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Physical Review E

Optimal control of a collection of parametric oscillators

K. H. Hoffmann, Bjarne Andresen, Peter Salamon

Published Jun 5, 2013• 22 citations
Publisher

Abstract

The problem of effectively adiabatic control of a collection of classical harmonic oscillators sharing the same time-dependent frequency is analyzed. The phase differences between the oscillators remain fixed during the process. This fact leads us to adopt the coordinates: energy, Lagrangian, and correlation, which have proved useful in a quantum description and which have the advantage of treating both the classical and quantum problem in one unified framework. A representation theorem showing that two classical oscillators can represent an arbitrary collection of classical or quantum oscillators is proved. An invariant, the Casimir companion, consisting of a combination of our coordinates, is the key to determining the minimum reachable energy. We present a condition for two states to be connectable using one-jump controls and enumerate all possible switchings for one-jump effectively adiabatic controls connecting any initial state to any reachable final state. Examples are discussed. One important consequence is that an initially microcanonical ensemble of oscillators will be transformed into another microcanonical ensemble by effectively adiabatic control. Likewise, a canonical ensemble becomes another canonical ensemble.

DOI
10.1103/physreve.87.062106

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The problem of effectively adiabatic control of a collection of classical harmonic oscillators sharing the same time-dependent frequency is analyzed. The phase differences between the oscillators remain fixed during the process. This fact leads us to adopt the coordinates: energy, Lagrangian, and correlation, which have proved useful in a quantum description and which have the advantage of treating both the classical and quantum problem in one unified framework. A representation theorem showing that two classical oscillators can represent an arbitrary collection of classical or quantum oscillators is proved. An invariant, the Casimir companion, consisting of a combination of our coordinates, is the key to determining the minimum reachable energy. We present a condition for two states to be connectable using one-jump controls and enumerate all possible switchings for one-jump effectively adiabatic controls connecting any initial state to any reachable final state. Examples are discussed. One important consequence is that an initially microcanonical ensemble of oscillators will be transformed into another microcanonical ensemble by effectively adiabatic control. Likewise, a canonical ensemble becomes another canonical ensemble.
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We design faster-than-adiabatic state transfers (switching of quantum numbers) in time-dependent coupled-oscillator Hamiltonians. The manipulation to drive the process is found using a two-dimensional invariant recently proposed in S. Simsek and F. Mintert, Quantum 5 (2021) 409, and involves both rotation and transient scaling of the principal axes of the potential in a Cartesian representation. Importantly, this invariant is degenerate except for the subspace spanned by its ground state. Such degeneracy, in general, allows for infidelities of the final states with respect to ideal target eigenstates. However, the value of a single control parameter can be chosen so that the state switching is perfect for arbitrary (not necessarily known) initial eigenstates. Additional 2D linear invariants are used to find easily the parameter values needed and to provide generic expressions for the final states and final energies. In particular we find time-dependent transformations of a two-dimensional harmonic trap for a particle (such as an ion or neutral atom) so that the final trap is rotated with respect to the initial one, and eigenstates of the initial trap are converted into rotated replicas at final time, in some chosen time and rotation angle.
Fast adiabatic cooling procedures have important implications for the attainability of absolute zero. While traditionally adiabatically cooling a system is associated with slow thermal processes, for the parametric quantum harmonic oscillator fast frictionless processes are known, which transfer a system from an initial thermal equilibrium at one temperature into thermal equilibrium at another temperature. This makes such systems special tools in analyzing the bounds on fast cooling procedures. Previous discussions of those systems used frictionless cooling assuming real frequencies of the oscillator. Using a control with imaginary frequencies (repulsive potential) revises previous implications for the possible operation of a quantum refrigerator. Here we discuss these requisite revisions in the context of the third law of thermodynamics. In addition to minimum time controls, which are always of the bang-bang form, fast frictionless processes with a continuous variation of the frequency have been presented previously in the literature. Such continuous variation controls have been experimentally verified by cooling a Bose-Einstein condensate, while minimum time controls still await verification. As some implementations may indeed not be able to implement the instantaneous jumps in frequency required by bang-bang controls, constraining the rate of change in the frequency calls for ramped bang-bang solutions. We present such solutions and compare their performance to the continuous controls used in the experiment.
The quantum harmonic oscillator with time-dependent frequency is a paradigmatic model of driven quantum dynamics and one of the few nontrivial systems that admits an exact analytical solution. In this review paper, we present a unified treatment of the time-dependent oscillator based on the Lewis-Riesenfeld invariant method, Bogoliubov transformations and the Ermakov-Pinney equation. We show how these approaches naturally connect to squeezing for the description of excitations production, and to the breakdown of adiabaticity under generic frequency protocols. Exact results for sudden quenches and smooth ramps are discussed in detail. By explicitly bridging invariant methods and squeezing formalism, this review is meant to provide a comprehensive framework for understanding nonequilibrium dynamics in quadratic potentials, with applications ranging from thermodynamics and condensed matter to quantum control theory.
Abstract In this work, we give a quantitative answer to the question: how sudden or how adiabatic is a frequency change in a quantum harmonic oscillator (HO)? We do that by studying the time evolution of a HO which is initially in its fundamental state and whose time-dependent frequency is controlled by a parameter (denoted by ϵ ) that can continuously tune from a totally slow process to a completely abrupt one. We extend a solution based on algebraic methods introduced recently in the literature that is very suited for numerical implementations, from the basis that diagonalizes the initial Hamiltonian to the one that diagonalizes the instantaneous Hamiltonian. Our results are in agreement with the adiabatic theorem and the comparison of the descriptions using the different bases together with the proper interpretation of this theorem allows us to clarify a common inaccuracy present in the literature. More importantly, we obtain a simple expression that relates squeezing to the transition rate and the initial and final frequencies, from which we calculate the adiabatic limit of the transition. Analysis of these results reveals a significant difference in squeezing production between enhancing or diminishing the frequency of a HO in a non-sudden way.
Abstract. For systems in an externally controllable time-dependent potential, the optimal protocol minimizes the mean work spent in a finite-time transition between given initial and final values of a control parameter. For an initially thermalized ensemble, we consider both Hamiltonian evolution for classical systems and Schrödinger evolution for quantum systems. In both cases, we show that for harmonic potentials, the optimal work is given by the adiabatic work even in the limit of short transition times. This result is counter-intuitive because the adiabatic work is substantially smaller than the work for an instantaneous jump. We also perform numerical calculations of the optimal protocol for Hamiltonian dynamics in an anharmonic quartic potential. For a two-level spin system, we give examples where the adiabatic work can be reached in either a finite or an arbitrarily short transition time depending on the allowed parameter space. PACS numbers: 05.70.-aOptimal protocols for Hamiltonian and Schrödinger dynamics 2 1.