Markovian quantum master equation beyond adiabatic regime

By introducing a temporal change time scale ${\tau }_{\text{A}}\left(t\right)$ for the time-dependent system Hamiltonian, a general formulation of the Markovian quantum master equation is given to go well beyond the adiabatic regime. In appropriate situations, the framework is well justified even if ${\tau }_{\text{A}}\left(t\right)$ is faster than the decay time scale of the bath correlation function. An application to the dissipative Landau-Zener model demonstrates this general result. The findings are applicable to a wide range of fields, providing a basis for quantum control beyond the adiabatic regime.

Figure 1

Schematic illustration of an open quantum system under the external control. The system, depicted by a (collective) spin, is in a bath and controlled by external field. The external field is in general time-dependent in order to make use of the quantum system as a physical resource, e.g., for the AQC and the QA.

Figure 2

The dissipative Landau-Zener model. The system is initially assumed to be $|\uparrow \rangle$ at $t=-\infty$. Then, the transition probability $P_{\uparrow \to \downarrow }$ ($P_{\uparrow \to \uparrow }$) to find the system in the ground (excited) state at $t=\infty$ is discussed. The difference between the two eigenenergies is given by $E\left(t\right)=\sqrt{{\mathrm{\Delta }}_0^2+{\left(vt\right)}^2}$. ${\tau }_{\text{LZ}}\equiv {\mathrm{\Delta }}_0/v$ denotes the characteristic time for the eigenstates to pass through the minimum gap ${\mathrm{\Delta }}_0$ around $t=0$. When the bath is at zero temperature ($T=0$), there is no thermal excitation and relaxation but only the spontaneous emission can occur due to the vacuum fluctuation.

Figure 3

$P_{\uparrow \to \downarrow }$ (blue) and $P_{\uparrow \to \uparrow }$ (red) at $T=0$, as a function of the LZ sweep velocity $v$. Symbols: the results of the Markovian QME. We remark that the imaginary part of $\mathrm{\Gamma }\left(\omega \right)$ (the Lamb shift) is not neglected in the calculation. Solid lines: the exact probabilities of $P_{\uparrow \to \downarrow }=1-P_{\uparrow \to \uparrow }=1-exp(-\pi W^2/2v)$ with $W^2={\{{\mathrm{\Delta }}_0-{\int }_0^{\infty }d\omega sin\theta cos\theta \frac{J\left(\omega \right)}{\omega }\}}^2+{\int }_0^{\infty }d\omega {sin}^2\theta J\left(\omega \right)$ [57, 58]. Note that $\theta =0$ (diagonal coupling) gives $W^2={\mathrm{\Delta }}_0^2$, which results in $P_{\uparrow \to \downarrow }=1-exp(-\pi {\mathrm{\Delta }}_0^2/2v)$. This is exactly the same as the LZ transition probability without dissipation, as pointed out in Ref. [57]. Parameters: $\eta =0.01$ and ${\omega }_{\text{c}}=30{\mathrm{\Delta }}_0$. The left arrows indicate the regimes by taking ${\tau }_{\text{X}}\left(t\right)\ll {\tau }_{\text{Y}}\left(t\right)$ as ${\tau }_{\text{X}}\left(t\right)\lesssim {\tau }_{\text{Y}}\left(t\right)/10$ for all $t$ ($\text{X}\in \{\text{E},\text{B}\}$, $\text{Y}\in \{\text{A}\text{S},\text{A}\}$). One is the ordinary adiabatic-evolution regime and the other is the adiabatic regime with respect to ${\tau }_{\text{B}}$. The green open squares correspond to the time evolutions in Fig. 4.

Figure 4

Time evolutions at $v=50{\mathrm{\Delta }}_0^2$ for $\theta =\pi /2$. (a) The probabilities of $P_{\text{G}}\left(t\right)$ and $P_{\text{E}}\left(t\right)$. The dotted lines are the corresponding probabilities without dissipation. The dashed lines show $P_{\uparrow \to \downarrow }$ and $P_{\uparrow \to \uparrow }$ for comparison. Panel (b) shows a zoom around $t=0$. (c) The time scales of ${\tau }_{\text{A}\text{S}}\left(t\right)$, ${\tau }_{\text{E}}\left(t\right)$, ${\tau }_{\text{A}}\left(t\right)$, ${\tau }_{\text{B}}$, and ${\tau }_{\text{R}}\left(t\right)$ in units of ${\mathrm{\Delta }}_0^{-1}$, according to Eqs. (56, 57, 58), Eq. (62), and Eq. (44). Panel (d) again shows a zoom around $t=0$. ${\tau }_{\text{A}\text{S}}\left(t\right)$ reaches its minimum value $2{\tau }_{\text{LZ}}$ at $t=0$; ${\tau }_{\text{A}\text{S}}\left(0\right)=2{\tau }_{\text{LZ}}$. The minimum value of ${\tau }_{\text{A}}\left(t\right)$ is also $2{\tau }_{\text{LZ}}$ but achieved not only at $t=0$ but also $t=\pm {\tau }_{\text{LZ}}$. We remark that panels (c) and (d) should be discussed with reference to Table 2. The parameters are the same as indicated by the green open squares in Fig. 3. The gray shaded area corresponds to $\left|t\right|\le {\tau }_{\text{LZ}}$.