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Spin-Bath Dynamics in a Quantum Resonator-Qubit System: Effect of a Mechanical Resonator Coupled to a Central Qubit

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Abstract

In line with an experimentally feasible protocol was proposed by A. Asadian et al. [ PRL 112, 190402 (2014)], we introduce a pure dephasing model where the interaction of the central qubit with a nano-mechanical resonator is affected by a spin-bath to study the dynamics of resonator-qubit entangled states. We show that how system-bath coupling as well as the coupling among the bath’s spins, initial semi-classical states of the phonon as well as the initial states of the bath, modify the entanglement of the system using measures like concurrence. To gain insight into the effect of these correlations, we study the dynamics of the mentioned setup both with and without initial correlations for arbitrary system-environment coupling strengths. It is also shown that stable entanglement, which is dependent on correlated initial states of the system-bath, their coupling strength, and bath temperature, occurs even in the presence of decoherence.

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Notes

  1. Without loss of generality, we employed β(τ) = β and φ(τ) = 0,π.

  2. The superscripts “u.c.” and “c.” instead of the letters uncorrelated and correlated, respectively, through the paper.

  3. It is worth mentioning that, the parameter t in the entangled state should be started after τ where we fixed it here as zero.

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Appendix: Derivation of Time Evolution Operator (2)

Appendix: Derivation of Time Evolution Operator (2)

The total Hamiltonian of the spin-resonator system interacts with a spin bath is H = HS + HSB + HB, where HS is the Hamiltonian of a central spin-resonator system, HSB is the interaction Hamiltonian between a central qubit and spin-bath, and HB refers to a bath Hamiltonian, too. Using the angular momentum addition theorem, i.e. \(J_{\pm }={\sum }^{N}_{i=1} \sigma ^{(i)}_{\pm }\), as well as the Holstein-Primakoff transformation, i.e. \(J_{+}=\left (J^{\dag }_{-}\right )=b^{\dag }\sqrt {N-b^{\dag }b}\), and \(J_{z}=b^{\dag }b-\frac {N}{2}\) we obtain the Hamiltonian of the system in as follows

$$ \begin{array}{@{}rcl@{}} &&H=\omega_{q} \sigma_{z}+\omega a^{\dag}a+\lambda(a+a^{\dag})\sigma_{z}+g\sigma_{z}\left( b^{\dag}b-\frac{N}{2}\right)+2g_{0}b^{\dag}b\left( 1-\frac{b^{\dag}b}{N}\right), \end{array} $$

then the time evolution operator reads \(U(t)=e^{-it\left (\omega _{q} \sigma _{z}+\omega a^{\dag }a+\lambda (a+a^{\dag })\sigma _{z}+g\sigma _{z}\left (b^{\dag }b-\frac {N}{2}\right )+2g_{0}b^{\dag }b\left (1-\frac {b^{\dag }b}{N}\right )\right )}\). Defining the operators A := −iωtaa and B = −iλt(a + a)σz, using the Zassenhaus formula [84], i.e.

$$ \begin{array}{@{}rcl@{}} &&e^{A+B}=e^{A}e^{B}e^{-\frac{[A,B]}{2}}e^{\frac{[A,[A,B]]+2[B,[A,B]]}{3!}}e^{-\frac{[A,[A,[A,B]]]+3[B,[A,[A,B]]]+3[B,[B,[A,B]]]}{4!}}...., \end{array} $$

, and applying the commutation relations:

$$ \begin{array}{@{}rcl@{}} &&[A,B]=-\lambda t^{2} \omega \sigma_{z} (a^{\dag}-a),\\ &&[A,[A,B]]=i\lambda t^{3} \omega^{2} \sigma_{z} (a^{\dag}+a),\\ &&[B,[A,B]]=2i\lambda^{2} t^{3} \omega,\\ &&[A,[A,[A,B]]]=\lambda t^{4} \omega^{3}\sigma_{z} (a^{\dag}-a),\\ &&[B,[A,[A,B]]]=[B,[B,[A,B]]]=0, \end{array} $$

