Dynamics of a mesoscopic charge quantum bit under continuous quantum measurement

Dynamics of a mesoscopic charge quantum bit under continuous quantum measurement

Hsi-Sheng Goan*and Gerard J. Milburn

Center for Quantum Computer Technology and Department of Physics, University of Queensland, Brisbane, Queensland 4072, Australia 共Received 1 March 2001; revised manuscript received 6 July 2001; published 16 November 2001兲

We present the conditional quantum dynamics of an electron tunneling between two quantum dots subject to a measurement using a low transparency point contact or tunnel junction. The double dot system forms a single qubit and the measurement corresponds to a continuous in time readout of the occupancy of the quantum dot.

We illustrate the difference between conditional and unconditional dynamics of the qubit. The conditional dynamics is discussed in two regimes depending on the rate of tunneling through the point contact: quantum jumps, in which individual electron tunneling current events can be distinguished, and a diffusive dynamics in which individual events are ignored, and the time-averaged current is considered as a continuous diffusive variable. We include the effect of inefficient measurement and the influence of the relative phase between the two tunneling amplitudes of the double dot/point contact system.

DOI: 10.1103/PhysRevB.64.235307 PACS number共s兲: 73.63.Kv, 85.35.Be, 03.65.Ta, 03.67.Lx I. INTRODUCTION

One of the key requirements for a physically implement- ing a quantum computational scheme is the ability to readout a single quantum bit共qubit兲 with high efficiency.¹ In an ion trap implementation this problem has already been solved using shelving spectroscopy.² However in solid state schemes implementing a high efficiency measurement of the charge or spin degree of freedom of a single electron 共or Cooper pair兲 will be very challenging. Various implementa- tions of quantum bits共qubits兲 and quantum gates for a solid- state quantum computer has been proposed.^3–7 The condi- tional dynamics of a single quantum particle 共qubit兲 in a single realization of continuous measurements is quite differ- ent from the ensemble average共unconditional兲 behavior that is more familiar to the condensed matter physics community.

An apparatus by its very nature as a measurement device, must at least cause decoherence of the measured system in the basis which diagonalizes the measured quantity. From this perspective, the measurement apparatus behaves like an environment, that is, a system with many degrees of freedom for which correlations between its subcomponents decay rap- idly with time. Indeed for a system to function as a measure- ment apparatus it must be composed of many degrees of freedom.⁸Thus every measured system is an open system. To understand the influence of the detector共environment兲 on the measured system, the conventional approach is to study the 共unconditional兲 master equation of the reduced density ma- trix. However, integrating or tracing out the environmental 共detector兲 degrees of the freedom to obtain the reduced den- sity matrix is equivalent to completely ignoring or averaging over the results of all measurement records. This averaging means the detector is treated as a pure environment for the system, rather than a measurement device which can provide information about the change of the state of the qubit. On the other hand, for the purpose of quantum computing, it is im- portant to understand how the quantum state of a single qu- bit, conditioned on a particular single realization of the mea- surement, evolves in time. A number of questions need to be answered that cannot be answered if we only determine the ensemble averaged behavior of the measured qubit. For ex-

ample, in the case of a continuous measurement it is neces- sary to determine how long it takes for a confident determi- nation of the state of the qubit at the start of the measurement, even if the qubit itself undergoes additional coherent evolution during the measurement process. Further- more it may be possible to consider adaptive measurement schemes which take a given time continuous measurement record, subject it to real-time signal processing, and then change the way in which the measurement acts through a feedback loop. Such schemes are already being implemented in quantum optics and offer the promise of reaching sensi- tivities at the quantum limit.^9,10

We illustrate, in this paper, the difference between condi- tional and unconditional 共ensemble average兲 dynamics by considering the problem of an electron tunneling between two coherently coupled quantum dots 共CQD’s兲, a two-state quantum system共qubit兲, using a low-transparency point con- tact共PC兲 or tunnel junction as a detector 共environment兲 con- tinuously measuring the position of the electron, schemati- cally illustrated in Fig. 1. We assume strong inner and inter

FIG. 1. Schematic representation of an electron tunneling be- tween two coupled quantum dots 共CQD’s兲, a two-state quantum system共qubit兲, using a low-transparency point contact 共PC兲 or tun- nel junction as a detector共environment兲 continuously measuring the position of the electron. Here ␮L and ␮R stand for the chemical potentials in the left and right reservoirs, respectively.

dot Coulomb repulsion, so only one electron can occupy this CQD system. The logical qubit states in this case are, respec- tively, the perfect localization of the electron charge states in one of the two CQD’s. A controlled-not-gate operation based on the charge qubit of two asymmetric CQD’s has been sug- gested in Ref. 7. Experimentally, coherent coupling between two CQD’s has been reported. It has been shown^11,12that if the inter-dot tunneling barrier is low and the strength of the coupling of two CQD’s is strong, the two CQD’s behave as a large single dot in a Coulomb blockade phenomenon. In ad- dition, the energy splitting between bonding and antibonding states of two CQD’s has been confirmed by microwave ab- sorption measurements.^13,14The CQD system studied here is similar to the superconducting Cooper-pair-box charge qubit^5,15,16in that they both use charge degrees of freedom as qubit basis states. For the superconducting Cooper-pair box, the charge on the island differs by the number of Cooper pairs times the charge 2e, compared to the electron charge e in one of the two dots in the CQD system. The PC, consid- ered here, is a charge-sensitive detector. The tunneling bar- rier height or the current through the tunneling junction of the PC detector depends on the proximity of an external charge. Hence the study of charge measurements by a PC detector is applicable to different types of charge qubit, such as the CQD’s or the Cooper-pair box. The problem of the CQD system measured by a low-transparency PC has been extensively studied in Refs. 17–26. The case of measure- ments by a general quantum point contact detector with ar- bitrary transparency has also been investigated in Refs. 27–

32. In addition, a similar system, a Cooper-pair-box qubit, measured by a single electron transistor has been studied in Refs. 33, 22, 20, 23, 25, 34, and 35.

Korotkov^19,21,25has obtained the Langevin rate equations for the CQD system measured by an ideal PC detector. These rate equations describe the random evolution of the density matrix that both conditions, and is conditioned by, the PC detector output. Recently, Ref. 26 presented a quantum trajectory^{36 – 46}measurement analysis of the same system. We found that the conditional dynamics of the CQD system can be described by the stochastic Schro¨dinger equation for the conditioned state vector, provided that the information car- ried away from the CQD system by the PC reservoirs can be recovered by the perfect detection of the measurements. We also analyzed the localization rates at which the qubit be- comes localized in one of the two states when the coupling frequency⍀ between the states is zero. We showed that the localization time discussed there is slightly different from the measurement time defined in Refs. 33,22,23. The mixing rate at which the two possible states of the qubit become mixed when ⍀⫽0 was calculated as well and found in agreement with the result in Refs. 22 and 23. In this paper, we focus on the qubit dynamics conditioned on a particular realization of the actual measured current through the PC device. Espe- cially, we take into account the effect of inefficient measure- ment on the conditional dynamics and illustrate the condi- tional quantum evolutions by numerical simulations.

