We apply the quantum Langevin equations approach to study nonlinear light propagation through one-dimensional interacting open quantum lattice models. We write a large set of quantum Langevin equations of lattice operators obtained after integrating out the light fields and use them to derive nonequilibrium features of the lattice models. We first consider a Heisenberg-type interacting spin-$\frac{1}{2}$ chain with nearest-neighbor coupling. The transient and steady-state transport properties of an incoming monochromatic laser light are calculated for this model. We find how the local features of the spin chain and the chain length dependence of light transport coefficient evolve with an increasing power of the incident light. The steady-state light transmission coefficient at a higher power depends nonmonotonically on the interaction in a finite chain. While the nonlinear light transmission in our studied model seems to be ballistic in the absence of interaction and for a high interaction, it shows an apparent system-size dependence at intermediate interactions. Later, we extend this method to the long-range interaction between spins of the driven-dissipative lattice model and to incorporate various losses typical in many atomic and solid-state systems.
We apply the quantum Langevin equations approach to study nonlinear light propagation through one-dimensional interacting open quantum lattice models. We write a large set of quantum Langevin equations of lattice operators obtained after integrating out the light fields and use them to derive nonequilibrium features of the lattice models. We first consider a Heisenberg-type interacting spin-21 chain with nearest-neighbor coupling. The transient and steady-state transport properties of an incoming monochromatic laser light are calculated for this model. We find how the local features of the spin chain and the chain length dependence of light transport coefficient evolve with an increasing power of the incident light. The steady-state light transmission coefficient at a higher power depends nonmonotonically on the interaction in a finite chain. While the nonlinear light transmission in our studied model seems to be ballistic in the absence of interaction and for a high interaction, it shows an apparent system-size dependence at intermediate interactions. Later, we extend this method to the long-range interaction between spins of the driven-dissipative lattice model and to incorporate various losses typical in many atomic and solid-state systems.
Abstract The role of optical nonlinearity in light propagation through two different 1D open QED lattices is investigated, namely a chain of qubits with direct coupling between the nearest neighbors and a chain of connected resonators to each of which a qubit is side‐coupled. Using the more accurate truncated Heisenberg‐Langevin equations method, a reduction of light transmission with increasing intensity in these lattices is shown due to effective photon–photon interactions and related photon blockade mediated by nonlinearity in qubits. In contrast to the direct‐coupled qubits, a revival in the light transmission is found in the side‐coupled qubits at relatively higher intensities due to saturation of qubits by photons. In absence of bulk dissipation, it is found that the standard quasi‐classical analysis fails to capture the reduction in light transmission due to effective photon–photon interaction. A systemic method is devised to modify the quasi‐classical analysis to give much better results.
We investigate the role of optical nonlinearity in light propagation through two different waveguide QED lattices, namely a chain of qubits with direct coupling between the nearest neighbors and a chain of connected resonators to each of which a qubit is side-coupled. We apply an efficient truncated Heisenberg-Langevin equations method to show loss of coherent light transmission with increasing intensity in these lattices due to effective photon-photon interactions and related photon blockade mediated by nonlinearity in qubits. In contrast to the direct-coupled qubits, we find a revival in the coherent light transmission in the side-coupled qubits at relatively higher intensities due to saturation of qubits by photons. We further study these lattices within the quasi-classical approximation, which fails for a broad set of parameters. We then devise a technique to modify the quasi-classical analysis to give much better results. Lastly, we examine light propagation in an inhomogeneous lattice of side-coupled qubits and observe non-monotonicity in light transmission with increasing light intensity.
Open quantum systems have drawn attention over decades due to its applicability in the foundation of theoretical physics, e.g. statistical mechanics, quantum mechanics and condensed matter physics. The dynamics of open quantum systems has been described as separate entities from their surrounding environment that consists of a very large number of modes, somehow coupled to the mode of the system. Even though the exact solution of the dynamical behavior of the system is impossible to calculate, we obtain a tentative solution using the crucial Markov approximation. The input-output formalism of the quantum Langevin equation (QLE) has been considered as a useful tool which provides a semi-classical description of the dynamics of the system, whereas the master equation provides a complete picture of the dynamics of the system expressed in terms of density matrix. While studying the dynamics of nonlinear system/environment coupling using QLE, we see, for a small value of external field, that the steady state system field does not change much from the steady state field obtained in the absence of nonlinear dissipation. However, in a case where a stronger external field is applied, we see that the deviation becomes substantial from the solution of linear system. We also see that the nonlinear coupling introduces significant difference in the cavity fluctuation spectrum. The description, therefore, provides a potential explanation of parametric effects in terms of nonlinear dissipation phenomena associated with the nonlinear coupling. Even though the theories developed in the context of open quantum systems have proven to be powerful tools, they do not provide a satisfactory platform to be implemented on non-linear Hamiltonians. We often approximate it by linearizing over nonlinear steady state field amplitude, and therefore, the interesting effects are often overlooked. The limitation of the analytics provokes us to simulate open quantum dynamics numerically. The numerical method consists of transformation of the environmental degrees of freedom to a one-dimensional many-body chain, and the computational technique includes numerical diagonalization and renormalization process. The time-adaptive density matrix renormalisation group (t-DMRG) is known as one of the most powerful techniques for the simulation of strongly-correlated many-body quantum systems. In this thesis, along with the theoretical modeling, we implement DMRG numerical scheme for the simulation of canonical S/B model by mapping it to one-dimensional harmonic chain with nearest neighbor interactions, and use the method to investigate the dynamics of the free dissipative system. The thermalization of open quantum systems is also studied by generating minimally entangled typical thermal states (METTS) through imaginary time evolution, and real-time evolving an empty system in the presence of the thermal bath. Further, we simulate coherently driven free dissipative Kerr nonlinear system numerically using Euler's method by solving Heisenberg equation of motion and t-DMRG algorithm, and demonstrate how the numerical results are analogous to classical bistability. By comparing with analytics, we see that the DMRG numerics is analogous to the quantum-mechanical exact solution obtained by mapping the equation of motion of the density matrix of the system to a Fokker-Plank equation. The comparison between two different numerical techniques shows that the semi-classical Euler's method determines the dynamics of the system field of one among two coherent branches, whereas DMRG numerics gives the superposition of both of them. Hence, DMRG-determined time dynamics undergoes generating non-classical states. Our approach of dealing with nonlinearity represents an important contribution in the developments of technique to study the dynamical and steady-state behavior of open quantum systems, which is a fundamental aspect of quantum physics.
