Dissipative dynamics of a driven qubit: Interplay between nonadiabatic dynamics and noise effects from the weak to strong coupling regime

We study the exact solution of the Schrödinger equation for the dissipative dynamics of a qubit, achieved by means of short iterative Lanczos method (SIL), which allows us to describe the qubit and the bath dynamics from weak to strong coupling regimes. We focus on two different models of a qubit in contact with the external environment: the first is the spin boson model (SBM), which gives a description of the qubit in terms of static tunneling energy and a bias field. The second model describes an externally driven qubit, where both the bias field and the tunneling rate are controlled by a time-dependent magnetic field, following a finite time protocol. We show that in the SBM case, our solution correctly describes the crossover from coherent to incoherent behavior of the magnetization, occurring at the Toulouse point. Furthermore, we show that the bath response dramatically changes during the system dynamics, going from nonresonant at small times to resonant behavior at long times. When the external driving field is present, for fixed values of the drive duration our results show that the bath can provide beneficial effects to the success of the protocol. We find evidence for a complex interplay between nonadiabaticity of the protocol due to the external drive and dissipation effects, which strongly depends on the detailed form of the qubit-bath interaction.

Figure 1

Magnetization $\langle {\sigma }_z\left(t\right)\rangle$ as a function of the rescaled time ${\mathrm{\Delta }}_{\text{r}}t$, in the case of Ohmic bath ($s=1$), $T=0, h_0=0$, for different values of the coupling strengths $\alpha$ in the range 0.10 to 0.50. The number of bath modes is $M=50$, the cutoff frequency ${\omega }_c=5\mathrm{\Delta }$ and the maximum number of excitations is $N_{\text{ph}}=6$.

Figure 2

Expectation values of $\mathrm{\Delta }{n}_k\left(t\right)$ computed for each bath mode $k$ at different times $\left\{t_{\text{1}},t_{\text{2}},t_{\text{3}},t_{\text{f}}\right\}=\left\{0.03,0.06,1.14,6.00\right\}$ (in units of ${\mathrm{\Delta }}_{\text{r}}^{-1}$), for fixed coupling strength $\alpha =0.40, {\omega }_c=5\mathrm{\Delta }, T=0, h_0=0, M=50$, and $N_{\text{ph}}=6$.

Figure 3

Expectation values of $\langle \mathrm{\Delta }{n}_k\rangle$ computed for each bath mode $k$ at rescaled time $t_{\text{sat}}=5.35$ (in units of ${\mathrm{\Delta }}_{\text{r}}^{-1}$), for different coupling strengths $\alpha$ in the range 0.10 to 0.50, ${\omega }_c=5\mathrm{\Delta }, T=0, h_0=0, M=50$, and $N_{\text{ph}}=6$.

Figure 4

Expectation values of $\langle H_{\text{B}}\left(t\right)\rangle$ (in units of ${\mathrm{\Delta }}_{\text{r}}$) computed as a function of the rescaled time ${\mathrm{\Delta }}_{\text{r}}t$, for different coupling strengths $\alpha$ in the range 0.10 to 0.50, ${\omega }_c=5\mathrm{\Delta }, T=0, h_0=0, M=50$, and $N_{\text{ph}}=6$.

Figure 5

Excess energy plotted as a function of the final time $t_{\text{f}}$ (in units of $h^{-1}$), for different coupling strengths $\alpha$ ranging from 0 to 0.20, in the case $\widehat{\mathit{n}}=\widehat{z}$. The number of modes has been fixed to $M=80$, the cutoff frequency ${\omega }_c=5h, N_{\text{ph}}=3$, and $T=0$. (Inset) Semilogarithmic plot of the same curves as in the main plot.

Figure 6

Plot of $\langle {\sigma }_z\left(t\right)\rangle$ as a function of time $t$ for the protocol in Eq. (9), with fixed final time $t_{\text{f}}=t_{\text{fmax}}=8.42/h$. The qubit couples to an Ohmic bath ($s=1$) along $\widehat{\mathit{n}}=\widehat{z}$, for different coupling strengths ranging from 0 to 0.40. The number of modes has been fixed to $M=70$, the cutoff frequency ${\omega }_c=5h, N_{\text{ph}}=5$, and $T=0$.