The problem of a ‘‘nonideal’’ detector was discussed in Refs. 19–21. There the nonideality of the detector is mod- eled as two ideal detectors ‘‘in parallel’’ with the output of

the second detector inaccessible. The information loss is due to the interaction with the second detector, treated as a ‘‘pure environment’’共which does not affect the observed detector current兲. As a consequence, the decoherence rate, ⌫tot, in that case is larger than the decoherence rate for the PC as an environment alone, ⌫tot⫺⌫d⫽␥d⬎0. Hence an extra deco- herence term, ⫺␥d␳ab, for example, is added in the rate equation␳^˙ab. However, this approach does not account for the inefficiency in the measurements, which arises when the detector sometimes misses detection. In that case, there is still only one PC detector共environment兲 and disregarding all measurement records leads to⌫tot⫽⌫d. Furthermore, the de- tector current is affected and in fact reduced by the ineffi- ciency in the measurements.

In this paper, we take into account the effect of inefficient measurement of the PC detector on the dynamics of the qu- bit. We also analyze the conditional qubit dynamics analyti- cally and numerically. The different behavior of uncondi- tional and conditional evolution is demonstrated. We present the conditional quantum dynamics over the full range of be- havior, from quantum jumps to quantum diffusion.²⁶In Refs.

17, 19, 21, and 25, the two tunneling amplitudes of the CQD–PC model were assumed to be real. In Ref. 26, the relative phase between them was taken into account. Here, we discuss and illustrate furthermore their influence on the qubit dynamics. In Sec. II, we describe the model Hamil- tonian and the unconditional master equation. We then obtain in Sec. III the quantum-jump and quantum-diffusive, condi- tional master equations for the case of inefficient measure- ments. Section IV is devoted to the analysis for the qubit dynamics. Numerical simulations of the conditional evolu- tion are presented in this section. Finally, a short conclusion is given in Sec. V. In the Appendix, the stationary noise power spectrum of the current fluctuations through the PC barrier is calculated in terms of the quantum-jump formal- ism.

II. UNCONDITIONAL MASTER EQUATION FOR THE CQD AND PC MODEL

Following the model of Refs. 17, 19, 21, and 26, we de- scribe the whole system共see Fig. 1兲 by the following Hamil- tonian:

H⫽HCQD⫹HPC⫹Hcoup, 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Tkqa_Lk^† a_Rq⫹Tqk*a_Rq^† a_Lk兲, 共3兲 Hcoup⫽

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 共mesoscopic charge qubit兲. The tun- neling Hamiltonian for the PC detector is represented by HPC. Here c_i (c_i^†) andប␻i represent the electron annihila- tion共creation兲 operator and energy for a single electron state in each dot, respectively. The coupling between these two dots is given byប⍀. Similarly, aLk,a_Rkandប␻k

L,ប␻k R are, respectively, the electron annihilation operators and energies for the left and right reservoir states at wave number k.

Hcoup, Eq. 共4兲, describes the interaction between the detec- tor and the measured system, depending on which dot is occupied. When the electron in the CQD system is located in dot 1, the effective tunneling amplitude of the PC detector changes from T_kq→Tkq⫹␹kq.

The 共unconditional兲 zero-temperature,⁴⁷ Markovian mas- ter equation of the reduced density matrix for the CQD sys- tem 共qubit兲 has been obtained in Refs. 17 and 26:

␳^˙共t兲⫽⫺ i

ប 关H^CQD,␳共t兲兴⫹D关T⫹Xn1兴␳共t兲 共5a兲

⬅L␳共t兲, 共5b兲

where n₁⫽c1

†c₁is the occupation number operator for dot 1 and the parameters T and X are given by D⫽兩T 兩²

⫽2␲^e兩T00兩²g_Lg_RV/ប and D⬘⫽兩T⫹X 兩²⫽2␲^e兩T00

⫹␹00兩²g_Lg_RV/ប. Here D and D⬘ are the average electron tunneling rates through the PC barrier without and with the presence of the electron in dot 1, respectively, eV⫽␮L

⫺␮R is the external bias applied across the PC (␮L and␮R

stand for the chemical potentials in the left and right reser- voirs, respectively兲, T00and␹00are energy-independent tun- neling amplitudes near the average chemical potential, and g_L and g_R are the energy-independent density of states for the left and right reservoirs. In Eq. 共5a兲, the superoperator^39,48,42D is defined as

D关B兴␳⫽J 关B兴␳⫺A关B兴␳^, 共6兲 where

J 关B兴␳⫽B␳^B†, 共7兲

A关B兴␳⫽共B^†B␳⫹␳^B†B兲/2. 共8兲 Finally, Eq.共5b兲 defines the Liouvillian operator L.

Evaluating the density matrix operator in the logical qubit charge states, 兩a典 ^and兩b典 共i.e., perfect localization state of the charge in dot 1 and dot 2, respectively兲, as in Ref. 17, we obtain

␳^˙aa共t兲⫽i⍀关␳ab共t兲⫺␳ba共t兲兴, 共9a兲

␳^˙ab共t兲⫽iE␳ab共t兲⫹i⍀关␳aa共t兲⫺␳bb共t兲兴⫺共兩X 兩²/2兲␳ab共t兲

⫹i Im 共T*_X兲␳ab共t兲, 共9b兲

whereបE⫽ប(␻2⫺␻1) is the energy mismatch between the two dots, ⌫d⫽兩X 兩²/2 is the decoherence rate, and ␳i j(t)

⫽具^i兩␳^(t)兩 j典. The relative phase between the two complex tunneling amplitudes (T and X ) 关the last term in Eq. 共9b兲兴,

cause an effective shift in the energy mismatch in the uncon- ditional dynamics. Physically, the presence of the electron in dot 1共state 兩a典) raises the effective tunneling barrier of the PC due to electrostatic repulsion. As a consequence, the effective 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).