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

$$ \begin{array}{@{}rcl@{}} &&U(t)=e^{-it\omega_{q} \sigma_{z}}e^{-i\omega t a^{\dag}a}e^{-it\lambda\sigma_{z}(a+a^{\dag})}e^{\frac{t^{2}\lambda\omega\sigma_{z}}{2}(a^{\dag}-a)}e^{\frac{it^{3}}{3!}[\lambda\omega^{2}\sigma_{z}(a^{\dag}+a)+4\lambda^{2}\omega]}\\ &&e^{-\frac{t^{4}\lambda\omega^{3}\sigma_{z}}{4!}(a^{\dag}-a)}e^{g\sigma_{z}\left( b^{\dag}b-\frac{N}{2}\right)+2g_{0}b^{\dag}b\left( 1-\frac{b^{\dag}b}{N}\right)} \end{array} $$

Now, using the Baker-Campbell-Hausdorff (BCH) formula [85], i.e.

$$ e^{A}e^{B}=e^{A+B+\frac{[A,B]}{2}+\frac{[A,[A,B]]-[B,[A,B]]}{12}+\frac{[A,[A,[A,B]]]+3[B,[A,[A,B]]]+3[B,[B,[A,B]]]}{24}+...}, $$

one can show that:

$$ \begin{array}{@{}rcl@{}} &&e^{-it\lambda\sigma_{z}(a+a^{\dag})}e^{\frac{t^{2}\lambda\omega\sigma_{z}}{2}(a^{\dag}-a)}=e^{-i\lambda t\sigma_{z}\left[a^{\dag}\left( 1+\frac{i\omega t}{2}\right)+a\left( 1-\frac{i\omega t}{2}\right)\right]}e^{-i\frac{\lambda^{2}t^{3}\omega}{2}},\\ &&e^{-it\lambda\sigma_{z}(a+a^{\dag})}e^{\frac{t^{2}\lambda\omega\sigma_{z}}{2}(a^{\dag}-a)}e^{\frac{it^{3}}{3!}[\lambda\omega^{2}\sigma_{z}(a^{\dag}+a)+4\lambda^{2}\omega]}=e^{-i\lambda t\sigma_{z}\left[a^{\dag}\left( 1+\frac{i\omega t}{2}+\frac{(i\omega t)^{2}}{3!}\right)+a\left( 1-\frac{i\omega t}{2}+\frac{(i\omega t)^{2}}{3!}\right)\right]}\\ &&\times e^{-i\frac{\lambda^{2}t^{3}\omega}{2}+4i\frac{\lambda^{2}t^{3}\omega}{3!}-i\frac{\lambda^{2}t^{5}\omega^{3}}{2*3!}},\\ &&e^{-it\lambda\sigma_{z}(a+a^{\dag})}e^{\frac{t^{2}\lambda\omega\sigma_{z}}{2}(a^{\dag}-a)}e^{\frac{it^{3}}{3!}[\lambda\omega^{2}\sigma_{z}(a^{\dag}+a)+4\lambda^{2}\omega]}e^{-\frac{t^{4}\lambda\omega^{3}\sigma_{z}}{4!}(a^{\dag}-a)}\\ &&=e^{-i\lambda t\sigma_{z}\left[a^{\dag}\left( 1+\frac{i\omega t}{2}+\frac{(i\omega t)^{2}}{3!}+\frac{(i\omega t)^{2}}{3!}+\frac{(i\omega t)^{3}}{4!}\right)+a\left( 1-\frac{i\omega t}{2}-\frac{(i\omega t)^{2}}{3!}+\frac{(i\omega t)^{3}}{4!}\right)\right]}\\ &&\times e^{-i\frac{\lambda^{2}t^{3}\omega}{2}+4i\frac{\lambda^{2}t^{3}\omega}{3!}-i\frac{\lambda^{2}t^{5}\omega^{3}}{2*3!}+i\frac{\lambda^{2}t^{5}\omega^{3}}{4!}\left( 1-\frac{\omega^{2} t^{2}}{2}\right)}, \end{array} $$

