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Quantum systems with balanced gain and loss, signatures of branch points, and dissociation effects

Holger Cartarius

Published Jan 1, 2014• 1 citation
Publisher

Abstract

Gain and loss to the wave function of quantum mechanics can in a convenient way be modelled by effective non-Hermitian Hamiltonians. Imaginary contributions to the potential introduce source and drain terms for the probability amplitude. A special class of non-Hermitian Hamiltonians are those which possess a parity-time symmetry. In spite of their non-Hermiticity these Hamiltonians allow for real energy eigenvalues, i.e. the existence of stationary states in the presence of balanced gain and loss. This effect has been identified theoretically in a large number of quantum systems. Its existence has also been proved experimentally in coupled optical wave guides. The wave guides are, however, only optical analogues of quantum systems. In the first part of this thesis it is shown from the theoretical side that Bose-Einstein condensates in a double-well setup are an ideal candidate for a first experimental realisation of a genuine quantum system with parity-time symmetry. When particles are removed from one well and coherently injected into the other the external potential is parity-time symmetric. To investigate the system the underlying time-independent and time-dependent Gross-Pitaevskii equations are solved numerically. It turns out that a subtle interplay between the nonlinearity of the Gross-Pitaevskii equation and the gain-loss effect leads to a complicated dynamics of the condensate wave function. However, the most important result is the existence of stationary states that are sufficiently stable to be observable in an experiment. Two suggestions for experimental realisations are presented. They are based on the idea of embedding the non-Hermitian parity-time-symmetric system into a larger structure described by a Hermitian Hamiltonian. A further effect of non-Hermitian Hamiltonians are so-called exceptional points, at which two resonances coalesce such that both their eigenvalues and wave functions become identical. It is shown that an exceptional point can unambiguously be identified by a characteristic non-exponential decay of the resonances. With numerically exact calculations for the hydrogen atom in crossed electric and magnetic fields this effect is verified in an experimentally accessible quantum system. The second part of the thesis is devoted to semiclassical Gaussian approximations to the Boltzmann operator, which have become an important tool for the investigation of thermodynamic properties of clusters of atoms at low temperatures. A numerically cheap frozen Gaussian approximation to the imaginary time propagator with a width matrix especially suited for the dynamics of clusters is developed. It is applied to the cases of Ar3 and Ar6. For these clusters classical-like transitions in one step from a bounded moiety to free particles are found for increasing temperatures. Additionally, the structure of the Ar6 cluster is studied in the bound configuration and during the dissociation. Quantum effects, i.e. differences with the purely classical case, manifest themselves in the low-temperature behaviour of the mean energy and specific heat as well as in a slight shift of the transition temperature. A first-order correction to the semiclassical propagator is used to improve the results of the calculation for Ar3, and it is shown how the correction can be used to objectively assess the validity of the frozen Gaussian approximation.