III. CONDITIONAL MASTER EQUATION FOR INEFFICIENT MEASUREMENT

Equation共5兲 describes the time evolution of reduced den- sity 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 evolution of the quantum state, conditioned on a particular measure- ment realization, the conditional master equation should be employed. The conditional master equations for a perfect detector in the quantum-jump and quantum diffusive cases have been derived in Refs. 25 and 26. In this paper, to take account the effect of the inefficiency in the measurements, which arises when the detector sometimes misses detection, we write first for the quantum-jump case that

关dNc共t兲兴²⫽dNc共t兲, 共10a兲

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10b兲 Here the subscript c indicates 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. In Eq. 共10兲, dNc(t) is a stochastic point process which represents the number 共either zero or one兲 of tunneling events seen in an infinitesimal time dt, 具ⁿ1典c(t)⫽Tr关n1␳c(t)兴, E关Y 兴 denotes an ensemble aver- age of a classical stochastic process Y, and

␳

˜_1c共t⫹dt兲⫽J 关T⫹Xn1兴␳c共t兲dt 共11兲 is the unnormalized density matrix²⁶ given the result of an electron tunneling through the PC barrier at the end of the time interval 关t,t⫹dt). The factor␨⭐1 represents the frac- tion of detections which are actually registered by the PC detector. The value␨⫽1 then corresponds to a perfect detec- tor or efficient measurement. By using the fact that current through the PC is i(t)⫽e dN(t)/dt, Eq. 共10b兲 with ␨⫽1 states that the average current is eD when dot 1 is empty, and is eD⬘ when dot 1 is occupied. In Ref. 25 the case of inefficient measurements is discussed in terms of insuffi- ciently small readout period. In other words, the bandwidth of the measurement device is not large enough to resolve and record every electron tunneling through the PC barrier.

By following the similar derivation as in Ref. 26, the sto- chastic quantum-jump master equation of the density matrix operator, conditioned on the observed event in the case of inefficient measurement in time dt can be obtained:

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⫹共1⫺␨兲J 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12兲}

where

P1c共t兲⫽D⫹共D⬘^⫺D兲具ⁿ¹典^c共t兲. 共13兲 In the quantum-jump case, in which individual electron tun- neling current events can be distinguished, the qubit state 关see Eq. 共12兲兴 undergoes a finite evolution 共a quantum jump兲 when there is a detection result 关dNc(t)⫽1兴 at randomly determined times共conditionally Poisson distributed兲.

The extension to the case of quantum diffusion can be carried out similarly as in Ref. 26. In this case, the electron counts or accumulated electron number in time␦t is consid- ered as a continuous diffusive variable satisfying a Gaussian white noise distribution^26,48

␦^N共t兲⫽兵␨^{兩T 兩2关1⫹2}⑀^cos␪具ⁿ1典c共t兲兴⫹

冑

␨兩T 兩␰共t兲其␦^t, 共14兲 where⑀⫽(兩X 兩/兩T 兩)Ⰶ1, ␪ is the relative phase between X andT, and␰(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15兲 Here E denotes an ensemble average. In obtaining Eq. 共14兲, we have assumed that 2兩T 兩兩X 兩cos␪Ⰷ兩X 兩². Hence, for the quantum-diffusive equations obtained later, we should re- gard, to the order of magnitude, that 兩cos␪兩⬃O(1)Ⰷ⑀

⫽(兩X 兩/兩T 兩) and 兩sin␪兩⬃O(⑀⁾Ⰶ1. The quantum-diffusive conditional master equation for the case of inefficient mea- surements can be found as

␳^˙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

冑

␨

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16兲 In arriving at Eq. 共16兲, we have used the stochastic Itoˆ calculus^49,50 for the definition of derivative as ␳^{˙ (t)}

⫽limdt→0关␳^(t⫹dt)⫺␳^(t)兴/dt. The conditional equations 共12兲 and 共16兲, under similar assumptions and approximations as in Ref. 26 but taking into account the effect of inefficient measurement, are the main results in this paper. We will analyze the qubit dynamics in detail in Sec. IV using these equations in terms of Bloch sphere variables关see Eqs. 共20兲 and 共21兲兴. In particular, the effect of inefficient measure- ments will be discussed in Sec. IV D. It is easy to see that the ensemble average evolution of Eq. 共16兲 reproduces the un- conditional master equation 共5a兲 by simply eliminating the white noise term using Eq. 共15兲. Similarly, averaging Eq.

共12兲 over the observed stochastic process, by setting E关dNc(t)兴 equal to its expected value Eq. 共10b兲, gives the

unconditional, deterministic master equation 共5a兲. It is also easy to verify that for zero efficiency ␨⫽0 关i.e., also dN_c(t)⫽0兴, the conditional equations 共12兲 and 共16兲, reduce to the unconditional one共5a兲. That is, the effect of averaging over all possible measurement records is equivalent to the effect of completely ignoring the detection records or the effect of no detection results being available.

To make the quantum-diffusive, conditional stochastic master equation共16兲 more transparent, we evaluate Eq. 共16兲 in the charge state basis as for Eq. 共9兲 and obtain

␳^˙aa共t兲⫽i⍀关␳ab共t兲⫺␳ba共t兲兴

⫹

冑

⁸␨⌫dcos␪␳aa共t兲␳bb共t兲␰共t兲, 共17a兲

␳^˙ab共t兲⫽i共E⫹兩T 兩兩X 兩sin␪兲␳ab共t兲⫹i⍀关␳aa共t兲⫺␳bb共t兲兴

⫺⌫d␳ab共t兲⫹

冑

²␨⌫d 兵^cos␪关␳bb共t兲⫺␳aa共t兲兴

⫹i sin␪其␳^ab共t兲␰共t兲, 共17b兲 where we have set兩X 兩⫽

冑

²⌫d. Again, either by taking en- semble average or for zero efficiency␨⫽0, Eq. 共17兲 reduces to Eq.共9兲.

IV. CONDITIONAL DYNAMICS UNDER CONTINUOUS MEASUREMENTS

As in Ref. 26, we represent the qubit density matrix ele- ments in terms of Bloch sphere variables in the charge state basis as

␳共t兲⫽关I⫹x共t兲␴x⫹y共t兲␴y⫹z共t兲␴z兴/2, 共18兲 where ␴i satisfies the properties of Pauli matrices. In this representation, the variable z(t) represents the population 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. Generally the product of the off-diagonal elements of ␳(t) is smaller than the product of the diagonal elements, leading to the relation x²(t)⫹y²(t)⫹z²(t)⭐1. When ␳(t) is represented by a pure state, the equal sign holds. In this case, the system state can be characterized by a point (x, y ,z) on the Bloch unit sphere.

