Quantum theory and stochastic processes — Some contact points

Hsi‐Sheng Goan, Gerard J Milburn, Howard M. Wiseman +1 more

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.

John R. Klauder

Giovanni Jona-Lasinio

Hsi‐Sheng Goan

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

Harish Parthasarathy