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Physical Review A

Significance of time-reversal symmetry for time-resolved measurements of hydrogenic and other atomic observables

G. Gabrielse

Published Jul 1, 1980• 15 citations
Publisher

Abstract

Time-resolved measurements of atomic observables are analyzed using a Liouville space formulation and a Hermitian unit tensor base. This approach makes it possible to distinguish cleanly the symmetries of a formation and/or excitation process completed by time $t=0$, a time evolution under experimental control between $t=0$ and $t=t$, and a measurement at $t=t$, even for hydrogenic observables. Each observable is labeled by a time-reversal quantum number, allowing exploration for the first time of the close relationship between time-reversal symmetry and the time evolution of atomic observables. The experimental reconstruction of atomic observables (at $t=0$) from subsequent time-resolved measurements of the anisotropy and polarization of emitted electric dipole photons is discussed. Hydrogenic observables are stressed and the use of strong fields is included, thus generalizing Fano and Macek's approach.

DOI
10.1103/physreva.22.138

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Time-resolved measurements of atomic observables are analyzed using a Liouville space formulation and a Hermitian unit tensor base. This approach makes it possible to distinguish cleanly the symmetries of a formation and/or excitation process completed by time t=0t=0t=0, a time evolution under experimental control between t=0t=0t=0 and t=tt=tt=t, and a measurement at t=tt=tt=t, even for hydrogenic observables. Each observable is labeled by a time-reversal quantum number, allowing exploration for the first time of the close relationship between time-reversal symmetry and the time evolution of atomic observables. The experimental reconstruction of atomic observables (at t=0t=0t=0) from subsequent time-resolved measurements of the anisotropy and polarization of emitted electric dipole photons is discussed. Hydrogenic observables are stressed and the use of strong fields is included, thus generalizing Fano and Macek's approach.
Time is used in non-relativistic quantum theory as a classical concept. On the one hand, time is the parameter entering Schrödinger's equation and measured by an external laboratory clock. But it could be considered that time also plays the role of an observable when a given event occurs (e.g., when a particle crosses through a potential barrier), and one would expect that like every observable, it should be represented in the theory by an operator whose properties are predictors of the outcome of time measurements made on physical systems. Several time operators have been proposed in different publications, but still, there is not a widely accepted consensus. That subject, among others concerned with time in quantum mechanics, is analyzed in the present chapter, where we focus on the analysis of time-reversal symmetry in non-relativistic quantum systems. The search for symmetries in Physics helps to study the structure of theories. Briefly speaking, a symmetry is a transformation that does not change the structure of the theory. We start with the analysis of Schrödinger’s Equation verifying that for obtaining a time-symmetric description of the dynamics of a particle, the operation of time-reversal must include conjugation represented by the action of an antiunitary operator acting over a wave function. In particular, we analyze quantum systems whose evolution is determined by a set of external parameters varying cyclically, discussing how even though these parameters return to their initial values, the state of the system acquires a geometric phase, which depends on the path followed by the external parameters in the parameter space and which in general can be associated with the lack of the time-reversal symmetry of the system.
We explore whether quantum field theory can be understood as the statistical mechanics of a time-reversal-invariant stochastic generalization of Hamiltonian dynamics. The motivation for this project, started with this paper, is to assign sharp values to all observables and thereby avoid the quantum measurement problem. In classical mechanics, motion is deterministic and corresponds to an evolution of the phase space probability density according to Liouville's equation that is governed by first derivatives of the Hamiltonian in phase space. We derive a generalization of the Liouville equation with natural constraints -- namely, reduction to classical Hamiltonian dynamics as the stochasticity parameter ℏ↦0\hbar\mapsto0ℏ↦0, Fokker-Planck form for the probability density evolution, local Hamiltonian dependence, time-reversal invariance, energy conservation, and minimality -- which turns out to be a Fokker-Planck equation with a generalized diffusion matrix that is symmetric, traceless, and constructed from the Hessian of the Hamiltonian. We then show that the Schrödinger equation in the coherent-state phase-space formulation of certain bosonic QFTs has precisely this form, with the Husimi function playing the role of the phase space probability density. The question to what extent this equation can be interpreted in terms of objective stochastic field theories is discussed in a companion paper.
The dynamics of Rydberg states of atomic hydrogen perturbed simultaneously by a static electric field and a resonant microwave field of elliptical polarization is analyzed in the quantum perturbative limit of small amplitudes. For some configurations, the secular motion (i.e., evolution in time of the elliptical electronic trajectory) may be classically predominantly chaotic. By changing the orientation of the static field with respect to the polarization of the microwave field, one can modify the global symmetries of the system and break any generalized time-reversal invariance. This has a dramatic effect on the statistical properties of the energy levels.
The multipole decomposition of the single-site density matrix provides a symmetry-adapted representation of local electronic degrees of freedom. Conventional, so-called fixed-shell, formulations do not span the full local single-particle operator space, as only operators mapping within the same orbital lll are resolved. Here we construct a complete orthogonal basis of real Hermitian multipole operators for the local density matrix by extending the existing formulation to inter-shell operators. We revisit the multipole decomposition as a decomposition of the operator space by SO(3)\mathrm{SO}(3)SO(3) by first decomposing the orbital and spin operator spaces. Then by coupling them we arrive at the spin-12\frac{1}{2}21​ local single-particle operator space, staying consistent with the existing fixed-shell formulations. We then classify the multipoles by parity and time-reversal symmetry, which allows for unique identification of multipole moments of the density matrix that contribute to expectation values of fully symmetry-resolved observables. As an application, we analyze the two enantiomers of chiral trigonal tellurium by computing the electric toroidal monopole moment selected by symmetry.
In quantum mechanics, measurements are dynamical processes and thus they should be capable of inducing currents. The symmetries of the Hamiltonian and measurement operator provide an organizing principle for understanding the conditions for such currents to emerge. The central role is played by the inversion and time-reversal symmetries. We classify the distinct behaviors that emerge from single and repeated measurements, with and without coupling to a dissipative bath. While the breaking of inversion symmetry alone is sufficient to generate currents through measurements, the breaking of time-reversal symmetry by the measurement operator leads to a dramatic increase in the magnitude of the currents. We consider the dependence on the measurement rate and find that the current is nonmonotonic. Furthermore, nondegenerate measurements can lead to current loops within the steady state even in the Zeno limit.
The formal time symmetry of the quantum measurement process is extensively discussed. Then, the origin of the alleged association between a fixed temporal direction and quantum measurements is investigated. It is shown that some features of such an association might arise from epistemological rather than purely physical assumptions. In particular, it is brought out that a sequence of statements bearing on quantum measurements may display intrinsic asymmetric properties, irrespective of the location of corresponding measurements in time t of the Schrödinger equation. The situation of an observer performing two measurements in two opposite directions of t is eventually investigated. Essential differences are found between two descriptions of this situation: the internal one (taking only into account what is recorded in the observer's memory) and the external one (whereby the observer is considered as a quantum system ruled by the Schrödinger equation). Finally, a method allowing several observers to establish a correspondence between their memory sizes is analyzed. The most important facts that usually lead to the associating of a preferential temporal direction with quantum measurements may be inferred from this correspondence.