The master equations written as a set of coupled stochas- tic differential equations in terms of the Bloch sphere vari- ables in Ref. 26 are under the assumptions of real tunneling amplitudes and perfect 共efficient兲 measurements. Here we include the effect of inefficient measurement and the influ- ence of the relative phase between the two tunneling ampli- tudes into the coupled equations. The unconditional master equation共5a兲 is equivalent to the following equations:

dx共t兲

dt ⫽⫺共E⫹兩T 兩兩X 兩sin␪兲y共t兲⫺⌫dx共t兲, 共19a兲 d y共t兲

dt ⫽共E⫹兩T 兩兩X 兩sin␪兲x共t兲⫺2⍀z共t兲⫺⌫dy共t兲, 共19b兲

dz共t兲

dt ⫽2⍀y共t兲. 共19c兲

We find that the quantum-diffusive, conditional master equa- tion 共16兲 can be written as

dx_c共t兲

dt ⫽⫺共E⫹兩T 兩兩X 兩sin␪兲yc共t兲⫺⌫dx_c共t兲

⫹

冑

²␨⌫d关⫺sin␪^yc共t兲⫹cos␪^zc共t兲xc共t兲兴␰共t兲, 共20a兲

d y_c共t兲

dt ⫽共E⫹兩T 兩兩X 兩sin␪兲xc共t兲⫺2⍀zc共t兲⫺⌫dy_c共t兲

⫹

冑

²␨⌫d关sin␪^xc共t兲⫹cos␪^zc共t兲yc共t兲兴␰共t兲, 共20b兲 dz_c共t兲

dt ⫽2⍀yc共t兲⫺

冑

²␨⌫dcos␪关1⫺zc

2共t兲兴␰共t兲.

共20c兲 For the quantum-jump, conditional master equation共12兲, we obtain

dx_c共t兲⫽dt

冉

^{⫺关E⫹共1⫺␨}兲兩T 兩兩X 兩sin␪兴yc共t兲⫺共1⫺␨兲⌫dx_c共t兲⫺␨共D⬘⫺D兲

2 z_c共t兲xc共t兲

冊

^{⫺dNc共t兲}

⫻

冉

²兩T 兩兩X 兩sin␪^yc共t兲⫹关2⌫d⫺共D⬘^⫺D兲zc共t兲兴xc共t兲

2D⫹共D⬘⫺D兲关1⫺zc共t兲兴

冊

^{, 共21a兲}

d y_c共t兲⫽dt

冉

^{关E⫹共1⫺␨}兲兩T 兩兩X 兩sin␪兴xc共t兲⫺共1⫺␨兲⌫dy_c共t兲⫺2⍀zc共t兲⫺␨共D⬘⫺D兲

2 z_c共t兲yc共t兲

冊

^{⫺dNc共t兲}

⫻

冉

⫺2兩T 兩兩X 兩sin␪^xc共t兲⫹关2⌫d⫺共D⬘^⫺D兲zc共t兲兴yc共t兲

2D⫹共D⬘⫺D兲关1⫺zc共t兲兴

冊

^{, 共21b兲}

dz_c共t兲⫽dt

冉

^{2⍀yc共t兲⫹␨共D⬘2⫺D兲关1⫺zc2共t兲兴}

冊

^{⫺dNc共t兲}

冉

^{2D共D⫹共D⬘⫺D兲关1⫺z}⬘⫺D兲关1⫺z^c2共t兲兴c共t兲兴

冊

^{. 共21c兲}

As expected, Eq.共20兲 averaged over the white noise reduces to Eq. 共19兲, provided that E关xc(t)兴⫽x(t) as well as similar replacements are performed for y_c(t) and z_c(t). Similarly, by using Eq.共10b兲, the ensemble average of Eq. 共21兲 reduces to the unconditional equation 共19兲. One can also observe that for zero efficiency␨⫽0, the conditional equations 共21兲 and 共20兲, reduce to the unconditional equation 共19兲 as well. Next we analyze the qubit dynamics in detail and present the nu- merical simulations for the time evolution using Eqs. 共20兲 and共21兲. Part of the results in Sec. IV A have been reported in Ref. 51.

A. From quantum jumps to quantum diffusion Figure 2共a兲 shows the unconditional 共ensemble average兲 time evolution of the population difference z(t) with the ini- tial qubit state being in state兩a典, i.e., dot 1 is occupied. The unconditional population difference z(t), rises from⫺1, un- dergoing some oscillations, and then tends towards zero, a steady共maximally mixed兲 state. On the other hand, the con- ditional time evolution, conditioned on one possible indi- vidual realization of the sequence of measurement results, behaves quite differently. We consider first the situation, where D⬘⫽兩T⫹X 兩²⫽0, discussed in Ref. 18. In this case, due to the electrostatic repulsion generated by the electron,

the PC is blocked共no electron is transmitted兲 when dot 1 is occupied. As a consequence, whenever there is a detection of an electron tunneling through the PC barrier, the qubit state is collapsed into state 兩b典, i.e., dot 2 is occupied. The quantum-jump conditional evolution shown in Fig. 2共b兲 关us- ing the same parameters and initial condition as in Fig. 2共a兲兴 is rather obviously different from the unconditional one in Fig. 2共a兲. The conditional time evolution is not smooth, but exhibits jumps, and it does not tend towards a steady state.

One can see that initially the system starts to undergo an oscillation. As the population difference z_c(t) changes in time, the probability for an electron tunneling through the PC barrier increases. This oscillation is then interrupted by the detection of an electron tunneling through the PC barrier, which bring z_c(t) to the value 1, i.e., the qubit state is col- lapsed into state 兩b典. Then the whole process starts again.

The randomly distributed moments of detections, dN_c(t), corresponding to the quantum jumps in Fig. 2共b兲 is illus- trated in Fig. 2共c兲. Although little similarity can be observed between the time evolution in Figs. 2共a兲 and 2共b兲, averaging over many individual realizations shown in Fig. 2共b兲 leads to a closer and closer approximation of the ensemble average in Fig. 2共a兲.

Next we illustrate how the transition from the quantum- jump picture to the quantum-diffusive picture takes place. In

Ref. 26 and Sec. III, we have seen that the quantum-diffusive equations can be obtained from the quantum-jump descrip- tion under the assumption of 兩T 兩Ⰷ兩X 兩. In Figs. 3共a兲–3共d兲 we plot conditional, quantum-jump evolution of z_c(t) and the corresponding moments of detections dN_c(t), with dif- ferent (兩T 兩/兩X 兩) ratios. Each jump 共discontinuity兲 in the z_c(t) curves corresponds to the detection of an electron through the PC barrier. One can clearly observes that with

increasing (兩T 兩/兩X 兩) ratio, the number of jumps increases.

