PT-symmetric theory is developed to extend quantum mechanics to a complex region, but it wins its great success first in classical systems, for example, optical waveguides and electric circuits, etc., because there are so many counterintuitive phenomena and striking applications, including unidirectional light transport, PT-enhanced sensors (one kind of exceptional-point-based sensor), and wireless power transfer. However, these phenomena and applications are mostly based on the ability to approach a PT-symmetric broken region, which makes it difficult to transfer them to the quantum regime, since the broken quantum PT-symmetric system has not been constructed effectively, until recently several methods have been raised. Here, we construct a quantum PT-symmetric system assisted by weak measurement, which can effectively transit from the unbroken region to the broken region. The full energy spectrum including the real and imaginary parts is directly measured using weak values. Furthermore, based on the ability of approaching a broken region, we for the first time translate the previously mentioned PT-enhanced sensor into the quantum version, and investigate its various features that are associated to the optimal conditions for sensitivity enhancement. In this experiment, we obtain an enhancement of 8.856 times over the conventional Hermitian sensor. Moreover, by separately detecting the real and imaginary parts of energy splitting, we can derive the additional information of the direction of perturbations. Our work paves the way of leading classical interesting PT phenomena and applications to their quantum counterparts. More generally, since the PT system is a subset of non-Hermitian systems, our work will be also helpful in the studies of general exception point in the quantum regime.
PT-symmetric theory is developed to extend quantum mechanics to a complex region, but it wins its great success first in classical systems, for example, optical waveguides and electric circuits, etc., because there are so many counterintuitive phenomena and striking applications, including unidirectional light transport, PT-enhanced sensors (one kind of exceptional-point-based sensor), and wireless power transfer. However, these phenomena and applications are mostly based on the ability to approach a PT-symmetric broken region, which makes it difficult to transfer them to the quantum regime, since the broken quantum PT-symmetric system has not been constructed effectively, until recently several methods have been raised. Here, we construct a quantum PT-symmetric system assisted by weak measurement, which can effectively transit from the unbroken region to the broken region. The full energy spectrum including the real and imaginary parts is directly measured using weak values. Furthermore, based on the ability of approaching a broken region, we for the first time translate the previously mentioned PT-enhanced sensor into the quantum version, and investigate its various features that are associated to the optimal conditions for sensitivity enhancement. In this experiment, we obtain an enhancement of 8.856 times over the conventional Hermitian sensor. Moreover, by separately detecting the real and imaginary parts of energy splitting, we can derive the additional information of the direction of perturbations. Our work paves the way of leading classical interesting PT phenomena and applications to their quantum counterparts. More generally, since the PT system is a subset of non-Hermitian systems, our work will be also helpful in the studies of general exception point in the quantum regime.
Comprehensive study on parity-time (PT) symmetric systems demonstrates the novel properties and innovative application of non-Hermitian physics in recent years. In the quantum regime, PT symmetric physics exhibits unique quantum dynamical behaviors such as spontaneous state-distinguishability oscillation. However, the construction and control of a PT symmetric quantum system are still challenging, that and restrict the experimental investigation of PT symmetric quantum nature and application. In this Letter, we propose and construct a recycling-structure PT symmetric quantum simulator for the first time, which can effectively simulate the discrete-time dynamical process of a PT symmetric quantum system in both unbroken and broken phases, to be different from our previous work [J.-S. Tang, et al., Nat. Photonics 10, 642 (2016)]. We investigate the dynamical features of quantum state distinguishability based on the PT symmetric simulator. Our results demonstrate the novel PT symmetric quantum dynamics characterized by the periodical oscillation of state distinguishability in the unbroken phase, and the monotonic decay of that in the broken phase. This work also provides a practical experimental platform for the future intensive study of PT symmetric quantum dynamics.
