Number-resolved master equation approach
to quantum measurement
and quantum transport
In addition to the well-known Landauer-Büttiker scattering theory and the nonequilibrium Green’s function technique for mesoscopic transports, an alternative (and very useful) scheme is quantum master equation approach. In this article, we review the particle-number ()-resolved master equation (-ME) approach and its systematic applications in quantum measurement and quantum transport problems. The -ME contains rich dynamical information, allowing efficient study of topics such as shot noise and full counting statistics analysis. Moreover, we also review a newly developed master equation approach (and its -resolved version) under self-consistent Born approximation. The application potential of this new approach is critically examined via its ability to recover the exact results for noninteracting systems under arbitrary voltage and in presence of strong quantum interference, and the challenging non-equilibrium Kondo effect.
Quantum master equation
is typically applied for the reduced state evolution of an open quantum system, e.g., in quantum optics and quantum dissipation studies Car93 (); Walls94 (). This formalism is also particularly appropriate for studying quantum measurements, where the measured subsystem is the system of interest, and the apparatus is an environment. In this context, in most cases, some internal degrees of freedom of the apparatus should be retained, which may result in certain back-action effects. Moreover, for quantum measurement, extra issues should be taken into account, such as the readout characteristics of the measurement and the stochastic evolution of the measured state conditioned on the stochastic results of measurement WM09 ().
An interesting solid-state application of quantum measurement is to measure charge qubits using a mesoscopic detector, which can be either a quantum-point-contact (QPC) detector Gur97 (); Win97 (); Moz02 (); Gur03 () or a single electron transistor (SET) Dev00 (); Sch98 (); Sch01 (); Ston02 (). For realistic applications of such measurement, the non-trivial correlation between the detector and qubit has been the focus of extensive studies Gur97 (); Gur03 (); Kor01a (); Goa01a (). For instance, for the qubit-QPC setup, if the energy transfer between the detector and qubit is ignored, the qubit may relax to invalid statistical mixture Shn02 (); Sta03 (); Ave04 (); Li04 ().
Moreover, for this type of measurement, we can construct a particle-number()-resolved master equation (-ME) scheme. That is, by properly clarifying the subspace of the apparatus states in association with the number of electrons transmitted, one can obtain Li05 (); Li05b ()
Here, is the (reduced) qubit state conditioned on the number of electrons “” transmitted in the detector. The Liouvillian is the well-known commutator defined by the system Hamiltonian . The superoperators are associated with the tunneling processes in the transport detector, which have explicit forms, as given in Li05 (). With the knowledge of , one is able to carry out the various readout characteristics of the measurement, by noting that the distribution function of the transmitted electrons is related with the -conditioned density matrix as , where the trace is over the system states.
For the quantum measurement discussed above, the detector itself is a transport device. Hence, the -ME approach, Eq. (1), is a natural tool for studying quantum transport through various mesoscopic (nano-scale) devices. In this context, however, the quantum coherence and/or many-body interaction effects may more significantly affect the transport properties and device functionalities. Simply, the master equation approach is appropriate for quantum transport mainly because we can regard the central device as the system of interest, and the transport leads (reservoirs) as the generalized environment.
Compared to the well-known Landauer-Büttiker theory Dat95 () and the nonequilibrium Green’s function formalism Jau96 (), the master equation approach (especially the -ME formulation Gur96b (); Gur96a ()) has been very useful for studying quantum noise in transport rev0003 (); Lev9396 (); Naz01 (); But02 (); Sam05 (); Sch0506 (); Bel05 (); Bk00 (); Bel04 (); Bla04 (); Jau04 (); Jau05 (); Bee06 (); Li07 (); Ens06 (). One may note that, beyond the usual (average) current, current fluctuations in mesoscopic transport can provide useful information for the relevant mechanisms. Moreover, a fascinating approach, known as full counting statistics (FCS) analysis Lev9396 (), can conveniently yield all the statistical cumulants of the number of transferred charges Naz01 (); But02 (); Sam05 (); Sch0506 (); Bel05 (); Bk00 (); Bel04 (); Bla04 (); Jau04 (); Jau05 (); Bee06 (); Li07 (). The FCS has been demonstrated experimentally for transport through quantum dots Ens06 ().
In essence, the -ME provides an important distribution function via , which contains rich information and allows for convenient calculation of not only the transport current, but also the noise spectrum and counting statistics. For instance, for the latter, all orders of the cumulants of the transmitted electrons can be calculated by using , where is the counting field and is the cumulant generating function (CGF).
