Quantum speed limit in the thermal spin-boson system with and without tunneling term

Abstract

In this paper, we study the spin-bosonic model, with and without tunneling terms, in detail. The spin-bosonic model without tunneling is studied by using the thermofield dynamics approach. Indeed, by considering temperature, we show that environmental states, while they become entangled with system, approach thermal coherent states with different phases. In addition, by considering the tunneling term, we study the interplay of the environmental cut-off frequency as well as the impacts of environmental temperature on the quantum speed limit in both cases, i.e., spin-boson system with and without tunneling term. In these studies, we indicate temperature play more important role in compare with cut-off frequency to control the quantumness of a spin system.

Appendix: Born Markov master equation for the spin-boson model

We assume that the central spin system moves in a symmetrical double-well potential, which lead to consider $\omega _{0}=0$. Hence, the master equation of spin-boson system is obtained as [9]:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\hat{\rho }_{S}(t)= & {} -i[\hat{H},\hat{\rho }_{S}]-\int _{0}^{\infty } \mathrm{d}\tau \Big \{ \nu (\tau ) [\hat{\sigma }_{z},[\hat{\sigma }_{z}(-\tau ),\hat{\rho }_{S}]]\nonumber \\&-i\eta ( \tau ) [\hat{\sigma }_{z}, \{ \hat{\sigma }_{z}(-\tau ),\hat{\rho }_{S}\}]\Big \}, \end{aligned}$$

(A1)

in which the noise kernel $\nu (\tau )$ and the dissipation one $\eta (\tau )$ are respectively defined by

$$\begin{aligned} \nu (\tau )= & {} \int _{0}^{\infty } \mathrm{\mathrm{d}}\omega J(\omega ) \coth \left[ \frac{\omega }{2k_{B}T}\right] \cos \left( \omega \tau \right) , \end{aligned}$$

(A2)

$$\begin{aligned} \eta (\tau )= & {} \int _{0}^{\infty } \mathrm{d}\omega J(\omega ) \sin \left( \omega \tau \right) . \end{aligned}$$

(A3)

By rewriting the relation (A1), we can obtain

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\hat{\rho }_{S}(t)= & {} -i[\hat{H}_{S}^{\prime },\hat{\rho }_{S}]-\tilde{\gamma }\{\sigma _{x},\rho _{S} \}-\tilde{D}[\hat{\sigma }_{z},[\hat{\sigma }_{z},\hat{\rho }_{S}]]\nonumber \\&-\zeta \hat{\sigma }_{z} \hat{\rho }_{S} \hat{\sigma }_{y}-\zeta ^{*} \hat{\sigma }_{y} \hat{\rho }_{S}\hat{\sigma }_{z}, \end{aligned}$$

(A4)

in which the re-normalized Lamb-shifted spin system Hamiltonian is given by $\hat{H}^{\prime }=\left( -\frac{1}{2}\Delta -\tilde{f}\right) \hat{\sigma }_{x}$; the normal-diffusion coefficient definition $\tilde{D}$, is achieved by

$$\begin{aligned} \tilde{D}= \int _{0}^{\infty } \mathrm{d}\tau \nu (\tau ) \cos \left( \omega \tau \right) , \end{aligned}$$

(A5)

and $\zeta =\tilde{f}+i\tilde{\gamma }$ is defined according to the anomalous-diffusion coefficient $\tilde{f}$ and damping coefficient $\tilde{\gamma }$ which are respectively defined as

$$\begin{aligned} \tilde{f}= & {} \int _{0}^{\infty } \mathrm{d} \tau \nu (\tau ) \sin \left( \omega \tau \right) , \end{aligned}$$

(A6)

$$\begin{aligned} \tilde{\gamma }= & {} \int _{0}^{\infty } \mathrm{d} \tau \eta (\tau ) \sin \left( \omega \tau \right) . \end{aligned}$$

(A7)

Hence, by considering the density matrix as $\rho _{S}=r_{0}(t)\sigma _{0}+ \sum _{i=1}^{3}r_{i}(t) \sigma _{i}$, we can obtain the dynamical equations as

$$\begin{aligned} \dot{r}_{0}= & {} 0, \quad \dot{r}_{x}=-2\tilde{\gamma }-4Dr_{x},\nonumber \\ \dot{r}_{y}= & {} (\Delta _{0}-4 \tilde{f})r_{z}-4D r_{y}, \quad \dot{r}_{z}=-\Delta _{0} r_{y}. \end{aligned}$$

(A8)

Now, by using the initial conditions of the spin system, we obtain:

$$\begin{aligned} \rho _{S}(t)=\left( \begin{array}{cc} \rho _{11} &{} \rho _{12}\\ \rho _{12}^{*} &{} \rho _{22} \end{array}\right) , \end{aligned}$$

(A9)

in which

$$\begin{aligned}&\rho _{11}= \frac{1}{2}+ \frac{1}{4 \tilde{f}-\Delta }\left( 2\varTheta -e^{-2Dt}\left( 2\varTheta \cosh \left[ \varTheta t\right] -2D\cosh \left[ \varTheta t\right] \right) \right) ,\\&\rho _{22}=\frac{1}{2}- \frac{1}{4 \tilde{f}-\Delta }\left( 2\varTheta -e^{-2Dt}\left( 2\varTheta \cosh \left[ \varTheta t\right] -2D\cosh \left[ \varTheta t\right] \right) \right) , \end{aligned}$$

with $\varTheta =\root \of {4D^{2}-\Delta _{0}^{2}+4\tilde{f}\Delta _{0}}$, and $\rho _{12}$ is given by the following relation:

$$\begin{aligned} \rho _{12}=\frac{1}{2}-e^{-2 D t} \left( \left( 1+\frac{\tilde{\gamma }}{D}\right) \sinh \left[ 2 D t\right] +i \sinh \left[ \varTheta t\right] \right) . \end{aligned}$$

Also, by using the spectral density, the normal-diffusion and the damping coefficient are respectively given by $D=\frac{\pi }{2}J(\Delta _{0}) \coth \left[ \frac{\Delta _{0}}{2 k_{B}T}\right]$ and $\tilde{\gamma }=\frac{\pi }{2}J(\Delta _{0})$; moreover, in the regime in which only the second-ordered approximation of $1/ \varLambda$ is valid, i.e., $\varLambda \gg 1$ and therefore the spectral density is given by $J(\omega )\approx 4J_{0} \omega \left( 1-\frac{\omega }{\varLambda }+ \frac{\omega ^{2}}{2\varLambda ^{2}}\right)$, by assuming $\coth [\frac{\Delta _{0}}{2k_{B}T}]\approx \frac{2k_{B}T}{\Delta _{0}}$, which means $k_{B}T\gg 1$, the anomalous-diffusion coefficient $\tilde{f}$ is obtained by $\tilde{f}=4 J_{0} k_{B}T\frac{\varLambda }{\Delta _{0}}$; Hence, the linear fidelity $\mathcal {F}(t)$ is given by

$$\begin{aligned} \mathcal {F}(t)=1-\left( 1+\frac{\tilde{\gamma }}{D}\right) e^{-2Dt}\sinh [2Dt]. \end{aligned}$$

(A10)