The amplitudes of the jumps of z_c(t), however, decreases from D⬘⫽0 with the certainty of the qubit being in state 兩b典 to the case of (D⫺D⬘⁾Ⰶ(D⫹D⬘) with a smaller probability of finding the qubit in state兩b典. Nevertheless, the population difference z_c(t) always jumps up since D⫽兩T 兩²⬎D⬘

⫽兩T ⫹X 兩². In other words, whenever there is a detection of an electron passing through PC, dot 2 is more likely occu- FIG. 2. Illustration for differ- ent behaviors between uncondi- tional and conditional evolutions.

The initial qubit state is兩a典^{. The} parameters are␨⫽1, E⫽0,␪⫽␲, 兩T 兩²⫽兩X 兩²⫽⍀, and time is in units of ⍀^⫺1. 共a兲 Unconditional, ensemble-averaged time evolution of z(t), which exhibits some os- cillation and then approaches a zero steady state value.共b兲 Condi- tional evolution of zc(t). The qu- bit starts an oscillation, which is then interrupted by a quantum jump 关corresponding to a detec- tion of an electron passing through the PC barrier in共c兲兴. Af- ter the jump, the qubit state is re- set to 兩b典 and a new oscillation starts. 共c兲 Randomly distributed moments of detections, which cor- respond to the quantum jumps in共b兲.

FIG. 3. Transition from quan- tum jumps to quantum diffusion.

The initial qubit state is兩a典^{. The} parameters are␨⫽1, E⫽0,␪⫽␲, 兩X 兩²⫽⍀, and time is in units of

⍀^⫺1. 共a兲–共d兲 are the quantum- jump, conditional evolutions of z_c(t), and corresponding detection moments with different 兩T 兩/兩X 兩 ratios: 共a兲 1, 共b兲 2, 共c兲 3, 共d兲 5.

With increasing 兩T 兩/兩X 兩 ratio, jumps become more frequent but smaller in amplitude. 共e兲 Repre- sents the conditional evolutions of zc(t) in the quantum diffusive limit. The variable␰(t), appearing in the expression of current through PC in quantum-diffusive limit, is a Gaussian white noise with zero mean and unit variance.

pied than dot 1. The case for quantum diffusion using Eq.

共20兲 is plotted in Fig. 3共e兲. In this case, very small jumps occur very frequently. We can see that the behavior of z_c(t) for兩T 兩⫽5兩X 兩 in the quantum-jump case shown in Fig. 3共d兲 is already very close to that of quantum diffusion shown in Fig. 3共e兲. To minimize the number of controllable variables, the same randomness is applied to produce the quantum- jump, conditional evolutions in Figs. 3共a兲–3共d兲. This, how- ever, does not mean that they would have had the same de- tection output, dN_c(t). The number of tunneling events in time dt, dN_c(t), does not depend on the randomness alone.

It also depends on 兩T 兩, 兩X 兩, and ␪, and has to satisfy Eq.

共10b兲 in a self-consistent manner. In fact, it both conditions and is conditioned by the conditional qubit density matrix.

Note that the unconditional evolution does not depend on the parameter 兩T 兩 when ␪⫽␲ 关see Eq. 共19兲兴. This implies that depending on the actual measured detection events, different measurement schemes 共measurement devices with different tunneling barriers or different values of 兩T 兩 when ␪⫽␲⁾ give different conditional quantum evolutions. But they would have the same ensemble average property if other pa- rameters and the initial condition are the same. Hence, aver- aging over all possible realizations, for each measurement scheme in Fig. 3, will lead to the same ensemble average behavior shown in Fig. 2共a兲.

B. Quantum Zeno effect

The quantum Zeno effect can be naturally described by the conditional dynamics. The case for quantum diffusion has been discussed in Refs. 19 and 21. Here, for complete- ness, we discuss the quantum-jump case. The quantum Zeno effect states that repeated observations of the system slow down transitions between quantum states due to the collapse

of the wave function into the observed state. Alternatively, the interaction with one measurement apparatus destroys the quantum coherence 共oscillations兲 between 兩a典 ^and兩b典 ^{at a} rate that is much faster than the tunneling rate⍀. For fixed

⍀, 兩T 兩, and ␪, by increasing the interaction with the PC detector兩X 兩⫽

冑

²⌫d, we increase the number and amplitude of jumps and hence the probability of the wave function being collapsed to the localized state. The time evolutions of the population difference z_c(t) for different ratios of (⌫d/⍀) are shown in Fig. 4. Here, the initial qubit state is兩a典^{, and} other parameters are ␨⫽1,E⫽0,␪⫽␲^,兩T 兩²⫽10⍀. We can observe that the period of coherent oscillations between the two qubit states increases with increasing (⌫d/⍀), while the time of a transition共switching time兲 decreases. In the limit of vanishing ⍀, a transition from one qubit state to the other state takes a time 共switching time兲 of order of localization time,²⁶ 1/␥loc

jump⫽(D⫹D⬘^)/关⌫d(

冑

^D⫹

冑

^D⬘⁾²兴. In the param- eter regime of Fig. 4共c兲 (⌫d/⍀⫽8), this time is still much smaller than the average time between state-changing transi- tions 共period of oscillations兲 due to ⍀, i.e., the mixing time,²⁶1/␥mix⫽⌫d/(4⍀²). Hence, we can already see from Fig. 4共c兲 for ⌫d/⍀⫽8 that very frequent repeated measure- ments would tend to localize the system.

The ensemble average behavior of z(t) is also shown in dashed line in Fig. 4. If E⫽0 and initially the electron is in dot 1, from the solution of Eq. 共9兲, the probability ␳aa(t)

⫽关1⫺z(t)兴/2 can be written as

␳aa共t兲⫽1

2

再

^{1⫹e⫺⌫dt/2}

冋

^cosh

冉

^⍀2⌫t

冊

^⫹⍀⌫d⌫sinh

冉

^⍀2⌫t

冊册冎

^,

共22兲 where ⍀_⌫⫽

冑

⌫d

2⫺(4⍀)². In the Appendix, the stationary noise power spectrum of the current fluctuations through the FIG. 4. Illustration of the quantum Zeno effect. Both condi- tional共in solid line兲 and uncondi- tional共in dashed line兲 evolutions of the population difference for different ratios of 共a兲 (⌫d/⍀)

⫽0.04, 共b兲 2, 共c兲 8, are shown.

The initial qubit state is兩a典^{. The} other parameters are␨⫽1, E⫽0,

␪⫽␲, 兩T 兩²⫽20⍀, and time is in units of (2⍀)^⫺1. Increasing (⌫d/⍀) ratio increases the period of coherent oscillations between the qubit states, while the time of a transition 共switching time兲 de- creases.

