹X*_T␳c共t兲n1⫺2 Re共T*_X兲

⫻具ⁿ1典c共t兲␳c共t兲兴. 共40兲

Thus the quantum-jump evolution of Eq. 共33兲 has been re- placed by quantum-diffusive evolution, Eq. 共40兲. Following the same reasoning in obtaining the SSE, Eq. 共35兲, for the

case of quantum-jump process, we find the quantum- diffusive, conditional master equation 共40兲 is equivalent to the following diffusive SSE:

d兩␺c共t兲典⫽

冋

^dt

冉

^{⫺បHi CQD⫺兩X兩22关n1⫺2n1具n1典c共t兲}

⫹具ⁿ1典c

2共t兲兴⫺i Im共T*_X兲n1

册冊

⫹␰共t兲dt 1

兩T 兩兵^T*_Xn1⫺X*_T具ⁿ1典c共t兲其^兩␺c共t兲典^. 共41兲 This equivalence can be verified using Eq.共36兲 and keeping terms up to order dt. Note, however, in this case^52,53 that terms of order␰(t)dt are to be regarded as the same order as dt, but关␰^(t)dt兴²⫽关dW(t)兴²⫽dt.

Our conditional master equation by its derivation is for- mulated in terms of Itoˆ calculus, while the stochastic rate equations in Refs. 18 and 20 are written in a Stratonovich calculus form.^52,53In contrast to the Stratonovich form of the rate equations, it is easy to see that the ensemble average evolution of our conditional master equation共40兲 reproduces the unconditional master equation共25a兲 by simply eliminat- ing the white-noise term using Eq. 共39兲. To show that our quantum-diffusive, conditional stochastic master equation 共40兲 reproduces the nonlinear Langevin rate equations ob- tained semiphenomenologically in Refs. 18 and 20, we evaluate Eq.共40兲 in the same basis as for Eq. 共13兲 and obtain

␳^˙aa共t兲⫽i⍀关␳ab共t兲⫺␳ba共t兲兴⫺

冑

⁸⌫d␳aa共t兲␳bb共t兲␰共t兲, 共42a兲

␳^˙ab共t兲⫽i␧␳ab共t兲⫹i⍀关␳aa共t兲⫺␳bb共t兲兴⫺⌫d␳ab共t兲

⫹

冑

²⌫d␳ab共t兲关␳aa共t兲⫺␳bb共t兲兴␰共t兲. 共42b兲 In obtaining Eq.共42兲, we have made the assumption of real tunneling amplitudes共i.e., 0_␲) as in Refs. 16, 18 and 20 in order to be able to compare the results directly. We have also set X⫽

冑

²⌫d. Again, the ensemble average of Eq. 共42兲 by eliminating the white-noise terms reduces to Eq. 共13兲. To further demonstrate the equivalence, we translate the sto- chastic rate equations of Refs. 18 and 20 into the Itoˆ formal- ism and compare them to Eq.共42兲. This is carried out in the Appendix. Indeed, Eq.共42兲 is equivalent to the Langevin rate equations in Refs. 18 and 20 for the ‘‘ideal detector.’’

V. ANALYTICAL RESULTS FOR CONDITIONAL DYNAMICS

To analyze the dynamics of a two-state system, such as the CQD system considered here, one can represent the sys- tem density-matrix elements in terms of Bloch sphere vari- ables. The Bloch sphere representation is equivalent to that of the rate equations. However, some physical insights into the dynamics of the system can sometimes be more easily visualized in this representation. Denoting the averages of the operators␴x,␴y,␴zby x, y, z, respectively, the density-

matrix operator for the CQD system can be expressed in terms of the Bloch sphere vector (x, y ,z) as

␳共t兲⫽关I⫹x共t兲␴x⫹y共t兲␴y⫹z共t兲␴z兴/2 共43a兲

⫽1

2

冉

^{x共t兲⫹iy共t兲1⫹z共t兲 x共t兲⫺iy共t兲1⫺z共t兲}

冊

^,

共43b兲 where the operators I and ␴i are defined using the fermion operators for the two dots:

I⫽c2

†c₂⫹c1

†c₁, 共44a兲

␴x⫽c2

†c₁⫹c1

†c₂, 共44b兲

␴y⫽⫺ic2

†c₁⫹ic1

†c₂, 共44c兲

␴z⫽c2

†c₂⫺c1

†c₁. 共44d兲

It is easy to see that Tr␳^(t)⫽1, I is a unit operator, and ␴i

defined above satisfies the properties of Pauli matrices. In this representation, the variable z(t) represents the popula- tion difference between the two dots. Especially, z(t)⫽1 and z(t)⫽⫺1 indicate that the electron is localized in dot 2 and dot 1, respectively. The value z(t)⫽0 corresponds to an equal probability for the electron to be in each dot.

The master equations共25a兲, 共40兲, and 共33兲, can be written as a set of coupled stochastic differential equations in terms of the Bloch sphere variables. For simplicity, in this section we assume that the tunneling amplitudes are real, i.e., 0⫽␲^, and we set 兩T 兩⫽T and X⫽

冑

²Td. By substituting Eq. 共43a兲 into Eq.共25a兲, and collecting and equating the coefficients in front of ␴x, ␴y, ␴z respectively, the unconditional master equation under the assumption of real tunneling amplitudes is equivalent to the following equations:

d

dt

冉

^{xy共t兲共t兲}

冊

^⫽

冉

^{⫺⌫␧ d ⫺⌫⫺␧d}

冊冉

^{xy共t兲共t兲}

冊

^⫹

冉

^{⫺2⍀z共t兲0}

冊

^,

共45a兲 dz共t兲

dt ⫽2⍀y共t兲. 共45b兲

Similarly for the quantum-diffusive, conditional master equation共40兲, we obtain

dx_c共t兲

dt ⫽⫺␧yc共t兲⫺⌫dx_c共t兲⫺

冑

²⌫dz_c共t兲xc共t兲␰共t兲, 共46a兲 d y_c共t兲

dt ⫽␧xc共t兲⫺2⍀zc共t兲⫺⌫dy_c共t兲⫺

冑

²⌫dz_c共t兲yc共t兲␰共t兲, 共46b兲 dz_c共t兲

dt ⫽2⍀yc共t兲⫹

冑

²⌫d关1⫺zc

2共t兲兴␰共t兲. 共46c兲 Again the c subscript is to emphasize that these variables refer to the conditional state. It is trivial to see that Eq. 共46兲 averaged over the white noise reduces to Eq. 共45兲, provided