Entanglement dynamics of a nano-mechanical resonator coupled to a central qubit

Abstract

We introduce a unitary operator which may be constructed conveniently by exploiting the properties of the Glauber displacement and parity operators. We show that it can be considered as a constant of motion of a quantum system including pure dephasing interaction of a central qubit with a nano-mechanical resonator. Using the eigenvectors of the Glauber displacement operator, we paves a way for an extensive study of the dynamics of resonator-qubit states. In addition, we show that the establishment of such a constant of motion, which includes the parity operator, provides a way to introduce a capable mechanical framework to generate some desired mechanical state as superposition of the Glauber coherent states. We investigate nonclassical properties of the generated mechanical states, by evaluating mechanical-squeezing, Wigner function, position–momentum entropic squeezing, and entanglement between spin-mechanical modes. Furthermore, the pairwise classical and quantum correlations are derived based on a necessary and sufficient condition for the zero-discord state. Finally, in order to study the mechanical state’s behavior in an environment, we consider that the output state is subject to amplitude (phase)-damping channels and their dissipative properties are analyzed.

Appendix: Derivation of time evolution operator (2)

The time evolution operator associated with the Hamiltonian (1) is given by

$$\begin{aligned}&U(\tau )=e^{-i\tau H}=e^{-i\tau \left( \omega _{0} \sigma _{z}+\omega a^{\dag }a+i\frac{\omega }{2}(a-a^{\dag })\sigma _{z}\right) }. \end{aligned}$$

Defining the operators $A:=-i\omega \tau a^{\dag }a$ and $B=\frac{\omega }{2} \tau (a-a^{\dag })\sigma _{z}$, using the Zassenhaus formula [44], i.e.

$$\begin{aligned}&e^{A+B}=e^{A}e^{B}e^{-\frac{[A,B]}{2}}e^{\frac{[A,[A,B]]+2[B,[A,B]]}{3!}}e^{-\frac{[A,[A,[A,B]]]+3[B,[A,[A,B]]]+3[B,[B,[A,B]]]}{4!}}\ldots , \end{aligned}$$

and applying the commutation relations:

$$\begin{aligned}&[A,B]=i\frac{(\omega \tau )^{2}}{2} \sigma _{z} (a^{\dag }+a),\nonumber \\&[A,[A,B]]=-(\omega \tau )^{2}B=-\frac{(\omega \tau )^{3}}{2}(a-a^{\dag })\sigma _{z},\nonumber \\&[B,[A,B]]=i\frac{(\omega \tau )^{3}}{2},\nonumber \\&[A,[A,[A,B]]]=-i\frac{(\omega \tau )^{4}}{2} \sigma _{z} (a^{\dag }+a),\nonumber \\&[B,[A,[A,B]]]=[B,[B,[A,B]]]=0, \end{aligned}$$

the time evolution operator U(t) can be recast into

$$\begin{aligned} U(t) =e^{-i\tau \omega _{0} \sigma _{z}}e^{-i\omega \tau a^{\dag }a}e^{\frac{\omega \tau }{2}(a-a^{\dag })\sigma _{z}}e^{-i\frac{(\omega \tau )^{2}}{2*2!} \sigma _{z} (a^{\dag }+a)}e^{i\frac{(\omega \tau )^{3}}{3!}}e^{-\frac{(\omega \tau )^{3}}{2*3!}(a-a^{\dag })\sigma _{z}}e^{i\frac{(\omega \tau )^{4}}{2*4!} \sigma _{z} (a^{\dag }+a)}\ldots . \end{aligned}$$

Now, using the Baker–Campbell–Hausdorff (BCH) formula [43], i.e.

