Control and manipulation of entanglement between two coupled qubits by fast pulses

Abstract

We have investigated the analytical and numerical dynamics of entanglement for two qubits that interact with each other via Heisenberg XXX-type interaction and subject to local time-specific external kick and Gaussian pulse-type magnetic fields in $x$–$y$ plane. The qubits have been assumed to be initially prepared in different pure separable and maximally entangled states and the effect of the strength and the direction of external fast pulses on concurrence has been investigated. The carefully designed kick or pulse sequences are found to enable one to obtain constant long-lasting entanglement with desired magnitude. Moreover, the time ordering effects are found to be important in the creation and manipulation of entanglement by external fields.

Appendices

Appendix A: Single positive kick

Here, we consider two qubits whose states are strongly perturbed by external fields which may be expressed as a sudden kick at $t=T$. The time dependent magnetic fields on qubits $1$ and $2$ may be expressed as $B_1(t)=\alpha \delta (t-T)$ and $B_2(t)=\beta \delta (t-T)$, respectively, where $\alpha$ and $\beta$ are called integrated magnetic strengths. For such a kick the integration over the time is trivial and the time evolution matrix in Eq. (5) becomes as [20, 23]:

$$\begin{aligned} \hat{U}^K(t)=e^{-i\hat{H}_0(t-T)}e^{-i\int \limits _{T-\epsilon }^{T+\epsilon }\hat{H}_{int}(t^{\prime })dt^{\prime }}e^{-i\hat{H}_0 T}, \end{aligned}$$

(12)

with matrix elements

$$\begin{aligned} U_{11}&= e^{-iJt}\cos \left(\frac{\alpha }{2}\right)\cos \left(\frac{\beta }{2}\right)=U_{44},\nonumber \\ U_{22}&= \frac{1}{2}e^{-iJt}\left(e^{4iJt}\cos \left(\frac{\varDelta }{2}\right)+\cos \left(\frac{\varOmega }{2}\right)\right)=U_{33},\nonumber \\ U_{23}&= -\frac{1}{2}e^{-iJt}\left(e^{4iJt}\cos \left(\frac{\varDelta }{2}\right)-\cos \left(\frac{\varOmega }{2}\right)\right)=U_{32},\nonumber \\ U_{12}&= \frac{1}{2}ie^{-iJt}e^{-i\theta }\left(e^{4iJT}\sin \left(\frac{\varDelta }{2}\right)-\sin \left(\frac{\varOmega }{2}\right)\right)=e^{-2i\theta }U_{43},\nonumber \\ U_{13}&= -\frac{1}{2}ie^{-iJt}e^{-i\theta }\left(e^{4iJT}\sin \left(\frac{\varDelta }{2}\right)+\sin \left(\frac{\varOmega }{2}\right)\right)=e^{-2i\theta }U_{42},\nonumber \\ U_{14}&= -e^{-iJt}e^{-2i\theta }\sin \left(\frac{\alpha }{2}\right)\sin \left(\frac{\beta }{2}\right)=e^{-4i\theta }U_{41},\nonumber \\ U_{21}&= -e^{iJ(t-2T)}e^{i\theta }\left(\cos \left(\frac{\beta }{2}\right)\sin \left(\frac{\alpha }{2}\right)\sin (\xi )+i\cos \left(\frac{\alpha }{2}\right)\sin \left(\frac{\beta }{2}\right)\cos (\xi )\right)\nonumber \\&= e^{2i\theta }U_{34},\nonumber \\ U_{24}&= -\frac{1}{2}ie^{-iJt}e^{-i\theta }\left(e^{2i\xi }\sin \left(\frac{\varDelta }{2}\right)+\sin \left(\frac{\varOmega }{2}\right)\right)=e^{-2i\theta }U_{31}, \end{aligned}$$

(13)

where $\varDelta =(\alpha -\beta ), \varOmega =(\alpha +\beta )$ and $\xi =2J(t-T)$. It should be noted that the propagator given by Eq. () is valid only at times $t>T$.

