We investigate the transition probabilities for the “flavor” eigenstates in the two-level quantum system, which is described by a non-Hermitian Hamiltonian with the parity and time-reversal (PT) symmetry. Particularly, we concentrate on the so-called PT-broken phase, where two eigenvalues of the non-Hermitian Hamiltonian turn out to be a complex-conjugate pair. In this case, we find that the transition probabilities will be unbounded in the limit of infinite time t → +∞. However, after performing a connection between a non-Hermitian system, which exhibits passive PT symmetry and global decay, and the neutral-meson system in particle physics, we observe that the diverging behavior of the transition probabilities is actually applicable to the gauge-transformed neutral-meson states, whereas the transition probabilities for physical states are exponentially suppressed by the global decay. We also present a brief review on the situation at the so-called exceptional point, where both the eigenvalues and eigenvectors of the Hamiltonian coalesce.
We investigate the transition probabilities for the “flavor” eigenstates in the two-level quantum system, which is described by a non-Hermitian Hamiltonian with the parity and time-reversal (PT) symmetry. Particularly, we concentrate on the so-called PT-broken phase, where two eigenvalues of the non-Hermitian Hamiltonian turn out to be a complex-conjugate pair. In this case, we find that the transition probabilities will be unbounded in the limit of infinite time t → +∞. However, after performing a connection between a non-Hermitian system, which exhibits passive PT symmetry and global decay, and the neutral-meson system in particle physics, we observe that the diverging behavior of the transition probabilities is actually applicable to the gauge-transformed neutral-meson states, whereas the transition probabilities for physical states are exponentially suppressed by the global decay. We also present a brief review on the situation at the so-called exceptional point, where both the eigenvalues and eigenvectors of the Hamiltonian coalesce.
We investigate how to define in a consistent way the probabilities of the transitions between the “flavor” states of the two-level quantum system, which is described by a non-Hermitian but parity and time-reversal (PT) symmetric Hamiltonian. Explicit calculations are carried out to demonstrate the conservation of probability if a proper definition of the final state is adopted. Finally, this formalism is applied to two-flavor neutrino oscillations νμ → νμ and νμ → ντ in vacuum, where the exact PT symmetry requires the vacuum mixing angle to be maximal, which is compatible with current neutrino oscillation experiments. A possible generalization to the three-flavor case is briefly discussed.
Non-Hermitian systems with parity-time reversal (PT) or anti-PT symmetry have attracted a wide range of interest owing to their unique characteristics and counterintuitive phenomena. One of the most extraordinary features is the presence of an exceptional point, across which a phase transition with spontaneously broken PT symmetry takes place. We implement a Floquet Hamiltonian of a single qubit with anti-PT symmetry by periodically driving a dissipative quantum system of a single trapped ion. With stroboscopic emission and quantum state tomography, we obtain the time evolution of density matrix for an arbitrary initial state, and directly demonstrate information retrieval, eigenstate coalescence, and topological energy spectra as unique features of non-Hermitian systems.
A parity-time (PT) transition is a spectral transition characteristic of non-Hermitian generators; it typically occurs at an exceptional point, where multiple eigenvectors coalesce. The concept of a PT transition has been extended to Markovian open quantum systems, which are described by the GKSL equation. Interestingly, the PT transition in many-body Markovian open quantum systems, the so-called \textit{Lindbladian PT (L-PT) phase transition}, is closely related to two classes of exotic nonequilibrium many-body phenomena: \textit{continuous-time crystals} and \textit{non-reciprocal phase transitions}. In this review, we describe the recent advances in the study of L-PT phase transitions. First, we define PT symmetry in three distinct contexts: non-Hermitian systems, nonlinear dynamical systems, and Markovian open quantum systems, highlighting the interconnections between these frameworks. Second, we develop mean-field theories of L-PT phase transitions for collective-spin systems and for bipartite bosonic systems with particle-number conservation. Within these classes of models, we show that L-PT symmetry can induce a breaking of continuous time-translation symmetry down to a discrete one, leading to persistent periodic dynamics. We further demonstrate that the L-PT phase transition point is typically \textit{a critical exceptional point}, where multiple collective excitation modes with zero excitation spectrum coalesce. These findings establish an explicit connection to continuous-time crystals and non-reciprocal phase transitions. Third, going beyond the mean-field theory, we analyze statistical and quantum properties, such as purity and quantum entanglement indicators of time-independent steady states for several specific models with the L-PT symmetry. Finally, we discuss future research directions for L-PT phase transitions.
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.
Currently, it has been claimed that certain Hermitian Hamiltonians have parity (P) and they are PT-invariant. We propose generalized definitions of time-reversal operator (T) and orthonormality such that all Hermitian Hamiltonians are P, T, PT, and CPT invariant. The PT-norm and CPT-norm are indefinite and definite respectively. The energy-eigenstates are either E-type (e.g., even) or O-type (e.g., odd). C mimics the charge-conjugation symmetry which is recently found to exist for a non-Hermitian Hamiltonian. For a Hermitian Hamiltonian it coincides with P. The Hamiltonians which are invariant under the joint transformation of Parity (x → −x) and Time-reversal (i → −i) are called PT-invariant. It has been conjectured [1] that such Hamiltonians possess real discrete energy-eigenvalues provided the PT symmetry is unbroken. PT-symmetry is called unbroken or exact if the energy-eigenstates are also simultaneous eigenstates of PT. On the other hand when PT-symmetry is spontaneously broken the energy-eigenvalues are complex conjugate pairs. Multipronged investigations supporting
In the presence of Lindblad decoherence, i.e., dissipative effects in an open quantum system due to interaction with an environment, we examine the transition probabilities between the eigenstates in the two-level quantum system described by non-Hermitian Hamiltonians with the Lindblad equation, for which the parity-time-reversal (PT) symmetry is conserved. First, the density matrix formalism for PT-symmetric non-Hermitian Hamiltonian systems is developed. It is shown that the Lindblad operators Lj are pseudo-Hermitian, namely, $\ensuremath{\eta}{L}_{j}^{}{\ensuremath{\eta}}^{\ensuremath{-}1}={L}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}$ with $\ensuremath{\eta}$ being a linear and positive-definite metric, and respect the PT symmetry as well. We demonstrate that the generalized density matrix ${\ensuremath{\rho}}_{\mathrm{G}}^{}(t)\ensuremath{\equiv}\ensuremath{\rho}(t)\ensuremath{\eta}$, instead of the normalized density matrix ${\ensuremath{\rho}}_{\mathrm{N}}^{}(t)\ensuremath{\equiv}\ensuremath{\rho}(t)/\mathrm{tr}\left[\ensuremath{\rho}(t)\right]$, should be implemented for the calculation of the transition probabilities in accordance with the linearity requirement. Second, the density matrix formalism is used to derive the transition probabilities in general cases of PT-symmetric non-Hermitian Hamiltonians. In some concrete examples, we calculate compact analytical formulas for the transition probabilities and explore their main features with numerical illustrations. We also make a comparison between our present results and our previous ones using state vectors in the absence of Lindblad decoherence.