Decoherence induced by a composite environment

It is not always justified to model the environment of a realistic quantum system as a collection of harmonic oscillators or two-level systems. To explore interesting physics associated with a composite environment, here we study the dynamics of a qubit coupled to a second two-level system, which is in turn coupled to a harmonic-oscillator bath. That is, the qubit of interest is in the presence of a composite environment consisting of the second two-level system and a conventional harmonic-oscillator bath. We investigate the issue of pointer states of decoherence for the qubit for different coupling strengths between the qubit and such a composite environment. It is shown that for weak-, intermediate-, or strong-coupling strengths, the qubit's reduced density matrix has a coupling-strength-dependent diagonal representation, thus yielding coupling-strength-dependent computational pointer states. Interestingly, the associated decoherence rate can decrease as the coupling strength increases.

Figure 1

(a) Behavior of ${\theta }_1$ (solid black line) and ${\theta }_2$ (dotted blue line) as a function of the coupling strength $\lambda$ for the coupling $H_{SQ}=\frac{\lambda }{2}{\sigma }_z^{\left(S\right)}{\sigma }_z^{\left(Q\right)}$. We work in dimensionless units throughout, with $\hslash =1$ and $k_B=1$, and we use an Ohmic spectral density for thermal bath $J\left(\omega \right)=G\omega e^{-\omega /{\omega }_c}$. We set ${\Delta }_S=1$, ${\varepsilon }_S=2$, ${\Delta }_Q=4$, ${\varepsilon }_Q=3$, $G=0.02$, $T=2$, and ${\omega }_c=50$. The initial system-environment state is chosen to be ${\rho }_{\text{tot}}=|0_S\rangle \langle 0_S|\otimes |0_Q\rangle \langle 0_Q|\otimes e^{-\beta H_{TB}}/Z_{TB}$, where ${\sigma }_z|0\rangle =|0\rangle$. From now on, we vary only $\lambda$; for all other figures, the parameters used are the same as those used here. (b) Same as (a), except that we now consider the coupling $H_{SQ}=\frac{\lambda }{2}{\sigma }_x^{\left(S\right)}{\sigma }_x^{\left(Q\right)}$.

Figure 2

(a) Behavior of the diagonal element ${\rho }_{11}$ of the RDM in the pointer basis (dotted magenta line) and the off-diagonal element $|{\rho }_{12}|$ (dotted blue line) for $\lambda =2$ with $H_{SQ}=\frac{\lambda }{2}{\sigma }_z^{\left(S\right)}{\sigma }_z^{\left(Q\right)}$. The evolution of ${\rho }_{11}$ and $|{\rho }_{12}|$ has been fitted with exponential functions of the form $a{e}^{-t/{\tau }_R}+b$ (solid black line) and $c{e}^{-t/{\tau }_{\text{dec}}}$ (solid red line), respectively. In this way, we obtain an estimate for ${\tau }_R$ and ${\tau }_{\text{dec}}$. (b) Same as (a), except that we now have $\lambda =8$. (c) Fluctuations of the off-diagonal elements $|{\rho }_{12}|$ in the CPS basis (solid black line) and the energy eigenbasis (dotted blue line) for $\lambda =2$.

Figure 3

(a) Behavior of ${\tau }_R$ (dotted blue line) and ${\tau }_{\text{dec}}$ (solid black line) as a function of $\lambda$ for $\frac{\lambda }{2}{\sigma }_z^{\left(S\right)}{\sigma }_z^{\left(Q\right)}$ coupling. The inset shows clearly the increase of ${\tau }_R$ and ${\tau }_{\text{dec}}$ as $\lambda$ is increased. The relation ${\tau }_{\text{dec}}\le 2{\tau }_R$ [29] is always satisfied. (b) Same as (a), except that we now have $H_{SQ}=\frac{\lambda }{2}{\sigma }_x^{\left(S\right)}{\sigma }_x^{\left(Q\right)}$.