Appendix B: Two positive kicks

The next example is the positive-positive kick sequence applied at times $t=T_1$ and $t=T_2$, namely, $B_1(t)=\alpha (\delta (t-T_1)+\delta (t-T_2))$ and $B_2(t)=\beta (\delta (t-T_1)+\delta (t-T_2))$. Following the procedure given in Eq. (), one obtains the time evolution matrix at times $t > T_2$ as [20, 23]:

$$\begin{aligned} \hat{U}^{K}(t)=e^{-i\hat{H}_0(t-T_2)}e^{-i\int \limits _{T_2-\epsilon }^{T_2+\epsilon }\hat{H}_{int}(t^{\prime })dt^{\prime }}e^{-i\hat{H}_0(T_2-T_1)}e^{-i\int \limits _{T_1-\epsilon }^{T_1+\epsilon }\hat{H}_{int}(t^{\prime })dt^{\prime }}e^{-i\hat{H}_0T_1},\nonumber \\ \end{aligned}$$

(14)

with matrix elements

$$\begin{aligned} U_{11}&= \frac{1}{4}e^{-iJt}\left(1+e^{4iJT}\left(\cos (\varDelta )-1\right)+\cos (\varDelta )+2\cos (\varOmega )\right)=U_{44},\nonumber \\ U_{22}&= \frac{1}{4}e^{-iJt}\left(e^{4iJt}+e^{4iJ(t-T)}\left(\cos (\varDelta )-1\right)+e^{4iJt}\cos (\varDelta )+2\cos (\varOmega )\right)=U_{33},\nonumber \\ U_{23}&= -\frac{1}{4}e^{-iJt}\left(e^{4iJt}+e^{4iJ(t-T)}\left(\cos (\varDelta )-1\right)+e^{4iJt}\cos (\varDelta )-2\cos (\varOmega )\right)=U_{32},\nonumber \\ U_{12}&= \frac{1}{2}ie^{-iJt}e^{-i\theta }\left(e^{6iJT}\cos (2JT)\sin (\varDelta )-\sin (\varOmega )\right)=e^{-2i\theta }U_{43},\nonumber \\ U_{13}&= -\frac{1}{2}ie^{-iJt}e^{-i\theta }\left(e^{6iJT}\cos (2JT)\sin (\varDelta )+\sin (\varOmega )\right)=e^{-2i\theta }U_{42},\nonumber \\ U_{14}&= \frac{1}{4}e^{-iJt}e^{-2i\theta }\left(e^{4iJT}-1-\left(1+e^{4iJT}\right)\cos (\varDelta )+2\cos (\varOmega )\right)=e^{-4i\theta }U_{41},\nonumber \\ U_{21}&= \frac{1}{4}ie^{-iJt}e^{i\theta }\left(e^{4iJ(t-2T)}\left(1+e^{4iJT}\right)\sin (\varDelta )-2\sin (\varOmega )\right)=e^{2i\theta }U_{34},\nonumber \\ U_{24}&= -\frac{1}{4}ie^{-iJt}e^{-i\theta }\left(e^{4iJ(t-2T)}\left(1+e^{4iJT}\right)\sin (\varDelta )+2\sin (\varOmega )\right)=e^{-2i\theta }U_{31}, \end{aligned}$$

(15)

where $\varDelta =(\alpha -\beta )$ and $\varOmega =(\alpha +\beta )$. Here, we have assumed equally distanced kicks applied at times $T_1=T$ and $T_2=2T$.

Appendix C: Three positive kicks

The final example is the sequence of three positive kicks applied at times $t=T_1, t=T_2$, and $t=T_3$ namely, $B_1(t)=\sum _{i=1}^3\alpha \delta (t-T_i)$ and $B_2(t)=\sum _{i=1}^3\beta \delta (t-T_i)$. Following the procedure given in Eq. (), one can obtain the time evolution matrix at times $t > T_3$ as [20]:

$$\begin{aligned} \hat{U}^{K}(t)&= e^{-i\hat{H}_0(t-T_3)}e^{-i\int \limits _{T_3-\epsilon }^{T_3+\epsilon }\hat{H}_{int}(t^{\prime })dt^{\prime }}e^{-i\hat{H}_0(T_3-T_2)}e^{-i\int \limits _{T_2-\epsilon }^{T_2+\epsilon }\hat{H}_{int}(t^{\prime })dt^{\prime }}\nonumber \\&\times e^{-i\hat{H}_0(T_2-T_1)}e^{-i\int \limits _{T_1-\epsilon }^{T_1+\epsilon }\hat{H}_{int}(t^{\prime })dt^{\prime }}e^{-i\hat{H}_0T_1}, \end{aligned}$$