Exceptional points (EPs) of a non-Hermitian Hamiltonian with parity-time-reversal (PT) symmetry have the potential to drastically enhance the capabilities of metrology and sensing through their power-law growing sensitivity to external perturbation. With the ability of generating and tuning dissipation in a single trapped ion system, we observe rich dynamics and detailed quantum phase transitions from the PT-symmetric phase to the symmetry-breaking phase. In this single qubit full quantum system, we develop a method to precisely determine the location of EP without any fitting parameter, and extract the eigenvalues in a unified way through all parameter regions. We can also obtain the full density matrix by quantum state tomography. Finally, we suggest from theoretical analysis that the periodically driving PT-symmetric non-Hermitian system can be used to measure the magnitude, frequency, and phase of time-dependent perturbation with EP enhancement.
Space-time reflection symmetry, or PT symmetry, first proposed in quantum mechanics by Bender and Boettcher in 1998 [2], has become an active research area in fundamental physics. This dissertation contains several research problems which are more or less related to this field of study. After an introduction on complementary topics for the main projects in Chap.1, we discuss about an idea which is originated from the remarkable paper by Chandrasekar et al in Chap.2. They showed that the (second-order constant-coefficient) classical equation of motion for a damped harmonic oscillator can be derived from a Hamiltonian having one degree of freedom. We gives a simple derivation of their result and generalizes it to the case of an nth-order constant-coefficient differential equation. In Chap.3 we studied the analytical continuation of the coupling constant g of a coupled quantum theory. We get to this conclusion that one can, at least in principle, arrive at a state whose energy is lower than the ground state of the theory. The idea is to begin with the uncoupled g = 0 theory in its ground state, to analytically continue around an exceptional point (square-root singularity) in the complex-coupling-constant plane, and finally to return to the point g = 0. In the course of this analytic continuation, the uncoupled theory ends up in an unconventional state whose energy is lower than the original ground-state energy. However, it is unclear whether one can use this analytic continuation to extract energy from the conventional vacuum state; this process appears to be exothermic but one must do work to vary the coupling constant g. PT-symmetric quantum mechanics began with a study of the Hamiltonian H=p^2+x^2(ix)^\epsilon. When \epsilon\geq 0, this portion of parameter space is known as the region of unbroken PT symmetry. The region of unbroken PT symmetry has been studied but the region of broken PT symmetry which is related to the negative epsilon has thus far been unexplored. In Chap.4 we present a detailed numerical and analytical examination of the behavior of the eigenvalues for _4<\epsilon<0. In particular, it reports the discovery of an infinite-order exceptional point at \epsilon =_1, a transition from a discrete spectrum to a partially continuous spectrum at \epsilon=_2, a transition at the Coulomb value \epsilon =_3, and the behavior of the eigenvalues as \epsilon approaches the conformal limit \epsilon =_4. Finally in Chap.5 we devised a simple and accurate numerical technique for finding eigenvalues, node structure, and expectation values of PT-symmetric potentials. The approach involves expanding the solution to the Schrdinger equation in series involving powers of both the coordinate and the energy. The technique is designed to allow one to impose boundary conditions in PT-symmetric pairs of Stokes sectors. The method is illustrated by using many examples of PT-symmetric potentials in both the unbroken- and broken-PT-symmetric regions.
