Universal dissipators for driven open quantum systems and the correction to linear response

TL;DR

The work derives a time-local master equation for driven open quantum systems by perturbing in both drive and system–bath coupling on equal footing. A universal dissipator emerges at second order, independent of the drive, while a memory-enabled third-order term provides a drive-dependent correction to linear response, signaling a non-Lindblad structure for non-Markovian baths. The framework is validated through stochastic and quantum-noise treatments, including qubit dephasing and qubits in a bosonic environment with pseudo-modes, where the third-order term markedly improves agreement with exact dynamics. This extends linear response theory to non-Markovian settings and suggests avenues for dissipative control and reservoir engineering in driven quantum technologies.

Abstract

We investigate in parallel two common pictures used to describe quantum systems interacting with their surrounding environment, i.e., the stochastic Hamiltonian description, where the environment is implicitly included in the fluctuating internal parameters of the system, and the explicit inclusion of the environment via the time-convolutionless projection operator method. Utilizing these two different frameworks, we show that the dissipator characterizing the dynamics of the reduced system, determined up to second order in the noise strength or bath-system coupling, is composed of two parts. One is universal, meaning that it keeps the same form regardless of the drive term. This form constitutes the relevant part of the dissipator only as long as the drive is weak. We thoroughly discuss the assumptions on which this treatment is based and its limitations. Then, by considering the first non-vanishing higher-order term in our expansion, we derive the other, drive-dependent, term completing the full dissipator. This part of the dissipator, originating from the third cumulant, is usually neglected when modeling the decoherent dynamics of controlled qubits. However, this further term constitutes the linear response correction due to memory-mediated environmental effects in driven-dissipative quantum systems. Also, it notably shows that the structure of our quantum master equation goes beyond the Lindblad form. The Lindblad form is recovered for memory-less baths. Finally, we demonstrate this technique to be highly accurate for the problems of dephasing in a driven qubit and for the theory of pseudo-modes for quantum environments.

Figures (5)

• Figure 1: Schematic representation of the driven open quantum system model. For ease of visualization, we depict a qubit (i.e., the system) interacting with a bath while undergoing a coherent operation via the red drive-signal. The time-local master equation corresponding to this model is displayed at the bottom. In black, we show the standard quantum-optical master equation for open systems without any time dependence in the system's Hamiltonian PetruccioneBK. In red, we highlight the additional terms that arise due to the simultaneous presence of a coherent drive acting on the system's DoFs. In particular, the drive term is added to the bare Hamiltonian of the isolated system in the von Neumann part of the equation, as it was anticipated by previous literature, but only by heuristic arguments. On top of this, the standard dissipator, $\mathbb{K}^{\mathrm{II}}(t)$, is modified by the term $\mathbb{K}^{\mathrm{III}}(t)$. We refer to this additional term as the third-order dissipator, second-order in the dissipation + first-order in the drive.
• Figure 2: Difference of the Laplace transforms of $\mathrm{R}_z(s)$ obtained from the exact but non-local in time master equation $\mathrm{R}^{\mathrm{NT}}_z(s)$ and the approximate third-order TL one $\mathrm{R}^{\mathrm{TL3}}_z(s)$. In (a) we plot the difference between Eq. \ref{['laplacezex']} and Eq. \ref{['laplacezII']}, while in (b) we plot the difference between Eq. \ref{['laplacezex']} and Eq. \ref{['laplacezIII']} in the same portion of the complex plane. The interesting behaviour of the resulting quantities is concentrated at the poles, as expected. From the comparison of the two plots is evident how the third order is notably closer to the exact solution. Interestingly, also, a smaller difference appears at $s=0$ than in (b) which is due to the memory effect introduced by the third order. This is the improvement due to the inclusion of the third-order, which captures the essentially non-Markovian part of the response.
• Figure 3: Components of the Bloch vector $\boldsymbol{\mathrm{r}}'(t)=\mathrm {Tr}[\boldsymbol{\sigma}\rho'(t)]$ in the rotating frame for $\varphi=\pi/4\,,\,\tau=0.1\,\Omega^{-1}\,,\,D=10^{-2}\,\Omega\,,\,g=4\times10^{-3}\,\Omega\,$. In (a) we plot the z component, for which $\mathrm{r}_z'=\mathrm{r}_z$ since it is invariant under rotations about $z$. In (b), the x component is plotted. Solid lines represent the exact curves that are solutions of \ref{['eqnovikov']} obtained through the inverse Laplace transform. Dashed lines represent the approximate solutions from the time-local master equation.
• Figure 4: Long-time behaviour of our TL approximation for $\varphi=\pi/4\,,\,\tau=0.1\,\Omega^{-1}\,,\,D=5\times10^{-2}\,\Omega\,,\,g=4\times10^{-3}\,\Omega\,$. Components of the Bloch vector $\boldsymbol{\mathrm{r}}(t)={\rm Tr}[\boldsymbol{\sigma}\rho(t)]$ are plotted. Solid lines represent the exact curves that are solutions of Eq. \ref{['eqnovikov']} obtained through the inverse Laplace transform. Dashed lines represent the approximate solutions from the second-order time-local master equation in (a) and (b) and from the third-order time-local master equation in (c) and (d). In (a) and (c), the $z$ component of the Bloch vector is plotted, while in (b) and (d) the $x$ component is plotted. It is evident that, going from the upper panels to the bottom ones, the accuracy is improved thanks to the inclusion of the third-order term in the master equation.
• Figure 5: Driven qubit in a quantum environment with a single PM (N=1) corresponding to a Lorentzian power spectrum. (a) The short-time dynamics of the Bloch vector component $\mathrm{r_x}$ showing a beating pattern. The second (red line) and third order (yellow line) approximations agree well with the exact pseudo mode (blue line) description. (b) The long-time dynamics of the same Bloch vector component reveal a decaying beating pattern in the exact solution, which is not captured by the TCL2 approximation. (c) The long-time dynamics is approximated very well with the inclusion of the third order correction. (d) The dynamics of the real part of the off-diagonal element of the density matrix in the interaction picture reveal the feature that is not captured by the second-order approximation. The parameters used for these plots are $\eta/\Omega=0.035,\xi/\Omega=0.75,\Gamma/\Omega=0.02,D/\Omega=0.04$.