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MASTER EQUATION APPROACH AND QUANTUM TRAJECTORIES OF A QUANTUM MEASUREMENT PROCESS

Hsi‐Sheng Goan

Published Jan 1, 2002• 0 citations
OpenAlex

Abstract

We provide a unified view for the master equation (ME) approach and the quantum trajectory (QT) approach to the measurement problem of an electron tunnelling between two coherently coupled quantum dots (CQD’s) using a low transparency point contact (PC) or tunnel junction as a detector. We show that the master equation for the reduced or “partially” reduced density matrix can be obtained when an average or “partial” average is taken on the conditional, stochastic density matrix over the possible outcomes of the measurement. Then we simulate ) , ( t N P , the probability of finding N electron that have tunnelled through the PC in time t , using 10000 QT’s and measurement records. The simulation results of ) , ( t N P are in very good agreement with those obtained from the Fourier analysis of the “partially” reduced density matrix. The theory of QT’s (also known as stochastic Schrodinger equation or stochastic ME) has been developed in last ten years mainly in the quantum optics community to describe open quantum system subject to continuous quantum measurements. But it was introduced to the context of solid-state mesoscopics only recently (e.g., Goan et. al. 2001 and references therein). The CQD system, considered here, forms a single charge quantum bit (qubit), and the measurement corresponds to a continuous in time readout of the occupancy of the quantum dots. The QT or stochastic quantum-jump ME of the density matrix operator ) (t c