More than a decade ago, it was shown that non-Hermitian Hamiltonians with combined parity (P) and time-reversal (T ) symmetry exhibit real eigenvalues over a range of parameters. Since then, the field of PT symmetry has seen rapid progress on both the theoretical and experimental fronts. These effective Hamiltonians are excellent candidates for describing open quantum systems with balanced gain and loss. Nature seems to be replete with examples of PT -symmetric systems; in fact, recent experimental investigations have observed the effects of PT symmetry breaking in systems as diverse as coupled mechanical pendula, coupled optical waveguides, and coupled electrical circuits. Recently, PT -symmetric Hamiltonians for tight-binding lattice models have been extensively investigated. Lattice models, in general, have been widely used in physics due to their analytical and numerical tractability. Perhaps one of the best systems for experimentally observing the effects of PT symmetry breaking in a one-dimensional lattice with tunable hopping is an array of evanescently-coupled optical waveguides. The tunneling between adjacent waveguides is tuned by adjusting the width of the barrier between them, and the imaginary part of the local refractive index provides the loss or gain in the respective waveguide. Calculating the time evolution of a wave packet on a lattice is relatively straightforward in the tight-binding model, allowing us to make predictions about the behavior of light propagating down an array of PT -symmetric waveguides. In this thesis, I investigate the the strength of the PT -symmetric phase (the region over which the eigenvalues are purely real) in lattices with a variety of PT - symmetric potentials. In Chapter 1, I begin with a brief review of the postulates of quantum mechanics, followed by an outline of the fundamental principles of PT - symmetric systems. Chapter 2 focuses on one-dimensional uniform lattices with a pair of PT -symmetric impurities in the case of open boundary conditions. I find that the PT phase is algebraically fragile except in the case of closest impurities, where the PT phase remains nonzero. In Chapter 3, I examine the case of periodic boundary conditions in uniform lattices, finding that the PT phase is not only nonzero, but also independent of the impurity spacing on the lattice. In addition, I explore the time evolution of a single-particle wave packet initially localized at a site. I find that in the case of periodic boundary conditions, the wave packet undergoes a preferential clockwise or counterclockwise motion around the ring. This behavior is quantified by a discrete momentum operator which assumes a maximum value at the PT -symmetry- breaking threshold. In Chapter 4, I investigate nonuniform lattices where the parity-symmetric hop- ping between neighboring sites can be tuned. I find that the PT phase remains strong in the case of closest impurities and fragile elsewhere. Chapter 5 explores the effects of the competition between localized and extended PT potentials on a lattice. I show that when the short-range impurities are maximally separated on the lattice, the PT phase is strengthened by adding short-range loss in the broad-loss region. Consequently, I predict that a broken PT symmetry can be restored by increasing the strength of the short-range impurities. Lastly, Chapter 6 summarizes my salient results and discusses areas which can be further developed in future research.
Inspired by the discovery of real energy spectra of non-Hermitian Hamiltonians obeying parity-time (PT) symmetry, many proposals and relevant experiments have been suggested and implemented for the realization of PT-symmetric Hamiltonians in various physical systems. Among them PT-symmetric atomic gases are of great interest as they possess many unique advantages, including: (i) the possession of active PT symmetry with balanced optical gain and loss, (ii) the ability to be extended to high-dimensional ones, and (iii) the feasibility to acquire enhanced Kerr nonlinearity based on the resonance character between light and atoms. Here, we review the new progresses made on the study of PT symmetry in coherent atomic gases since our last review paper [Advances in Physics: X, 2, 737–783 (2017)]. Particularly, we focus on the newly emerged research topics in this direction, including nonlocal nonlinear effect, non-Hermitian quantum effect, exceptional point and associated phenomenon, and non-Hermitian skin effect with PT symmetry. Perspectives and potential applications of the subject are also discussed.
Abstract Non-Hermitian parity-time ( PT P T ) and anti-parity-time ( APT APT )-symmetric systems exhibit novel quantum properties and have attracted increasing interest. Although many counterintuitive phenomena in PT P T - and APT APT -symmetric systems were previously studied, coherence flow has been rarely investigated. Here, we experimentally demonstrate single-qubit coherence flow in PT P T - and APT APT -symmetric systems using an optical setup. In the symmetry unbroken regime, we observe different periodic oscillations of coherence. Particularly, we observe two complete coherence backflows in one period in the PT P T -symmetric system, while only one backflow in the APT APT -symmetric system. Moreover, in the symmetry broken regime, we observe the phenomenon of stable value of coherence flow. We derive the analytic proofs of these phenomena and show that most experimental data agree with theoretical results within one standard deviation. This work opens avenues for future study on the dynamics of coherence in PT P T - and APT APT -symmetric systems.