non-Hermitian Hamiltonian Approach Research Articles

Formula omitted]symmetric tight-binding 1D chains: Coupling to the continuum and superradiance transition

By the use of the transfer matrix (TM) approach and effective non-Hermitian Hamiltonian (EnHH) approach we study transport and scattering properties of PT−symmetric tight-binding one-dimensional (1D) chains. In the TM approach we derive an exact expression for the transmission T through 1D chains that incorporates all the scattering-system parameters: incident-wave energy E, chain length L, gain/loss amplitude γ, and coupling strength to perfect leads χ. We show that the behavior of T vs. L strongly depends on the value of γ with respect to the critical value γcr, defined through γcr2=4−E2. For γ<γcr, the transmission T is a periodic function of L, however, for γ>γcr, T decreases exponentially with L. We also predict the unforeseen dependence T∝L−2 at γ=γcr. By the use of the EnHH approach to the same model which is now connected to continuum through left and write ends, we focus on the onset of the superradiance transition as a function of E, L, γ, and χ. The main interest here is in the competition between the balanced gain/loss terms in the bulk of the chain and the coupling to continuum through the ends of the chain. Our results demonstrate that the presence of the gain/loss in the bulk strongly modifies the width of non-superradiant states, however, did not change the width of the superradiant ones. The analytical predictions are supported by detailed numerical simulations.

Vibrational response functions for multidimensional electronic spectroscopy in the adiabatic regime: A coherent-state approach.

Multi-dimensional spectroscopy represents a particularly insightful tool for investigating the interplay of nuclear and electronic dynamics, which plays an important role in a number of photophysical processes and photochemical reactions. Here, we present a coherent state representation of the vibronic dynamics and of the resulting response functions for the widely used linearly displaced harmonic oscillator model. Analytical expressions are initially derived for the case of third-order response functions in an N-level system, with ground state initialization of the oscillator (zero-temperature limit). The results are then generalized to the case of Mth order response functions, with arbitrary M. The formal derivation is translated into a simple recipe, whereby the explicit analytical expressions of the response functions can be derived directly from the Feynman diagrams. We further generalize to the whole set of initial coherent states, which form an overcomplete basis. This allows one, in principle, to derive the dependence of the response functions on arbitrary initial states of the vibrational modes and is here applied to the case of thermal states. Finally, a non-Hermitian Hamiltonian approach is used to include in the above expressions the effect of vibrational relaxation.

Persistence of topological phases in non-Hermitian quantum walks

Discrete-time quantum walks are known to exhibit exotic topological states and phases. Physical realization of quantum walks in a lossy environment may destroy these phases. We investigate the behaviour of topological states in quantum walks in the presence of a lossy environment. The environmental effects in the quantum walk dynamics are addressed using the non-Hermitian Hamiltonian approach. We show that the topological phases of the quantum walks are robust against moderate losses. The topological order in one-dimensional split-step quantum walk persists as long as the Hamiltonian respects exact {{mathcal {P}}}{{mathcal {T}}}-symmetry. Although the topological nature persists in two-dimensional quantum walks as well, the {{mathcal {P}}}{{mathcal {T}}}-symmetry has no role to play there. Furthermore, we observe topological phase transition in two-dimensional quantum walks that is induced by losses in the system.

Photoprotecting Uracil by Coupling with Lossy Nanocavities.

We analyze how the photorelaxation dynamics of a molecule can be controlled by modifying its electromagnetic environment using a nanocavity mode. In particular, we consider the photorelaxation of the RNA nucleobase uracil, which is the natural mechanism to prevent photodamage. In our theoretical work, we identify the operative conditions in which strong coupling with the cavity mode can open an efficient photoprotective channel, resulting in a relaxation dynamics twice as fast as the natural one. We rely on a state-of-the-art chemically detailed molecular model and a non-Hermitian Hamiltonian propagation approach to perform full-quantum simulations of the system dissipative dynamics. By focusing on the photon decay, our analysis unveils the active role played by cavity-induced dissipative processes in modifying chemical reaction rates, in the context of molecular polaritonics. Remarkably, we find that the photorelaxation efficiency is maximized when an optimal trade-off between light–matter coupling strength and photon decay rate is satisfied. This result is in contrast with the common intuition that increasing the quality factor of nanocavities and plasmonic devices improves their performance. Finally, we use a detailed model of a metal nanoparticle to show that the speedup of the uracil relaxation could be observed via coupling with a nanosphere pseudomode, without requiring the implementation of complex nanophotonic structures.

