Anisotropic Landau-Lifshitz-Gilbert models of dissipation in qubits

We derive a microscopic model for dissipative dynamics in a system of mutually interacting qubits coupled to a thermal bath that generalizes the dissipative model of Landau-Lifshitz-Gilbert to the case of anisotropic bath couplings. We show that the dissipation acts to bias the quantum trajectories towards a reduced phase space. This model applies to a system of superconducting flux qubits whose coupling to the environment is necessarily anisotropic. We study the model in the context of the D-Wave computing device and show that the form of environmental coupling in this case produces dynamics that are closely related to several models proposed on phenomenological grounds.

Figure 1

Dynamics of spins in the presence of anisotropic dissipation. The direction and rate of evolution over the sphere surface are indicated by the streamlines and grayscale density, darker regions indicate faster evolution. (a) For weak anisotropic coupling the state precesses similarly to isotropic coupling. (b) For strong anisotropic coupling dynamics is markedly different. The system rapidly relaxes to a reduced $O\left(2\right)$ manifold where it undergoes constrained dynamics.

Figure 2

Typical oscillatory stochastic dynamics of spins in the presence of anisotropic dissipation with a Drude bath. (a) A trajectory (solid) with parameters $\gamma =400, B=50/3T=50{\omega }_d$, plotted on the Bloch sphere. (b) The same trajectory plotted with the unphysical choice of $T=0$ to illustrate the deterministic part of the dynamics: The system oscillates around the $O\left(2\right)$ manifold with decaying amplitude. The lines $\varphi =0$ and $\theta ={\theta }^{*}$ (dashed) are shown, and an arrowhead indicates the initial state.

Figure 3

Different timescales of relaxation in the stochastic dynamics of (a) $\theta$ and (b) $\varphi$ when coupled to a Drude bath (note the different plot ranges). An ensemble of 1000 spins ($s=1/2$) initially at $\theta =3\pi /4, \varphi =3\pi /4$ when $t=0$ evolve with a magnetic field in the ${\theta }^{*}=\pi /4,{\varphi }^{*}=0$ direction. The coupling is $\gamma =5\times {10}^3$, with energy scales $B=10T=100{\omega }_d$, satisfying $B\gg T\gg {\omega }_d\gg B/\gamma s$. On the timescales of $\theta$ dynamics the fast oscillations in the trajectories [see Fig. 4] means that the trajectories are best characterised by an envelope with upper bound ${\theta }_U$, lower bound ${\theta }_L$ and mid-point ${\theta }_M$, these are simply read off the oscillating trajectory as shown in Fig. 4. Since the initial conditions is an extrema of the fast oscillations the initial point lies on ${\theta }_U$ and ${\varphi }_U$. The ensemble averages $\langle {\theta }_M\rangle$ (solid) and $\langle {\theta }_U\rangle ,\langle {\theta }_L\rangle$ (dashed) are shown with $\langle {\theta }_U\rangle +{\sigma }_U=\langle {\theta }_U\rangle +\sqrt{\langle {\theta }_U^2\rangle -{\langle {\theta }_U\rangle }^2}$, and $\langle {\theta }_L\rangle -{\sigma }_L=\langle {\theta }_L\rangle -\sqrt{\langle {\theta }_L^2\rangle -{\langle {\theta }_L\rangle }^2}$ (both dot–dashed) illustrating the ensemble width. (a) The slow $\theta$ coordinate relaxes towards the equilibrium value ${\theta }^{*}=\pi /4$ on a timescale $\gamma s/B$, approaching it at $t\approx {10}^4/B$. The vertical line indicates the range of plot (b). (b) The same statistics are presented for the $\varphi$ dynamics: The $\varphi$ dynamics relaxes to its equilibrium distribution much faster on a characteristic timescale $\tau \sim 1/{\omega }_d$ and is fully relaxed by $t\approx 500/B$.

Figure 4

Different dynamical timescales of typical trajectories in $\theta$ and $\varphi$ when coupled to a Drude bath. A plot of three sample trajectories from the ensemble studied in Fig. 3. Being drawn from the ensemble these spins ($s=1/2$) evolved in the same conditions: initially prepared at $\theta =3\pi /4, \varphi =3\pi /4$ at $t=0$ and evolve with a magnetic field in the $\theta =\pi /4, \varphi =0$ direction. The coupling is $\gamma =5\times {10}^3$, with energy scales $B=10T=100{\omega }_d$, satisfying $B\gg T\gg {\omega }_d\gg B/\gamma s$. (a) Trajectories in $\theta$ relax on a long timescale. The vertical dashed line indicates the range of plot (b). (b) $\varphi$ relaxes on a shorter timescale, the confinement of $\varphi$ is evidenced by the typically small excursions from $\varphi =0$. (c) Oscillatory behavior induced by the bath occurring on shorter timescales $\tau \sim \sqrt{{\tau }_{\varphi }/{\omega }_d}$ is plotted for $\theta$, similar behavior occurs for $\varphi$. This behavior is expanded on in Appendix pp2, where the oscillations appear in Eq. (B5). Each oscillatory trajectory can be characterized by the upper and lower edges ${\theta }_U$ and ${\theta }_L$ of its envelope, and its midpoint ${\theta }_M$. The ensemble statistics of these quantities are studied in Fig. 3.

Figure 5

Comparison of Metropolis-Hastings and anisotropic LLG dynamics. The ensemble averaged value $\langle \theta \rangle$ of Metropolis Hastings (dashed) dynamics is compared with the anisotropic LLG dynamics (solid) data from Fig. 3. The LLG dynamics have small oscillations (see Fig. 4 for further detail), thus we plot the ensemble averaged midpoint of these small oscillations, $\langle {\theta }_M\rangle$, as previously in Fig. 3. Both simulation were performed using an ensemble of 1000 spins ($s=1/2$) initialised at $\theta =3\pi /4$, evolving in a magnetic field in the ${\theta }^{*}=\pi /4, {\varphi }^{*}=0$ direction, and a temperature $T=B/10$. For the LLG dynamics the coupling is $\gamma =5\times {10}^3$, with energy scales $B=10T=100{\omega }_d$, satisfying $B\gg T\gg {\omega }_d\gg B/\gamma s$.