Symmetric Two-level Systems Research Articles

Quantum Simulation of Pseudo-Hermitian-φ-Symmetric Two-Level Systems.

Non-Hermitian (NH) quantum theory has been attracting increased research interest due to its featured properties, novel phenomena, and links to open and dissipative systems. Typical NH systems include PT-symmetric systems, pseudo-Hermitian systems, and their anti-symmetric counterparts. In this work, we generalize the pseudo-Hermitian systems to their complex counterparts, which we call pseudo-Hermitian-φ-symmetric systems. This complex extension adds an extra degree of freedom to the original symmetry. On the one hand, it enlarges the non-Hermitian class relevant to pseudo-Hermiticity. On the other hand, the conventional pseudo-Hermitian systems can be understood better as a subgroup of this wider class. The well-defined inner product and pseudo-inner product are still valid. Since quantum simulation provides a strong method to investigate NH systems, we mainly investigate how to simulate this novel system in a Hermitian system using the linear combination of unitaries in the scheme of duality quantum computing. We illustrate in detail how to simulate a general P-pseudo-Hermitian--symmetric two-level system. Duality quantum algorithms have been recently successfully applied to similar types of simulations, so we look forward to the implementation of available quantum devices.

Time Evolution of Bath Properties in Spin-Boson Dynamics.

The dynamical behaviors of a two-level system (TLS) coupled to a harmonic dissipative bath has been studied extensively using a variety of analytical and numerical methods. The focus of the vast majority of these studies has been on the properties of the TLS, averaged with respect to the bath degrees of freedom. In this work, we use real-time path integral methods to probe the behavior of select bath degrees of freedom during the dynamics of a symmetric two-level system (TLS) coupled to a dissipative bath by calculating system-bath densities (SBD) and coordinate expectation values. Overall, the SBD motion on each diabatic state is simpler than the motion of the total density. In the weak coupling regime, which characterizes the parameters of oscillators that comprise such a bath, the SBD on each TLS state remains primarily compact and Gaussian-like, such that its peak is well characterized by the mode expectation value. In the absence of a dissipative environment, nonadiabatic density depletion leads to spikes in coordinate expectation values. The evolution of the SBD peak trajectory for two discrete modes exhibits Lissajous patterns with frequency-dependent shapes that strongly resemble classical trajectory motion on a torus. These patterns become more complex when the coupling of the mode to the TLS is increased outside of this regime, leading to persistent small amplitude oscillations in the TLS populations characterized by a very slow decay and SBD trajectories that exhibit behaviors reminiscent of chaotic classical systems. Indirect coupling to a dissipative bath has a stabilizing effect on the dynamics, eliminating spikes, synchronizing the SBD motion on the two diabatic states and regularizing the SBD trajectory to simple rectangular Lissajous-like shapes with a slowly shrinking boundary, regardless of the mode frequencies.

Temporal correlation beyond quantum bounds in non-Hermitian PT- symmetric dynamics of a two level system

We study the dynamics of a non-Hermitian PT- symmetric Hamiltonian of a two level system (TLS) which has real eigenvalues. Within the framework of Hermitian quantum mechanics, it is known that maximal violation of Leggett–Garg inequality (LGI) is bounded by K 3 = 3/2 (Luder’s bound). We show that this absolute bound can be evaded when dynamics is governed by non-Hermitian PT- symmetric Hamiltonians with real eigenvalues. Moreover, the extent of violation can be optimized to asymptotically approach the algebraic maximum of , which is otherwise observed for Hermitian Hamiltonian with infinite dimensional Hilbert space. The extreme violation of LGI is shown to be directly related to the two basic ingredients: (i) the Bloch equation for the TLS having a non-linear terms which allow for accelerated dynamics of states on the Bloch sphere exceeding all known quantum speed limits of state evolution; and (ii) quantum trajectory of states lie on a great circle (geodesic path) on the Bloch sphere at all times. We demonstrate that such extreme temporal correlation of TLS can be simulated in realistic system by embedding the TLS into a higher dimensional Hilbert space such that the composite system obeys unitary dynamics. Specifically we show that a four dimensional embedding of non-Hermitian PT- symmetric TLS is enough to host K 3 → 3 limit. We also discuss the effect of random noise on our results. Finally we conclude with a comparative study of our results with existing experimental realization of embedding.

Density matrix and purity evolution in dissipative two-level systems: I. Theory and path integral results for tunneling dynamics.

The time evolution of the purity (the trace of the square of the reduced density matrix) and von Neumann entropy in a symmetric two-level system coupled to a dissipative harmonic bath is investigated through analytical arguments and accurate path integral calculations on simple models and the singly excited bacteriochlorophyll dimer. A simple theoretical analysis establishes bounds and limiting behaviors. The contributions to purity from a purely incoherent term obtained from the diagonal elements of the reduced density matrix, a term associated with the difference of the two eigenstate populations, and a third term related to the square of the time derivative of a site population, are discussed in various regimes. In the case of tunneling dynamics from a localized initial condition, the complex interplay among these contributions leads to the recovery of purity under low-temperature, weakly dissipative conditions. Memory effects from the bath are found to play a critical role to the dynamics of purity. It is shown that the strictly quantum mechanical decoherence process associated with spontaneous phonon emission is responsible for the long-time recovery of purity. These analytical and numerical results show clearly that the loss of quantum coherence during the evolution toward equilibrium does not necessarily imply the decay of purity, and that the time scales relevant to these two processes may be entirely different.

