The concept of parity-time symmetry has firmly established non-Hermiticity as a versatile degree of freedom on a variety of physical platforms. In general, the non-Hermitian dynamics of open systems are perceived to be inextricably linked to complex-valued potentials facilitating the local attenuation and coherent amplification in wave mechanics. Along these lines, time reversal symmetry is associated with a complex conjugation of the potential landscape, in essence swapping gain and loss. Here we leverage nonorthogonal coupled-mode theory to synthesize genuinely non-Hermitian dynamics without either gain or loss, and experimentally demonstrate parity-time symmetry via fluorescence measurements in femtosecond-laser-written arrays. Our projective approach allows features of non-Hermiticity to be utilized in scenarios where actual amplification and/or attenuation may disrupt the desired physics, e.g. in nonlinear systems or quantum optics.
Unbroken PT-symmetry in the absence of gain or loss
The concept of parity-time symmetry has firmly established non-Hermiticity as a versatile degree of freedom on a variety of physical platforms. In general, the non-Hermitian dynamics of open systems are perceived to be inextricably linked to complex-valued potentials facilitating the local attenuation and coherent amplification in wave mechanics. Along these lines, time reversal symmetry is associated with a complex conjugation of the potential landscape, in essence swapping gain and loss. Here we leverage nonorthogonal coupled-mode theory to synthesize genuinely non-Hermitian dynamics without either gain or loss, and experimentally demonstrate parity-time symmetry via fluorescence measurements in femtosecond-laser-written arrays. Our projective approach allows features of non-Hermiticity to be utilized in scenarios where actual amplification and/or attenuation may disrupt the desired physics, e.g. in nonlinear systems or quantum optics.
Non-Hermitian optical systems with parity-time (PT) symmetry have recently revealed many intriguing prospects that outperform conservative structures. The prevous works are mostly rooted in complex arrangements with controlled gain-loss interplay. Here, we demonstrate anti-PT symmetry inherent in nonlinear optical interactions based upon forward optical four-wave mixing in a laser-cooled atomic ensemble with negligible linear gain and loss. We observe the pair of frequency modes undergo a nontrivial anti-PT phase transition between coherent power oscillation and optical parametric amplification in presence of a large phase mismatch.
Analysis of fundamentally open systems which exhibit both spatial reflection (parity) and time-reversal symmetry, called PT-symmetric [1] systems, has played an important role in understanding the dynamics of a broad class of non-Hermitian systems. Such systems support distinct regimes of stability and instability respectively, and the transition between these plays an important role in the system dynamics. We investigate a variety of non-Hermitian systems which exhibit a PT-symmetry breaking transition in both classical and quantum contexts using an array of theoretical and numerical approaches which are directly amenable to physical experiments. Organizationally, we divide these studies into those done on classical and quantum systems. Specifically for classical systems we analyze several interesting phenomena in the context of optical waveguide arrays. These provide an important testing ground for much of the discrete, PT-symmetric theory. In our case, we analyze PT-symmetric extensions of several interesting lattice models. Firstly, we explore single gain/loss pair, PT-symmetric extensions of the Su-Schreiffer-Heeger (SSH) model [2] and the Aubry-Andre-Harper (AAH) model [3,4]. In both cases, we find a veiled symmetry which is responsible for the PT-symmetry. Secondly we expand upon the Rudner-Levitov (RL) model [5,6], a non-Hermitian extension of the SSH model with multiple gain/loss pairs, and we explore the dependence of topological phenomena on changes in the initial state as well as their robustness in the presence of gain-induced nonlinearity. For quantum systems, we explore the rich PT-symmetric phase diagram resulting from time-periodic, Floquet PT-symmetry [7,8]. This is experimentally implementable in ultracold atomic systems [9], and features a reentrant PT-symmetric phase. Secondly, we examine quantum multi-photon statistics for a simple non-Hermitian model which is transformed and mapped to a system with unitary evolution via a unitary dilation, and is fully realizable on a small-scale quantum simulator [10].
The exploration of non-Hermitian physics has predominantly been a frontier in quantum mechanics, revealing exotic phenomena not observed in conventional Hermitian systems. However, its extension to classical mechanics offers a rich and largely untapped avenue for understanding open and active systems characterized by dissipation and gain. This paper systematically investigates the theoretical foundations of non-Hermitian classical mechanics, focusing on the emergence and implications of dissipation and gain symmetries. We demonstrate how concepts like parity-time (PT) symmetry and pseudo-Hermiticity, originally quantum mechanical, find natural classical analogues in systems with balanced energy loss and amplification. The study delves into the modification of classical Hamiltonian and Lagrangian formalisms to accommodate non-conservative forces, leading to complex-valued potentials and phase space trajectories. We explore the role of exceptional points as spectral degeneracies in these classical systems, showing how they govern the dynamics and give rise to unique phenomena such as unidirectional transport and enhanced sensitivity. Furthermore, we discuss the practical implications for designing classical mechanical systems with tailored dissipative and amplification properties, opening new possibilities for classical sensing, energy harvesting, and wave manipulation. This work aims to establish a robust theoretical framework for non-Hermitian classical mechanics, highlighting its potential to unveil emergent symmetries and complex dynamics in macroscopic systems.
Non-Hermitian optical systems with parity-time (PT) symmetry have recently revealed many intriguing prospects that outperform conservative structures. The previous works are mostly rooted in complex arrangements with controlled gain-loss interplay. Here, we demonstrate anti-PT symmetry inherent in the nonlinear optical interaction based upon forward optical four-wave mixing in a laser-cooled atomic ensemble with negligible linear gain and loss. We observe that the pair of frequency modes undergo a nontrivial anti-PT phase transition between coherent power oscillation and optical parametric amplification in presence of a large phase mismatch.
The non-Hermitian models, which are symmetric under parity ( P ) and time-reversal ( T ) operators, are the cornerstone for the fabrication of new ultra-sensitive optoelectronic devices. However, providing the gain in such systems usually demands precise control of nonlinear processes, limiting their application. In this paper, to bypass this obstacle, we introduce a class of time-dependent non-Hermitian Hamiltonians (not necessarily Floquet) that can describe a two-level system with temporally modulated on-site potential and couplings. We show that implementing an appropriate non-Unitary gauge transformation converts the original system to an effective one with a balanced gain and loss. This will allow us to derive the evolution of states analytically. Our proposed class of Hamiltonians can be employed in different platforms such as electronic circuits, acoustics, and photonics to design structures with hidden PT -symmetry potentially without imaginary onsite amplification and absorption mechanism to obtain an exceptional point.
Abstract A wide class of non-Hermitian Hamiltonians can possess entirely real eigenvalues when they have parity-time (PT) symmetric potentials. Recently, this family of non-Hermitian systems has attracted considerable attention in diverse areas of physics due to their extraordinary properties, especially in optical systems based on solid-state materials, such as coupled gain–loss waveguides and microcavities. Considering the desired refractive index can be effectively manipulated through atomic coherence, it is important to realize such non-Hermitian optical potentials and further investigate their distinct properties in atomic systems. In this paper, we review the recent theoretical and experimental progress of non-Hermitian optics with coherently prepared multi-level atomic configurations. The realizations of (anti-) PT symmetry with different schemes have extensively demonstrated the special optical properties of non-Hermitian optical systems with atomic coherence.