In this article, we briefly review the -ME approach and its applications in quantum measurement and quantum transport problems. In Sec. II, for the qubit-QPC setup and general quantum transport system, we first review the key idea and main procedures for constructing the -ME formalism, and then outline the methods of calculating the measurement/transport current and noise spectrum (using MacDonald’s formula). Particular attention will be given to decomposition of the -dependent subspaces of the reservoir states and the consequences of the closed circuit nature, which would significantly affect the reservoir state averages. In Sec. III, we discuss the application of the -ME to two measurement setups in detail, i.e., a qubit measured by QPC and SET detectors. In Sec. IV, we further discuss the application of the -ME to quantum transport by using the double-dot Aharonov-Bohm (DDAB) interferometer and Majorana fermion (MF) probe as examples. In Sec. V, we review the newly proposed self-consistent Born approximation based master equation (SCBA-ME) approach to quantum transport; the SCBA-ME scheme goes beyond the usual master equation approach under the standard Born approximation which, for instance, can recover the exact results of quantum transport through noninteracting systems and predict the challenging non-equilibrium Kondo effect for transport through Anderson impurity (interacting dots). Finally, in Sec. VI we present our concluding remarks.
Ii General Formalism
In this section,
we review the construction of the -ME formalism, and outline the methods of applying it to calculate the measurement/transport current and noise spectrum.
ii.1 Number()-Resolved Master Equation
ii.1.1 Set-up (I): Qubit measurement using QPC detector
For the sake of generality,
we formally consider an arbitrary quantum system measured using a QPC detector. The whole setup can be described by the Hamiltonian as follows Li05 ():
In this decomposition, the free part of the total Hamiltonian contains the Hamiltonians of the measured system and the QPC reservoirs (the last two terms). The Hamiltonian describes electron tunneling through the QPC, e.g., from state in the left reservoir to state in the right one, with a tunneling amplitude of which may depend on the state of the observable.
Regarding the tunneling Hamiltonian as a perturbation, on the basis of the second-order Born expansion, we can derive a formal equation for the reduced density matrix as Yan98 ()
Here, the Liouvillian superoperators are defined as , , and with the usual propagator (Green’s function) associated with . The reduced density matrix is , resulting from tracing all the detector degrees of freedom from the entire density matrix. However, for quantum measurement where specific readout information is to be recorded, the average should be taken over the unique class of states of the detector that is being kept track of.
The Hilbert space of the detector can be classified as follows. First, define the subspace in the absence of electron tunneling through the detector as , which is spanned by the product of the many-particle states of the two isolated reservoirs, formally denoted as . Then, introduce the tunneling operator , and denote the Hilbert subspace corresponding to -electrons tunneled from the left to the right reservoirs as , where . The entire Hilbert space of the detector is .
Here, , which is the reduced density matrix of the measured system conditioned on the number of electrons tunnelled through the detector until time . Now, we transform the Liouvillian product in Eq. (4) to the conventional form:
For simplicity, we rewrite the interaction Hamiltonian as . Here, we have assumed the tunneling amplitude to be real and independent of the reservoir-state “”, and denoted it by which depends on the state of the measured system. The detector fluctuation is described by , with and . Two physical considerations are further made, as follows: (i) Instead of the conventional Born approximation for the entire density matrix , we propose the ansatz , where is the density operator of the detector reservoirs with -electrons tunnelled through the detector. With the ansatz of the density operator, tracing over the subspace yields
Here, we have utilized the orthogonality between states in different subspaces, which leads to term selection from the entire density operator . (ii) Due to the closed nature of the detector circuit, the extra electrons tunneled into the right reservoir will flow back into the left reservoir via the external circuit. In addition, the rapid relaxation processes in the reservoirs will quickly bring the reservoirs to the local thermal equilibrium state determined by the chemical potentials. As a consequence, after the procedure (i.e., the state selection) as expressed by Eq. (6), the detector density matrices and in Eq. (6) can be well approximated by , i.e., the local thermal equilibrium reservoir state. Under this consideration, the correlation functions become, , , and . Here, stands for .