DOI
10.18419/opus-5130

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Gain and loss to the wave function of quantum mechanics can in a convenient way be modelled by effective non-Hermitian Hamiltonians. Imaginary contributions to the potential introduce source and drain terms for the probability amplitude. A special class of non-Hermitian Hamiltonians are those which possess a parity-time symmetry. In spite of their non-Hermiticity these Hamiltonians allow for real energy eigenvalues, i.e. the existence of stationary states in the presence of balanced gain and loss. This effect has been identified theoretically in a large number of quantum systems. Its existence has also been proved experimentally in coupled optical wave guides. The wave guides are, however, only optical analogues of quantum systems. In the first part of this thesis it is shown from the theoretical side that Bose-Einstein condensates in a double-well setup are an ideal candidate for a first experimental realisation of a genuine quantum system with parity-time symmetry. When particles are removed from one well and coherently injected into the other the external potential is parity-time symmetric. To investigate the system the underlying time-independent and time-dependent Gross-Pitaevskii equations are solved numerically. It turns out that a subtle interplay between the nonlinearity of the Gross-Pitaevskii equation and the gain-loss effect leads to a complicated dynamics of the condensate wave function. However, the most important result is the existence of stationary states that are sufficiently stable to be observable in an experiment. Two suggestions for experimental realisations are presented. They are based on the idea of embedding the non-Hermitian parity-time-symmetric system into a larger structure described by a Hermitian Hamiltonian. A further effect of non-Hermitian Hamiltonians are so-called exceptional points, at which two resonances coalesce such that both their eigenvalues and wave functions become identical. It is shown that an exceptional point can unambiguously be identified by a characteristic non-exponential decay of the resonances. With numerically exact calculations for the hydrogen atom in crossed electric and magnetic fields this effect is verified in an experimentally accessible quantum system. The second part of the thesis is devoted to semiclassical Gaussian approximations to the Boltzmann operator, which have become an important tool for the investigation of thermodynamic properties of clusters of atoms at low temperatures. A numerically cheap frozen Gaussian approximation to the imaginary time propagator with a width matrix especially suited for the dynamics of clusters is developed. It is applied to the cases of Ar3 and Ar6. For these clusters classical-like transitions in one step from a bounded moiety to free particles are found for increasing temperatures. Additionally, the structure of the Ar6 cluster is studied in the bound configuration and during the dissociation. Quantum effects, i.e. differences with the purely classical case, manifest themselves in the low-temperature behaviour of the mean energy and specific heat as well as in a slight shift of the transition temperature. A first-order correction to the semiclassical propagator is used to improve the results of the calculation for Ar3, and it is shown how the correction can be used to objectively assess the validity of the frozen Gaussian approximation.
A quantum system exhibiting PT\mathcal{PT}PT symmetry is a Bose-Einstein condensate in a double-well potential with balanced particle gain and loss, which is described in the mean-field limit by a Gross-Pitaevskii equation with a complex potential. A possible experimental realization of such a system by embedding it into a Hermitian time-dependent four-mode optical lattice was proposed by Kreibich et al. [Phys. Rev. A 87, 051601(R) (2013)], where additional potential wells act as reservoirs and particle exchange happens via tunneling. Since particle influx and outflux have to be controlled explicitly, a set of conditions on the potential parameters had to be derived. In contrast to previous work, our focus lies on a full many-body description beyond the mean-field approximation using a Bose-Hubbard model with time-dependent potentials. This gives rise to additional quantum effects, such that the differences between mean-field and many-body dynamics are of special interest. We further present stationary analytical solutions for the embedded wells in the mean-field limit, different approaches for the embedding into a many-body system, and a very efficient method for the evaluation of hopping terms to calculate exact Bose-Hubbard dynamics.
As one of the essential features in non-Hermitian systems coupled with environment, the exceptional point has attracted much attention in many physical fields. The phenomena that eigenvalues and eigenvectors of the system simultaneously coalesce at the exceptional point are also one of the important properties to distinguish from Hermitian systems. In non-Hermitian systems with parity-time reversal symmetry, the eigenvalues can be continuously adjusted in parameter space from all real spectra to pairs of complex-conjugate values by crossing the phase transition from the parity-time reversal symmetry preserving phase to the broken phase. The phase transition point is called an exceptional point of the system, which occurs in company with the spontaneous symmetry broken and many novel physical phenomena, such as sensitivity-enhanced measurement and loss induced transparency or lasing. Here, we focus on a two-qubit quantum system with parity-time reversal symmetry and construct an experimental scheme, prove and verify the features at its third-order exceptional point, including high-order energy response induced by perturbation and the coalescence of eigenvectors. We first theoretically study a two-qubit non-Hermitian system with parity-time reversal symmetry, calculate the properties of eigenvalues and eigenvectors, and prove the existence of a third-order exceptional point. Then, in order to study the energy response of the system induced by perturbation, we introduce an Ising-type interaction as perturbation and quantitatively demonstrate the response of eigenvalues. In logarithmic coordinates, three of the eigenvalues are indeed in the cubic root relationship with perturbation strength, while the fourth one is a linear function. Moreover, we study the eigenvectors around exceptional point and show the coalescence phenomenon as the perturbation strength becomes smaller. The characterization of the response of eigenvalues at high-order exceptional points is a quite diffcult task as it is in gneneral diffcult to directly measure eigenenergies in a quantum system composed of a few qubits. In practice, the time evolution of occupation on a particular state is used to indirectly fit the eigenvalues. In order to make the fitting of experimental data more reliable, we want to determine an accurate enough expressions for the eigenvalues and eigenstates. To this aim, we employ a perturbation treatment and show good agreement with the numerical results of states occupation obtained by direct evolution. Moreover, we find that after the system evolves for a long enough time, it will end up to one of the eigenstates, which gives us a way to demonstrate eigenvector coalescence by measuring the density matrix via tomography and parity-time reversal transformation. To show our scheme is experimentally applicable, we propose an implementation using trapped 171Yb+ ions. We can map the parity-time reversal symmetric Hamiltonian to a purely dissipative two-ion system:use microwave to achieve spin state inversion, shine a 370 nm laser to realize dissipation of spin-up state, and apply Raman operation for Mølmer-Sørensen gates to implement Ising interaction. By adjusting the corresponding microwave and laser intensity, the spin coupling strength, the dissipation rate and the perturbation strength can be well controlled. We can record the probability distribution of the four product states of the two ions and measure the density matrix by detecting the fluorescence of each ion on different Pauli basis.
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 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.
Realizing PT{\mathcal{P}}{\mathcal{T}}PT-symmetric BEC subsystems in closed Hermitian systems
In open double-well Bose–Einstein condensate systems which balance in- and outfluxes of atoms and which are effectively described by a non-Hermitian -symmetric Hamiltonian -symmetric states have been shown to exist. -symmetric states obey the combined parity and time reversal symmetry. We tackle the question of how the in- and outfluxes can be realized and introduce a Hermitian system in which two -symmetric subsystems are embedded. This system no longer requires an in- and outcoupling to and from the environment. We show that the subsystems still have -symmetric states. In addition we examine what degree of detail is necessary to correctly model the -symmetric properties and the bifurcation structure of such a system. We examine a four-mode matrix model and a system described by the full Gross–Pitaevskii equation in one dimension. We see that a simple matrix model correctly describes the qualitative properties of the system. For sufficiently isolated wells there is also quantitative agreement with the more advanced system descriptions. We also investigate which properties the wave functions of a system must fulfil to allow for -symmetric states. In particular the requirements for the phase difference between different parts of the system are examined.