Dynamics of a quantum two-state system in a linearly driven quantum bath

When an open quantum system is driven by an external time-dependent force, the coupling of the driving to the central system is usually included, whereas the impact of the driving field on the bath is neglected. We investigate the effect of a quantum bath of linearly driven harmonic oscillators on the relaxation dynamics of a quantum two-level system which is not directly driven. In particular, we calculate the frequency-dependent response of the system when the bath is subject to Dirac and Gaussian driving pulses. We show that a time-retarded effective force on the system is induced by the driven bath which depends on the full history of the perturbation and the spectral characteristics of the underlying bath. In particular, when a structured Ohmic bath with a pronounced Lorentzian peak is considered, the dynamical response of the system to a driven bath is qualitatively different than that of the undriven bath. Specifically, additional resonances appear which can be directly associated with a Jaynes-Cummings-like effective energy spectrum.

Figure 1

General setup of a pulse-shaped bath drive. The bath (B) is in equilibrium until time $t_a$, when the bath driving force $F\left(t\right)$ (orange, shown as a generic Gaussian pulse) is activated. Subsequently, the bath is driven (orange stripes), with the perturbation centered at some time $t_g$. At time $t_0$ the system (S) is coupled to the driven bath. We consider a two-level system with the ground state $g$ and excited state $e$ coupled to the harmonic bath. Bath driving leads to an additional effective force $F_{\mathrm{eff}}\left(t\right)$, as shown in this work.

Figure 2

Normalized effective force $F_{\text{eff}}\left(t\right)$ (blue solid line) and direct driving force $\overline{\eta }F\left(t\right)$ (red line with squares) for an Ohmic bath driven by (a) a Dirac $\delta$ pulse and (b) a Gaussian pulse. The bath is characterized by the parameters $\eta =0.05\mathrm{\Delta }$ and ${\omega }_c=5\mathrm{\Delta }$. The Dirac pulse in (a) occurs at $t_a=5{\mathrm{\Delta }}^{-1}$ with interaction strength $\overline{\eta }=2\mathrm{\Delta }$. The Gaussian pulse in (b) is centered at $t_g=5{\mathrm{\Delta }}^{-1}$ and starts at $t_a=0$ with interaction strength $\overline{\eta }=6\mathrm{\Delta }$ and width $\sigma ={\mathrm{\Delta }}^{-1}$. Both quantities are normalized with respect to the maximum of the effective force to allow for a comparison of relative strengths. Notice that the height of the Dirac pulse has also been chosen to correspond to its effective strength as well.

Figure 3

Time-dependent relaxation rate (blue solid line) and momentary energy (red line with squares) for the Ohmic bath driven by (a) a Dirac pulse and (b) a Gaussian pulse at zero temperature. The parameters are the same as in Fig. 2. The normalization has been chosen with respect to the undriven (equilibrium) relaxation rate ${\gamma }_1^{\mathrm{eq}}=J\left(\mathrm{\Delta }\right)/2$ and the bare, undriven system energy scale $\mathrm{\Delta }$.

Figure 4

Dynamics of the expectation values $r_i\left(t\right)=-{\langle {\tau }_i\rangle }_t$ (green solid line with open circles, dash-dotted blue line, and dotted red line) of the two-level system in an Ohmic bath at zero temperature, driven by (a) a Dirac pulse and (b) a Gaussian pulse. The (dimensionless) external driving force $F\left(t\right)$ is also shown (red solid line with squares) as a comparison. The dynamics are generated by the forces and rates shown in Figs. 2 and 3, with parameters being the same as in Fig. 2. The system is set to be initially in equilibrium, i.e., in the ground state.

Figure 5

Normalized effective force (blue solid line) and direct driving force $\overline{\kappa }F\left(t\right)$ (red line with squares) for a driven Lorentzian bath with (a) a Dirac driving pulse and (b) a Gaussian driving pulse. The bath is characterized by the parameters $\kappa =0.05\mathrm{\Delta },\mathrm{\Omega }=1.5\mathrm{\Delta }$, and $\mathrm{\Gamma }=0.1\mathrm{\Delta }$. The Dirac pulse in (a) occurs at $t_a=5{\mathrm{\Delta }}^{-1}$ with interaction strength $\overline{\kappa }=2\mathrm{\Delta }$. The Gaussian pulse in (b) is centered at $t_g=5{\mathrm{\Delta }}^{-1}$ and starts at $t_a=0$ with interaction strength $\overline{\kappa }=6\mathrm{\Delta }$ and width $\sigma ={\mathrm{\Delta }}^{-1}$. Both quantities are normalized with respect to the maximum of the effective force to allow for a comparison of relative strengths. Notice that the height of the Dirac pulse has also been chosen to correspond to its effective strength.

Figure 6

Time-dependent relaxation rates (blue solid line) and momentary energy (red line with squares) for a Lorentzian bath driven by (a) a Dirac pulse or (b) a Gaussian pulse at zero temperature. The parameters are the same as in Fig. 5. The normalization has been chosen with respect to the undriven (equilibrium) relaxation rate ${\gamma }_1^{\mathrm{eq}}=J\left(\mathrm{\Delta }\right)/2$ and the bare, undriven system energy scale $\mathrm{\Delta }$.

Figure 7

Dynamics of the expectation values $r_i\left(t\right)=-{\langle {\tau }_i\rangle }_t$ (green solid line with open circles, dash-dotted blue line, and dotted red line) of the two-level system in a Lorentzian bath at zero temperature, driven by (a) a Dirac pulse and (b) a Gaussian pulse. The (dimensionless) external driving force $F\left(t\right)$ is also shown (red solid line with squares) as a comparison. The dynamics are generated by the forces and rates shown in Figs. 5 and 6, with parameters being the same as in Fig. 5. The system is set to be initially in equilibrium, i.e., in the ground state.

Figure 8

Frequency-dependent response (red solid line) of a quantum two-level system to (a) and (b) an Ohmic bath and (c) and (d) a Lorentzian bath at zero temperature driven by (a) and (c) a Dirac pulse or (b) and (d) a Gaussian pulse. For comparison, the response to an undriven bath is also shown (blue dashed line). The parameters used are given below Fig. 2 for the Ohmic bath and below Fig. 5 for the Lorentzian bath. Both quantities have been normalized with respect to the maximum of the driven frequency response to allow for a comparison of relative strengths.

Figure 9

Frequency-dependent response (red solid line) to a Lorentzian bath at zero temperature driven by (a) a Dirac pulse or (b) a Gaussian pulse away from the fundamental frequency $\mathrm{\Delta }$. For comparison, the response to an undriven bath is also shown (blue line with circles). The parameters used are given below Fig. 5. Both quantities have been normalized with respect to the maximum of the driven frequency response to allow for a comparison of relative strengths. The emerging peaks correspond well to energy gaps (blue dotted lines) obtained numerically from the Hamiltonian in Eq. (43) with $g=\sqrt{\kappa \mathrm{\Omega }/8}\approx 0.1\mathrm{\Delta } \left(\mathrm{\Omega }=1.5\mathrm{\Delta }\right)$. (c) The level diagram for $g=0$ shows the corresponding transitions (blue arrows).