PC barrier is calculated for the case of E⫽0 and the result can be written as:²¹

S共␻兲⫽S0⫹ 4⍀²共⌬i兲²⌫d

共␻²⫺4⍀²兲²⫹⌫d

2␻^{2. 共23兲}

where S₀⫽2ei_⬁⫽e²␨^(D⬘⫹D) represents the shot noise, i_⬁⫽e␨^(D⬘⫹D)/2 is the steady-state current and ⌬i⫽e␨^(D

⫺D⬘) represents the difference between the two average currents. For⌫d⬍4⍀, ␳aa(t) shows the damped oscillatory behavior in the immediate time regime 关see dashed line in Figs. 4共a兲 and 4共b兲兴. In this case, the spectrum has a double peak structure, indicating that coherent tunneling is taking place between the two qubit states. This is illustrated in Figs.

5共a兲 and 5共b兲. When ⌫d⭓4⍀, ␳aa(t) does not oscillate but decays in time purely exponentially, saturating at the prob- ability 1/2关see dashed line in Fig. 4共c兲兴. This corresponds to a classical, incoherent behavior. In this case, only a single peak, centering at ␻⫽0, appears in the noise spectrum, as illustrated in Fig. 5共c兲. The evolution of zc(t) in Fig. 4共c兲, is one of the possible conditional evolutions in this parameter regime (⌫d/⍀⫽8). In this parameter regime ⌫d⭓4⍀, the conditional evolution z_c(t) behaves very close to a probabi- listic jumping or random telegraph process. After ensemble averaging over all possible realizations of such conditional evolutions, one would then obtain the classical, incoherent behavior.

C. Relative phase of the tunneling amplitudes The relative phase between the two complex tunneling amplitudes produces effects on both conditional and uncon- ditional dynamics of the qubit. In the following, we consider

the case that␨⫽1 and E⫽0. From Eq. 共21兲, after each jump the imaginary part of the product (T*X) seems to cause an additional rotation around the z axis in the Bloch sphere, but does not directly change the population probability z_c(t) of the qubit. However, the actual conditional evolution of the Bloch sphere variables is complicated. It is stochastic and nonlinear, and depends on the relative phase of the tunneling amplitudes in a nontrivial way. Nevertheless, after ensemble average, the imaginary part of (T*X) generates an effective shift in the energy mismatch of the qubit states关see Eq. 共9兲兴.

There are situations in which the effect of the relative phase of the tunneling amplitudes can be easily seen. For ␨

⫽1 and E⫽0, if the tunneling amplitudes are real, i.e., ␪

⫽␲, and the initial condition x_c(0)⫽0, then the time evo- lution of x_c(t), from Eq.共21兲, does not change and remains at the value 0 at all times. But if ␪⫽␲ ^{or sin}␪⫽0, the conditional evolution of x_c(t) behaves rather differently. It changes after the first detection共quantum jump兲 takes place.

Figure 6 shows the evolutions of the Bloch variables x_c(t),y_c(t),z_c(t) with the same initial condition 共the qubit being in 兩a典) and parameters but different relative phases:

␪⫽␲^for共a兲–共c兲 and␪⫽cos^⫺1(兩X 兩/兩T 兩) for 共d兲–共f兲. We can clearly see quite different behaviors of x_c(t) in these two cases. The asymmetry of the electron population in z_c(t), due to effectively generated energy mismatch in the second case in Fig. 6共f兲, can be roughly observed. The effect of the relative phase is small in the case of quantum diffusion. As noted in Sec. III, in order for the quantum-diffusive equa- tions to be valid, we should regard, to the order of magni- tude, that 兩cos␪兩⬃O(1) and 兩sin␪兩⬃O(⑀). This implies that in this case ␪⬇␲. Hence the effect of the relative phase is small and the conditional dynamics does not deviate much from the case that the tunneling amplitudes are assumed to be real.^19,21,25

FIG. 5. A plot of the noise power spectrum of the current, normalized by the shot noise level for different ratios of共a兲 (⌫d/⍀)

⫽0.04, 共b兲 2, 共c兲 8. All the param- eters are the same as the corre- sponding ones in Fig. 4. For small (⌫d/⍀) ratio, two sharp peaks ap- pear in the noise power spectrum, as shown in共a兲. In 共b兲, a double peak structure is still visible, indi- cating that coherent tunneling be- tween the two qubit states still ex- ists. In the classical, incoherent regime ⌫d⭓4⍀, only one single peak appears, as shown in共c兲.

D. Inefficient measurement and non-ideality

We have shown²⁶that for␨⫽1, the conditional time evo- lution of the qubit can be described by a ket state vector satisfying the stochastic Schro¨dinger equation. It is then ob- vious that perfect detection or efficient measurement pre- serves state purity for a pure initial state. However, the inef- ficiency and nonideality of the detector spoils this picture.

The decrease in our knowledge of the qubit state leads to partial decoherence for the qubit state. We next find the par- tial decoherece rate introduced in this way.

The stochastic differential equations in the form of Itoˆ calculus^49,50have the advantage that it is easy to see that the ensemble average of the conditional equations over the ran- dom process␰(t) leads to the unconditional equations. How- ever, it is not a natural physical choice. For example, for ␨

⫽1, the term ⫺⌫d␳ab(t) in Eq.共17b兲 does not really cause decoherence of the conditional qubit density matrix. It sim- ply compensates the noise term due to the definition of de- rivative in Itoˆ calculus. Hence, in this case the conditional evolution of␳ab(t) does not really decrease in time exponen- tially. To find the partial decoherence rate generated by inef- ficiency␨⬍1, we transform Eq. 共17b兲 into the form of Stra- tonovich calculus.^49,50We then obtain for␪⫽␲^:

␳^˙ab共t兲⫽iE␳ab共t兲⫹i⍀关␳aa共t兲⫺␳bb共t兲兴

⫺关␳bb共t兲⫺␳aa共t兲兴

冑

²⌫d

e兩T 兩 关i共t兲⫺i0兴␳ab共t兲

⫺共1⫺␨兲⌫d␳ab共t兲, 共24兲

where i(t)⫺i0⫽e兩T 兩兵␨

冑

²⌫d关1⫺2␳aa(t)兴⫹

冑

␨␰^(t)其^{. Here} we have used the following relations: the conditional current

i(t)⫽e␦^N(t)/␦^{t with}␦N(t) given by Eq. 共14兲 and the av- erage current i₀⫽e␨^(D⫹D⬘)/2, where D⫽兩T 兩² and D⬘

⫽兩T 兩²⫺2兩T 兩兩X 兩 in the quantum-diffusive limit. In this form, Eq.共24兲 elegantly shows how the qubit density matrix is conditioned on the measured current. We find that the last term in Eq. 共24兲 is responsible for decoherence. In other words, the partial decoherence rate for an individual realiza- tion of inefficient measurements is (1⫺␨⁾⌫d. For a perfect detector␨⫽1, this decoherence rate vanishes and the condi- tional ␳ab(t), as expected, does not decay exponentially in time. Similar conclusion could be drawn from Eq. 共21兲 for the quantum-jump case. For␪⫽␲, the off-diagonal variables x_c(t) and y_c(t) seem to decrease in time with the rate (1

⫺␨⁾⌫d.

