Conditional counting statistics of electrons tunneling through quantum dot systems measured by a quantum point contact
We theoretically study the conditional counting statistics of electron transport through a system consisting of a single quantum dot (SQD) or coherently coupled double quantum dots (DQD’s) monitored by a nearby quantum point contact (QPC) using the generating functional approach with the maximum eigenvalue of the evolution equation matrix method, the quantum trajectory theory method (Monte Carlo method), and an efficient method we develop. The conditional current cumulants that are significantly different from their unconditional counterparts can provide additional information and insight into the electron transport properties of mesoscopic nanostructure systems. The efficient method we develop for calculating the conditional counting statistics is numerically stable, and is capable of calculating the conditional counting statistics for a more complex system than the maximum eigenvalue method and for a wider range of parameters than the quantum trajectory method. We apply our method to investigate how the QPC shot noise affects the conditional counting statistics of the SQD system, going beyond the treatment and parameter regime studied in the literature. We also investigate the case when the interdot coherent coupling is comparable to the dephasing rate caused by the back action of the QPC in the DQD system, in which there is considerable discrepancy in the calculated conditional current cumulants between the population rate (master-) equation approach of sequential tunneling and the full quantum master-equation approach of coherent tunneling.
The time-resolved measurement of electron charges through a single quantum dot (SQD) by a nearby quantum point contact (QPC) detector has been demonstrated experimentally Elzerman04 (); Schleser04 (); Vandersypen04 (); Gustavsson06 (); Gustavsson09 (); Ubbelohde12 (). The ability to detect individual charges in real time makes it possible to count electrons one by one as they pass through the quantum dot(QD) Elzerman04 (); Schleser04 (); Vandersypen04 (); Gustavsson06 (); Gustavsson09 (); Ubbelohde12 (); Lu03 (); Fujisawa04 (); Bylander05 (); Fujisawa06 (); Fricke07 (). The time-resolved charge detection has allowed the precise measurement of the QD shot noise at subfemtoampere current levels, and the full counting statistics (FCS) of the current Gustavsson06 (); Gustavsson09 (); Ubbelohde12 ().
FCS in quantum transport provides information of quantum statistical properties of transport phenomena and is studied mostly based on the computation of its moment or cumulant generating function Levitov96 (); key-4 (); Nazarov03 (). Computing the generating function is more convenient in practice than the direct calculation of the probability distribution function and then performing average over the powers of electron number or current. A theoretical approach called number-resolved master-equation approach has been applied to calculate the generating functions and unconditional FCS for the nanostructure electron transport problems key-4 (); Nazarov03 (); Gurvitz97 (); Schoell04 (); key-7 (); key-8 (); key-6 ().
When a measurement is made on a single quantum system and the result is available, the state or density matrix of the system is a conditional state conditioned on the measurement result Korotkov01 (); Goan01a (); Goan01b (). Thus, the conditional state of the system is important when its subsequent time evolution is concerned. If a single system is under continuous monitoring and one wants to map out the system state evolution conditioned on the continuous in time measurement results, the conditional (Bayesian) stochastic Schrd̈inger or stochastic master equation approach or the quantum trajectory theory (quantum Monte Carlo method) can be employed Korotkov01 (); Goan01a (); Goan01b (); Goan03 (); Goan04 (). Each quantum trajectory can mimic the stochastic system state evolution conditioned on the continuous in time measurement outcomes in a single run of a realistic experiment. The stochastic element in the quantum trajectory corresponds exactly to the consequence of the random outcomes of the measurement record Korotkov01 (); Goan01a (); Goan01b (); Goan03 (); Goan04 (). Thus, the quantum trajectories have the full information of the statistical properties about the measured system and can give insight to the unconditional quantities.
