We introduce and characterize a scheme for emulating non-Markovian quantum system-bath dynamics using the discrete electromagnetic field modes of a parametrically driven cavity. In this scheme, the character of a bosonic bath (the form of the bath spectral density) is tied directly to the shape of the real-time waveform describing periodic parametric cavity modulation. As an explicit example, we demonstrate the feasibility of this scheme for a fiber-loop experiment with currently achievable parameters, supported by numerical simulations and analytical estimates. In particular, we show that a localization transition of the spin-boson model can be accurately emulated in this fiber-loop cavity system. This localization effect is characterized by a sharp transition from partial decay to complete decay for a two-level system coupled to a bosonic bath as the bath spectral density $J(ω)\propto ω^{s}$ is continously tuned from the sub-ohmic ($s<1$) to the super-ohmic ($s>1$) regime. The transition is only exactly realized in the thermodynamic limit (for an infinite number of bath modes) and for sufficiently weak system-bath coupling. We highlight a competition between the weak-coupling and thermodynamic limits in this problem and show that challenges in approximately realizing the transition can nevertheless be overcome. Finally, we provide bounds on the systematic error introduced when emulating non-Markovian dynamics for arbitrary system observables. The scheme presented here can enable the realization of a modular platform for engineering custom non-Markovian baths, with potential applications in quantum thermodynamics, thermalization of many-body systems, and resource-efficient quantum simulations of open quantum systems.
We introduce and characterize a scheme for emulating non-Markovian quantum system-bath dynamics using the discrete electromagnetic field modes of a parametrically driven cavity. In this scheme, the character of a bosonic bath (the form of the bath spectral density) is tied directly to the shape of the real-time waveform describing periodic parametric cavity modulation. As an explicit example, we demonstrate the feasibility of this scheme for a fiber-loop experiment with currently achievable parameters, supported by numerical simulations and analytical estimates. In particular, we show that a localization transition of the spin-boson model can be accurately emulated in this fiber-loop cavity system. This localization effect is characterized by a sharp transition from partial decay to complete decay for a two-level system coupled to a bosonic bath as the bath spectral density J(ω)∝ωs is continously tuned from the sub-ohmic (s<1) to the super-ohmic (s>1) regime. The transition is only exactly realized in the thermodynamic limit (for an infinite number of bath modes) and for sufficiently weak system-bath coupling. We highlight a competition between the weak-coupling and thermodynamic limits in this problem and show that challenges in approximately realizing the transition can nevertheless be overcome. Finally, we provide bounds on the systematic error introduced when emulating non-Markovian dynamics for arbitrary system observables. The scheme presented here can enable the realization of a modular platform for engineering custom non-Markovian baths, with potential applications in quantum thermodynamics, thermalization of many-body systems, and resource-efficient quantum simulations of open quantum systems.
The spin-boson model, describing a two-level system coupled to a bosonic bath, is extensively studied as a paradigmatic dissipative quantum system, exhibiting rich dynamical behavior and even a localization transition in the strong coupling regime. Here, we additionally consider dephasing as a source of Markovian dissipation on top of the non-Markovian dynamics due to an Ohmic bath, and investigate the dynamics of the spin. We show that the characteristic frequency of the spin dynamics, while strongly renormalized by the bosonic bath, changes in a simple fashion (or does not change at all) with dephasing. To obtain these results, we develop an exact nonperturbative method known as the stochastic Schr\"odinger equation, mimicking the Ohmic bath via a stochastic magnetic field combined with the Lindblad quantum master equation due to dephasing, which allows us to numerically compute the dynamics. Furthermore, we derive weak-coupling analytic results utilizing the well-known noninteracting blip approximation. Our findings are relevant to quantum simulation of the spin-boson model in the regime of strong coupling in trapped ions and circuit QED architectures among others.
