Fluctuating parametric drive of coupled classical oscillators can simulate dissipative qubits

TL;DR

The paper tackles the problem of simulating dissipative quantum two-level system (TLS) dynamics using a purely classical setup of two coupled oscillators. By introducing stochastic fluctuations in the parametric drive and employing a cumulant/Redfield treatment, it derives relaxation rates $T_1$ and $T_2$ and a pure dephasing time $T_φ$, showing that $T_2^{-1} = (2T_1)^{-1} + T_φ^{-1}$ and that dephasing vanishes at zero bias ($ε_0=0$). The authors also report an infinite-temperature stationary state arising from the stochastic driving and outline experimental tests with levitated nanoparticles and nanostring resonators to realize the predicted decoherence dynamics. While this classical approach provides a tractable platform to study quantum dissipation, it cannot capture spontaneous emission and scales exponentially with the number of simulated qubits, limiting its applicability to larger quantum systems.

Abstract

We investigate a system composed of two coupled oscillators subject to stochastic fluctuations in its internal parameters. In particular, we answer the question whether the well-known classical analogy of the quantum dynamics of two-level systems (TLS), i.e. qubits, provided by two coupled oscillators can be extended to simulate the dynamics of dissipative quantum systems. In the context of nanomechanics, the analogy in the dissipation free case has already been tested in multiple experimental setups, e.g., doubly clamped or cantilever string resonators and optically levitated particles. A well-known result of this classical analogy is that the relaxation and decoherence times of the analog quantum system must be equal, i.e. $T_1=T_2$, in contrast to the general case of quantum TLS. We show that this fundamentally quantum feature, i.e. $T_1\neq T_2$, can be implemented as well in the aforementioned classical systems by adding stochastic fluctuations in their internal parameters. Moreover, we show that these stochastic contributions can be engineered in the control apparatus of those systems, discussing, in particular, the application of this theory to levitated nanoparticles and to nanostring resonators.

Figures (3)

• Figure 1: Schematics of the classical-mechanical model under consideration. Two masses $m$ are connected through a spring with the time-independent spring constant $h$, and each of them is connected to the neighbouring wall by springs with time-dependent spring constants $k_1(t)$ and $k_2(t)$, with $k_2(t)-k_1(t)=\Delta k(t)\,$. We also include friction by means of a damping coefficient $\gamma$ equal for both masses. Moreover, we include an external driving force $f(t)$. This external driving has the purpose of initially feeding energy into the system that is then driven parametrically Rugar91DykmanRMP2022, via the modulation of the spring constants.
• Figure 2: Decay of the polarization $\mathrm{r}'_z={\rm Tr}(\sigma_z\rho)$ of the Bloch vector (orange), and of the coherences $\rm{Re}(\mathrm{r}'_+)={\rm Tr}(\sigma_+\rho)$ (purple), for $\varepsilon_0=1.5\Delta, G=0.5\Delta$. Solid lines are the plot of the numerical solution to the Redfield equation \ref{['EQRedfield']}. Dashed colored lines picture the exponential decay with times given by Eqs. \ref{['EQTime1']},\ref{['EQTime2']} ($\mathrm{r}'_i(0)=\mathrm{r}_i(0)$). Dotted gray and black lines represent the corresponding plot neglecting the oscillating terms on the right hand side of Eq. \ref{['EQBlochInt']} (secular approximation). (a) Correlated noise, $G\tau_c=0.25$, and (b) white noise limit, $G\tau_c=0$.
• Figure 3: Trajectory of the BV on the unit Bloch sphere for different initial conditions, i.e., $\boldsymbol{\mathrm{r}}(0)=\bigl[1/\sqrt{3}\,,\,1/\sqrt{3}\,,\,1/\sqrt{3}\bigr]^T$ (purple), $\boldsymbol{ \mathrm{r}}(0)=\bigl[1/\sqrt{2}\,,\,1/\sqrt{2}\,,\,0\bigr]^T$ (yellow), and $\boldsymbol{ \mathrm{r}}(0)=\bigl[0\,,\,0\,,\,-1\bigr]^T$ (blue). Both are plotted in the case of resonant driving $\omega=\Delta$ and for (a) $G=0.3\Delta$, $D=0.5\Delta$ and $\Delta\tau_c=0.5$, and (b) $G=0.3\Delta$ and $D=\Delta$, $\tau_c=0$. All BV trajectories decay to the infinite temperature state at the center of the Bloch sphere.