In Bloch sphere variable representation, we can use the quantity P_c(t)⫽xc

2(t)⫹yc 2(t)⫹zc

2(t) as a measure of the pu- rity of the qubit state, or equivalently as a measure of how much information the conditional measurement record gives about the qubit state. If the conditional state of the qubit is a pure state then P_c(t)⫽1; if it is a maximally incoherent mixed state then P_c(t)⫽0. We plot in Fig. 7 the quantum- jump, conditional evolution of the purity P_c(t) for different inefficiencies, ␨⫽1,0.6,0.2 共in solid line兲, and 0 共in dotted line兲. Figure 7共a兲 is for an initial qubit state being in a pure state 兩a典, while Fig. 7共b兲 is for a maximally mixed initial state. We can see from Fig. 7共a兲 that the purity Pc(t)⫽1 at all times for␨⫽1, while it hardly or not at all reaches 1 for almost all time for␨⬍1. This means that partial information about the changes of the qubit state is lost irretrievably in inefficient measurements. In addition, roughly speaking, the overall behavior of P_c(t) decreases with decreasing␨^{. This} indicates that after being averaged over a long period of time, 具^Pc(t)典t would also decrease with decreasing ␨^{. For} FIG. 6. Effect of relative phase on the qubit dynamics. The condi- tional evolutions of x_c(t), y_c(t), and z_c(t) with the same initial condition共the qubit being in 兩a典⁾ and parameters (␨⫽1, E⫽0, ␪

⫽␲, 兩T 兩²⫽4兩X 兩²⫽4⍀), but dif- ferent relative phases are shown:

共a兲–共c兲 for ␪⫽␲ and 共d兲–共f兲 for

␪⫽cos^⫺1(兩X 兩/兩T 兩). The relative phase causes quite different evolu- tions for x_c(t).

␨⫽0, the evolution of P(t) becomes smooth and tends to- ward the value zero共the maximally mixed steady state兲. For a nonpure initial state关see Fig. 7共b兲兴, the qubit state is even- tually collapsed towards a pure state and then remains in a pure state for␨⫽1. But the complete purification of the qubit state cannot be achieved for␨⬍1. As in Figs. 3共a兲–3共d兲, the same randomness has been applied to generate the quantum- jump, conditional evolution in Figs. 7共a兲 and 7共b兲. Note that the only difference between evolution in Fig. 7共a兲 and the corresponding one in Fig. 7共b兲 is the different initial states.

So when the qubit density matrix in Fig. 7共b兲 gradually evolves into the same state as in Fig. 7共a兲, the corresponding P_c(t) in Fig. 7共b兲 would then follow the same evolution as in Fig. 7共a兲. This behavior can be observed in Fig. 7. The purity-preserving conditional evolution for a pure initial state, and gradual purification for a nonpure initial state for an ideal detector have been discussed in Refs. 19–21,24 in the quantum-diffusive limit.

The nonideality of the PC detector is modeled in Refs.

19–21,24 by another ideal detector ‘‘in parallel’’ to the origi- nal one but with inaccessible output. We can add, as in Refs.

19–21,24, an extra term,⫺␥d␳ab(t), to Eq. 共24兲 to account for the ‘‘nonideality’’ of the detector. The ideal factor␩ ^in- troduced there^19–21,24can be modified to take account of in- efficient measurement discussed here. We find

␩⫽1⫺ ⌫

⌫tot⫽ ␨⌫d

⌫d⫹␥d

, 共25兲

where⌫⫽(1⫺␨⁾⌫d⫹␥d and⌫tot⫽⌫d⫹␥d. For␥d⫽0, we have ␩⫽␨. In Ref. 25, inefficient measurement is discussed in terms of insufficiently small readout period. As a result, the information about the tunneling times of the electrons passing through the PC barrier is partially lost.

V. CONCLUSION

We have obtained the quantum-jump and quantum- diffusive, conditional master equations, taking into account the effect of inefficient measurements ␨⭐1 under the weak system-environment coupling and Markovian approxima- tions. These conditional master equations describe the ran- dom evolution of the measured qubit density matrix, which both conditions and is conditioned on, a particular realization of the measured current. If and only if detections are perfect 共efficient measurement兲, i.e., ␨⫽1, are the stochastic master equations for the conditioned density matrix operators 共12兲 and共16兲, equivalent to the stochastic Schro¨dinger equations 关Eqs. 共35兲 and 共41兲 of Ref. 26, respectively兴 for the condi- tioned states. If the detection is not perfect and some infor- mation about the system is unrecoverable, the evolution of the system can no longer be described by a pure state vector.

For the extreme case of zero efficiency detection, the infor- mation 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 equations 共12兲 and 共16兲 after being averaged over all possible measurement records re- duces to the unconditional, deterministic master equation 共5a兲, leading to decoherence for the system.

We have used the derived conditional equations to ana- lyze the conditional qubit dynamics in detail and illustrate the conditional evolution by numerical simulations. Specifi- cally, the conditional qubit dynamics evolving from quantum jumps to quantum diffusion has been presented. Further- more, we have described the quantum Zeno effect in terms of the quantum-jump conditional dynamics. We have calculated the stationary noise power spectrum of the current fluctua- tions through the PC barrier in terms of the quantum-jump formalism. We have also discussed the effect of inefficient FIG. 7. Effect of inefficiency on the state purity. The quantum- jump, conditional evolution of the purity P_c(t) for different ineffi- ciencies, ␨⫽1,0.6,0.2 共in solid line兲, and 0 共in dotted line兲 are plotted in 共a兲 for an initial qubit state being in a pure state兩a典^,共b兲 for a maximally mixed initial state. The other parameters are E

⫽0, ␪⫽␲, 兩T 兩²⫽4兩X 兩²⫽4⍀.

The purity-preserving conditional evolution for a pure initial state, and gradual purification for a non- pure initial state for␨⫽1 are il- lustrated. However, the complete purification of the qubit state can- not be achieved for␨⬍1.

measurement and the influence of relative phase between the two tunneling amplitudes on the qubit dynamics.

