One-dimensional many-body entangled open quantum systems with tensor network methods

We present a collection of methods to simulate entangled dynamics of open quantum systems governed by the Lindblad equation with tensor network methods. Tensor network methods using matrix product states have been proven very useful to simulate many-body quantum systems and have driven many innovations in research. Since the matrix product state design is tailored for closed one-dimensional systems governed by the Schr\"odinger equation, the next step for many-body quantum dynamics is the simulation of open quantum systems. We review the three dominant approaches to the simulation of open quantum systems via the Lindblad master equation: quantum trajectories, matrix product density operators, and locally purified tensor networks. Selected examples guide possible applications of the methods and serve moreover as a benchmark between the techniques. These examples include the finite temperature states of the transverse quantum Ising model, the dynamics of an exciton traveling under the influence of spontaneous emission and dephasing, and a double-well potential simulated with the Bose-Hubbard model including dephasing. We analyze which approach is favorable leading to the conclusion that a complete set of all three methods is most beneficial, push- ing the limits of different scenarios. The convergence studies using analytical results for macroscopic variables and exact diagonalization methods as comparison, show, for example, that matrix product density operators are favorable for the exciton problem in our study. All three methods access the same library, i.e., the software package Open Source Matrix Product States, allowing us to have a meaningful comparison between the approaches based on the selected examples. For example, tensor operations are accessed from the same subroutines and with the same optimization eliminating one possible bias in a comparison of such numerical methods.Comment: 24 pages, 8 figures. Small extension of time evolution section and moving quantum simulators to introduction in comparison to v

Confining the state of light to a quantum manifold by engineered two-photon loss

Physical systems usually exhibit quantum behavior, such as superpositions and entanglement, only when they are sufficiently decoupled from a lossy environment. Paradoxically, a specially engineered interaction with the environment can become a resource for the generation and protection of quantum states. This notion can be generalized to the confinement of a system into a manifold of quantum states, consisting of all coherent superpositions of multiple stable steady states. We have experimentally confined the state of a harmonic oscillator to the quantum manifold spanned by two coherent states of opposite phases. In particular, we have observed a Schrodinger cat state spontaneously squeeze out of vacuum, before decaying into a classical mixture. This was accomplished by designing a superconducting microwave resonator whose coupling to a cold bath is dominated by photon pair exchange. This experiment opens new avenues in the fields of nonlinear quantum optics and quantum information, where systems with multi-dimensional steady state manifolds can be used as error corrected logical qubits

Localized Faraday patterns under heterogeneous parametric excitation

Faraday waves are a classic example of a system in which an extended pattern emerges under spatially uniform forcing. Motivated by systems in which uniform excitation is not plausible, we study both experimentally and theoretically the effect of heterogeneous forcing on Faraday waves. Our experiments show that vibrations restricted to finite regions lead to the formation of localized subharmonic wave patterns and change the onset of the instability. The prototype model used for the theoretical calculations is the parametrically driven and damped nonlinear Schr\"odinger equation, which is known to describe well Faraday-instability regimes. For an energy injection with a Gaussian spatial profile, we show that the evolution of the envelope of the wave pattern can be reduced to a Weber-equation eigenvalue problem. Our theoretical results provide very good predictions of our experimental observations provided that the decay length scale of the Gaussian profile is much larger than the pattern wavelength.Comment: 10 pages, 9 figures, Accepte

A two-step trigonometrically fitted semi-implicit hybrid method for solving special second order oscillatory differential equation

In this paper, we derived a semi-implicit hybrid method (SIHM) which is a two-step method to solve special second order ordinary differential equations (ODEs). The SIHM which is three-stage and fourth-order is then trigonometrically fitted and denoted by TF-SIHM3(4). The method is constructed using trigonometrically fitted properties instead of using phase-lag and amplification properties. Numerical integration show that TF-SIHM3(4) is more accurate in term of accuracy compared to the existing explicit and implicit methods of the same order

Interaction instability of localization in quasiperiodic systems

Integrable models form pillars of theoretical physics because they allow for full analytical understanding. Despite being rare, many realistic systems can be described by models that are close to integrable. Therefore, an important question is how small perturbations influence the behavior of solvable models. This is particularly true for many-body interacting quantum systems where no general theorems about their stability are known. Here, we show that no such theorem can exist by providing an explicit example of a one-dimensional many-body system in a quasiperiodic potential whose transport properties discontinuously change from localization to diffusion upon switching on interaction. This demonstrates an inherent instability of a possible many-body localization in a quasiperiodic potential at small interactions. We also show how the transport properties can be strongly modified by engineering potential at only a few lattice sites.Comment: 10 pages; (v2: additional explanations, data, and references

