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arXiv:2108.04360 · Quantum Reports

Effective and Efficient Resonant Transitions in Periodically Modulated Quantum Systems

Isabel Sainz, Andrés García Sandoval, Andrei B. Klimov

Published Feb 27, 2021• 6 citations
arXivHTMLPDF

Abstract

We analyze periodically modulated quantum systems with SU(2) and SU(1,1) symmetries. Transforming the Hamiltonian into the Floquet representation we apply the Lie transformation method, which allows us to classify all effective resonant transitions emerging in time-dependent systems. In the case of a single periodically perturbed system, we propose an explicit iterative procedure for the determination of the effective interaction constants corresponding to every resonance both for weak and strong modulation. For coupled quantum systems we determine the efficient resonant transitions appearing as a result of time modulation and intrinsic non-linearities.

DOI
10.3390/quantum3010011

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We analyze periodically modulated quantum systems with SU(2) and SU(1,1) symmetries. Transforming the Hamiltonian into the Floquet representation we apply the Lie transformation method, which allows us to classify all effective resonant transitions emerging in time-dependent systems. In the case of a single periodically perturbed system, we propose an explicit iterative procedure for the determination of the effective interaction constants corresponding to every resonance both for weak and strong modulation. For coupled quantum systems we determine the efficient resonant transitions appearing as a result of time modulation and intrinsic non-linearities.
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Abstract The dynamics of qubits coupled to a harmonic oscillator with time-periodic coupling is investigated in the framework of Floquet theory. This system can be used to model nonadiabatic phenomena that require a periodic modulation of the qubit/oscillator coupling. The case of a single qubit coupled to a resonator populated with n = 0, 1 photon is explicitly treated. The time-dependent Schrödinger equation describing the system’s dynamics is solved within the Floquet formalism and compared to a solution obtained with perturbative approaches in the time- and Laplace-domain. Excellent quantitative agreement is found between the analytical and numerical calculations within the Floquet approach, demonstrating the effectiveness of the technique for the study of the dynamical Lamb effect.
Using the time-dependent unitary transformation instead of the invariant operator, the solutions of SU(1, 1) and SU(2) time-dependent quantum systems are obtained. It is shown that the evolution operator is decomposed into the product form of two unitary operators in such a way that one of them has the same periodicity as the Hamiltonian and the other correspond to the Floquet operator which gives the cyclic states and their associated phases over one period of the evolution. The non-adiabatic Berry's (or Aharonov-Anandan) phases are determined totally by such a unitary transformation.
We study a non-Hermitian two-level system with square-wave modulated dissipation and coupling. Based on the Floquet theory, we achieve an effective Hamiltonian from which the boundaries of the <?CDATA PT{\mathscr{P}}{\mathscr{T}}PT?> P T phase diagram are captured exactly. Two kinds of <?CDATA PT{\mathscr{P}}{\mathscr{T}}PT?> P T symmetry broken phases are found, whose effective Hamiltonians differ by a constant ω/2. For the time-periodic dissipation, a vanishingly small dissipation strength can lead to the <?CDATA PT{\mathscr{P}}{\mathscr{T}}PT?> P T symmetry breaking in the (2k – 1)-photon resonance (Δ = (2k – 1) ω), with k = 1,2,3… It is worth noting that such a phenomenon can also happen in 2k-photon resonance (Δ = 2kΔ), as long as the dissipation strengths or the driving times are imbalanced, namely γ 0 ≠ – γ 1 or T 0 ≠ T 1. For the time-periodic coupling, the weak dissipation induced <?CDATA PT{\mathscr{P}}{\mathscr{T}}PT?> P T symmetry breaking occurs at Δ eff = kω, where Δ eff = (Δ 0 T 0 + Δ 1 T 1)/T. In the high frequency limit, the phase boundary is given by a simple relation γ eff = ±Δ eff.
We apply the Floquet theorem and the resonating averages method to solve the Schrödinger equation of the harmonic oscillator whose frequency varies periodically with time. We determine its Floquet states and its corresponding wave functions. The formalism provides an efficiency tool for analyzing the various transition probabilities as well as the uncertainty relation of the system. The effect of a periodic external force on the Floquet states and the two-photon algebra realization are shortly discussed.
We describe a general method to solve a time-dependent Hamiltonian system exhibiting dynamical symmetry under the Lie algebra SU(1,1). The instantaneous eigenstates can be obtained by applying a displacement operator which is constructed explicitly. The construction proves to be equivalent to the solution of the classical equations of motion. One can also obtain the exact solution of the quantum Hamiltonian using this displacement operator. We define and discuss the differences between the cyclic evolution in the space of the physical states and in the parameter space. This leads to a method to define and find the T-cyclic and periodic states and their phases for a system with dynamical symmetry SU(1,1) just by using its time evolution operator. For SU(1,1) one can show that T-cyclic states only exist for some values of T which can be determined through an inequality which is found explicitly and discussed. The example used to check these results has been the degenerate optical parametric oscillator with dynamical symmetry SU(1,1). The system has no periodic states but it does possess T-cyclic states. The characteristic interaction times are distributed in forbidden and allowed regions as in the band theory of particles in a periodic potential. We also define exact phases and adiabatic phases along the lines of the above discussion on T-cyclic and periodic states. For sufficiently low frequencies the initial eigenstates of the system acquire non-trivial Berry phases in each cycle of the evolution of the functional parameters on which H depends. Such a phase can be measured experimentally and from the discussion of its possible values under various circumstances emerges an interesting relationship with the golden mean.