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arXiv:2108.01324 · Entropy

Evanescent Wave Approximation for Non-Hermitian Hamiltonians

Benedetto Militello, Anna Napoli

Published Jun 4, 2020• 1 citation
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Abstract

The counterpart of the rotating wave approximation for non-Hermitian Hamiltonians is considered, which allows for the derivation of a suitable effective Hamiltonian for systems with some states undergoing decay. In the limit of very high decay rates, on the basis of this effective description we can predict the occurrence of a quantum Zeno dynamics, which is interpreted as the removal of some coupling terms and the vanishing of an operatorial pseudo-Lamb shift.

DOI
10.3390/e22060624

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The counterpart of the rotating wave approximation for non-Hermitian Hamiltonians is considered, which allows for the derivation of a suitable effective Hamiltonian for systems with some states undergoing decay. In the limit of very high decay rates, on the basis of this effective description we can predict the occurrence of a quantum Zeno dynamics, which is interpreted as the removal of some coupling terms and the vanishing of an operatorial pseudo-Lamb shift.
Effective non-Hermitian Hamiltonians describing decaying systems are considered and analyzed in connection with the occurrence of possible Hilbert space partitioning. This fact can be interpreted as Zeno effect or Zeno dynamics, according to the dimension of the subspace one focuses on. Depending on the complex phases of the diagonal terms of the Hamiltonian, the system reacts in different ways, requiring larger moduli for the Zeno phenomena to occur when the complex phase is close to π/2\pi/2π/2. Unperturbed dynamics are also shown to persist in some especial conditions where interaction-free subspaces are present.
We show that the counterrotating, neglected in the previous studies of the quantum Zeno effect (QZE) in atomic decay, can have a large impact on the short-time evolution. We calculate the electron self-energy, the Lamb shift, and the QZE without making the rotating-wave approximation (RWA) and show that, for hydrogen in free space, the Zeno time is longer by 2 orders of magnitude than that obtained from the RWA. We also show that there is no anti-Zeno effect as the counterrotating terms and rotating terms represent the opposite processes in the higher frequency region. Consequently, the experimental measurement of the QZE may be much easier than what was determined with the RWA results.
In this article, we systematically study the spontaneous decay phenomenon of a two-level system under the influences of both its environment and repetitive measurements. In order to clarify some well-established conclusions about the quantum Zeno effect (QZE) and the quantum anti-Zeno effect (QAZE), we do not use the rotating wave approximation (RWA) in obtaining an effective Hamiltonian. We examine various spectral distributions by making use of our present approach in comparison with other approaches. It is found that with respect to a bare excited state even without the RWA, the QAZE can still happen for some cases, for example, the interacting spectra of hydrogen. However, for a physical excited state, which is a renormalized dressed state of the atomic state, the QAZE disappears and only the QZE remains. These discoveries inevitably show a transition from the QZE to the QAZE as the measurement interval changes.
This thesis studies counter-rotating terms part of an Hamiltonian usually neglected in the so-called Rotating Wave Approximation. Generalising the usual description of light-matter interactions and open quantum systems it is demonstrated that normally neglected counter-rotating terms have the potential to allocate energy among different system degrees of freedom. It is pointed out in examples that this aspect can affect the energy concentration in quantum systems. Initially, a composite quantum system is considered, i.e. bipartite systems like atom-cavity systems and coupled optical resonators without decay. By resorting to methods from quantum field theory it is shown that for such bosonic systems, the Rotating Wave Approximation cannot be applied far off resonance. In fact, the counter-rotating terms are related to an entropy operator that is capable of generating an irreversible time evolution. The vacuum state of the system is shown to evolve into a generalised coherent state exhibiting entanglement of the modes in which the counter-rotating terms are expressed. Furthermore, it is demonstrated that a non-trivial behaviour of the photon emission rate of such composite quantum system can occur when the counter-rotating terms are not dropped. In such a system, there is a coupling to infinitely many modes with most of them being far off resonance. An energy concentrating mechanism is discussed which cannot be described by the Rotating Wave Approximation. Its result is the continuous leakage of photons from open quantum system, even in the absence of external driving. Finally, a model is proposed to explain the origin of the sudden energy concentration in the intriguing phenomenon of sonoluminescence. The model is based on the quantum dynamics of trapped particles and assumes the presence of a weak but highly inhomogeneous electric field. It is shown that the counter-rotating terms can significantly contribute to the energy focussing mechanism in terms of quantum coherences.
Diagonalization of the effective Hamiltonian describing an open quantum system is the usual method of tracking its exceptional points (EPs). Although such a method is successful for tracking EPs in Markovian systems, it may be problematic in non-Markovian systems where a closed expression of the effective Hamiltonian describing the open system may not exist. In this work we provide an alternative method of tracking EPs in open quantum systems, using an experimentally measurable quantity, namely, the effective decay rate of a qubit. The quantum system under consideration consists of two nonidentical interacting qubits, one of which is coupled to an external environment. We develop a theoretical framework in terms of the time-dependent Schr\"odinger equations of motion, which provides analytical closed-form solutions of the Laplace transforms of the qubit amplitudes, enabling the study of various cases of environmental spectral densities. The link between the peaked structure of the effective decay rate of the qubit that interacts indirectly with the environment and the onset of the quantum Zeno effect is investigated, revealing the connections between the latter and the presence of exceptional points. Our treatment and results in addition reveal an intricate interplay between non-Markovian dynamics, the quantum Zeno effect, and non-Hermitian physics.