Dynamics of entanglement creation between two spins coupled to a chain

Abstract

We study the dynamics of entanglement between two spins which is created by the coupling to a common thermal reservoir. The reservoir is a spin-$\frac{1}{2}$ Ising transverse field chain thermally excited; the two defect spins couple to two spins of the chain which can be at a macroscopic distance. In the weak-coupling and low-temperature limit, the spin chain is mapped onto a bath of linearly interacting oscillators using the Holstein–Primakoff transformation. We analyze the time evolution of the density matrix of the two defect spins for transient times and deduce the entanglement which is generated by the common reservoir. We discuss several scenarios for different initial states of the two spins and for varying distances.

A Time evolution of the reduced density matrix

Under the assumption of vanishing defects free Hamiltonian, the full dynamics is described by

$$\begin{aligned} {\widetilde{H}}= & {} \frac{1}{2}(\widetilde{{\mathbf {p}}}^{S})^2 +\frac{1}{2}(\widetilde{{\mathbf {x}}}^{S})^TD^S\widetilde{{\mathbf {x}}}^{S} +S_x^S({\widetilde{\gamma }}^{S})^T\widetilde{{\mathbf {x}}}^S \nonumber \\&+\,\frac{1}{2}(\widetilde{{\mathbf {p}}}^{A})^2+\frac{1}{2} (\widetilde{{\mathbf {x}}}^{A})^TD^A\widetilde{{\mathbf {x}}}^{A} +S_x^A({\widetilde{\gamma }}^{A})^T\widetilde{{\mathbf {x}}}^A \nonumber \\= & {} {\widetilde{H}}_b^S +{\widetilde{H}}_i^S + {\widetilde{H}}_b^A + {\widetilde{H}}_i^A. \end{aligned}$$

(32)

In the interaction picture with respect to the bath Hamiltonian, we find

$$\begin{aligned} {\widetilde{H}}^{(I)}=e^{i{\widetilde{H}}_b^St} {\widetilde{H}}_i^Se^{-i{\widetilde{H}}_b^St}+e^{i{\widetilde{H}}_b^At} {\widetilde{H}}_i^Ae^{-i{\widetilde{H}}_b^At}. \end{aligned}$$

(33)

Using Baker–Campbell–Hausdorff expansion, one easily finds

$$\begin{aligned} {\widetilde{H}}^{(I)}= & {} \sum _i\bigg (\gamma _i^S\cos (\widetilde{\omega }_i^St) S_x^S{\widetilde{x}}_i^S+\frac{\gamma _i^S}{\widetilde{\omega }_i^S}S_x^S{\widetilde{p}}_i^S \nonumber \\&+\gamma _i^A\cos (\widetilde{\omega }_i^At)S_x^A{\widetilde{x}}_i^A +\frac{\gamma _i^A}{\widetilde{\omega }_i^A}S_x^A{\widetilde{p}}_i^A\bigg ). \end{aligned}$$

(34)

Neglecting the free defects Hamiltonian permits us to write the full time evolution operator as ${\widetilde{U}}(t)={\widetilde{U}}^S(t){\widetilde{U}}^A(t)$. Operators ${\widetilde{U}}^{(S,A)}(t)$ can now be obtained with the Ansatz

$$\begin{aligned} {\widetilde{U}}^{S,A}(t)= & {} \exp {\left( \displaystyle i\sum _i\delta _i^{S,A}(t)\right) } \nonumber \\&\times \exp {\left( i\displaystyle \sum _i\big (\phi _i^{S,A}(t) {\widetilde{p}}_i^{S,A}+(\phi _i^{S,A})'(t){\widetilde{x}}_i^{S,A}\big )\right) }. \end{aligned}$$

(35)

Deriving ${\widetilde{U}}(t)$ with respect to time

$$\begin{aligned} \dot{{\widetilde{U}}}= & {} i\sum _i\bigg ( {\dot{\delta }}_i^S +{\dot{\phi }}_i^S{\widetilde{p}}_i^S-\frac{1}{2} \frac{d}{dt} \big (\phi _i^S(\phi _i^{S})'\big )\bigg )\widetilde{U}^S\widetilde{U}^A \nonumber \\&+\,i\sum _i\bigg ( {\dot{\phi }}_i^S{\widetilde{x}}_i^S +({\dot{\phi }}_i^{S})'\phi _i^S\bigg )\widetilde{U}^S\widetilde{U}^A \nonumber \\&+\,i\sum _i\bigg ({\dot{\delta }}^A_i+{\dot{\phi }}_i^A{\widetilde{p}}_i^A -\frac{1}{2}\frac{d}{d t}\big (\phi _i^A(\phi _i^{A})'\big )\bigg ) \widetilde{U}^S\widetilde{U}^A \nonumber \\&+\,i\sum _i\bigg ( {\dot{\phi }}_i^A{\widetilde{x}}_i^A+({\dot{\phi }}_i^{A})'\phi _i^A\bigg ) \widetilde{U}^S\widetilde{U}^A, \end{aligned}$$

