Pancharatnam Phase and Quantum Correlation for Two-qubit System in Correlated Dephasing Environment

Abstract

The time-dependent Pancharatnam phase and the quantum correlations are studied for the two-qubit system in correlated dephasing environment. We find that in both X-state and Y-state, the rapid changes of the time-dependent Pancharatnam phase are intrinsic in the died and alive phenomena of quantum entanglement described by the concurrence under the exactly same parameters, while the slow changes of the time-dependent Pancharatnam phase correspond to the sudden death of entanglement. The results show that the time-dependent Pancharatnam phase includes the information of quantum correlations.

Appendix

Under correlated dephasing environment,The analytical solutions of the master (3) are given by

$$\begin{array}{@{}rcl@{}} \rho_{11}(t)&=&\frac{1-a}{3}, \end{array} $$

(41)

$$\begin{array}{@{}rcl@{}} \rho_{14}(t)&=&\frac{1}{9}e^{i\delta} e^{-2({\Gamma}+{\Gamma}_0+i\omega)t}, \end{array} $$

(42)

$$\begin{array}{@{}rcl@{}} \rho_{22}(t)&=&\frac{1}{3}+\frac{1}{2}c e^{x t}+c_r\cos(x_i t) e^{x_r t}\\&&-c_i\sin(x_i t) e^{x_r t}), \end{array} $$

(43)

$$\begin{array}{@{}rcl@{}} \rho_{23}(t)&=&\frac{1}{2}c \eta e^{x t}-(c_i \eta_i-c_r \eta_r)\cos(x_i t) e^{x_r t}-(c_r \eta_i\\&&+c_i \eta_r)\times\sin(x_i t) e^{x_r t}+i\frac{1}{4J}(-c x e^{x t}\\&&+2 e^{x_r t}(c_i x_i-c_r x_r)\cos(x_i t), \end{array} $$

(44)

$$\begin{array}{@{}rcl@{}} \rho_{32}(t)&=&\frac{1}{2}c \eta e^{x t}-(c_i \eta_i-c_r \eta_r)\cos(x_i t) e^{x_r t}-(c_r \eta_i\\&&+c_i \eta_r)\times\sin(x_i t) e^{x_r t}-i\frac{1}{4J}(-c x e^{x t}\\&&+2 e^{x_r t}(c_i x_i-c_r x_r)\cos(x_i t), \end{array} $$

(45)

$$\begin{array}{@{}rcl@{}} \rho_{33}(t)&=&\frac{1}{3}-\frac{1}{2}c e^{x t}-c_r\cos(x_i t) e^{x_r t}\\&&+c_i\sin(x_i t) e^{x_r t}), \end{array} $$

(46)

$$\begin{array}{@{}rcl@{}} \rho_{41}(t)&=&\frac{1}{9}e^{-i\delta} e^{-2({\Gamma}+{\Gamma}_0-i\omega)t}, \end{array} $$

(47)

$$\begin{array}{@{}rcl@{}} \rho_{44}(t)&=&\frac{a}{3}, \end{array} $$

(48)

where $x=-\frac {4({\Gamma }-{\Gamma }_0)}{3}-\frac {2^{1/3}m_1}{3\xi _1}+\frac {\xi _1}{3\times 2^{1/3}}, x_{r}=-\frac {4({\Gamma }-{\Gamma }_0)}{3}+\frac {m_1}{3\times 2^{2/3}\xi _1}-\frac {\xi _1}{6\times 2^{1/3}}, x_{i}=\frac {\sqrt {3}m_1}{3\times 2^{2/3}\xi _1}+\frac {\sqrt {3}\xi _1}{6\times 2^{1/3}}, \xi _1=(16({\Gamma }-{\Gamma }_0)^3-72J^2({\Gamma }-{\Gamma }_0)+576n^2({\Gamma }-{\Gamma }_0)\chi ^2_r +(4(12J^2-4({\Gamma }-{\Gamma }_0)^2+48n^2\chi ^2_r)^3+(16({\Gamma }-{\Gamma }_0)^3-72J^2({\Gamma }-{\Gamma }_0) +576n^2({\Gamma }-{\Gamma }_0)\chi ^2_r)^2)^{1/2})^{1/3},m_1=12J^2-4({\Gamma }-{\Gamma }_0)^2+48n^2\chi ^2_r, \eta =(4J^2+2({\Gamma }-{\Gamma }_0) x+x^2)/(8Jn\chi _r),\eta _r=(4J^2-x_i^2+x_r^2+2x_r({\Gamma }-{\Gamma }_0) )/(8J n\chi _r),\eta _i=(x_i x_r+x_i ({\Gamma }-{\Gamma }_0)/(4J n\chi _r).$ The constants c,c _r and c _i are determined by the initial conditions (5) for the X-state and (15) for the Y-state. Thus we have c = (−2x _i cosχ−4η _i J sinχ)/(3(η _r −η)x _i +3η _i (x _i −x _r )), c _r = (−x _i cosχ−2η _i J sinχ)/(3(η−η _r )x _i +3η _i (x _r −x)), c _i = ((x−x _r ) cosχ−2(η _r −η)J sinχ)/(3(η−η _r )x _i +3η _i (x _r −x)).

In case of the initial Y-state, the ρ ₁₄(0)=ρ ₄₁(0)=0.