Continuous Variable Structured Collision Models

TL;DR

The work develops continuous-variable collision models with structured environmental units, modeling each unit as $N_E$ oscillators in a ring with spring-like couplings. It analyzes two interaction paradigms: beamsplitter-like couplings that admit a continuous-time Lyapunov description and spring-like couplings that generally require discrete-time propagation, with a special continuous-time limit under a zero-sum coupling condition. A key finding is that thermal environments with internal structure can induce effective squeezing and a modified bath temperature $T_{ ext{hat E}}$, enabling dynamics analogous to squeezed baths without external squeezing resources. The authors verify the first and second laws of thermodynamics for both examples, and show that a secondary environment can drive the system to a steady state in the spring-like case, highlighting the practical relevance for quantum thermodynamics and potential experimental realizations in trapped ions or photonics.

Abstract

Quantum collision models allow for the dynamics of open quantum systems to be described by breaking the environment into small segments, typically consisting of non-interacting harmonic oscillators or two-level systems. This work introduces structure within these environmental units via spring-like interactions between N coupled oscillators in a ring structure, initially prepared in a thermal state. Two models of interest are examined. The first highlights a case in which a continuous time evolution can be obtained, wherein the system interacts with the environment via a beam-splitter-like, energy-preserving, interaction. The resulting dynamics are analogous to those due to interactions with unstructured units prepared as squeezed thermal states. The second model highlights a case in which the continuous time limit for the evolution cannot be taken generally, requiring instead discrete-time propagation. Special cases in which the continuous time limit can be taken are also investigated, alongside the addition of a secondary environment to induce a steady state. The first and second laws of thermodynamics are verified for both examples.

Figures (7)

• Figure 1: A system composed of a harmonic oscillator interacting with an environment composed of structured units, each containing $N_E$ interacting oscillators arranged in a ring, via a collision model. Each environmental unit interacts with the system for a timestep $\delta t$ before being replaced by an identical unit.
• Figure 2: Effective temperature of the environmental units $T_{\hat{E}}$ interacting with the system as a function of the internal coupling strength $\lambda_I$ within the environment. The blue (solid), red (dashed) and green(dotted) lines represent the effective temperature of an environmental unit composed of $3$, $4$ and $5$ oscillators respectively, where the system interacts solely with the normal mode frequency of the environmental units $\tilde{\omega}_{E_{\text{Max}}}$ corresponding to the largest wavevector within the Brillouin zone. Parameters: $\omega_E = 1$, $T_E = 1$, $\tilde{\gamma}_{E_{m'}} = 0.5$.
• Figure 3: The steady-state energy $H_{\rm{Steady}}$ (left) and work $\dot W$ (right) applied to a system interacting with the normal mode frequency of the environmental units $\tilde{\omega}_{E_{\text{Max}}}$ corresponding to the largest wavevector within the Brillouin zone. Each unit is composed of $4$ oscillators prepared in a thermal state at a temperature $T_E$ as a function of the internal coupling strength $\lambda_I$ (solid line). The horizontal dashed line represents the energy of a thermal state $E_{\rm eq}$. The vertical lines act as a guide to the eye, with the dashed line at $\lambda_I=0$ and the dotted line at $\lambda_{I} = \lambda_{I_C}$. The steady state internal energy is independent of internal coupling, such that $\dot{U}=0$ for all $\lambda_I$, while the heat flow can be found through $\dot{Q} = - \dot{W}$. Parameters: $\omega_S = \omega_E = 1$, $T_E = 1$, $\tilde{\gamma}_{E_{m'}} = 0.5$.
• Figure 4: Dynamics of the energy of a single oscillator system prepared in the thermal state as a function of time, with discrete time-steps $\delta t = 0.01$, interacting with the center of mass frequency of environmental units composed of $N_E=4$ oscillators. Parameters: $\omega_S=\omega_E = 1$, $T_S=T_E=1$, $\lambda_I=0.67$, $\tilde{\lambda}_{{E_{1}}} = 15$.
• Figure 5: The heat current $\Delta Q$ (left) and work $\Delta W$ (right) applied to the system the system with same parameters as Fig. \ref{['fig:Discrete Spring Dynamics']}. The change in the internal energy of the system $\Delta U$ can be found as the sum $\Delta Q$ and $\Delta W$.