(36)

where the dot represents time derivative and the explicit time dependence has been omitted. Schrödinger’s equation for the evolution operator $\dot{{\widetilde{U}}}(t)=-i{\widetilde{H}}^{(I)}(t){\widetilde{U}}(t)$ can be now used to find $\phi _i^{A(B)}$ and $\phi _i^{'A(B)}$. Using the initial condition ${\widetilde{U}}(0)=\mathbb {1}$, we find

$$\begin{aligned} \phi _i^{S,A}( t)= & {} -\frac{\gamma _i^{S,A}S_x^{S,A}}{\widetilde{\omega }_i^{S,A}}\sin \left( \widetilde{\omega }_i^{S,A} t\right) \nonumber \\ (\phi _i^{S,A})'( t)= & {} -\frac{\gamma _i^{S,A}S_x^{S,A}}{\widetilde{\omega }_i^{S,A}} \left( \cos \left( \widetilde{\omega }_i^{S,A} t\right) -1\right) \nonumber \\ \delta _i^{S,A}( t)= & {} \frac{(\gamma _i^{S,A})^2(S_x^{S,A})^2}{\widetilde{\omega }_i^{S,A}}\bigg ( t-\frac{\sin \left( \widetilde{\omega }_i^{S,A} t\right) }{ \widetilde{\omega }_i^{S,A}}\bigg ). \end{aligned}$$

(37)

The evolution operator is seen to be a displacement operator

$$\begin{aligned} \widetilde{U}( t)=e^{i \sum _i\big (\delta _i^S(t)+\delta _i^A(t)\big )} e^{i({\mathbf {Q}}^S\mathbf {{\widetilde{x}}}^S-{\mathbf {R}}^S \mathbf {{\widetilde{p}}}^S)}e^{i({\mathbf {Q}}^A\mathbf {{\widetilde{x}}}^A -{\mathbf {R}}^A\mathbf {{\widetilde{p}}}^A)}, \end{aligned}$$

(38)

with

$$\begin{aligned}&{\mathbf {Q}}^{S,A}=-\sum _i\frac{{\gamma }_i^{S,A}}{{{\widetilde{\omega }}}_i^{S,A}}\sin \left( {\widetilde{\omega }}_i^{S,A} t \right) S_x^{S,A}{\mathbf {e}}_i^{S,A}, \nonumber \\&\quad {\mathbf {R}}^{S,A}=-\sum _i\frac{{\gamma }_i^{S,A}}{{\big ({\widetilde{\omega }}}_i^{S,A}\big )^2}\left( \cos \left( {\widetilde{\omega }}_i^{S,A} t\right) -1\right) S_x^{S,A}{\mathbf {e}}_i^{S,A} \end{aligned}$$

(39)

and ${\mathbf {e}}_j$ are unit vectors on the j-th direction.

The evolved reduced density matrix of the defects can now be obtained by assuming a thermal environment and integrating over the bath degrees of freedom. After a lengthy but straightforward calculation, the matrix elements in the eigenbasis $\{{|s_i\rangle } \}$ of the $S_x^{S,A}$ operator are given by

$$\begin{aligned} {\langle s_i|} \rho _d(t){|s_j\rangle } \!= & {} \!\exp \! \left\{ \!-\!\big [f^S(t)(s_i^{S}\!-\!s_j^{S})^2\!+\!f^A(t)(s_i^{A}\!-\!s_j^{A})^2\big ]\right. \nonumber \\&\left. +\,i\big [\varphi ^S(t)((s_i^{S})^2\!-\!(s_j^{S})^2)\!+\!\varphi ^A(t)((s_i^{A})^2\!-(\!s_j^{A})^2)\big ]\right\} \nonumber \\&\times {\langle s_i|} \rho _d(0){|s_j\rangle }, \end{aligned}$$

(40)

where the coefficients $f^{S,A}(t)$ and $\varphi ^{S,A}(t)$ are given by

$$\begin{aligned} f^{S,A}(t)= & {} \sum _{i} \frac{\left( {\widetilde{\gamma }}_i^{S,A}\right) ^2(2{\widetilde{n}}^{S,A}_i-1)}{2\left( {\widetilde{\omega }}_i^{(S,A)}\right) ^3}\left( 1- \cos \left( {\widetilde{\omega }}_i^{S,A}t\right) \right) , \end{aligned}$$

(41)

$$\begin{aligned} \varphi ^{S,A}(t)= & {} \sum _{i} \frac{\left( {\widetilde{\gamma }}_i^{S,A}\right) ^2}{2\left( {\widetilde{\omega }}_i^{(S,A)}\right) ^2}\left( t -\frac{\sin \left( {\widetilde{\omega }}_i^{S,A}t\right) }{{\widetilde{\omega }}_i^{S,A}} \right) . \end{aligned}$$

(42)

and ${\widetilde{n}}^S_i$ (${\widetilde{n}}^A_i$) is the thermal occupation number of mode i of the symmetric (antisymmetric) bath defined by ${\widetilde{n}}^{S,A}_i=\displaystyle (e^{{\widetilde{\omega }}_i^{S,A}/T}-1)^{-1}$.