Speeding up and slowing down the relaxation of a qubit by optimal control

We consider a two-level quantum system prepared in an arbitrary initial state and relaxing to a steady state due to the action of a Markovian dissipative channel. We study how optimal control can be used for speeding up or slowing down the relaxation towards the fixed point of the dynamics. We analytically derive the optimal relaxation times for different quantum channels in the ideal ansatz of unconstrained quantum control (a magnetic field of infinite strength). We also analyze the situation in which the control Hamiltonian is bounded by a finite threshold. As by-products of our analysis, we find that (i) if the qubit is initially in a thermal state hotter than the environmental bath, quantum control can not speed up its natural cooling rate; (ii) if the qubit is initially in a thermal state colder than the bath, it can reach the fixed point of the dynamics in finite time if a strong control field is applied; (iii) in the presence of unconstrained quantum control, it is possible to keep the evolved state indefinitely and arbitrarily close to special initial states which are far away from the fixed points of the dynamics.

Figure 1

Density plot of $T_{\mathrm{free}}^{\mathrm{AD}}({\mathbf{r}}_i,\epsilon )$ of Eq. (13) as a function of the initial state ${\mathbf{r}}_i=(r_{ix},r_{iy},r_{iz})$. As the system is invariant under rotations around the $z$ axis, we set $r_y=0$ without loss of generality. Here, $\epsilon =0.04$ and the noise parameters have been set equal to $\beta =2$ and $\gamma =e^{\beta }-1\approx 6.39$. The fixed point is indicated with a green star.

Figure 2

Schematic diagram showing the optimal paths in the case of (a) cooling (path A) and (b) heating (path B) on the $x\text{−}z$ plane of the Bloch sphere. We start from an initial state ${\rho }_i$ with radius $r_i$. The fixed point is given by ${\rho }_f$ with radius $r_f$ (green star). The solid vertical line is the $z$ axis.

Figure 3

Density plots of the minimal time $T_{\mathrm{fast}}^{\mathrm{AD}}({\mathbf{r}}_i;\epsilon )$ of Eq. (18) for a generalized amplitude-damping channel as a function of the initial state ${\mathbf{r}}_i=(r_{ix},r_{iy},r_{iz})$. The parameters $\beta$ and $\epsilon$ are as in Fig. 1 in (a), while $\beta =0.7$ and $\epsilon =0.04$ in (b). The fixed point is indicated by a green star. The inset shows a section of the density plot along the $x$ axis.

Figure 4

Plot of the optimal-time evolution $T_m$ needed to bring the initial state ${\rho }_i$ with ${\mathbf{r}}_i=(0.38,-0.22,-0.46)$ towards the fixed point of a generalized amplitude-damping channel with $\beta =2$ and $\gamma =e^{\beta }-1$. Data obtained via numerical optimization of the control parameters $h_{\mu ,n}$ of Eq. (25) setting $\tau =N_c=10$ and $\epsilon =0.04$. In the limit of $m\to \infty$ and of $N_c\to \infty$, we expect $T_m$ to saturate to the corresponding value of the function $T_{\mathrm{fast}}^{\mathrm{AD},\mathrm{cool}}({\mathbf{r}}_i,\epsilon )$ given in Eq. (19) (dashed line).

Figure 5

Comparison between the numerical (solid line) and the analytical bound (32) (dashed line) values of the slope $A$ as a function of $\beta$ for $N_c=\tau =10$ and $\epsilon =0.04$. The initial point ${\mathbf{r}}_i$ is the same as in Fig. 4. $m/{\dot{\theta }}_0$ decreases for larger values of $\beta$, thus resulting in a better match between the numerical and analytical values in this regime. Inset: variation of $T_m$ as a function of $m$ for small $m$ for $\tau =N_c=10$, $\beta =2$, and $\epsilon =0.04$. As expected, $T_m$ decreases linearly with $m$.

Figure 6

Plot of the speed $v_{\mathrm{AD}}$ on the $x\text{−}z$ plane of the Bloch sphere when the fixed point is the thermal state corresponding to $\beta =2$. The curve $v_{\mathrm{AD}}=0$ (dashed line) is an ellipse which passes through the origin of coordinates (black dot) and the fixed point (green star).