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Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields

Oscillators with constrained energy difference: Coherent states and a quantum clock

Yoshiaki Ohkuwa

Published Aug 2, 2000• 2 citations
Publisher

Abstract

In order to study the ``problem of time in quantum gravity,'' Rovelli proposed a model of a two harmonic oscillator system where one of the oscillators can be thought of as a ``clock'' for the other oscillator. In this paper we examine a model where the Hamiltonian is a difference between two harmonic oscillators, and we consider one of them which has the minus sign as a `'clock,'' since the gravitational degree of freedom has a minus sign in the Hamiltonian of quantum cosmology. Klauder's projection operator approach to generalized coherent states is used to define physical states and operators. The resolution of unity is derived in terms of a gauge invariant coordinate. We investigate the ``quantum clock'' and show that the evolution described by it is identical to the classical motion when the energy becomes large.

DOI
10.1103/physrevd.62.064008

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In order to study the problem of time in quantum gravity,'' Rovelli proposed a model of a two harmonic oscillator system where one of the oscillators can be thought of as a clock'' for the other oscillator. In this paper we examine a model where the Hamiltonian is a difference between two harmonic oscillators, and we consider one of them which has the minus sign as a `'clock,'' since the gravitational degree of freedom has a minus sign in the Hamiltonian of quantum cosmology. Klauder's projection operator approach to generalized coherent states is used to define physical states and operators. The resolution of unity is derived in terms of a gauge invariant coordinate. We investigate the ``quantum clock'' and show that the evolution described by it is identical to the classical motion when the energy becomes large.
In order to study the "problem of time", Rovelli proposed a model of a two harmonic oscillator system where one of the oscillators can be thought of as a 'clock' for the other oscillator. In this paper we examine a model where the Hamiltonian is a difference between two harmonic oscillators, and we consider one of them which has the minus sign as a 'clock'. Klauder's projection operator approach to generalized coherent states is used to define physical states and operators. The resolution of unity is derived in terms of a gauge invariant coordinate. We investigate the 'quantum clock' and show that the evolution described by it is identical to the classical motion when the energy becomes large.
The ``problem of time'' has been a pressing issue in quantum gravity for some time. To help understand this problem, Rovelli proposed a model of a two harmonic oscillator system where one of the oscillators can be thought of as a ``clock'' for the other oscillator, thus giving a natural time reference frame for the system. Recently, the author has constructed an explicit form for coherent states on the reduced phase space of this system in terms of Klauder's projection operator approach. In this paper, I will discuss the features of this model, and then by using coherent state representations and other tools from coherent state quantization, I investigate the construction of gauge-invariant operators on this reduced phase space and the ability to use a quantum oscillator as a ``clock.''
No physical clock is free: every clock is a quantum system that interacts with what it measures. This note studies the exact relationship between clock precision and the decoherence induced by universal time-dilation coupling in solvable constrained quantum systems. Using a clock-system interaction determined by energy coupling, the work derives the exact coherent-overlap decay law associated with time-dilation decoherence and shows how relational quantum dynamics emerges from constrained systems. The global time translation remains a gauge symmetry, while the time evolution recovered by conditioning on the clock exhibits the corresponding deformed relational dynamics. The main result is an exact resolution-decoherence tradeoff: the product of the clock's time resolution and the induced decoherence rate is independent of the clock state and depends only on the coupling strength and the system energy difference. This shows that improving the precision of a quantum clock necessarily increases the disturbance it causes when universally coupled to the system it measures. The result provides an exact model-level realization of a fundamental limitation of physical reference frames: a reference system that allows more precise access to information must pay an unavoidable quantum cost. The work is formulated within finite constrained quantum models with declared assumptions: idealized clocks, a specified universal dilation coupling, and no dynamical gravitational field. The results are accompanied by verification and reproducibility materials.
The classical and quantum dynamics of simple time-reparametrization-invariant models containing two degrees of freedom are studied in detail. Elimination of one ``clock'' variable through the Hamiltonian constraint leads to a description of time evolution for the remaining variable which is essentially equivalent to the standard quantum mechanics of an unconstrained system. In contrast with a similar proposal of Rovelli, evolution is with respect to a geometrical proper time, and the Heisenberg equation of motion is exact. The physical phase space contains no coordinate associated with the eliminated ``clock'' variable. Therefore, time evolution is not with respect to the observable readings of a physical clock. Rather, the eliminated variable can be construed as intrinsic to a particular observer, and, hence, as unobservable, in principle, by this observer. The possibility of a ``test clock,'' which would reveal time evolution while contributing negligibly to the Hamiltonian constraint, is examined, and found to be viable in the semiclassical limit of large quantum numbers. \textcopyright{} 1996 The American Physical Society.
Coherent states in the harmonic oscillator have long been thought of as a bridge from quantum mechanics to classical mechanics. Many of their important properties (most notably those involving time evolution) are only valid in the harmonic oscillator and are at best approximations elsewhere. This thesis is an investigation into several means of generalizing coherent states for other systems and examines the resultant states for some systems. One system of interest is the harmonic oscillator with centripetal barrier for which coherent states of several disparate definitions coincide. Also studied is this spherical rotator, a system which is particularly amenable to defining annihilation operator coherent states. A third system under investigation is the hydrogen atom. This system serves as an arena for the development of an extension to a generalization due to Klauder [J. Phys. A. 29(12):L293-L298. 1996]. Klauder's construction is only applicable to systems without energy degeneracies and must be extended for application where degeneracies are present. The author provides a means for this extension and applies the complete construction to the hydrogen atom problem. As a demonstration of how this construction may be adapted, the author constructs Rydberg wave packets which are initially localized and exhibit full and fractional revivals in the long time evolution.
A fundamental description of time can be consistent not only with the usual monotonic behavior but also with an oscillating physical clock variable, coupled to the degrees of freedom of a system evolving in time. Generically, one would in fact expect some kind of oscillating motion of a system that is dynamical and interacts with its surroundings, as required for a fundamental clock that can be noticed by any other system. Unitary evolution does not require a monotonic clock variable and can be achieved more generally by formally unwinding the oscillating clock movement, keeping track not only of the value of the clock variable but also of the number of cycles it has gone through at any moment. The specific treatment used here employs concatenations of evolution operators, alternating between forward and backward evolution of the oscillating clock. Because the clock and an evolving system have a common conserved energy, the clock is in different cycles for different energy eigenstates of the system state. As a result, the clock is generically in a quantum state with a superposition of different clock cycles, a key feature that distinguishes oscillating clocks from monotonic time. In addition, coherence could be lost faster than observed, for instance if a system that would be harmonic in isolation is made anharmonic by interactions with a fundamental clock, implying observational bounds on fundamental clocks. Numerical computations show that coherence is maintained over long timescales provided the clock period is much smaller than the system period. A small loss of coherence nevertheless remains and, measured in terms of the relative standard deviation of the system period, is proportional to the ratio of the system period and the clock period. Since the precision of atomic clocks could not be achieved if atomic frequencies would be subject to additional variations from coupling to a fundamental clock, an upper bound on the clock period can be obtained that turns out to be much smaller than currently available direct or indirect measurements of time.