Figure 7

Plot of $\langle {\sigma }_y\left(t\right)\rangle$ as a function of time $t$ for the protocol in Eq. (9), with fixed final time $t_{\text{f}}=t_{\text{fmax}}=8.42/h$. The qubit couples to an Ohmic bath ($s=1$) along $\widehat{\mathit{n}}=\widehat{z}$, for different coupling strengths ranging from 0 to 0.40. The number of modes has been fixed to $M=70$, the cutoff frequency ${\omega }_c=5h, N_{\text{ph}}=5$, and $T=0$.

Figure 8

Plot of $\langle {\sigma }_x\left(t\right)\rangle$ as a function of time $t$ for the protocol in Eq. (9), with fixed final time $t_{\text{f}}=t_{\text{fmax}}=8.42/h$. The qubit couples to an Ohmic bath ($s=1$) along $\widehat{\mathit{n}}=\widehat{z}$, for different coupling strengths ranging from 0 to 0.40. The number of modes has been fixed to $M=70$, the cutoff frequency ${\omega }_c=5h, N_{\text{ph}}=5$, and $T=0$.

Figure 9

Fidelity at final time $t_{\text{f}}F\left(t_{\text{f}}\right)$, plotted against the coupling strength $\alpha$ in the range 0.01 to 0.45, for two fixed final times $t_{\text{f}}=\left\{t_{\text{fmax}},t_{\text{fmin}}\right\}=\left\{8.42/h,12.17/h\right\}$ corresponding to the first second order maximum and the second minimum of Eq. (12). The number of modes has been fixed to $M=70$, the cutoff frequency ${\omega }_c=5h, N_{\text{ph}}=5$, and $T=0$.

Figure 10

Excess energy plotted as a function of the final time $t_{\text{f}}$ (in units of $h^{-1}$), for different coupling strengths $\alpha$ ranging from 0 to 0.20, in the case of $\widehat{\mathit{n}}=\widehat{x}$. The number of modes has been fixed to $M=80$, the cutoff frequency ${\omega }_c=5h, N_{\text{ph}}=3$, and $T=0$.

Figure 11

Excess energy plotted as a function of the final time $t_{\text{f}}$ chosen in the range 0.2 to 1.5 (in units of $h^{-1}$), for fixed $\alpha =0.10, M=80, N_{\text{ph}}=3, T=0$, and different cutoff frequencies ${\omega }_c$.

Figure 12

Plot of $\langle {\sigma }_z\left(t\right)\rangle$ as a function of time $t$ for the biased SBM, having fixed ${\omega }_c=5\mathrm{\Delta }, \alpha =0.001, h_0=0.5\mathrm{\Delta }, T=0$, and $N_{\text{ph}}=3$. SIL results (red points) compared with theoretical curve from Eq. (A1) (solid blue curve).

Figure 13

Plot of $\langle {\sigma }_x\left(t\right)\rangle$ as a function of time $t$ for the biased SBM, having fixed ${\omega }_c=5\mathrm{\Delta }, \alpha =0.001, h_0=0.5\mathrm{\Delta }, T=0$, and $N_{\text{ph}}=3$. SIL results (red points) compared with theoretical curve from Eq. (A1) (solid blue curve).

Figure 14

Plot of the quality factor $Q$ against the coupling strength $\alpha$ at $T=0$: numerical estimate derived from a fit of the numerical curves showed in Sec. 3 (red points), compared with the theoretical result in Eq. (B1) known from CFT and NIBA predictions (blue curve).

Figure 15

Magnetization $\langle {\sigma }_z\left(t\right)\rangle$ as a function of the rescaled time, computed at the Toulouse point ($\alpha =1/2$), for $h_0=0$ (unbiased case), at $T=0$; we plot of the numerical SIL result (red curve), compared with the theoretical curve in Eq. (7) (solid blue curve), which is valid in the limit ${\omega }_c\to \infty$. As in the main text, $M=50, {\omega }_c=5\mathrm{\Delta }$, and $N_{\text{ph}}=6$.

Figure 16

Magnetization $\langle {\sigma }_z\left(t\right)\rangle$ as a function of the rescaled time, computed at the Toulouse point ($\alpha =1/2$), with fixed bias $h_0=3\mathrm{\Delta }$, and $T=0$; we plot the numerical SIL result (red points), compared with the theoretical curve in Eq. (7) (solid blue curve). As in the main text, $M=50, {\omega }_c=5\mathrm{\Delta }$, and $N_{\text{ph}}=6$.