Here, , , and . Under the wide-band approximation for the detector reservoirs, the spectral function can be explicitly expressed as Li04 (): , where , and is the temperature. (Here, and in the following, we use the unit system of ). In Eq. (7), the terms in describe the effect of fluctuation of forward and backward electron tunneling through the detector on the measured system. In particular, the Liouvillian operator “” in contains the information of energy transfer between the detector and the measured system, which correlates the energy (spontaneous) relaxation of the measured system with the inelastic electron tunneling in the detector. At high-voltage limit, formally , the spectral function , and Eq. (7) reduce to the result derived by Gurvitz et al Gur97 (); Moz02 (); Gur03 (); Goa01a ().
ii.1.2 Set-up (II): Quantum transport
quantum transport, schematically shown in Fig. 2, can be described by the following Hamiltonian:
Here, is the central system (device) Hamiltonian, which can be quite general (e.g., includes many-body interactions). () is the creation (annihilation) operator of electrons in state , which indicates both the orbital and spin degrees of freedom. The second and third terms describe, respectively, the two (left and right) leads (reservoirs) and the tunneling between them and the central system. The lead electrons are also attached here by index “” to characterize their possible correlation with the system states. For instance, this will be the typical situation for spin-dependent transport.
If we introduce and express the tunneling Hamiltonian as
then, by considering this tunneling Hamiltonian as a perturbation, the second-order Born expansion results in an unconditional master equation for the reduced density matrix of the same form of Eq. (3).
To construct a conditional (i.e. particle-number-resolved) master equation, one should keep track of the number of electrons that arrive at the collector. Let us classify the Hilbert space of the electrodes as follows. First, we define the subspace in the absence of electrons at the collector as “”, which is spanned by the product of all many-particle states of the two isolated reservoirs, formally denoted as . Then, we introduce the Hilbert subspace “” ( ), corresponding to -electrons at the collector. The entire Hilbert space of the two electrodes is .
With this type of classification for the reservoir states, the average over states in the entire Hilbert space “” is replaced with states in the subspace “”, leading to a conditional master equation Li05b ()
Here, , where is the state of the whole system. is the reduced density matrix of the central system conditioned on the number of electrons arriving at the collector until time .
As for the qubit-QPC problem, two important considerations are made, as follows: (i) Instead of the conventional Born approximation for the entire density matrix , we propose the ansatz , where is the density operator of the electron reservoirs associated with -electrons arrived at the collector. (ii) Due to the closed nature of the transport circuit, the extra electrons arriving at the collector will flow back into the emitter (left reservoir) via the external circuit. In addition, the rapid relaxation processes in the reservoirs will quickly bring the reservoirs to the local thermal equilibrium state determined by the chemical potentials.
Here, and . The spectral functions are defined in terms of the Fourier transform of the reservoir correlation functions, i.e., , where and .
The “”-dependence of Eq. (11) is somehow similar to the usual rate equation, despite its operator feature. Each term in Eq. (11) can be interpreted similarly using the conventional c-number rate equation. Unlike in the Bloch equation derived by Gurvitz et al Gur96b (), in Eq. (11), is also coupled to , which is not present in Ref. Gur96b, . This difference originates from the fact that Eq. (11) is valid for non-zero temperatures.
ii.2 Current and Noise Spectrum
ii.2.1 Qubit measurement by QPC
With the knowledge of
, one can carry out the various readout characteristics of the measurement. In particular, for a qubit with and comparable to or smaller than the measurement time Sch98 (), the qubit oscillation cannot be read out using the conventional single shot measurement. In this regime, continuous weak measurement is an alternative scheme to record the qubit oscillations, e.g., in the power spectrum of the output current.
First, for the ensemble-average current, simple expression is related to the unconditional density matrix . The derivation is from the fact that the current is associated with the probability distribution , via , where . By considering the Summation over “” and making use of the cyclic property under trace, we obtain Li05 ()
where is the average current over time and . It can be proved that Li05 ()
where , which can be calculated via its equation of motion Li05 ()
Combining the above three equations, the noise spectrum can be easily obtained via Laplace transform in terms of simple algebraic manipulations.
ii.2.2 Quantum transport
For quantum transport
Here, the unconditional density matrix satisfies the usual master equation (which can be obtained by summing up Eq. (11) over “”)
Eqs. (II.2.2) and (17) can serve as a convenient starting point to compute the transport current. In practice, one can first diagonalize the central system Hamiltonian, then perform the Liouvillian operations in the eigen-state representation.
For transport, the current noise spectrum can provide additional dynamic information beyond the current itself. We know that, for time-dependent transport, the currents across the left and right junctions (between the central system and the two leads) are not necessarily equal to each other. This requires a definition for the power spectrum using the “average” current , where and are two coefficients determined by the junction capacitances Li053 (), which satisfy . This leads to the noise spectrum of the current consisting of three parts Li053 (): , where is the noise spectrum of the left (right) junction current and is the fluctuation spectrum of the electron number on the central device.