Universal Tool for Single-Photon Circuits: Quantum Router Design.

We demonstrate that the non-Hermitian Hamiltonian approach can be used as a universal tool to design and describe a performance of single photon quantum electrodynamical circuits (cQED). As an example of the validity of this method, we calculate a novel six port quantum router, constructed from four qubits and three open waveguides. We have obtained analytical expressions, which describe the transmission and reflection coefficients of a single photon in general form taking into account the spread qubit’s parameters. We show that, due to naturally derived interferences, in situ tuning the probability of photon detection in desired ports.

A comparative study on different non-Hermitian approaches for modeling open quantum systems

In this paper, we test and compare the performance of two different theoretical methods that use anti-Hermitian terms for modeling open quantum systems. It is, the non-Hermitian quantum mechanics method (NHQM) and the recent methodology known as corrected non-Hermitian effective Hamiltonian (corrected NHEH) approach. In particular, we provide a comprehensive analysis of the performance of these two theoretical approaches by describing the dynamics of the pumped-dissipative Jaynes–Cummings model. Our results demonstrate that whereas the NHQM method fails for describing correctly the time evolution of the density matrix elements as well as the quantum fidelity measure. The corrected NHEH approach shows an excellent agreement with simulations based on the Lindblad master equation formalism.

A study on the role of the initial conditions and the nonlinear dissipation in the non-Hermitian effective Hamiltonian approach

This work examines the initial conditions dependence and the influence of nonlinear dissipation in the quantum dynamics originated by the non-Hermitian effective Hamiltonian approach. In particular, we test the performance of this theoretical approach for modeling the dynamics of the pumped-dissipative Jaynes Cummings model under different initial states such as a Fock, Gaussian and maximally mixed states, and as well as for a polaritonic light-matter state. It is demonstrated unequivocally that whereas the non-Hermitian effective Hamiltonian approach fails in to reproduce correctly the time evolution of the density matrix elements and observables of the system, the corrected version of this theoretical approach recently developed by us shows an excellent agreement with simulations based on the Lindblad master equation formalism.

A comparative study on the reliability of non-Hermitian effective Hamiltonian approach for modeling open quantum systems

In this paper, we conduct a comprehensive analysis on the reliability of non-Hermitian effective Hamiltonian approach for modeling quantum dissipative dynamics. It is found that one of the major drawbacks to adopting this theoretical approach is that it fails to reproduce correctly both the dynamics and the time evolution of observables of the system. With this in mind, we propose a method that allows correcting the non-Hermitian effective Hamiltonian approach when there exists a conserved quantity in the system. In order to illustrate the performance of our corrected non-Hermitian effective Hamiltonian approach, we consider a two-level system interacting with an electromagnetic cavity mode where the effects of the environment are included in a phenomenological manner. More precisely, the dissipation and pump in the system are considered through non-Hermitian Hamiltonian terms. Our numerical results have been focused on the time evolution of density matrix elements, observables as well as the emission spectrum of the system, and they demonstrate that our proposed methodology is in excellent agreement with the predictions of the Lindblad master equation approach.

Transfer of excited state between two qubits in an open waveguide

The probability amplitudes of the processes related to the transfer of the excited state from one qubit to another are calculated using an indirect interaction in an open waveguide. The system consists of two qubits located at an arbitrary distance from each other. The non-Hermitian effective Hamiltonian approach used herein makes it possible to bypass using the Markovchain approximation method. Analytic expressions describing the probability of transfer of the excited state from one qubit to another under different initial states of the system were obtained.

