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arXiv:1804.08151 · Annals of Physics

Quantum dynamics of a small symmetry breaking measurement device

Hylke C. Donker, Hans A. De Raedt, Mikhail I. Katsnelson

Published Jul 10, 2018• 10 citations
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Abstract

A quantum measuring instrument is constructed that utilises symmetry breaking to enhance a microscopic signal. The entire quantum system consists of a system–apparatus–environment triad that is composed of a small set of spin-1/2 particles. The apparatus is a ferromagnet that measures the z-component of a single spin. A full quantum many-body calculation allows for a careful examination of the loss of phase coherence, the formation and amplification of system–apparatus correlations, the irreversibility of registration, the fault tolerance, and the bias of the device.

DOI
10.1016/j.aop.2018.07.010

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A quantum measuring instrument is constructed that utilises symmetry breaking to enhance a microscopic signal. The entire quantum system consists of a system–apparatus–environment triad that is composed of a small set of spin-1/2 particles. The apparatus is a ferromagnet that measures the z-component of a single spin. A full quantum many-body calculation allows for a careful examination of the loss of phase coherence, the formation and amplification of system–apparatus correlations, the irreversibility of registration, the fault tolerance, and the bias of the device.
In a quantum measurement, a coupling g between the system S and the apparatus A triggers the establishment of correlations, which provide statistical information about S. Robust registration requires A to be macroscopic, and a dynamical symmetry breaking of A governed by S allows the absence of any bias. Phase transitions are thus a paradigm for quantum measurement apparatuses, with the order parameter as pointer variable. The coupling g behaves as the source of symmetry breaking. The exact solution of a model where S is a single spin and A a magnetic dot (consisting of N interacting spins and a phonon thermal bath) exhibits the reduction of the state as a relaxation process of the off‐diagonal elements of S + A, rapid due to the large size of N. The registration of the diagonal elements involves a slower relaxation from the initial paramagnetic state of A to either one of its ferromagnetic states. If g is too weak, the measurement fails due to a “Buridan’s ass” effect. The probability distribution for the magnetization then develops not one but two narrow peaks at the ferromagnetic values. During its evolution it goes through wide shapes extending between these values.
The measurement of a spin-$\half$ is modeled by coupling it to an apparatus, that consists of an Ising magnetic dot coupled to a phonon bath. Features of quantum measurements are derived from the dynamical solution of the measurement, regarded as a process of quantum statistical mechanics. Schrödinger cat terms involving both the system and the apparatus, die out very quickly, while the registration is a process taking the apparatus from its initially metastable state to one of its stable final states. The occurrence of Born probabilities can be inferred at the macroscopic level, by looking at the pointer alone. Apparent non-unitary behavior of the measurement process is explained by the arisal of small many particle correlations, that characterize relaxation.
A Hamiltonian model is solved, which satisfies all requirements for a realistic ideal quantum measurement. The system S is a spin-½, whose z -component is measured through coupling with an apparatus A = M + B, consisting of a magnet M formed by a set of N >> 1 spins with quartic infinite-range Ising interactions, and a phonon bath B at temperature T . Initially A is in a metastable paramagnetic phase. The process involves several time-scales. Without being much affected, A first acts on S, whose state collapses in a very brief time. The mechanism differs from the usual decoherence. Soon after its irreversibility is achieved. Finally, the field induced by S on M, which may take two opposite values with probabilities given by Born's rule, drives A into its up or down ferromagnetic phase. The overall final state involves the expected correlations between the result registered in M and the state of S. The measurement is thus accounted for by standard quantum-statistical mechanics and its specific features arise from the macroscopic size of the apparatus.
An exactly solvable model for a quantum measurement is discussed, that integrates quantum measurements with classical measurements. The z-component of a spin-1/2 test spin is measured with an apparatus, that itself consists of magnet of N spin-1/2 particles, coupled to a bath. The initial state of the magnet is a metastable paramagnet, while the bath starts in a thermal, gibbsian state. Conditions are such that the act of measurement drives the magnet in the up or down ferromagnetic state, according to the sign of s_z of the test spin. The quantum measurement goes in two steps. On a timescale 1/\sqrt{N} the collapse takes place due to a unitary evolution of test spin and apparatus spins; on a larger but still short timescale this collapse is made definite by the bath. Then the system is in a `classical' state, having a diagonal density matrix. The registration of that state is basically a classical process, that can already be understood from classical statistical mechanics.
An exactly solvable model for a quantum measurement is discussed which is governed by hamiltonian quantum dynamics. The z‐component ŝz of a spin −12 is measured with an apparatus, which itself consists of magnet coupled to a bath. The initial state of the magnet is a metastable paramagnet, while the bath starts in a thermal, gibbsian state. Conditions are such that the act of measurement drives the magnet in the up or down ferromagnetic state according to the sign of sz of the tested spin. The quantum measurement goes in two steps. On a timescale 1/N the off‐diagonal elements of the spin’s density matrix vanish due to a unitary evolution of the tested spin and the N apparatus spins; on a larger but still short timescale this is made definite by the bath. Then the system is in a ‘classical’ state, having a diagonal density matrix. The registration of that state is a quantum process which can already be understood from classical statistical mechanics. The von Neumann collapse and the Born rule are derived rather than postulated.
A quantum measurement involves energy exchanges between the system to be measured and the measuring apparatus. Some of them involve energy losses, for example because energy is dissipated into the environment or is spent in recording the measurement outcome. Moreover, these processes take time. For this reason, these exchanges must be taken into account in the analysis of a quantum measurement engine, and set limits to its efficiency and power. We propose a quantum engine based on a spin 1/2 particle in a magnetic field and study its limitations due to the quantum nature of the evolution. The coupling with the electromagnetic vacuum is taken into account and plays the role of a measurement apparatus. We fully study its dynamics, work, power, and efficiency.