Single-qubit decoherence under a separable coupling to a random matrix environment

This paper describes the dynamics of a quantum two-level system (qubit) under the influence of an environment modeled by an ensemble of random matrices. In distinction to earlier work, we consider here separable couplings and focus on a regime where the decoherence time is of the same order of magnitude as the environmental Heisenberg time. We derive an analytical expression in the linear response approximation, and study its accuracy by comparison with numerical simulations. We discuss a series of unusual properties, such as purity oscillations, strong signatures of spectral correlations (in the environment Hamiltonian), memory effects, and symmetry-breaking equilibrium states.

Figure 1

The reduced dynamics for the symmetric eigenstate of ${\sigma }_x$ as initial state, for $\Delta =1$ and $\lambda =0.1$ as a function of dimensionless time $t$, where the Heisenberg time is $t=2\pi$. Numerical simulations (red [gray] points) are compared to the linear response theory (green [gray] solid line) and its exponentiated version (blue [gray] dashed line). In panel (a) we show the real part of $\langle {\varrho }_{\mathrm{c}}^{21}\left(t\right)\rangle$ and in panel (b) the purity of $\langle {\varrho }_{\mathrm{c}}\left(t\right)\rangle$.

Figure 2

The reduced dynamics for the symmetric eigenstate of ${\sigma }_y$ as initial state, for $\Delta =1$ and $\lambda =0.1$ as a function of dimensionless time, where the Heisenberg time is at $t=2\pi$. Numerical simulations (red [gray] points) are compared to the linear response theory (green [gray] solid line) and its exponentiated version (blue [gray] dashed line). Panel (a) shows the imaginary part of $\langle {\varrho }_{\mathrm{c}}^{21}\left(t\right)\rangle$ and panel (b) shows the purity of $\langle {\varrho }_{\mathrm{c}}\left(t\right)\rangle$.

Figure 3

The reduced dynamics of the qubit for the symmetric eigenstate of ${\sigma }_z$ as initial state, for $\Delta =1$ and $\lambda =0.1$ as a function of dimensionless time $t$, where the Heisenberg time is at $t=2\pi$. Numerical simulations (red [gray] points) are compared to the linear response theory (green [gray] solid line) and its exponentiation (blue [gray] dashed line). Panel (a) shows the average diagonal element $\langle {\varrho }_{\mathrm{c}}^{11}\left(t\right)\rangle$ and panel (b) shows the purity of $\langle {\varrho }_{\mathrm{c}}\left(t\right)\rangle$.

Figure 4

The purity of the average reduced qubit state, as a function of dimensionless time $t$, for spectra with GUE correlations (red [gray] triangles and blue [gray] squares) and without correlations (dashed red [gray] line and blue [gray] solid line). For the coupling strength and the level splitting, we choose $\mu =0.1$ and $\Delta =0.25$. As initial condition, we consider an eigenstate of ${\sigma }_z$ (red [gray] triangles and dashed red [gray] line) and of ${\sigma }_x$ (blue [gray] squares and solid blue [gray] line).

Figure 5

Trace distance between ${\varrho }_{\mathrm{c}}^{\mathrm{ns}}\left(t\right)$ and ${\varrho }_{\mathrm{c}}^{\mathrm{ELR}}\left(t\right)$ as a function of dimensionless time $t$, for the initial state being the symmetric eigenstate of ${\sigma }_y$ and $\mu =0.25$. The solid line corresponds to $\Delta =1$ and the dotted line to $\Delta =1.5$. The inset shows the modulus squared of the average nondiagonal matrix element of the qubit state. There, the triangles (squares) and the solid line (dotted line) correspond to the numerical simulation and the ELR, respectively.

Figure 6

Maximum trace distance between ELR approximation and numerical simulations as a function of the coupling strength $\mu$ and the level splitting $\Delta$, for the initial state being an eigenstate of ${\sigma }_y$.

Figure 7

Decoherence process for $\Delta =0.25$ and $\mu =0.1$ for the initial state being an eigenstate of ${\sigma }_x$ (red [gray] solid curves) and of ${\sigma }_y$ (blue [gray] dashed curves). (a) The trajectory of the average qubit state $\langle {\varrho }_{\mathrm{c}}\left(t\right)\rangle$ remains in the $xy$ plane of the Bloch sphere. (b) Purity as a function of dimensionless time $t$ for the same cases.