$$\begin{aligned}&e^{A}e^{B}=e^{A+B+\frac{[A,B]}{2}+\frac{[A,[A,B]]-[B,[A,B]]}{12}+\frac{[A,[A,[A,B]]]+3[B,[A,[A,B]]]+3[B,[B,[A,B]]]}{24}+\cdots }, \end{aligned}$$

one can show that:

$$\begin{aligned}&e^{\frac{\omega \tau }{2}(a-a^{\dag })\sigma _{z}}e^{-i\frac{(\omega \tau )^{2}}{2*2!} \sigma _{z} (a^{\dag }+a)}=e^{-i\frac{(\omega \tau )^{3}}{2*2!}}e^{-\frac{\omega \tau }{2}\sigma _{z}\left[ \left( 1+i\frac{\omega \tau }{2!}\right) a^{\dag }-\left( 1-i\frac{\omega \tau }{2!}\right) a\right] }\\&\quad =e^{-i\frac{(\omega \tau )^{3}}{2*2!}}e^{\frac{i\sigma _{z}}{2}\left[ \left( (i\omega \tau )+\frac{(i\omega \tau )^2}{2!}\right) a^{\dag }+\left( (-i\omega \tau )+\frac{(-i\omega \tau )^2}{2!}\right) a\right] },\\&e^{\frac{\omega \tau }{2}(a-a^{\dag })\sigma _{z}}e^{-i\frac{(\omega \tau )^{2}}{2*2!} \sigma _{z} (a^{\dag }+a)}e^{-\frac{(\omega \tau )^{3}}{2*3!}(a-a^{\dag })\sigma _{z}}\\&\quad =e^{-i\frac{(\omega \tau )^{3}}{4}\left( 1+\frac{(\omega \tau )^{2}}{3!}\right) }e^{-\frac{\omega \tau }{2}\sigma _{z}\left[ \left( 1+i\frac{\omega \tau }{2!}-\frac{(\omega \tau )^2}{3!}\right) a^{\dag }-\left( 1-i\frac{\omega \tau }{2!}-\frac{(\omega \tau )^2}{3!}\right) a\right] }\\&\quad =e^{-i\frac{(\omega \tau )^{3}}{4}\left( 1+\frac{(\omega \tau )^{2}}{3!}\right) }e^{\frac{i\sigma _{z}}{2}\left[ \left( (i\omega \tau )+\frac{(i\omega \tau )^2}{2!}+\frac{(i\omega \tau )^3}{3!}\right) a^{\dag }+\left( (-i\omega \tau )+\frac{(-i\omega \tau )^2}{2!}+\frac{(-i\omega \tau )^3}{3!}\right) a\right] }. \end{aligned}$$

By exploiting the above relations, the time evolution operator U(t) gets

$$\begin{aligned} U(t)= & {} e^{-it\omega _{0} \sigma _{z}}e^{-i\omega t a^{\dag }a}e^{if(\tau )}e^{\frac{i\sigma _{z}}{2}\left[ \left( (i\omega \tau )+\frac{(i\omega \tau )^2}{2!}+\frac{(i\omega \tau )^3}{3!}+\cdots \right) a^{\dag }+\left( (-i\omega \tau )+\frac{(-i\omega \tau )^2}{2!}+\frac{(-i\omega \tau )^3}{3!}+\cdots \right) a\right] }\nonumber \\= & {} e^{-it\omega _{0} \sigma _{z}}e^{-i\omega t a^{\dag }a}e^{if(\tau )}e^{\frac{i\sigma _{z}}{2}\left[ \left( e^{i\omega \tau }-1\right) a^{\dag }+\left( e^{-i\omega \tau }-1\right) a\right] }\nonumber \\= & {} e^{if(\tau )}e^{-it\omega _{0} \sigma _{z}}e^{\frac{-i\sigma _{z}}{2}\left[ \left( e^{-i\omega \tau }-1\right) a^{\dag }+\left( e^{i\omega \tau }-1\right) a\right] }e^{-i\omega t a^{\dag }a}\nonumber \\= & {} e^{if(\tau )}e^{-it\omega _{0} \sigma _{z}}e^{\frac{-i\sigma _{z}}{2}(\eta a^{\dag }+{\overline{\eta }} a)}e^{-i\omega t a^{\dag }a}, \end{aligned}$$

where we set $\eta =e^{-i\omega \tau }-1$ and $f(t)=-\frac{(i\omega \tau )^2}{2*2!}-\frac{(\omega \tau )^3}{2*3!}-i\frac{(\omega \tau )^{3}}{4}\left( 1+\frac{(\omega \tau )^{2}}{3!}\right) +\cdots$