Accessing the bath information in open quantum systems with the stochastic $c$-number Langevin equation method

In traditional open quantum systems, the baths are usually traced out so that only the system information is left in the equations of motion. However, recent studies reveal that using only the system degrees of freedom can be insufficient. In this work, we develop a stochastic $c$-number Langevin equation method which can conveniently access the bath information. In our approach, the studied quantities are the expectation values of operators which can contain both system operators and bath operators. The dynamics of the operators of interest is formally divided into separate system and bath parts, with auxiliary stochastic fields. After solving the independent stochastic dynamics of the system part and the bath part, we can recombine them by taking the average over these stochastic fields to obtain the desired quantities. Several applications of the theory are highlighted, including the pure dephasing model, the spin-boson model, and an optically excited quantum dot coupled to a bath of phonons.

Figure 1

Results for the pure dephasing model. Here, the cutoff frequency is ${\omega }_c=0.5$ and the coupling strength $\mathrm{\Gamma }=1$. In (a) and (b), the stochastic results are calculated taking an average over $45\times {10}^6$ trajectories, and the results of the stochastic method and the analytical expression are denoted by the subscript $s$ and $a$, respectively. The accumulated errors are shown in (c) and (d).

Figure 2

Expectation values of different operators. The cutoff frequency is ${\omega }_c=0.5$ and the coupling strength $\mathrm{\Gamma }=1$. The stochastic results are calculated taking an average over $5\times {10}^6$ trajectories. (a) Comparison between the stochastic and analytical results for the system operators. (b) Comparison between the stochastic and analytical results for coupling energy.

Figure 3

Expectation values of different operators with classical control. The cutoff frequency is ${\omega }_c=0.5$, the coupling strength is $\mathrm{\Gamma }=1$, and the inverse of the temperature is $\beta =1$. The stochastic results are calculated with the average over $40\times {10}^6$ trajectories. The control field is ideal $\pi$ pulses in the $y$ direction.

Figure 4

Expectation values of different operators. The cut-off frequency is ${\omega }_c=0.5$, and the coupling strength is $\mathrm{\Gamma }=1$. The stochastic results are calculated with the average over $40\times {10}^6$ trajectories for (a) and (b). (a) The results at the low temperature $\beta =1000$. (b) The results at hight temperature $\beta =1$. The pumped cases, shown in (c) and (d), are calculated for different detunings $\delta$ at high temperature $\beta =1$ with the pumping intensity $\mathrm{\Omega }=0.5$ (in the units of ${\omega }_0$), and $10\times {10}^6$ trajectories. The expectation values of ${\sigma }_z$ are shown in (c), and the coupling energies are shown in (d).

Figure 5

Quantum dot systems with different pump intensity. The displacement of the bath is calculated using $\langle x\left(t\right)\rangle ={\sum }_k\langle [g_k^{*}{a}_k^{†}\left(t\right)+g_k{a}_k\left(t\right)]\rangle$. The cutoff frequency is ${\omega }_c=2.2{\mathrm{ps}}^{-1}$ and the coupling strength is $\alpha =0.027{\mathrm{ps}}^{-2}$. The temperatures are (a)–(e) $T=50\mathrm{K}$ and (f) $4.2\mathrm{K}$. The detunings for (a)–(d) are $\delta =0$, and for (e) and (f), the detunings are $\delta =-1.26{\mathrm{ps}}^{-1}$. The stochastic results are calculated with the average over $1\times {10}^6$ trajectories. The pump intensities are (a) $\mathrm{\Omega }=\pi /6{\mathrm{ps}}^{-1}$, (b) $\mathrm{\Omega }=\pi /2{\mathrm{ps}}^{-1}$, (c) $\mathrm{\Omega }=\pi {\mathrm{ps}}^{-1}$, and (d) $\mathrm{\Omega }=4\pi {\mathrm{ps}}^{-1}$. For (e) and (f), the pump intensities are time dependent, $\mathrm{\Omega }\left(t\right)=1.28exp[-{(t/\tau )}^2]{\mathrm{ps}}^{-1}$ with $\tau =20.2\mathrm{ps}$, corresponding to the pulse area $\mathrm{\Theta }=14.6\pi$.