then the time evolution operator U(t) can be rewritten as

$$ \begin{array}{@{}rcl@{}} &&U(t)=e^{-it\omega_{q} \sigma_{z}}e^{-i\omega t a^{\dag}a}e^{-i\lambda t\sigma_{z}\left[a^{\dag}\left( 1+\frac{i\omega t}{2}+\frac{(i\omega t)^{2}}{3!}+\frac{(i\omega t)^{2}}{3!}+\frac{(i\omega t)^{3}}{4!}+...\right)+a\left( 1-\frac{i\omega t}{2}-\frac{(i\omega t)^{2}}{3!}+\frac{(i\omega t)^{3}}{4!}+...\right)\right]}\\ &&\times e^{-i\frac{\lambda^{2}t^{3}\omega}{2}+4i\frac{\lambda^{2}t^{3}\omega}{3!}-i\frac{\lambda^{2}t^{5}\omega^{3}}{2*3!}+i\frac{\lambda^{2}t^{5}\omega^{3}}{4!}\left( 1-\frac{\omega^{2} t^{2}}{2}\right)+...}e^{g\sigma_{z}\left( b^{\dag}b-\frac{N}{2}\right)+2g_{0}b^{\dag}b\left( 1-\frac{b^{\dag}b}{N}\right)}\\ &&=e^{if(t)}e^{-it\omega_{q} \sigma_{z}}e^{-i\omega t a^{\dag}a}e^{-\frac{\lambda}{\omega}\sigma_{z}\left[a^{\dag}\left( e^{i\omega t}-1\right)-a\left( e^{-i\omega t}-1\right)\right]}e^{g\sigma_{z}\left( b^{\dag}b-\frac{N}{2}\right)+2g_{0}b^{\dag}b\left( 1-\frac{b^{\dag}b}{N}\right)}\\ &&=e^{if(t)}e^{-it\omega_{q} \sigma_{z}}e^{-i\omega t a^{\dag}a}e^{-\frac{\lambda}{\omega}\sigma_{z}\left( \overline{\eta} a^{\dag}-\eta a\right)}e^{g\sigma_{z}\left( b^{\dag}b-\frac{N}{2}\right)+2g_{0}b^{\dag}b\left( 1-\frac{b^{\dag}b}{N}\right)}\\ &&=e^{if(t)}e^{-it\omega_{q} \sigma_{z}}e^{-\frac{\lambda}{\omega}\sigma_{z}\left( \overline{\eta} a^{\dag}e^{-i\omega t}-\eta ae^{i\omega t}\right)}e^{-i\omega t a^{\dag}a}e^{g\sigma_{z}\left( b^{\dag}b-\frac{N}{2}\right)+2g_{0}b^{\dag}b\left( 1-\frac{b^{\dag}b}{N}\right)}\\ &&=e^{if(t)}e^{-it\omega_{q} \sigma_{z}}e^{\frac{\lambda}{\omega}\sigma_{z}\left( \eta a^{\dag}-\overline{\eta} a\right)}e^{-i\omega t a^{\dag}a}e^{g\sigma_{z}\left( b^{\dag}b-\frac{N}{2}\right)+2g_{0}b^{\dag}b\left( 1-\frac{b^{\dag}b}{N}\right)} \end{array} $$
(15)

where we set η = eiωt − 1 and \(f(t)=-\frac {\lambda ^{2}t^{3}\omega }{2}+4\frac {\lambda ^{2}t^{3}\omega }{3!}-\frac {\lambda ^{2}t^{5}\omega ^{3}}{2*3!}-\frac {\lambda ^{2}t^{5}\omega ^{3}}{4!}\left (1-\frac {\omega ^{2} t^{2}}{2}\right )+...\)

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Dehghani, A., Mojaveri, B. & Vaez, M. Spin-Bath Dynamics in a Quantum Resonator-Qubit System: Effect of a Mechanical Resonator Coupled to a Central Qubit. Int J Theor Phys 59, 3107–3123 (2020). https://doi.org/10.1007/s10773-020-04565-3

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