For (), we use the MacDonald’s formula
where denotes the “number” matrix and is the stationary state. Here, we also introduce
The final expression for is Li053 ()
For the charge fluctuations on the central system, the symmetrized noise spectrum can be expressed using Li053 ()
where , where and is the electron-number operator of the central system. Using the cyclic property under trace, we have , where . Desirably, satisfies the equation of the usual reduced density matrix. Its Fourier transform can be easily solved using Li053 ()
Then, we have Li053 ()
ii.3 Counting Statistics and Large-Deviation Analysis
In order to get
information in addition to the average current and current fluctuation spectrum, with the knowledge of (and thus ), one can perform FCS Lev9396 () and large-deviation (LD) analysis LD9809 (); SM8792 (); Gar10 (). For FCS analysis, the current CGF can be constructed using FCS07 ()
where is the so-called counting field. Based on the CGF, , the cumulant can be readily carried out via . As a result, one can easily confirm that the first two cumulants, and , give rise to the mean value and the variance of the transmitted electrons, while the third one (skewness), , characterizes the asymmetry of the distribution. Here, . Moreover, one can relate the cumulants to measurable quantities, e.g., the average current by , and the zero-frequency shot noise by . In addition, the important Fano factor is simply given by , which characterizes the extent of current fluctuations: indicates a super-Poisson fluctuating behavior, while indicates a sub-Poisson process.
The real nature of the transform factor , in contrast to the complex one, , makes the resultant somewhat resemble the partition function in statistical mechanics. Using analogous terms in statistical mechanics, in Eq. (28), the trajectories are categorized by a dynamical order parameter “” or its conjugate field “”. This is realized by an exponential weight similar to the Boltzmann factor, with the dynamical order parameter representing the energy or magnetization and the conjugate field representing the temperature or magnetic field.
is called LD function in LD analysis. In statistical mechanics, the partition function measures the number of microscopic configurations accessible to the system under given conditions. For the mesoscopic transport under consideration, if we are interested in the dynamical aspects of the transport electrons, the above insight can be used for LD analysis in the time domain. That is, the LD function is a measure of the number of trajectories accessible to the “counter”, which favorably characterizes the trajectory space from multiple angles according to the effect of the conjugate field. In particular, it allows one to inspect the rare fluctuations or extreme events by tuning the conjugate field “”.
We emphasize that, if one performs the conventional FCS analysis only, using either a complex transform factor or a real one would make no difference since the limit will be considered at the end. However, for the LD study, we must use the real factor , which plays a role in categorizing (selecting) the trajectories. This type of selection would enables us to perform statistical analysis for the fluctuations of sub-ensembles of trajectories. For instance, implies mainly selecting the inactive trajectories (with small ), while prefers the active trajectories (with large ). In particular, by varying , the -dependent statistics can reveal interesting dynamical behaviors in the time-domain. In other words, based on the distribution function , which contains complete information of all the trajectories, the LD approach, beyond the conventional FCS analysis, captures more information from via the -dependent cumulants.
This equation allows us to carry out the LD function via , where the trace is over the central system states. Accordingly, we obtain the generating function for arbitrary counting time . Further, we can prove LD11 ()
|and more generally,|
Here, for brevity, we used the notation for . Using these cumulants, we can define a finite-counting-time average current and the shot noise . In addition, the generalized Fano factor, , will be of interest to characterize the fluctuation properties.
We notice that, in order to obtain , we only need to determine the various -th order derivatives of , . This is an efficient method to compute the -dependent cumulants for finite counting time. That is, by performing the derivatives on Eq. (29) and defining , we obtain a set of coupled equations for , whose solution then gives .
In long counting time limit, it can be proved that . Desirably, the asymptotic form, , allows one to identify the LD function for the smallest eigenvalue of the counting matrix, i.e., the r.h.s of Eq. (29).
Iii Application to Quantum Measurement
In this section,
we illustrate the application of the -ME to two examples of quantum measurement, i.e., for a charge qubit measured respectively by QPC and SET detectors.
iii.1 Qubit Measured by QPC
Let us specify
the charge qubit as a pair of coupled quantum dots, described by the Hamiltonian . We then introduce and set as the reference energy. The qubit eigen-energies are obtained as and . Correspondingly, the eigenstates are for the excited state and for the ground state, where is introduced using and . The coupling between the qubit and detector is characterized by , where and .
Here, , , and , with . At zero temperature, Eq. (31) can be further simplified. Compared with previous results Shn02 (); Sta03 (), we find that under low voltage (), Eq. (31) is reduced to the same result given by Shnirman et al. Shn02 (), but under it differs from the results in Refs. Shn02, and Sta03, .