Mollow triplet through pump-probe single-photon spectroscopy of artificial atoms

We analyze a photon transport through an 1D open waveguide side coupled to the $N$-photon microwave cavity with embedded artificial two- level atom (qubit). The qubit state is probed by a weak signal at the fundamental frequency of the waveguide. Within the formalism of projection operators and non-Hermitian Hamiltonian approach we develop a one-photon approximation scheme to obtain the photon wavefunction which allows for the calculation of the probability amplitudes of the spontaneous transitions between the levels of two Rabi doublets in $N$- photon cavity. We obtain analytic expressions for the transmission and reflection factors of the microwave signal through a waveguide, which contain the information of the qubit parameters. We show that for small number of cavity photons the Mollow spectrum consists of four spectral lines which is a direct manifestation of quantum nature of light. The results obtained in the paper are of general nature and can be applied to any type of qubits. The specific properties of the qubit are only encoded in the two parameters: the energy $\Omega$ of the qubit and its coupling $\lambda$ to the cavity photons.

Parity-time symmetry breaking in magnetic systems

The understanding of out-of-equilibrium physics, especially dynamic instabilities and dynamic phase transitions, is one of the major challenges of contemporary science, spanning the broadest wealth of research areas that range from quantum optics to living organisms. Focusing on nonequilibrium dynamics of an open dissipative spin system, we introduce a non-Hermitian Hamiltonian approach, in which non-Hermiticity reflects dissipation and deviation from equilibrium. The imaginary part of the proposed spin Hamiltonian describes the effects of Gilbert damping and applied Slonczewski spin-transfer torque. In the classical limit, our approach reproduces Landau-Lifshitz-Gilbert-Slonczewski dynamics of a large macrospin. We reveal the spin-transfer torque-driven parity-time symmetry-breaking phase transition corresponding to a transition from precessional to exponentially damped spin dynamics. Micromagnetic simulations for nanoscale ferromagnetic disks demonstrate the predicted effect. Our findings can pave the way to a general quantitative description of out-of-equilibrium phase transitions driven by spontaneous parity-time symmetry breaking.

Non-Hermitian Hamiltonian approach to the microwave transmission through a one-dimensional qubit chain

We investigate the propagation of microwave photons in a one-dimensional open waveguide interacting with a number of artificial atoms (qubits). Within the formalism of projection operators and non-Hermitian Hamiltonian approach we develop a one-photon approximation scheme for the calculation of the transmission and reflection factors of the microwave signal in a waveguide which contains an arbitrary number \emph{N} of non-interacting qubits. We considered in detail the resonances and photon mediated entanglement for two and three qubits in a chain. We showed that in non Markovian case the resonance widths, which define the decay rates of the entangled state, can be much smaller than the decay width of individual qubit. It is also shown that for identical qubits in the long wavelength limit a coherent superradiant state is formed with the width being equal to the sum of the widths of spontaneous transitions of \emph{N} individual qubits. The results obtained in the paper are of general nature and can be applied to any type of qubits. The specific properties of the qubit are only encoded in the two parameters: the qubit energy $\Omega$ and the rate of spontaneous emission $\Gamma$.

Gate controlled resonant widths in double-bend waveguides: bound states in the continuum

We consider quantum transmission through double-bend - and Z-shaped waveguides controlled by the finger gate potential. Using the effective non-Hermitian Hamiltonian approach we explain the resonances in transmission. We show a difference in transmission in the short waveguides that is the result of different chirality in Z and waveguides. We demonstrate that the potential selectively affects the resonant widths resulting in the occurrence of bound states in the continuum.

Non-Hermitian Hamiltonian approach to quantum transport in disordered networks with sinks: Validity and effectiveness

We investigate the validity of the non-Hermitian Hamiltonian approach in describing quantum transport in disordered tight-binding networks connected to external environments, acting as sinks. Usually, non-Hermitian terms are added, on a phenomenological basis, to such networks to summarize the effects of the coupling to the sinks. Here, we consider a paradigmatic model of open quantum network for which we derive a non-Hermitian effective model, discussing its limit of validity by a comparison with the analysis of the full Hermitian model. Specifically, we consider a ring of sites connected to a central one-dimensional lead. The lead acts as a sink that absorbs the excitation initially present in the ring. The coupling strength to the lead controls the opening of the ring system. This model has been widely discussed in literature in the context of light-harvesting systems. We analyze the effectiveness of the non-Hermitian description both in absence and in presence of static disorder on the ring. In both cases, the non-Hermitian model is valid when the energy range determined by the eigenvalues of the ring Hamiltonian is smaller than the energy band in the lead. Under such condition, we show that results about the interplay of opening and disorder, previously obtained within the non-Hermitian Hamiltonian approach, remain valid when the full Hermitian model in presence of disorder is considered. The results of our analysis can be extended to generic networks with sinks, leading to the conclusion that the non-Hermitian approach is valid when the energy dependence of the coupling to the external environments is sufficiently smooth in the energy range spanned by the eigenstates of the network.