Time-periodic quantum systems are present all over science and technology. They have been successfully studied using Floquet theory, and a broad diversity of interesting phenomena such as coherent destruction of tunnelling, dynamical localization and creation of new phases of matter have been discovered. Furthermore, periodic driving has also been introduced as a tool for engineering novel quantum systems that have no static analogue. Besides these remarkable achievements for isolated systems, in reality, systems always interact with their surroundings. Taking the effect of the environment into account, time-periodic quantum systems have been mainly described using the Floquet-Markov approach, which consists in deriving a weak coupling master equation in the Floquet basis. In this thesis, we focus on how to overcome these standard approximations in the description of driven open quantum systems. We start by providing a general review on the topics of isolated driven systems, open quantum systems and the Floquet-Markov approach, before presenting the main results of this work. Under the effect of high-frequency driving, we provide an analytic solution to a large class of open quantum systems. Our method relies on discrete symmetries of the system-bath Hamiltonian and grants access to the time evolution operator of the full system, including bath degrees of freedom. The solution can be interpreted as the stroboscopic evolution of a family of observables under the influence of an effective static Hamiltonian. This provides a flexible simulation procedure of nontrivial Hamiltonians. The results are exemplified with the driven spin-boson model, where a coherent control of dissipation is observed. By a novel combination of Floquet theory, full counting statistics and a Markovian embedding strategy (reaction coordinate mapping), more intricate scenarios with additional driven system-environment interactions are addressed. Hereby, we can access the dynamics and thermodynamics of a periodically driven thermal machine. For the particular case where the working medium is a two-level system, we identify four different operating regimes, including a heat engine and a refrigerator. As the coupling strength is increased, their efficiency and coefficient of performance decrease and the refrigerator regime disappears. Notably, our model reproduces the setup of laser cooling of trapped ions in a specific parameter limit. Following the same combination of methods, we investigate electron pumping through a single quantum dot with periodically modulated energy and tunnelling amplitudes. In the high-frequency regime, the direction of the current can be controlled with the driving phase and a rectification effect that is exclusive to the non-Markovian character of the model is observed. In the low-frequency regime, we apply an additional transformation to view the model as three independent transport channels and observe that the coupling strength also controls the current direction. Studying charge fluctuations we validate that our model behaves as a single-electron source.
We present an error-controlled framework for the quantum simulation of non-Markovian open-system dynamics in Gaussian environments. For a system coupled linearly to a Gaussian environment with zero mean, the bath's influence on the reduced dynamics is completely determined by its two-time correlation kernel. This allows us to replace the microscopic environment with an equivalent bath that has the same kernel. On a finite simulation interval, a rank-N approximation of the kernel yields an effective bath of N bosonic modes with time-dependent couplings to the system. To encode these modes in qubits, we first truncate each mode to its lowest d occupation levels. When stronger couplings lead to large occupations, we introduce coupling dilution, which replaces each mode with several more weakly coupled copies while preserving the bath correlation kernel. This distributes excitations across more modes and trades additional modes for smaller local dimensions. We derive explicit reduced-state error bounds and establish polynomial circuit width and depth in simulation time and inverse accuracy under stated spectral-decay and coupling-regularity assumptions, provided elementary unitaries admit polynomial-cost implementations.
For many open quantum systems, a master equation approach employing the Markov approximation cannot reliably describe the dynamical behaviour. This is the case, for example, in a number of solid state or biological systems, and it has motivated a line of research aimed at quantifying the amount of non-Markovian behaviour in a given model. Within this framework, we investigate the dynamics of a quantum harmonic oscillator linearly coupled to a bosonic bath. We focus on Gaussian states, which are suitably treated using a covariance matrix approach. Concentrating on an entanglement based non-Markovian behaviour quantifier (NMBQ) proposed by Rivas et. al. [1], we consider the role that near resonant and off-resonant modes play in affecting the NMBQ. By using a large but finite bath of oscillators for both Ohmic and super Ohmic spectral densities we find, by systematically increasing the coupling strength, initially the near resonant modes provide the most significant non-Markovian effects, while after a certain threshold of coupling strength the off-resonant modes play the dominant role. We also consider the NMBQ for two other models where we add a single strongly coupled oscillator to the model in extra bath mode and 'buffer' configurations, which affects the modes that determine non-Markovian behaviour.
Dissipation caused by nonhomogeneous bosonic finite chain environments (implementable in, e.g., segmented ion traps) is investigated through an exact diagonalization approach, avoiding common approximations used to describe open systems. Different spectral densities, including band gaps, can be engineered to separately assess different factors leading to memory effects, namely, the environment's size, temperature, proximity of the cutoff frequency, the spectral density's shape (sub-Ohmic, Ohmic, super-Ohmic), and the strength of its coupling to the system. Non-Markovianity is quantified with two recently introduced measures related to information backflow and nondivisibility of the system dynamical map. By sweeping the bath spectrum via tuning of the system frequency we show strongest memory effects at band-gap edges and provide an interpretation based on energy flow between system and environment. A system weakly coupled to a stiff chain ensures a Markovian dynamics, while the size of the environment as well as the local density of modes are not substantial factors. We show an opposite effect when increasing the temperature inside or outside the spectral band gap. Further, non-Markovianity arises for larger (negative and positive) powers of algebraic spectral densities, being the Ohmic case not always the most Markovian one.