The traditional assumption of Hermiticity in quantum theories has been remarkably successful both in enabling a formalism that is mathematically elegant and yielding excellent agreement with experiment. However, the physical meaning of this assumption has never been entirely obvious. Further, it arguably introduces a degree of inconsistency into the theory, since it demands that the energy eigenvalues of a given system must be real, yet well-understood physical quantities such as atomic relaxation rates are determined by the imaginary part of a complex eigenvalue, which appears via an analytic continuation of the original Hermitian theory. However, the Hermiticity requirement can be enforced in an entirely self-consistent manner if we restrict ourselves to the case of purely discrete, finite quantum systems, which suggests the following physical interpretation: imposing Hermiticity requires that a given system be isolated from its surrounding environment.

But, in reality, no physical system can ever be truly isolated. Hence, a number of essentially non-Hermitian approaches have been developed throughout the years to describe the interaction of quantum systems with their surrounding environment. Under one approach, the environment is described by a large number of quantum modes, which collectively interact with the central system. When we take the limit such that the number of modes goes to infinity, these microscopic degrees of freedom give rise to an energy continuum. The resonant interaction between the central quantum system and the continuum then gives rise to the complex eigenvalue, which corresponds to the so-called resonance state. This resonance state resides outside of the usual Hilbert space, which is the origin of the non-Hermiticity in this picture.

Our studies of quantum dynamics and resonance phenomena in open quantum systems are aimed at revealing fundamental dynamical principles as well as developing concepts useful from a quantum control perspective. For example, we have studied a variety of scenarios in which the exponential decay process associated with the resonance can be partially or even completely suppressed by the formation of quasi-bound states in continuum [1, 2] or bound states in continuum [3]. Bound states in continuum (BICs) are states with discrete eigenvalues that counter-intuitively reside within the energy continuum associated with the environment. Such states can exist as they decouple from the continuum due to quantum interference; hence they illustrate one of the subtlest features of the quantum realm [See Fig. 1, taken from Ref. 3]. Useful applications of bound states in continuum are recently being explored in experiment such as the enhancement of laser modes in cavities [4].

Taking the idea of using BICs to suppress exponential decay a step further, we have recently shown that the dynamics in the subspace adjacent to the bound state in continuum are intrinsically non-Markovian (i.e. reversible). Specifically, by initializing a state that lies orthogonal to the bound state in continuum, one can achieve complete non-Markovian decay, despite the usual resonance condition being satisfied [5]. This can be viewed as an opportunity to observe deviations from exponential decay that theoretically appear on very long timescales in quantum systems, but in practice are very difficult to detect in experiment because they only occur after many lifetimes of the exponential decay have passed. Because these deviations occur due to the existence of a lower threshold (or band edge) on the energy continuum, we were able to show that an anti-localized virtual state with eigenvalue appearing near the threshold introduces a timescale T_Q that characterizes the dynamics, which takes the form of a power law decay 1/t^p [see Fig. 2]. As a quantum control mechanism, we note this timescale is inversely proportional T_Q ~1/Δ to the energy gap Δ between the virtual state and the threshold, which the experimentalist can potentially manipulate to control the exponent appearing in the power law [5, 6]. In Ref. [5] we proposed to observe these effects in a photonic lattice experiment.

Many of the phenomena that we study in quantum dynamics and non-Hermitian physics can find applications in the realm of photonics and cavity quantum electrodynamics (cavity QED), in which interactions between light and matter can be manipulated at the mesoscopic scale. In a recent work, we obtained a new result relevant to a well-known problem involving photonic band gap materials [7, 8]. Further, this work gives an application of the so-called exceptional points, which are non-Hermitian degeneracies at which not only do eigenvalues coincide but the eigenstates themselves coalesce [9–13].

Previously, it was known that an atom with transition frequency tuned near the edge of a photonic band gap would give rise to a complex non-Markovian decay scheme, incorporating fractional decay due to the presence of a bound state in the gap [see Fig. 3]. We obtained deeper insight into this problem by revealing that the dynamics are largely determined by an exceptional point occurring directly at the threshold [9]. Further, we generalized the problem by revealing general conditions such that an eigenvalue triplet— consisting of a bound state, a resonance, and an anti-resonance— would converge on the threshold as the coupling g between a generic quantum emitter and a 1-D system were shut off [see Fig. 4]. We showed that the dynamics in such a situation is determined by a cascading series of time domains; in the first and most important domain, we revealed that the quantum emitter evolves according to an unusual decay of the form 1 – Ct^3/2. However, because the decay is non-Markovian (reversible), over time, some of the emitted particle is reabsorbed into the bound state and the initial state is partially restored. This gives way to the next time domain in which the emitter experiences a decaying oscillation due to quantum interference between the individual contributions from the bound state, resonance and the continuum threshold. Finally, the emitter settles down to a survival probability of almost exactly 4/9, which is attributable to the occupation of the bound state [see Fig. 3]. Finally, we proposed to observe these effects in a circuit QED experiment [9].

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[2] S. Garmon, H. Nakamura, N. Hatano, and T. Petrosky, Phys. Rev. B 80, 115318 (2009).

[3] S. Tanaka, S. Garmon, G. Ordonez, and T. Petrosky, Phys. Rev. B 76, 153308 (2007).

[4] A. Kodigala, T. Lepetit, Q. Gu, B. Bahari, Y. Fainman, and B. Kanté, Nat. 541, 196 (2017).

[5] S. Garmon, K. Noba, G. Ordonez, and D. Segal, Phys. Rev. A 99, 010102(R) (2019).

[6] S. Garmon, T. Petrosky, L. Simine, and D. Segal, Fortschr. Phys. 61, 261 (2013).

[7] A. G. Kofman, G. Kurizki, and B. Sherman, J. Mod. Opt. 41, 353 (1994).

[8] S. John and T. Quang, Phys. Rev. A 50, 1764 (1994).

[9] S. Garmon, G. Ordonez, and N. Hatano, Phys. Rev. Research 3, 033029 (2021).

[10] S. Garmon and G. Ordonez, J. Math. Phys. 58, 062101 (2017).

[11] S. Garmon, I. Rotter, N. Hatano, and D. Segal, Int. J. Theor. Phys. 51, 3536 (2012).

[12] K. Kanki, S. Garmon, S. Tanaka, and T. Petrosky, J. Math. Phys. 58, 092101 (2017).

[13] S. Garmon, M. Gianfreda, and N. Hatano, Phys. Rev. A 92, 022125 (2015).