Recovery of Purity in Dissipative Tunneling Dynamics.

The time evolution of purity for an initially localized state of a symmetric two-level system coupled to a dissipative bath is investigated using numerically exact real-time path integral methods. With strong system-bath coupling and high temperature, the purity decays monotonically to its fully mixed value, with a short-time Gaussian behavior, which is subsequently followed by exponential evolution. However, under low-temperature and weak coupling conditions, a substantial recovery of purity is observed. A simple theoretical analysis reveals three contributions that correspond to a completely incoherent, eigenstate population difference and rate terms. The last two of these terms can counter the early drop of purity and are responsible for its rebound. These findings caution against using purity as a measure of decoherence in the dynamics of quantum dissipative systems.

Analytic solutions for generalized -symmetric Rabi models**Project supported in part by the National Natural Science Foundation of China (Grant No. 11874251), the Scientific Research Starting Foundation of Shaanxi Normal University, China, and the Scientific Research Plan Funded by the Education Department of Shaanxi Province, China (Grant No. 17JK0786).

We theoretically investigate the exact solutions for generalized parity–time()-reversal-symmetric Rabi models driven by external fields with monochromatic periodic, linear, and parabolic forms, respectively. The corresponding exact solutions are presented in terms of the confluent Heun equations without any approximation. In principle, the analytic solutions derived here are valid in the whole parameter space. Such a kind of study may offer potential coherent control schemes of the -symmetric two-level systems.

Analytical results for a parity-time-symmetric two-level system under synchronous combined modulations

We propose a simple method of combined synchronous modulations to generate the analytically exact solutions for a parity-time symmetric two-level system. Such exact solutions are expressible in terms of simple elementary functions and helpful for illuminating some generalizations of appealing concepts originating in the Hermitian system. Some intriguing physical phenomena, such as stabilization of a non-Hermitian system by periodic driving, non-Hermitian analogs of coherent destruction of tunneling (CDT) and complete population inversion (CPI), are demonstrated analytically and confirmed numerically. In addition, by using these exact solutions we derive a pulse area theorem for such non-Hermitian CPI in the parity-time symmetric two-level system. Our results may provide an additional possibility for pulse manipulation and coherent control of the parity-time symmetric two-level system.

Time-dependent -symmetric quantum mechanics

The parity-time-reversal ()-symmetric quantum mechanics (QM) (PTQM) has developed into a noteworthy area of research. However, to date, most known studies of PTQM focused on the spectral properties of non-Hermitian Hamiltonian operators. In this work, we propose an axiom in PTQM in order to study general time-dependent problems in PTQM, e.g., those with a time-dependent -symmetric Hamiltonian and with a time-dependent metric. We illuminate our proposal by examining a proper mapping from a time-dependent Schrödinger-like equation of motion for PTQM to the familiar time-dependent Schrödinger equation in conventional QM. The rich structure of the proper mapping hints that time-dependent PTQM can be a fruitful extension of conventional QM. Under our proposed framework, we further study in detail the Berry-phase generation in a class of -symmetric two-level systems. It is found that a closed path in the parameter space of PTQM is often associated with an open path in a properly mapped problem in conventional QM. In one interesting case, we further interpret the Berry phase as the flux of a continuously tunable fictitious magnetic monopole, thus highlighting the difference between PTQM and conventional QM despite the existence of a proper mapping between them.

Inversion symmetric two-level systems and the low-temperature universality in disordered solids

The low-temperature universal properties in disordered and amorphous solids are considered. We introduce a model that includes two types of two-level systems (TLSs), which, based on their local symmetry, interact weakly or strongly with the phonon field. This accounts well for the experimental results and addresses some long-standing questions: the nature of the TLSs, the smallness and universality of the phonon attenuation, and the energy scale of 3 K below which universality is observed. Our model describes disordered lattices; we also discuss its application to amorphous solids.

Geometric phase inPT-symmetric quantum mechanics

Unitary evolution in $\mathcal{P}$$\mathcal{T}$-symmetric quantum mechanics (QM) with a time-dependent metric is found to yield an interesting class of adiabatic processes. As an explicit example, a Berry-like phase associated with a $\mathcal{P}$$\mathcal{T}$-symmetric two-level system is derived and is interpreted as the flux of a fictitious monopole with a tunable charge plus a singular string component with nontrivial phase contributions. To gain more insight, the Hermitian analog of our non-Hermitian problem is also analyzed, which results in an intriguing class of geometric-phase problems in conventional QM as well, where the Hamiltonian includes a perturbative term that is proportional to the rate of change in adiabatic parameters.