(16)

with matrix elements for $T_1=T, T_2=2T$ and $T_3=3T$,

$$\begin{aligned} U_{11}&= \frac{1}{8}e^{-iJt}\left(\left(3-2e^{4iJT}-e^{8iJT}\right)\cos \left(\frac{\varDelta }{2}\right)+\left(1+e^{4iJT}\right)^2\cos \left(\frac{3\varDelta }{2}\right)+4\cos \left(\frac{3\varOmega }{2}\right)\right)\nonumber \\&= U_{44},\nonumber \\ U_{22}&= \frac{1}{2}e^{-iJt}\left(\cos \left(\frac{3\varOmega }{2}\right)+e^{4iJ(t-T)}\cos \left(\frac{\varDelta }{2}\right)((1+\cos (4JT))\cos (\varDelta )+i\sin (4JT)-1)\right)\nonumber \\&= U_{33},\nonumber \\ U_{23}&= \frac{1}{2}e^{-iJt}\left(\cos \left(\frac{3\varOmega }{2}\right)-e^{4iJ(t-T)}\cos \left(\frac{\varDelta }{2}\right)((1+\cos (4JT))\cos (\varDelta )+i\sin (4JT)-1)\right)\nonumber \\&= U_{32},\nonumber \\ U_{12}&= \frac{1}{2}ie^{-iJt}e^{-i\theta }\left(e^{8iJT}(\cos (4JT)+2\cos (2JT)^2\cos (\varDelta ))\sin \left(\frac{\varDelta }{2}\right)-\sin \left(\frac{3\varOmega }{2}\right)\right)\nonumber \\&= e^{-2i\theta }U_{43},\nonumber \\ U_{13}&= -\frac{1}{2}ie^{-iJt}e^{-i\theta }\left(e^{8iJT}(\cos (4JT)+2\cos (2JT)^2\cos (\varDelta ))\sin \left(\frac{\varDelta }{2}\right)+\sin \left(\frac{3\varOmega }{2}\right)\right)\nonumber \\&= e^{-2i\theta }U_{42},\nonumber \\ U_{14}&= \frac{1}{8}e^{-iJt}e^{-2i\theta }\left(\left(2e^{4iJT}+e^{8iJT}-3\right)\cos \left(\frac{\varDelta }{2}\right)-\left(1+e^{4iJT}\right)^2\cos \left(\frac{3\varDelta }{2}\right)+4\cos \left(\frac{3\varOmega }{2}\right)\right)\nonumber \\&= e^{-4i\theta }U_{41},\nonumber \\ U_{21}&= \frac{1}{2}ie^{-iJt}e^{i\theta }\left(e^{4iJ(t-2T)}(\cos (4JT)+2\cos (2JT)^2\cos (\varDelta ))\sin \left(\frac{\varDelta }{2}\right)-\sin \left(\frac{3\varOmega }{2}\right)\right)\nonumber \\&= e^{2i\theta }U_{34},\nonumber \\ U_{24}&= -\frac{1}{2}ie^{-iJt}e^{-i\theta }\left(e^{4iJ(t-2T)}(\cos (4JT)+2\cos (2JT)^2\cos (\varDelta ))\sin \left(\frac{\varDelta }{2}\right)+\sin \left(\frac{3\varOmega }{2}\right)\right)\nonumber \\&= e^{-2i\theta }U_{31}, \end{aligned}$$

(17)

where $\varDelta =(\alpha -\beta )$ and $\varOmega =(\alpha +\beta )$.