ACKNOWLEDGMENTS

H.S.G. is grateful for useful discussions with A. N. Ko- rotkov, G. Scho¨n, D. Loss, Y. Hirayama, J. S. Tsai, and G. P.

Berman. H.S.G. would like to thank H. B. Sun and H. M.

Wiseman for their assistance and discussions in the early stage of this work.

APPENDIX: CALCULATION OF THE NOISE POWER SPECTRUM OF THE CURRENT FLUCTUATIONS In this Appendix, we calculate the stationary noise power spectrum of the current fluctuations through the PC when there is the possibility of coherent tunneling between the two qubit states. Usually one can calculate this noise power spec- trum using the unconditional, deterministic master equation approach, which gives only the average characteristics. We, however, calculate it through the stochastic formalism pre- sented here. The fluctuations in the observed current, i(t), are quantified by the two-time correlation function:

G共␶兲⫽E关i共t⫹␶兲i共t兲兴⫺E关i共t⫹␶兲兴E关i共t兲兴. 共A1兲 The noise power spectrum of the current is then given by

S共␻兲⫽2

冕

⫺⬁

⬁

d␶^G共␶兲e^⫺i␻␶. 共A2兲 The ensemble expectation values of the two-time correlation function for the current in the case of quantum diffusion has been calculated in Ref. 21. Here we will present the quantum-jump case. The current in this case is given by i(t)⫽e dN(t)/dt. We will follow closely the calculation in the Appendix of Ref. 39 to calculate the two-time correlation function, E关dNc(t⫹␶^)dN(t)兴. First we consider the case when␶Ⰷdt⬎0, where dt is the minimum time step consid- ered. Since dN(t) is a classical point process, it is either zero or one. As a result, E关dNc(t⫹␶^)dN(t)兴 is nonvanishing only if there is an electron-tunneling event inside each of these two infinitesimal time intervals, 关t,t⫹dt兴 and 关t⫹␶^,t

⫹␶⫹dt兴. Hence, we can write

E关dNc共t⫹␶兲dN共t兲兴⫽Prob关dN共t兲

⫽1兴E关dNc共t⫹␶兲兩dN(t)⫽1兴, 共A3兲 where the subscript to the vertical line is the condition for which the subscript on dN_c(t⫹␶) exists. From Eqs.

共10b兲 and 共11兲, we have Prob关dN(t)⫽1兴⫽␨^Tr关^˜␳1(t⫹dt)兴 and E关dNc(t⫹␶⁾兩dN(t)⫽1兴 ⫽␨^Tr兵J 关T ⫹ Xn1兴E关␳1c(t

⫹␶⁾兩dN(t)⫽1兴其. Using the fact that E关␳c(t)兴⫽␳(t) and Eqs.

共5b兲 and 共11兲, we can write

E关␳1c共t⫹␶兲兩dN(t)⫽1兴⫽e^{L(␶⫺dt)˜}␳1共t⫹dt兲/Tr关␳^˜1共t⫹dt兲兴

⫽␨^eL(␶⫺dt)兵J 关T ⫹Xn1兴␳共t兲dt其^/ Tr关^˜␳1共t⫹dt兲兴. 共A4兲

Hence, to leading order in dt, we obtain for␶⬎0:

E关dNc共t⫹␶兲dN共t兲兴⫽␨^2dt2Tr关J 关T ⫹Xn1兴

⫻e^L␶兵J 关T ⫹Xn1兴␳共t兲其^兴.

共A5兲 For␶⫽0, we have, from Eq. 共10兲, that

E关dN共t兲dN共t兲兴⫽E关dN共t兲兴⫽␨关D⫹共D⬘^⫺D兲具ⁿ¹典共t兲兴dt.

共A6兲 For short times, this term dominates and we may regard dN(t)/dt as␦-correlated noise for a suitably defined␦ ^func- tion. Thus the current-current two-time correlation function for ␶⭓0 can be written as

E关i共t⫹␶兲i共t兲兴⫽E

冋

^{dNc共t⫹dt ␶兲 dNdt共t兲}

册

⫽e²␨兵^D⫹共D⬘⫺D兲Tr关n1␳共t兲兴其

⫻␦共␶兲⫹␨^2Tr关J 关T ⫹Xn1兴

⫻e^L␶兵J 关T ⫹Xn1兴␳共t兲其兴. 共A7兲 In this form, we have related the ensemble averages of clas- sical random variable to the quantum averages with respect to the qubit density matrix. The case␶⭐0 is covered by the fact that the current–current two-time correlation function or G(␶) is symmetric in␶^{, i.e., G(}␶⁾⫽G(⫺␶^).

Next we calculate steady-state G(␶^{) and S(}␻^{). We can} simplify Eq. 共A7兲 using the following identities for an arbi- trary operator B: Tr关J 关n1兴B兴⬅Tr关n1B兴, Tr关e^L␶B兴⫽Tr关B兴, and Tr关Be^L␶␳_⬁兴⫽Tr关B␳_⬁兴, where the ⬁ subscript indicates that the system is at the steady state and the steady-state density matrix ␳_⬁ is a maximally mixed state. Hence we obtain the steady-state G(␶^{) for}␶⭓0 as

G共␶兲⫽ei_⬁␦共␶兲⫹e²␨²共D⬘^⫺D兲2

⫻兵^{Tr关n1eL␶关n1}␳⬁兴⫺Tr关n1␳⬁兴²其^{, 共A8兲} where the steady-state average current i_⬁⫽e␨^(D⫹D⬘^)/2.

The first term in Eq. 共A8兲 represents the shot noise compo- nent. It is easy to evaluate Eq. 共A8兲 analytically for E⫽0 case. The case for the asymmetric qubit,E⫽0, can be calcu- lated numerically. Evaluating Eq.共A8兲 for E⫽0, we find

G共␶兲⫽ei_⬁␦共␶兲⫹共⌬i兲²

4

冉

^{␮⫹e␮␮⫺}⫹^␶⫺⫺^␮␮^⫺_⫺^e␮⫹␶

冊

^{, 共A9兲}

where␮_⫾⫽⫺(⌫d/2)⫾

冑

⁽⌫d/2)²⫺4⍀², and we have repre- sented ⌬i⫽e␨^(D⫺D⬘) as the difference between the two average currents. After Fourier transform following from Eq.

共A2兲, the power spectrum of the noise is then obtained as the expression of Eq. 共23兲. Note that from Eq. 共23兲, the noise spectrum at␻⫽2⍀ for␪⫽␲, i.e., real tunneling amplitudes, can be written as

S共2⍀兲⫺S0

S₀ ⫽2␨^共

冑

^D⫹

冑

^D⬘兲²

共D⫹D⬘兲 , 共A10兲