In some cases, one is concerned with the system state or physical observables conditioned on some average quantities (e.g., average current) in a given period of time rather than instantaneous and continuous in time measurement results. For example, the conditional counting statistics of electron transport through a SQD coupling to a QPC has been measured in the experiment by Sukhorukov et al. Sukhorukov07 (). The conditional FCS that is the statistical current cumulants of one system given the observation of a particular average current in time in the other system could be substantially different from its unconditional counterpart. A theoretical approach that utilizes the number-resolved rate (master) equation of a bistable SQD system and neglects the QPC shot noise was put forward to calculate the steady-state conditional FCS for the SQD-QPC system Sukhorukov07 (); Jordan04 ().
One of the purposes of this paper is to provide a connection with, and a unified picture of, the quantum trajectory, the (partially reduced) number-resolved master-equation and the unconditional (reduced) master-equation approaches. We show that the master equations for the reduced or partially reduced density matrix can be simply obtained when an average or partial average is taken on the conditional, stochastic density matrix (or quantum trajectories) over the possible outcomes of the measurements Goan01a (); Goan01b (); Goan03 ().
Another purpose of this paper is to investigate the effect of QPC shot noise on the conditional FCS of the SQD-QPC system as well as to develop an efficient and systematic way to calculate the conditional FCS for more complex systems of interacting nanoscale conductors. Our investigation goes beyond the analysis presented in Ref. Sukhorukov07, . In Ref. Sukhorukov07, , the number-resolved population master (rate) equation for bistable system was first transformed into the counting field (inverse Fourier transform) space and then the eigenvalue with the smallest absolute real part (or maximum eigenvalue) in the matrix of the transformed master equation was found analytically. To evaluate the integral in partial or mixed Fourier transform space analytically with the analytic form of the eigenvalue to obtain the conditional steady-state current moment (cumulant) generating function, a further approximation to neglect the QPC shot noise was madeSukhorukov07 (). For the experimental parameters used in Ref. Sukhorukov07, , the QPC shot noise as compared to the noise contribution of the random telegraph signal in the QPC current trace induced by random electrons tunneling on and off the QD is indeed small and can be neglected. On the other hand, for the parameter regimes where the QPC shot noise cannot be ignored, obtaining analytical expressions for the conditional steady-state current moments or cumulants is very difficult. Furthermore, for more complicated interacting nanoscale conductors with the dimension of the matrix equation of the master equation growing up, to find analytical solution of the maximum eigenvalue becomes very hard, not to mention to obtain the analytical forms of the conditional steady-state current moment or current cumulant generating function. Besides, direct numerical evaluation of the conditional cumulant generation function in the same way as in Ref. Sukhorukov07, and then taking partial derivatives to obtain conditional cumulants are quite numerically unstable. In these cases, the quantum trajectory approach may give the conditional states or conditional current cumulants by simultaneously simulating an ensemble of current outcomes and corresponding quantum trajectories, and then categorizing and averaging the current outcomes of one system (e.g., the QD system) for each of the observed average current value in the other system (e.g., the QPC). However, in some parameter regimes where the probabilities to observe the average QPC current in certain values are very small, it is then computationally expensive to simulate and map out the conditional current cumulant in the whole parameter space of the average QPC current by the quantum trajectory method as an extremely large number of trajectories are required to have enough statistical samples in those very low probability domains. Thus developing a method to evaluate the conditional counting statistics directly and effectively for more complex systems and for a wide range of parameter space is desirable. It is one of the aims of the paper to develop such an efficient method.
The paper is organized as follows. In Sec. II, we introduced the model and Hamiltonian of the QD-QPC system that will be considered. In Sec. III, we present the unconditional master equation for the reduced density matrix of the QD system. We then derive the conditional, stochastic master equation (or quantum trajectory equation) that mimics the dynamics of the QD system conditioned on the observed random outcomes in Sec. IV. Then the number-resolved master equation and its inverse Fourier transform in the counting field space are discussed in Sec. V. The procedure to calculate the unconditional and conditional FCS are described in terms of generating functional approach in Sec. VI. Here we also introduce our efficient method to calculate the moments and cumulants of the conditional FCS. Section VII is devoted to the presentation and discussion of the results we obtain. Specifically, we provide a thorough analysis using the method of Ref. Sukhorukov07, , the quantum trajectory theory and the efficient method we develop to simulate and calculate the conditional current and noise of the SQD-QPC and DQD-QPC systems. We also investigate how the QPC shot noise affects the conditional QD current cumulants. Finally, a short conclusion is given in Sec. VIII. The detailed procedure of the semiempirical method used to count the number of tunneling electrons through the QD system in each random current trace of quantum trajectories is described in Appendix A.