Scalar and vector Keldysh models in the time domain

The exactly solvable Keldysh model of disordered electron system in a random scattering field with extremely long correlation length is converted to the time-dependent model with extremely long relaxation. The dynamical problem is solved for the ensemble of two-level systems (TLS) with fluctuating well depths having the discrete Z_2 symmetry. It is shown also that the symmetric TLS with fluctuating barrier transparency may be described in terms of the planar Keldysh model with dime-dependent random planar rotations in xy plane having continuous SO(2) symmetry. The case of simultaneous fluctuations of the well depth and barrier transparency is subject to non-abelian algebra. Application of this model to description of dynamic fluctuations in quantum dots and optical lattices is discussed.

Dynamics of a qubit coupled to a broadened harmonic mode at finite detuning

We study the dynamics of a symmetric two-level system strongly coupled to a broadened harmonic mode. Upon mapping the problem on to a spin–boson model with peaked spectral density, we show how analytic solutions can be obtained, at arbitrary detuning and finite temperatures, in the case of large Q-factors of the oscillator. One, two or more dominating oscillation frequencies of the two-level particle can be observed as a consequence of the entanglement with the oscillator. Our approximated analytical solution agrees well with numerical predictions obtained within the non-interacting blip approximation.

Two‐level systems in glasses under high pressure: temperature cycling effect

According to our studies of polymeric glasses, the pressure of several kilobars remarkably reduces the width of spectral holes and the magnitude of their residual broadening due to the temperature cycling, both effects being roughly linear in pressure. These effects are treated within the scope of the same modified soft anharmonic potential (SAP) model. In this model the sign of the coordinate x is chosen so that the pressure contribution to a SAP is positive for the positive x. In such a case, a decrease of x corresponds to the decrease of the glass volume. The observed pressure effects can be explained if one takes into account the fundamental distinction between compression and expansion of the condensed matter. For two-level systems (TLSs) this inequality gives rise to the asymmetry in the distribution of the energy differences between the two wells: the most numerous are TLSs for which the well, that is associated with a larger glass volume, is located at a higher energy than the other well. As a result, an applied pressure reduces the number of almost symmetric TLSs, which provide the greatest contribution to the hole width. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Cooperativity and resonances in periodically driven spin-boson systems.

We present an analytical recursive formulation for the quantum transport in symmetric two-level systems under the influence of both dissipation and periodic driving. The rate-matching condition for quantum stochastic resonance despite its different appearance is found to be physically the same as that in the classical case. Analyzed are also the Rabi resonance and its implication to quantum stochastic resonance. We demonstrate that no matter how weak the driving field is, transport can involve about 70% population in the vicinities of the third-harmonic as well as the fundamental-harmonic Rabi resonance. Recovered is also an adiabatic passage condition in which the transport carries a nearly 100% population in the low frequency and strong driving limit.

Stabilization of localized states in dissipative tunneling systems interacting with monochromatic fields

We study the dynamics of an initially localized symmetric two-level system coupled to high-temperature dissipative environments and driven by a strong time-periodic force which corresponds to high-frequency monochromatic light. Qualitative arguments based on the quantized representation of the radiation field predict a wealth of intriguing behaviors which are confirmed and quantified via accurate numerical path integral calculations. With the exception of very strong friction we find that high-frequency driving always helps stabilize localized states. At intermediate friction the delocalization rate approaches a “universal’’ limiting value which is largely independent of the parameters of the environment and of the specifics of the driving force, depending only on its overall strength. This robust behavior implies that localized states can be stabilized in these systems without much finetuning of external conditions. In the weak friction regime the interplay between phase interference and dissipation results in nonmonotonic variation of the decay rate with friction and driving frequency. The path integral results are compared to those obtained earlier via analytical treatments.

Low-temperature properties of a two-level system interacting with conduction electrons: An application of the overcompensated multichannel Kondo model.

We consider an atom in a double-well potential [parametrized by a two-level system (TLS)] interacting with screened conduction electrons. The tunneling of the TLS is assisted by the Fermi gas. Close to the strong-coupling fixed point the renormalization-group equations for the noncommutative model in the case of spinless fermions, but an arbitrary number of relevant orbital channels, lead to the n-channel Kondo problem with S=1/2. By solving the thermodynamic Bethe-ansatz equations for the n-channel Kondo problem numerically in the presence of a magnetic field, we discuss the low-temperature properties of the TLS close to the fixed point. The Zeeman splitting corresponds to the energy difference between the two positions of the tunneling atom. The atom is not localized at one of the potential minima. The susceptibility diverges as H\ensuremath{\rightarrow}0 and T\ensuremath{\rightarrow}0, indicating that the symmetric TLS is unstable against a local lattice deformation via coupling to phonons. The lattice distortion disappears above a critical temperature ${\mathit{T}}_{\mathit{c}}$. The ground-state equilibrium situation of the TLS corresponds to H\ensuremath{\ne}0 and a Fermi-liquid picture applies. The values for the specific heat become very large as H\ensuremath{\rightarrow}0. For small fields the specific heat shows a double-peak structure, which is particularly pronounced for n=2.