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arXiv:2507.18420

The Quantum Foucault Modes

Samuel N. Alperin

Published Jul 24, 2025• 0 citations
arXivHTMLPDF

Abstract

The driven quantum harmonic oscillator is fundamental to a number of important physical systems. Here, we consider the quantum harmonic oscillator under non-Hermitian, PT-symmetric driving, showing that the resulting set of Wigner-space trajectories of an initial coherent state is identical to the set of real-space trajectories of the classical Foucault pendulum. Remarkably, in the case mapped from the trivial 1D pendulum, the corresponding quantum dynamics are those of an oscillator with periodically evolving momentum but fixed position, a novel type of dynamics which are forbidden in classical systems.

DOI
10.48550/arxiv.2507.18420

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The driven quantum harmonic oscillator is fundamental to a number of important physical systems. Here, we consider the quantum harmonic oscillator under non-Hermitian, PT-symmetric driving, showing that the resulting set of Wigner-space trajectories of an initial coherent state is identical to the set of real-space trajectories of the classical Foucault pendulum. Remarkably, in the case mapped from the trivial 1D pendulum, the corresponding quantum dynamics are those of an oscillator with periodically evolving momentum but fixed position, a novel type of dynamics which are forbidden in classical systems.
Summary form only given. The quantum driven pendulum can be implemented using laser cooled rubidium atoms which are exposed to a far detuned amplitude modulated standing wave. This system allows controlled variation of the effective Planck's constant and the amount of coupling to the environment (decoherence). Therefore it is possible to undertake experiments in both classical and quantum regimes of the dynamics of these atoms. The potential can be extremely well approximated as one-dimensional, making the system ideal for simulation using a master equation or quantum trajectories. The dynamics of the system feature distinct regions of regular motion (second order resonances) bounded by a sea of chaos. The resonances are observed as side peaks in the atomic momentum distribution, which each contain up to 30% of the atoms initially in the trap. We report on the results of our experiments to achieve quantum phase space tunneling. After adiabatic elimination of the excited state, the Hamiltonian describing the system is given. The evolution of the quantum driven pendulum can be described with the Floquet operator. Eigenstates of the Floquet operator may be identified with fixed points in phase space. Starting the evolution in a superposition of two Floquet states which is localised in one resonance, time evolution leads to localisation in the other fixed point. This is a coherent tunneling process forbidden by classical dynamics. To observe this tunneling process it is important to localise the superposition of Floquet states in one fixed point. We discuss the effectiveness of alternative methods of loading most of the atoms into a single resonance in order to observe quantum phase space tunneling.
The non-Hermitian quadratic oscillator known as the Swanson oscillator is one of the popular PT -symmetric model systems. Here a full classical description of its dynamics is derived using recently developed metriplectic flow equations, which combine the classical symplectic flow for Hermitian systems with a dissipative metric flow for the anti-Hermitian part. Closed form expressions for the metric and phase-space trajectories are presented which are found to be periodic in time. Since the Hamiltonian is only quadratic the classical dynamics exactly describe the quantum dynamics of Gaussian wave packets. It is shown that the classical metric and trajectories as well as the quantum wave functions can diverge in finite time even though the PT -symmetry is unbroken, i.e., the eigenvalues are purely real.
In recent years, there have been intensive studies of non-Hermitian physics and parity–time (PT) symmetry due to their fundamental importance in theory and outstanding applications. A distinctive character in PT-symmetric system is phase transition (spontaneous PT-symmetry breaking), i.e. an all-real energy spectrum changes into an all-complex one when the non-Hermitian parameter exceeds a certain threshold. However, the conditions for PT-symmetric system with real energy spectrum to occur are rather restrictive. The generalization of PT-symmetric potentials to wider classes of non-PT-symmetric complex potentials with all-real energy spectra is a currently important endeavor. A simple PT-symmetric two-level Floquet quantum system is now being actively explored, because it holds potential for the realization of non-unitary single-qubit quantum gate. However, studies of the evolution dynamics of non-PT-symmetric two-level non-Hermitian Floquet quantum system are still relatively rare. In this paper, we investigate the non-Hermitian physics of a periodically driven non-PT-symmetric two-level quantum system. By phase-space analysis, we find that there exist so-called pseudo fixed points in phase space representing the Floquet solutions with fixed population difference and a time-dependent relative phase between the two levels. According to these pseudo fixed points, we analytically construct a non-unitary evolution operator and then explore the dynamic behaviors of the non-PT-symmetric two-level quantum system in different parameter regions. We confirm both analytically and numerically that the two-level non-Hermitian Floquet quantum system, although it is non-parity-time-symmetric, still features a phase transition with the quasienergy spectrum changing from all-real to all-complex energy spectrum, just like the PT symmetric system. Furthermore, we reveal that a novel phenomenon called quasi-PT symmetric dynamics occurs in the time evolution process. The quasi-PT symmetric dynamics is so named in our paper, in the sense that the time-evolution of population probabilities in the non-PT-symmetric two-level system satisfies fully the time-space symmetry (PT symmetry), while time-evolution of the quantum state (containing the phase) does not meet such a PT symmetry, due to the fact that time-evolution of the phases of the probability amplitudes on the two levels violates the requirement for the PT symmetry.
We explore analytically the quantum dynamics of a point mass pendulum using the Heisenberg equation of motion. Choosing as variables the mean position of the pendulum, a suitably defined generalised variance and a generalised skewness, we set up a dynamical system which reproduces the correct limits of simple harmonic oscillator like and free rotor like behaviour. We then find the unexpected result that the quantum pendulum released from and near the inverted position executes oscillatory motion around the classically unstable position provided the initial wave packet has a variance much greater than the variance of the well known coherent state of the simple harmonic oscillator. The behaviour of the dynamical system for the quantum pendulum is a higher dimensional analogue of the behaviour of the Kapitza pendulum where the point of support is vibrated vertically with a frequency higher than the critical value needed to stabilize the inverted position. A somewhat similar phenomenon has recently been observed in the non equilibrium dynamics of a spin - 1 Bose-Einstein Condensate.
We show theoretically how a driven harmonic oscillator can be used as a quantum simulator for the non-Markovian damped harmonic oscillator. In the general framework, our results demonstrate the possibility to use a closed system as a simulator for open quantum systems. The quantum simulator is based on sets of controlled drives of the closed harmonic oscillator with appropriately tailored electric field pulses. The non-Markovian dynamics of the damped harmonic oscillator is obtained by using the information about the spectral density of the open system when averaging over the drives of the closed oscillator. We consider single trapped ions as a specific physical implementation of the simulator, and we show how the simulator approach reveals physical insight into the open system dynamics, e.g., the characteristic quantum mechanical non-Markovian oscillatory behavior of the energy of the damped oscillator, usually obtained by the non-Lindblad-type master equation, can have a simple semiclassical interpretation.
The first application of quantum ideas to a system with a nonuniform potential energy is to the harmonic oscillator: a pendulum, or a mass on a spring. The result is a quantization of energy with equally spaced energy levels. The wave function is localized in the potential well but can do quantum tunneling beyond the turning points into classically forbidden regions. A wave function can be a negative quantity, but the related position probability is always positive. The wave function shows the particle is not in any single place; in a sense it is in many places simultaneously. The fact that the lowest energy quantum state does not vanish is an example of zero-point energy. States of very high energy show a strong analogy to classical oscillator behavior, an example of the Bohr correspondence principle.