Ii Quantum-dot system measured by a QPC
We consider a system consisting of either a SQD [see Fig. 1(a)] or coherently coupled DQD’s [Fig. 1(b)] measured by a QPC Gurvitz97 (); Schoell06 (); Gustavsson06 (); Sukhorukov07 (); Gustavsson09 (). The QD system is connected to two leads (reservoirs) biased so that electrons can tunnel onto the SQD (onto the left dot of the coherently coupled DQD’s) from the left lead and off the SQD (off the right dot of the DQD’s) onto the right lead. The QPC serves as a sensitive electrometer since its tunneling barrier can be modulated by the charge on a nearby QD. In our setup, as the electron moves into the SQD (the right dot of the DQD’s), it changes the tunnel barrier of the nearby QPC. In this way the modulated current through the QPC can be used to continuously monitor the occupation of the QD. We will follow the treatment given in Ref. Goan01a, ; Goan01b, ; Goan03, to describe the dynamics of the system.
The Hamiltonian for the QD system coupling to the QPC can be written as
where here is the Hamiltonian for the QD system consisting of the left lead, right lead and the central part system and the tunneling between them. The symbols , and , are respectively the electron annihilation operators and energies for the left and right reservoir states for the QD system at wave number . For a SQD system, we have the indices in and the Hamiltonian of the central part system is just
and for a DQD system, we have , in and
Here () and represent the electron annihilation (creation) operator and energy for a single electron state in dot , respectively. In other words, dot 2 denotes the central QD in the SQD system, and dot 1 and dot 2 stand for the left dot and right dot, respectively, in the DQD system. The tunneling Hamiltonian for the QPC detector is represented by . Similarly, , and , are respectively the electron annihilation operators and energies for the source and drain reservoir states for the QPC at wave number . [Eq. (4)] describes the interaction between the QPC detector and dot . When the electron is located in dot , the effective tunneling amplitude of the QPC detector changes from . We investigate here a simpler case of electrons transport through the DQD-QPC system in which the QPC couples only to the right dot (dot 2) of the DQD system Gurvitz97 (); Schoell06 (); Gustavsson09 (); Marcus04 () to illustrate the usage of our method and discuss the effects of QPC shot noise and interdot coupling on the conditional current cumulants. Our approach can be straightforwardly generalized to the case where the QPC couples to both dots with different coupling strengths Fujisawa06 (); Fricke07 (); Gustavsson09 (); key-36 (); key-38 (); key-39 ().
Iii Unconditional master equation
By following the treatment in Refs. Gurvitz97, ; Goan01a, ; Goan01b, , the (unconditional) zero-temperature, Markovian master equation of the reduced density matrix for the quantum dot (QD) system can be obtained as:
where is the occupation number operator of dot measured by the QPC. In Eq. (7) and the rest of the paper, the Hamiltonian of the QD system takes the form of Eq. (5) for the SQD system and Eq. (6) for the DQD system, and the subscripts for the SQD system and and for the DQD system. The parameters and are given by , and . Here and are the average electron tunneling rates through the QPC barrier without and with the presence of the electron in dot respectively, is the external bias applied across the QPC ( and stand for the chemical potentials in the source and drain reservoirs, respectively), and are energy-independent tunneling amplitudes near the average chemical potential, and and are the energy-independent density of states for the source and drain reservoirs. and are the tunneling rates from the left lead to the QD system and from the QD system to the right lead, respectively. In Eq. (7), the superoperator is defined as:
where . Finally, Eq. (8) defines the Liouvillian operator .