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We provide a unified view for the master equation (ME) approach and the quantum trajectory (QT) approach to the measurement problem of an electron tunnelling between two coherently coupled quantum dots (CQD’s) using a low transparency point contact (PC) or tunnel junction as a detector. We show that the master equation for the reduced or “partially” reduced density matrix can be obtained when an average or “partial” average is taken on the conditional, stochastic density matrix over the possible outcomes of the measurement. Then we simulate ) , ( t N P , the probability of finding N electron that have tunnelled through the PC in time t , using 10000 QT’s and measurement records. The simulation results of ) , ( t N P are in very good agreement with those obtained from the Fourier analysis of the “partially” reduced density matrix. The theory of QT’s (also known as stochastic Schrodinger equation or stochastic ME) has been developed in last ten years mainly in the quantum optics community to describe open quantum system subject to continuous quantum measurements. But it was introduced to the context of solid-state mesoscopics only recently (e.g., Goan et. al. 2001 and references therein). The CQD system, considered here, forms a single charge quantum bit (qubit), and the measurement corresponds to a continuous in time readout of the occupancy of the quantum dots. The QT or stochastic quantum-jump ME of the density matrix operator ) (t c
We provide a unified picture for the master equation approach and the quantum trajectory approach to a measurement problem of a two-state quantum system (a qubit), an electron coherently tunneling between two coupled quantum dots (CQD's) measured by a low transparency point contact (PC) detector. We show that the master equation of ``partially'' reduced density matrix can be derived from the quantum trajectory equation (stochastic master equation) by simply taking a ``partial'' average over the all possible outcomes of the measurement. If a full ensemble average is taken, the traditional (unconditional) master equation of reduced density matrix is then obtained. This unified picture, in terms of averaging over (tracing out) different amount of detection records (detector states), for these seemingly different approaches reported in the literature is particularly easy to understand using our formalism. To further demonstrate this connection, we analyze an important ensemble quantity for an initial qubit state readout experiment, P(N,t), the probability distribution of finding N electron that have tunneled through the PC barrier(s) in time t. The simulation results of P(N,t) using 10000 quantum trajectories and corresponding measurement records are, as expected, in very good agreement with those obtained from the Fourier analysis of the ``partially'' reduced density matrix. However, the quantum trajectory approach provides more information and more physical insights into the ensemble and time averaged quantity P(N,t). Each quantum trajectory resembles a single history of the qubit state in a single run of the continuous measurement experiment. We finally discuss, in this approach, the possibility of reading out the state of the qubit system in a single-shot experiment.
We provide a unified picture for the master equation approach and the quantum trajectory approach to a measurement problem of a two-state quantum system (a qubit), an electron coherently tunneling between two coupled quantum dots (CQD's) measured by a low transparency point contact (PC) detector. We show that the master equation of ``partially'' reduced density matrix can be derived from the quantum trajectory equation (stochastic master equation) by simply taking a ``partial'' average over the all possible outcomes of the measurement. If a full ensemble average is taken, the traditional (unconditional) master equation of reduced density matrix is then obtained. This unified picture, in terms of averaging over (tracing out) different amount of detection records (detector states), for these seemingly different approaches reported in the literature is particularly easy to understand using our formalism. To further demonstrate this connection, we analyze an important ensemble quantity for an initial qubit state readout experiment, P(N,t)P(N,t)P(N,t), the probability distribution of finding NNN electrons that have tunneled through the PC barrier in time ttt. The simulation results of P(N,t)P(N,t)P(N,t) using 10000 quantum trajectories and corresponding measurement records are, as expected, in very good agreement with those obtained from the Fourier analysis of the ``partially'' reduced density matrix. However, the quantum trajectory approach provides more information and more physical insights into the ensemble and time averaged quantity P(N,t)P(N,t)P(N,t). Each quantum trajectory resembles a single history of the qubit state in a single run of the continuous measurement experiment. We finally discuss, in this approach, the possibility of reading out the state of the qubit system in a single-shot experiment.
We obtain the finite-temperature unconditional master equation of the density matrix for two coupled quantum dots (CQD's) when one dot is subjected to a measurement of its electron occupation number using a point contact (PC). To determine how the CQD system state depends on the actual current through the PC device, we use the so-called quantum trajectory method to derive the zero-temperature conditional master equation. We first treat the electron tunneling through the PC barrier as a classical stochastic point process (a quantum-jump model). Then we show explicitly that our results can be extended to the quantum-diffusive limit when the average electron tunneling rate is very large compared to the extra change of the tunneling rate due to the presence of the electron in the dot closer to the PC. We find that in both quantum-jump and quantum-diffusive cases, the conditional dynamics of the CQD system can be described by the stochastic Schr\"odinger equations for its conditioned state vector if and only if the information carried away from the CQD system by the PC reservoirs can be recovered by the perfect detection of the measurements.
We derive the quantum trajectory or stochastic (conditional) master equation for a single superconducting Cooper-pair box (SCB) charge qubit measured by a single-electron transistor (SET) detector. This stochastic master equation describes the random evolution of the measured SCB qubit density matrix which both conditions and is conditioned on a particular realization of the measured electron tunneling events through the SET junctions. Hence it can be regarded as a Monte Carlo method that allows us to simulate the continuous quantum measurement process. We show that the master equation for the ``partially'' reduced density matrix [Y. Makhlin et al., Phys. Rev. Lett. 85, 4578 (2000)] can be obtained when a ``partial'' average is taken on the stochastic master equation over the fine grained measurement records of the tunneling events in the SET. Finally, we present some Monte Carlo simulation results for the SCB/SET measurement process. We also analyze the probability distribution P(m,t)P(m,t)P(m,t) of finding mmm electrons that have tunneled into the drain of the SET in time ttt to demonstrate the connection between the quantum trajectory approach and the ``partially'' reduced density matrix approach.
A quantum trajectory represents a single history of the state vector of an open quantum system. The stochastic nature of its evolution can be viewed as resulting from the quantum probabilities for the outcomes of measurements performed on the bath. We derive stochastic Schrödinger equations for various trajectory models, starting from the full Schrödinger equation for the system and the bath combined. The state reduction of the bath at a measurement also projects the state of the system on a pure state, thereby destroying the entanglement. The master equation for the system's density matrix is obtained as a result of the discussion, and it is not taken as its starting point. The general concepts are illustrated in the specific case of an atom coupled to the radiation field.
We propose a quantum trajectory analysis of a scheme to measure the states of a coupled-dot device (qubit) where there is a fluctuating energy gap Δ between the two states. The system consists of the qubit and a readout dot coupled to source and drain leads. The tunnel rate through the detector is conditioned by the occupation number of the nearer quantum dot (target) of the qubit and therefore probes the states of the qubit. We derive a Lindblad-form master equation to calculate the unconditional evolution of the qubit and a conditional stochastic master equation calculating the conditional evolution for different tunneling rates. The results show the effects of various device parameters and provide the optimum selection and combination of the system structure.