We investigate collective behaviour that appears in open, many-body systems of two- and four-level atoms. Here, ``open" refers to the system interacting with an external environment that causes dissipation. We derive a general open quantum master equation to describe the system dynamics only, independent of this environment. We identify processes such as coherent exchange of virtual photons and modified decay rates caused by long-range interactions between all pairs of atoms that scale with distance by inverse power laws. We explore excitation transport within a one-dimensional chain of atoms where the atomic transition dipoles are coupled to the free radiation field. When the interatomic spacing is smaller than or comparable to the wavelength of light associated with the photon emitted from a given transition, virtual photon exchange interactions facilitate excitation transport through the chain. Atomic systems coupled to an environment display dissipative dynamics, however subradiant transport is exhibited from a variety of initial states; spontaneous emission from the chain occurs at a rate much slower than that for an individual atom. In particular, we find a region within the decay spectrum that consists entirely of subradiant states with a corresponding linear dispersion relation in the interaction energy. Identifying this subspace allows for the dispersionless transport of wave packets over long distances with near-zero decay. Moreover, the group velocity of the wave packet and direction of the transport can be controlled via an external uniform magnetic field while preserving its subradiant character. We discuss a number of experimental considerations to justify the feasibility and robustness of this protocol. Initial state preparation is outlined, utilising external laser driving to excite the system into the single excitation sector. Furthermore, we consider positional disorder by explicitly accounting for the external atomic degrees of freedom -- position and momentum -- which allows us to model the positions of all atoms by a motional state representing the occupation of a lattice well with a given width. These discussions are made in the low-temperature limit, where the atomic motion is essentially frozen, and we identify that subradiant transport is indeed robust. Finally, we explore the experimental limits of interatomic spacing and imperfect filling within an experimentally achievable optical lattice. We calculate the photon emission rate -- an experimentally measurable quantity -- and compare the emission spectra to our analysis. By limiting parameters to those achieved experimentally, we observe a reduction, yet not an absence, of collective behaviour. The simplicity and versatility of this system, together with the robustness of subradiance against disorder, makes it relevant for a range of applications such as lossless energy transport and long-time light storage. The lifetime of an atomic excitation could be increased by a factor of thousands to millions for a chain of atoms under the conditions that we explore in this thesis.
Based on the references [P1–P6], we investigate in this cumulative PhD thesis the dynamics and transport in closed and open interacting spin chain models. We focus on (i) the connection between the open and the closed system dynamics [P1, P2], (ii) the influence of various system parameters on the transport behavior in closed and open systems [P3, P5], and in addtion (iii) the dynamics and transport in closed models [P4, P6]. As for (i), we explore the connection of the dynamics in open systems, modeled by the Lindblad equation, and the correlation functions in the closed system [P1]. Building on this connection for magnetization transport, we study whether the time evolution of the open quantum system can be described in terms of classical correlation functions generated by the Hamiltonian equations of motion for real vectors [P2]. Second, we study in (ii) the influence of different parameters on the diffusion constants in closed and open systems [P3, P5]. In Ref. [P3], we study the impact of integrability-breaking perturbations in both closed and open spin chain systems. While we find a continuous change of the diffusion constant with perturbation strength in the closed system, in the open system, in the weak-perturbation limit, we indicate the possibility of a diverging diffusion constant. Also, we investigate the influence of the system-bath coupling on the transport behavior in boundary-driven open systems [P5]. We observe that the diffusion constants depend on the system-bath coupling and that in the nonequilibrium steady state, they inevitably retain finite-size effects, limiting the computation of long-time transport coefficients. Additional for (iii), in Ref. [P4], we first examine the impact of perturbations on the dynamics in closed spin chain models. We employ projector-operator techniques to understand, on perturbative grounds, the nontrivial damping of the standard correlation function. Second, we also discuss a dynamical quantum typicality approach for studying the energy transport in bipartite systems at low temperatures [P6].
We study nonequilibrium steady states of the one-dimensional discrete nonlinear Schrödinger equation. This system can be regarded as a minimal model for the stationary transport of bosonic particles such as photons in layered media or cold atoms in deep optical traps. Due to the presence of two conserved quantities, namely, energy and norm (or number of particles), the model displays coupled transport in the sense of linear irreversible thermodynamics. Monte Carlo thermostats are implemented to impose a given temperature and chemical potential at the chain ends. As a result, we find that the Onsager coefficients are finite in the thermodynamic limit, i.e., transport is normal. Depending on the position in the parameter space, the "Seebeck coefficient" may be either positive or negative. For large differences between the thermostat parameters, density and temperature profiles may display an unusual nonmonotonic shape. This is due to the strong dependence of the Onsager coefficients on the state variables.