The conditional dynamics is quite different from its unconditional counterpart. For example, the unconditional dynamics of the number of electrons on the SQD system follows immediately from Eqs. (7) and (5) as
where . Clearly the average current through the SQD does not depend at all on the current through the QPC in this model. This is because the Hamiltonian describing the interaction between the SQD and the QPC commutes with the number operator . However if we ask for the conditional dynamics of the SQD given an observed averaged current in time or given an instantaneous current in time through the QPC, we need a different equation or even a stochastic equation for .
Iv Conditional master equation and quantum trajectories
There are two classical stochastic currents through this system: the current, , through the QPC and the current, , through the QD. Equation (7) describes the time evolution of the reduced density matrix when these classical stochastic processes are averaged over. To make contact with a single realization of the random outcomes of the measurement records and study the stochastic evolution of the QD state, conditioned on a particular measurement realization, we need the conditional master equation. We first define the relevant point processes that are the source of the classically observed stochastic currents.
where is a stochastic point process which represents the number (either zero or one) of tunneling events in the QPC seen in an infinitesimal time ,
and denotes an ensemble average of a classical stochastic process . The subscript indicates that the quantity to which it is attached is conditioned on previous observations of the the occurrences (detection records) of the electrons tunneling through the QPC barrier in the infinitesimal time in the past. The factor represents the fraction of tunneling events which are actually registered by the circuit containing the QPC detector. The value then corresponds to a perfect detector or efficient measurement. By using the fact that current through the QPC is , Eq. (12) with states that the average current is when the dot is empty, and is when the dot is occupied.
Similarly, we can specify the quantum jump conditional dynamics through the QD system by defining two stochastic point processes and which represent, respectively, the numbers (either zeros or ones) of tunneling events from the left lead to dot and from dot to the right lead seen in an infinitesimal time : Goan01a (); Goan01b (); Goan03 (); Goan04 ():
Unraveling both the QPC and the QD equations, we write the conditional master equation at zero temperature as:
We now focus on the conditional dynamics of the QD as the QPC current, , is continuously monitored. In the experiment the observed values of the random telegraph process are not fixed at the average values, ,, but are themselves stochastic processes as electrons tunnel through the QPC. We average over the jump process onto and off the QD. The stochastic quantum-jump master equation of the density matrix operator, conditioned on the observed event in QPC current in the case of inefficient measurement in time can be obtained as,
In the quantum-jump case, in which individual electron QPC tunneling current events can be distinguished, the QD system state [see Eq. (19)] undergoes a finite evolution (a quantum jump) when there is a detection result  at randomly determined times (conditionally Poisson distributed).
As Fig. 1(c) of Ref. Sukhorukov07, suggests, the current through the QPC could be quite large and while we may be able to resolve the random telegraph signal jumps between the two average values, and , we may not have sufficient bandwidth in the circuit to resolve the jump events through the QPC. The individual tunnel events through the QPC are too rapid to be resolved in the external circuit, resulting in a process more like a white noise stochastic process. This leads us to make the diffusive approximation to the quantum-jump stochastic master equation for describing the conditional QPC current dynamics. We now replace the quantum-jump master equation for the QPC with the quantum diffusion stochastic master equation. In this case, the total number of electrons that tunnel through the QPC in a time , large compared to the inverse of the jump rate, but small compared to the typical circuit response time, is considered as a continuous diffusive variable satisfying a Gaussian white noise distribution Goan01a (); Goan01b ():
where , is the relative phase between and , and is a Gaussian white noise characterized by
Here denotes an ensemble average. In stochastic calculus, is known as the infinitesimal Wiener increment. In obtaining Eq. (20), we have assumed that . Hence, for the quantum-diffusive equations obtained later, we should regard, to the order of magnitude, that and .
