Exactly solvable PT\mathcal{PT}-symmetric models in two dimensions

Non-hermitian, $\mathcal{PT}$-symmetric Hamiltonians, experimentally realized in optical systems, accurately model the properties of open, bosonic systems with balanced, spatially separated gain and loss. We present a family of exactly solvable, two-dimensional, $\mathcal{PT}$ potentials for a non-relativistic particle confined in a circular geometry. We show that the $\mathcal{PT}$ symmetry threshold can be tuned by introducing a second gain-loss potential or its hermitian counterpart. Our results explicitly demonstrate that $\mathcal{PT}$ breaking in two dimensions has a rich phase diagram, with multiple re-entrant $\mathcal{PT}$ symmetric phases.Comment: 6 pages, 6 figure

Conserved quantities, exceptional points, and antilinear symmetries in non-Hermitian systems

Over the past two decades, open systems that are described by a non-Hermitian Hamiltonian have become a subject of intense research. These systems encompass classical wave systems with balanced gain and loss, semiclassical models with mode selective losses, and minimal quantum systems, and the meteoric research on them has mainly focused on the wide range of novel functionalities they demonstrate. Here, we address the following questions: Does anything remain constant in the dynamics of such open systems? What are the consequences of such conserved quantities? Through spectral-decomposition method and explicit, recursive procedure, we obtain all conserved observables for general -symmetric systems. We then generalize the analysis to Hamiltonians with other antilinear symmetries, and discuss the consequences of conservation laws for open systems. We illustrate our findings with several physically motivated examples

PT -symmetry breaking in a Kitaev chain with one pair of gain-loss potentials

Parity-time (PT) symmetric systems are classical, gain-loss systems whose dynamics are governed by non-Hermitian Hamiltonians with exceptional-point (EP) degeneracies. The eigenvalues of a PT-symmetric Hamiltonian change from real to complex conjugates at a critical value of gain-loss strength that is called the PT breaking threshold. Here, we obtain the PT threshold for a one-dimensional, finite Kitaev chain—a prototype for a p-wave superconductor—in the presence of a single pair of gain and loss potentials as a function of the superconducting order parameter, on-site potential, and the distance between the gain and loss sites. In addition to a robust, nonlocal threshold, we find a rich phase diagram for the threshold that can be qualitatively understood in terms of the band structure of the Hermitian Kitaev model. In particular, for an even chain with zero on-site potential, we find a re-entrant PT-symmetric phase bounded by second-order EP contours. Our numerical results are supplemented by analytical calculations for small system sizes

Conserved quantities in non-hermitian systems via vectorization method

Open classical and quantum systems have attracted great interest in the past two decades. These include systems described by non-Hermitian Hamiltonians with parity-time (PT) symmetry that are best understood as systems with balanced, separated gain and loss. Here, we present an alternative way to characterize and derive conserved quantities, or intertwining operators, in such open systems. As a consequence, we also obtain non-Hermitian or Hermitian operators whose expectations values show single exponential time dependence. By using a simple example of a PT-symmetric dimer that arises in two distinct physical realizations, we demonstrate our procedure for static Hamiltonians and generalize it to time-periodic (Floquet) cases where intertwining operators are stroboscopically conserved. Inspired by the Lindblad density matrix equation, our approach provides a useful additionto the well-established methods for characterizing time-invariants in non-Hermitian systems

Raising the PT -transition threshold by strong coupling to neutral chains

The PT-symmetry-breaking threshold in discrete realizations of systems with balanced gain and loss is determined by the effective coupling between the gain and loss sites. In one-dimensional chains, this threshold is maximum when the two sites are closest to each other or the farthest. We investigate the fate of this threshold in the presence of parallel, strongly coupled, Hermitian (neutral) chains and find that it is increased by a factor proportional to the number of neutral chains. We present numerical results and analytical arguments for this enhancement. We then consider the effects of adding neutral sites to PT-symmetric dimer and trimer configurations and show that the threshold is more than doubled, or tripled by their presence. Our results provide a surprising way to engineer the PT threshold in experimentally accessible samples

On-Demand Parity-Time Symmetry in a Lone Oscillator through Complex Synthetic Gauge Fields

What is the fate of an oscillator when its inductance and capacitance are varied while its frequency is kept constant? Inspired by this question, we propose a protocol to implement parity-time (PT ) symmetry in a lone oscillator. Different forms of constrained variations lead to static, periodic, or arbitrary balanced gain and loss profiles, that can be interpreted as purely imaginary gauge fields. With a state-of-the-art, dynamically tunable LC oscillator comprising synthetic circuit elements, we demonstrate static and Floquet PT breaking transitions, including those at vanishingly small gain and loss, by tracking the circuit energy. Concurrently, we derive and observe conserved quantities in this open, balanced gain-loss system, both in the static and Floquet cases. Lastly, by measuring the circuit energy, we unveil a giant dynamical asymmetry along exceptional-point contours that emerge symmetrically from the Hermitian degeneracies at Floquet resonances. Distinct from material or parametric gain and loss mechanisms, our protocol enables on-demand parity-time symmetry in a minimal classical system—a single oscillator—and may be ported to other realizations including metamaterials and optomechanical systems

Manifestations of Varying Grading Level in CdSe/ZnSe Core–Shell Nanocrystals

Graded interface core–shell nanocrystals (NCs) are being investigated to attain highly luminescent systems and are also anticipated to be “blinking-free” at the single-particle level. In the present work, CdSe/ZnSe-graded core–shell NCs with varying confinement potential profile are studied at the single-particle level. The internal structure of NCs is determined with the aid of optical, structural, and chemical probes. Notably, the radiative lifetime for different nanostructures decays monoexponentially. The variation in the radiative lifetime due to differing internal structure is understood on the basis of recently reported first-principles study on different interfaces of core–shell NCs, in particular attributed to varying overlap of electron–hole wave functions. The single NC measurements reveal that the percentage of nonblinking NCs is higher for slowly varying confinement potential. Statistically suppressed blinking (∼80%) and a single exponential PL decay curve, accompanied by a very narrow (∼30 meV) emission line at the single NC level, are observed in graded CdSe/ZnSe core–shell NCs