Coupled mode theory for acoustic resonators

We develop the effective non-Hermitian Hamiltonian approach for open systems with Neumann boundary conditions. The approach can be used for calculating the scattering matrix and the scattering function in open resonator–waveguide systems. In higher than one dimension the method represents acoustic coupled mode theory in which the scattering solution within an open resonator is found in the form of expansion over the eigenmodes of the closed resonator decoupled from the waveguides. The problem of finding the transmission spectra is reduced to solving a set of linear equations with a non-Hermitian matrix whose anti-Hermitian term accounts for coupling between the resonator eigenmodes and the scattering channels of the waveguides. Numerical applications to acoustic two-, and three-dimensional resonator–waveguide problems are considered.

Stimulated Raman adiabatic passage in an open quantum system: Master equation approach

A master equation approach to the study of environmental effects in the adiabatic population transfer in three-state systems is presented. A systematic comparison with the non-Hermitian Hamiltonian approach [N. V. Vitanov and S. Stenholm, Phys. Rev. A {\bf 56}, 1463 (1997)] shows that in the weak coupling limit the two treatments lead to essentially the same results. Instead, in the strong damping limit the predictions are quite different: in particular the counterintuitive sequences in the STIRAP scheme turn out to be much more efficient than expected before. This point is explained in terms of quantum Zeno dynamics.

Superradiance transition in one-dimensional nanostructures: An effective non-Hermitian Hamiltonian formalism

Using an energy-independent non-Hermitian Hamiltonian approach to open systems, we fully describe transport through a sequence of potential barriers as external barriers are varied. Analyzing the complex eigenvalues of the non-Hermitian Hamiltonian model, a transition to a superradiant regime is shown to occur. Transport properties undergo a strong change at the superradiance transition, where the transmission is maximized and a drastic change in the structure of resonances is demonstrated. Finally, we analyze the effect of the superradiance transition in the Anderson localized regime.

Photon emission from a driven single-molecule source: A renormalization group approach

The photon emission from a single molecule driven simultaneously by a laser and a slow electric radio frequency (rf) field is studied. We use a non-Hermitian Hamiltonian approach which accounts for the radiative decay of a two-level system modeling the single-molecule source. We apply the renormalization group method for differential equations to obtain long time solution of the corresponding Schrodinger equation, which allows us to calculate the average waiting time for the first photon emission. Then, we analyze the conditions for suppression and enhancement of photon emission in this dissipative two-level system. In particular we derive a transcendental equation, which yields the nontrivial rf field control parameters, for which enhancement and suppression of photon emission occurs. For finite values of radiative decay rate an abrupt transition to the state when both situations are indistinguishable is found for certain values of the rf field parameters. Our results are shown to be in agreement with the available experiments [Ch. Brunel et al., Phys. Rev. Lett. 81, 2679 (1998)].

Photon emission from a single-molecule source driven by an rf field

The behavior of a single molecule driven simultaneously by a laser and by an electric radio frequency field is investigated using a non-Hermitian Hamiltonian approach. Employing the renormalization group method for differential equations we calculate the average waiting time for the first photon emission event to occur, and determine the conditions for the suppression and enhancement of photon emission. An abrupt transition from localization-like behavior to delocalization behavior is found.

Imaginary phases in two-level model with spontaneous decay

We study a two-level model coupled to the electromagnetic vacuum and to an external classic electric field with fixed frequency. The amplitude of the external electric field is supposed to vary very slow in time. Garrison and Wright [{\it Phys. Lett.} {\bf A128} (1988) 177] used the non-hermitian Hamiltonian approach to study the adiabatic limit of this model and obtained that the probability of this two-level system to be in its upper level has an imaginary geometric phase. Using the master equation for describing the time evolution of the two-level system we obtain that the imaginary phase due to dissipative effects is time dependent, in opposition to Garrison and Wright result. The present results show that the non-hermitian hamiltonian method should not be used to discuss the nature of the imaginary phases in open systems.