By taking the diffusive limit on the QPC, the quantum-diffusive conditional master equation for the case of inefficient measurements can be found as:
We will now make the simplifying assumption that . In that case and are real and . This corresponds to as in the experiment of Ref. Sukhorukov07, . The conditional current through the QPC, , conditioned on the dot occupation, satisfies the stochastic differential equation
We can now find from Eq. (22) the conditional dynamics of the dot occupation conditioned on the observed instantaneous QPC current in time . For the SQD-QPC system, we have
Note that the noise “turns off” when the dot (dot 2) is either occupied or empty. This can be understood if we regard the QPC current as a measurement of the dot occupation. Suppose that , and , in which case an electron will eventually tunnel onto the dot. The QPC current must eventually revel this fact, as the current through the QPC will increase. After a small interval of time we will be confident that this is a real effect and not a random fluctuation and the conditional mean becomes locked on unity with no further fluctuation. A parallel argument can be made in the case that . We thus see that this feature of the noise is a reflection of the fact that monitoring the QPC current gives us information on the state of the QD.
Similarly, we obtain from Eq. (22) the equations of motion to determine the DQD coherence and occupations conditioned on the observed instantaneous QPC current in time as
The quantum trajectory theory provides us with full information of the statistical properties about the measured system as that of an experimentalist who actually performs a time-resolved transport experiment. One can use Eq. (25) for the SQD system and Eqs. (26)–(30) for the DQD system to calculate the conditional charge occupation number and then use Eq. (23) to mimic the measured QPC current record continuously in time in a single run of a realistic experiment. We show in Fig. 2(a) a typical realization of the trajectories of for the SQD-QPC system obtained by the quantum trajectory theory and in Fig. 2(b) its corresponding conditional QPC current taking into account the detection bandwidth in experiment Sukhorukov07 (). The simulated QPC current shows random switchings between two average currents, which correspond to the single-electron tunneling onto and off the QD. It indeed resembles the typical measured QPC current shown in Fig. 1(c) of Ref. Sukhorukov07, . Simulating a great amount of trajectories by many different random realizations of , one is able to calculate all the statistical transport quantities of the QD systems, such as the conditional counting statistics. One can obtain the time average QPC current in time by integrating the instantaneous QPC current and acquire the average QD current conditional on QPC current in time in its corresponding trajectory (see Sec. VII.2 for details). We simulate a great amount of trajectories and sort by different , and use the formulas of the conditional moments and cumulants to obtain the conditional counting statistics from the data collected from these trajectories.
It is easy to see that the ensemble average evolution of the conditional master equation, Eq. (22), reproduces the unconditional master equation (7) by simply eliminating the white noise term using Eq. (21). Similarly, averaging Eq. (19) over the observed stochastic process, by setting equal to its expected value Eq. (12), gives the unconditional, deterministic master equation (7). It is also easy to verify that for zero efficiency [i.e., also ], the conditional equations (19) and (22) reduce to the unconditional one, (7). 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.
V Number-resolved master equation
To study the current cumulants of one conductor (e.g., the QD system) conditioned on the average current of the other conductor (e.g. the QPC), we turn to the number-resolved master equation Gurvitz97 (); Korotkov01 (); Shnirman98 (); Makhlin00 () or the master equation for the partially reduced density matrix Goan03 (); Goan04 () of the joint QD and QPC system. If electrons have tunneled through the QPC and electrons have tunneled through the right junction of the QD at time , then the accumulated number of electrons in the drain of the QPC at the earlier time , due to the contribution of the jump term of the QPC, should be for electron in the drain of the QD, and it should be in the drain (right lead) of the QD system due to the contribution of the jump term of the QD for electron in the drain of the QPC Goan03 (); Goan04 (). Hence, after writing out the number dependence , , , or explicitly for the density matrix in Eq. (7), we obtain the master equation for the “partially” reduced density matrix as:
For simplicity, in the following we set the QPC detection efficiency corresponding to perfect detections or efficient measurements. We deal with the case of the SQD-QPC system first. After evaluating Eq. (31) in the occupation number basis and of the SQD, we obtain the rate equations as:
where with referring to the QD occupation number states.
To obtain the solution of with in the number-resolved or the “partially” reduced density matrix approach, we can first apply a two-dimensional Fourier transform (to the counting field space) Shnirman98 (); Makhlin00 (); Goan03 (); Goan04 ()
We note here again that we have set and to be real and their relative phase angle so that .
