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Australian Journal of Physics

Quantum Measurement and Stochastic Processes in Mesoscopic Conductors

Gerard J Milburn

Published Feb 23, 2001• 9 citations
Publisher

Abstract

In quantum measurement theory it is necessary to show how a quantum source conditions a classical stochastic record of measured results. We discuss mesoscopic conductance using quantum stochastic calculus to elucidate the quantum nature of the measurement taking place in these systems. To illustrate the method we derive the current fluctuations in a two terminal mesoscopic circuit with two tunnel barriers containing a single quasi bound state on the well. The method enables us to focus on either the incoming/ outgoing Fermi fields in the leads, or on the irreversible dynamics of the well state itself. We show an equivalence between the approach of BÜttiker and the Fermi quantum stochastic calculus for mesoscopic systems.

DOI
10.1071/ph00003

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In quantum measurement theory it is necessary to show how a quantum source conditions a classical stochastic record of measured results. We discuss mesoscopic conductance using quantum stochastic calculus to elucidate the quantum nature of the measurement taking place in these systems. To illustrate the method we derive the current fluctuations in a two terminal mesoscopic circuit with two tunnel barriers containing a single quasi bound state on the well. The method enables us to focus on either the incoming/ outgoing Fermi fields in the leads, or on the irreversible dynamics of the well state itself. We show an equivalence between the approach of BÜttiker and the Fermi quantum stochastic calculus for mesoscopic systems.
We show an equivalence between the approach of Buttiker and the Fermi quantum stochastic calculus for mesoscopic systems. To illustrate the method we derive the current fluctuations in a two terminal mesoscopic circuit with two tunnel barriers containing a single quasi bound state on the well. The method enables us to focus on either the incoming/outgoing Fermi fields in the leads, or on the irreversible dynamics of the well state itself. The quantum stochastic calculus we use is the Fermi analogue of the input/output methods of quantum optics.
Mesoscopic physics and nanoelectronics concentrate on systems with dimensions somewhere between atomic and everyday macroscopic scale. Modern technology enables construction of submicron nanostructures where modeling based on classical physics has proven inadequate. It is possible to design electric circuits where dynamics of single electrons and photons are controlled using state-of-the-art experimental methods. For quantitative understanding of these systems it is necessary to resort to a quantum-mechanical description. Quantum phenomena, such as tunneling and a wave-like interference of particles, are essential ingredients of physics in mesoscopic systems. Field of mesoscopic physics contains a rich variety of topics ranging from fundamental condensed matter physics to quantum information processing and possible future technological applications. \n \nThis thesis presents theoretical studies of mesoscopic quantum phenomena in nanostructures and small electronic devices. We have focused on effects of environment fluctuations and investigated connections between fluctuations and transport phenomena. Decoherence in quantum bits (qubits) and quantum state engineering in superconducting circuits are also studied. The theoretical analysis in each case requires an open-system treatment. \n \nEffects of current fluctuations on quantum probe systems have been studied in detail. We have calculated transitions induced by current noise and discussed how these could be used for characterization of fluctuations. We have also shown that electric fluctuations play a key role in radiation and photon heat transport in nanostructures. Motivated by recent advances in mesoscopic electron-photon systems, we have studied a response of a coupled resonator-qubit system, squeezing of quantum fluctuations in small superconducting circuits and investigated decoherence in Josephson flux qubits.
Superconducting circuits, photonic cavities, and tiny mechanical oscillators are versatile systems in which quantum dynamics can be observed and controlled. These various technologies can be combined, and the resulting “circuits” are often referred to as nano-electromechanical or optomechanical systems. In the context of quantum physics, such devices are referred to as “mesoscopic” because they contain a large number of atoms. This terminology distinguishes them from the “microscopic” systems of atom-optics, such as single ions [364] and atoms [545, 233, 26], and the spins of individual nuclei [412]. Nano-engineered systems provide an especially powerful arena for exploring quantum measurement and control because of their potential to realize complex circuits. In this chapter we explain how to treat these systems and describe in detail a common method used to measure them. Superconducting circuits The electrical circuits that we describe in this chapter are quite different from those that use normal conductors, such as copper wire at room temperature. Normal conductors are subject to strong dissipation and decoherence from the environment. As a result the dynamical degrees of freedom of the circuits, the currents and voltages, behave as classical variables. In superconducting circuits, the resistance, and thus the dissipation, is so small that the currents and voltages behave as quantum mechanical degrees of freedom. Of course, the current in a wire is actually composed of many elementary particles that each obey quantum mechanics. Why then should the current act like a single quantum degree of freedom? A mechanical analogy is useful here. Recall that even though a macroscopic pendulum is composed of many particles, when one talks about the motion of the pendulum one means the motion of the center-of-mass of its constituent particles. Since the center of mass is merely a linear combination of the coordinates of each particle, it is itself a valid quantum mechanical coordinate with a conjugate momentum. In a similar way the total current in a superconducting wire is a collective degree of freedom that obeys quantum mechanics.
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.
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 study noninvasive measurement of stationary currents in mesoscopic systems. It is shown that the measurement process is fully described by the Schrödinger equation without any additional reduction postulate and without the introduction of an observer. Nevertheless the possibility of observing a particular state out of coherent superposition leads to collapse of the wave function, even though the measured system is not distorted by interaction with the detector. Experimental consequences are discussed.