Here, three currents are defined as , , and , with and being the detector currents corresponding to qubit states and , respectively. The other quantities in Eq. (32) are , and . The three noise spectrum components are, respectively, (i) the zero-frequency noise , (ii) the Lorentzian spectral function with a peak around the qubit Rabi frequency , and (iii) , originating from the qubit-relaxation-induced inelastic tunnelling effect in the detector. In addition to , the qubit relaxation effect is also manifested in and , i.e., giving rise to the second term of and reducing the pre-factor in from unity. If the qubit-relaxation-induced inelastic effect is neglected or at the limit of the high bias voltage , Eq. (32) returns to the result obtained in previous work Kor01a (); Goa01a ().
The measurement-induced relaxation effects of the qubit are shown in Fig. 3. The major effect of the qubit relaxation shown in Fig. 3(a) is the lowering of the entire noise spectrum in qualitative consistence with the findings of Gurvitz et al Gur03 (), where an external thermal bath is introduced to cause qubit relaxation. However, the spontaneous relaxation discussed here does not diminish the telegraph noise peak near zero frequency in the incoherent case, which implies the presence of the Zeno effect, in contrast to the major conclusion of Ref. Gur03, . In addition, the transition behavior from the coherent to the incoherent regime is different. Figure 3(b) shows the voltage effect where the coherent peak around reduces as the measurement voltage decreases. Interestingly, this effect alters the fundamental bound of 4 for the signal-to-noise ratio, , which was determined by Korotkov et al. at the high voltage limit (see the inset) Kor01a ().
The voltage effect is further shown in Fig. 4 by the 3D plot of the scaled spectra for different qubit symmetries. In contrast to the present result, we notice that in Ref. Sta03, , no spectral structure was found, i.e., in the range of for the symmetric qubit (). However, Shnirman et al showed the existence of the coherent peaks at for voltage greater than Shn02 (). For an asymmetric qubit, as shown in Fig. 4(b), the coherent peaks at are destroyed and a peak around is formed. This transition originates from the breakdown of the resonant condition, which replaces the Rabi oscillation of the qubit with incoherent jumping.
iii.2 Qubit Measured by SET
As second example of quantum measurement
, we consider a charge qubit measured by an SET Dev00 (); Sch98 (); Sch01 (), as schematically shown in Fig. 5. The SET is a sensitive charge-state detector, which is suitable for fast qubit read-out in solid-state quantum computation. For single-shot measurement, i.e., where the qubit state is unambiguously determined in one run, an important figure of merit is the detector’s efficiency, defined as the ratio of information gained time and the measurement-induced dephasing time Sch98 (); Sch01 (). In the weakly responding regime, it was found that the SET has a rather poor quantum efficiency Sch01 (); Kor01 (); Moz04 (). However, later study showed that, for a strong-response SET, the quantum limit of an ideal detector can be reached, resulting in an almost pure conditioned state Wis06 ().
As mentioned earlier for the QPC detector, a more implementable approach is continuous weak measurement rather than single-shot measurement. This type of measurement allows one to determine the ensemble average of detector and qubit states, and the qubit coherent oscillation is read out from the spectral density of the detector. In continuous weak measurement, an interesting generic result is the so-called Korotkov-Averin (K-A) bound, i.e., the signal-to-noise ratio (SNR) bounded by a fundamental limit of “4” K-A01 (), which can be broken only in cases such as when performing quantum nondemolition (QND) measurement Ave02 (), adding quantum feedback control Wang07 (), or using two detectors But06 (). We consider continuous weak measurement of qubits using strongly responding SETs Wis06 (); Gur05 () and show that, for both models in Refs. Wis06, and Gur05, , the SNR can violate the universal Korotkov-Averin bound AK ().
For simplicity, we assumed spinless electrons. The system Hamiltonian, , contains a qubit, SET central dot, and their Coulomb interaction (the -term). For the qubit, we assumed that each dot has only one bound state, i.e., the logic states and with energies and and a coupling amplitude . is the number operator of qubit state , which is 1 for being occupied and 0 otherwise. For the SET, and are the electron creation (annihilation) operators of the central dot and reservoirs, respectively. is introduced as the number operator of the SET dot. Similar to previous work, we assumed that the SET works in the strong Coulomb-blockade regime, with only a single level involved in the measurement process. Finally, describes the tunnel coupling of the SET dot to the leads, with amplitude