Similarly for the case of coherently coupled DQD’s measured by a QPC [see Fig. 1(b)], the number-resolved master equation in the Fourier space (counting field space) can also be written in the form of Eq. (35) with given in matrix form as
and the column vector density matrix defined as . Here the matrix elements with indices denote the Fock states , , and of the system,i.e. no electron, one electron in the first dot (left dot), one electron in the second dot (right dot), and one in each dot, respectively. The functions , and , where , , , , .
appears in the diagonal elements of the last two rows of the resultant matrix of Eq. (38) and thus plays the role of dephasing rate for the unconditional dynamics of the DQD’s. As becomes larger, the QPC tends to localize the electron on the dot and thus reduces the coherent tunneling that changes the DQD states between and . When , one expects and from the last two rows of the master equation, Eq. (35) with defined in Eq. (38), will decay much faster than other density matrix elements. As a result, one can set the last two rows of Eq. (38)) equal to zero and then substitute the solution of and back to the coupled equation. Thus we obtain an effective tunneling rate between the two dots as
In this case, the coherent tunneling matrix of in the master equation in the Fourier space (counting field space) reduces to a sequential tunneling matrix
with defined in Eq. (39), and the column vector density matrix becomes involved only the population elements.
In principle, one can solve the resultant coupled first-order differential equations obtained from Eq. (35) for the column elements of and then perform an inverse Fourier transform to obtain . The probability distribution of finding electron that have tunneled through the QPC and electrons that have tunneled into the drain of the QD during time can then be obtained as:
From this distribution function , all orders of unconditional and conditional cumulants (counting statistics) of transmitted electrons can be in principle calculated.
Vi Counting statistics: generating functional approach
vi.1 Unconditional counting statistics
In practice, a more efficient method is the generating functional technique. One may define the moment generating function as Nazarov03 ()
From this definition, it is easy to check that the th moment of and the th moment of can be written as
The cross-cumulants can be calculated through the cumulant generating function as
For example, , , , , etc.
and the cumulant generating function is then
Thus the solution of the number-resolved master equation in the Fourier space (counting field space) has a direct connection with the generating function approach to calculate the FCS.
vi.2 Conditional counting statistics
Having described the joint statistical properties of both the QPC and QD currents, we discuss the conditional counting statistics: the statistical current fluctuations (cumulants) of one system given the observation of a given average current in the other system in time .
In QD-QPC transport system, the probability of having electrons tunneling into the drain of the QD system conditioned on electrons passing through QPC in time can be written as
By defining the conditional moment generating function as
the -th moment of electrons number passing through the QD system, conditioned on the number of electrons in the drain of the QPC is given by
where the subscript “” denotes the quantity it attaches to being conditional. The conditional cumulant could be found by taking partial derivatives with respect to on the conditional cumulant generating function .
In obtaining Eq. (53), we have used the fact that can also be expressed as the inverse Fourier transform of with respect to the counting field variable . Since , one can calculate the conditional counting statistics once having the solution of the number-resolved master equation in the Fourier space (counting field space) .
vi.3 FCS in the stationary state
Unconditional current cumulant. In the stationary or steady state (), the calculation of moments or cumulants can be simplified. The solution of Eq. (35) can be symbolically written as
There is a unique eigenvalue of which develops from the zero eigenvalue of with the smallest absolute real part. The rest of the eigenvalue(s) has (have) larger finite negative real parts that make their contributions considerably much smaller for large times. As a consequence, the long-time dynamics of the moment generating functional Eq. (46) near the stationary state can be well approximated as Nazarov03 (); Sukhorukov07 ()
For the SQD-QPC system, the eigenvalue can be found from Eq. (37) to be:
We can define the QPC current and QD current (setting ) in time . Replacing and , we then obtain the distribution function of the two current
In the long-time (stationary) limit where the time should be much larger than , we may thus evaluate the integral (58) in the stationary phase approximation. The dominant contribution to the joint probability distribution then takes the form of a Legendre transform Sukhorukov07 ():
Since the long-time charge-number cumulant generating function from Eq. (55) is linear in time, we may define the long-time (stationary-state) current cumulant generating function as , which is time-independent. The stationary-state current cumulant can then be calculated through