エネルギー関数を持つ発展方程式に対する幾何学的数値計算法

学位の種別: 課程博士審査委員会委員 : (主査)東京大学教授 松尾 宇泰, 東京大学教授 中島 研吾, 東京大学准教授 鈴木 秀幸, 東京大学准教授 長尾 大道, 東京大学准教授 齋藤 宣一University of Tokyo(東京大学

Numerical Simulations of Water Waves\u27 Modulational Instability Under the Action of Wind and Dissipation

Since the work of Benjamin & Feir (1967), water waves propagating in infinite depth are known to be unstable to modulational instability. The evolution of such wave trains is well described through fully nonlinear simulations, but also by means of simplified models, such as the nonlinear Schrödinger equation. Segur et al. (2005) and Wu et al. (2006) studied theoretically and numerically the evolution of this instability, and both concluded that a long term restabilization occurs in these conditions. More recently, Kharif et al. (2010) considered wind forcing and viscous dissipation within the framework of a forced and damped nonlinear Schrödinger equation, and discussed the range of parameters for which this behavior is still valid. This work aims to demonstrate how numerical simulations are useful to analyze their theoretical predictions. Since we are dealing with long term stability, results are especially complicated to obtain experimentally. Thus, numerical simulations of the fully nonlinear equations turn out to be a very useful tool to provide a validation for the model. Here, the evolution of the modulational instability is investigated within the framework of the two-dimensional fully non linear potential equations, modified to include wind forcing and viscous dissipation. The wind model corresponds to the Miles theory. The introduction of dissipation in the equations is briefly discussed. The marginal stability curve derived from the fully nonlinear numerical simulations coincides with the curve obtained by Kharif et al. (2010) from a linear stability analysis. Furthermore, the long term evolution of the wave trains can be obtained through the numerical simulations, and it is found that the presence of wind forcing promotes the occurrence of a permanent frequency-downshifting without invoking damping due to breaking wave phenomenon

Quantum annealing and advanced optimization strategies of closed and open quantum systems

Adiabatic quantum computation and quantum annealing are powerful methods designed to solve optimization problems more efficiently than classical computers. The idea is to encode the solution to the optimization problem into the ground state of an Ising Hamiltonian, which can be hard to diagonalize exactly and can involve long-range and multiple-body interactions. The adiabatic theorem of quantum mechanics is exploited to drive a quantum system towards the target ground state. More precisely, the evolution starts from the ground state of a transverse field Hamiltonian, providing the quantum fluctuations needed for quantum tunneling between trial solution states. The Hamiltonian is slowly changed to target the Ising Hamiltonian of interest. If this evolution is infinitely slow, the system is guaranteed to stay in its ground state. Hence, at the end of the dynamics, the state can be measured, yielding the solution to the problem. In real devices, such as in the D-Wave quantum annealers, the evolution lasts a finite amount of time, which gives rise to Landau-Zener diabatic transitions, and occurs in the presence of an environment, inducing thermal excitations outside the ground state. Both these limitations have to be carefully addressed in order to understand the true potential of these devices. The present thesis aims to find strategies to overcome these limitations. In the first part of this work, we address the effects of dissipation. We show that a low-temperature Markovian environment can improve quantum annealing, compared with the closed-system case, supporting other previous results known in the literature as thermally-assisted quantum annealing. In the second part, we combine dissipation with advanced annealing schedules, featuring pauses and iterated or adiabatic reverse annealing, which, in combination with low-temperature environments, can favor relaxation to the ground state and improve quantum annealing compared to the standard algorithm. In general, however, dissipation is detrimental for quantum annealing especially when the annealing time is longer than the typical thermal relaxation and decoherence time scales. For this reason, it is essential to devise shortcuts to adiabaticity so as to reach the adiabatic limit for relatively short times in order to decrease the impact of thermal noise on the performances of QA. To this end, in the last part of this thesis we study the counterdiabatic driving approach to QA. In counterdiabatic driving, a new term is added to the Hamiltonian to suppress Landau-Zener transitions and achieve adiabaticity for any finite sweep rate. Although the counterdiabatic potential is nonlocal and hardly implementable on quantum devices, we can obtain approximate potentials that dramatically enhance the success probability of short-time quantum annealing following a variational formulation