Vol.07 No.01(2018), Article ID:23714,5 pages
10.12677/CMP.2018.71004

Geometric Phase in a Quantum Coherent Tunable Coupling of Superconducting Qubits

Yuanxin Qiao, Zhaoxian Yu

Department of Physics, Beijing Information Science and Technology University, Beijing

Received: Jan. 21st, 2018; accepted: Feb. 2nd, 2018; published: Feb. 9th, 2018

ABSTRACT

By using the Lewis-Riesenfeld invariant theory, we have studied the geometric phase in a quantum coherent tunable coupling of superconducting qubits. The geometric phase has nothing to do with the tunnel splitting, but also the dc energy bias and the HF signal. This result has certain significance to the quantum computation.

Keywords:Geometric Phase, Continuous Monitoring of Rabi Oscillations, Josephson Flux Qubit

1. 引言

2. 模板

$\stackrel{^}{H}=-\frac{1}{2}\Delta {\sigma }_{x}-\frac{1}{2}\epsilon {\sigma }_{z}-W\left(t\right){\sigma }_{z},$ (1)

$A=-\frac{1}{2}\Delta ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}B=-\frac{1}{2}\epsilon ,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\beta \left(t\right)=W\left(t\right),$ (2)

$\stackrel{^}{H}=A{\sigma }_{x}+B{\sigma }_{z}+\beta \left(t\right){\sigma }_{z},$ (3)

$\left[{\sigma }_{+},{\sigma }_{-}\right]={\sigma }_{z},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[{\sigma }_{z},{\sigma }_{+}\right]=2{\sigma }_{+}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\left[{\sigma }_{z},{\sigma }_{-}\right]=-2{\sigma }_{-}.$ (4)

3. 动力学与几何相位

$i\frac{\partial \stackrel{^}{I}\left(t\right)}{\partial t}+\left[\stackrel{^}{I}\left(t\right),\stackrel{^}{H}\left(t\right)\right]=0.$ (5)

$\stackrel{^}{I}\left(t\right)|{\lambda }_{n},t〉={\lambda }_{n}|{\lambda }_{n},t〉,$ (6)

$\frac{\partial {\lambda }_{n}}{\partial t}=0$ 。此系统的与时间有关的薛定谔方程是

$i\frac{\partial {|\psi \left(t\right)〉}_{s}}{\partial t}=\stackrel{^}{H}\left(t\right){|\psi \left(t\right)〉}_{s}$ (7)

${|{\lambda }_{n},t〉}_{s}=\mathrm{exp}\left[i{\delta }_{n}\left(t\right)\right]|{\lambda }_{n},t〉$ (8)

${|\psi \left(t\right)〉}_{s}=\underset{n}{\sum }{C}_{n}\mathrm{exp}\left[i{\delta }_{n}\left(t\right)\right]|{\lambda }_{n},t〉,$ (9)

${\delta }_{n}\left(t\right)={\int }_{0}^{t}\text{d}{t}^{\prime }〈{\lambda }_{n},{t}^{\prime }|i\frac{\partial }{\partial {t}^{\prime }}-\stackrel{^}{H}\left({t}^{\prime }\right)|{\lambda }_{n},{t}^{\prime }〉,$ (10)

${C}_{n}={〈{\lambda }_{n},0|\psi \left(0\right)〉}_{s}.$

$\stackrel{^}{I}\left(t\right)=\alpha \left(t\right){\sigma }_{+}+{\alpha }^{*}\left(t\right){\sigma }_{-}+\beta \left(t\right){\sigma }_{z}$ (11)

$i\stackrel{˙}{\alpha }\left(t\right)+2\left(A\beta -\alpha \beta \right)=0,\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\stackrel{˙}{\beta }+A\left(\alpha -{\alpha }^{*}\right)=0.$ (12)

$\beta \mathrm{cos}\left(2|\xi |\right)+\alpha \xi \sqrt{2|\xi |}\left[\mathrm{sin}\sqrt{2|\xi |}+\frac{1}{\sqrt{2|\xi |}}-1\right]=1$ (13)

$\alpha +\frac{\xi |\beta |}{|\xi |}\left[\mathrm{sin}\left(2|\xi |+2|\xi |-1\right)\right]=1,$ (14)

${\stackrel{^}{I}}_{V}\equiv {\stackrel{^}{V}}^{†}\left(t\right)\stackrel{^}{I}\left(t\right)\stackrel{^}{V}\left(t\right)={\sigma }_{z}.$ (15)

${\stackrel{^}{V}}^{†}\left(t\right)\frac{\partial \stackrel{^}{V}\left(t\right)}{\partial t}=\frac{\partial \stackrel{^}{L}}{\partial t}+\frac{1}{2!}\left[\frac{\partial \stackrel{^}{L}}{\partial t},\stackrel{^}{L}\right]+\frac{1}{3!}\left[\left[\frac{\partial \stackrel{^}{L}}{\partial t},\stackrel{^}{L}\right],\stackrel{^}{L}\right]+\frac{1}{4}\left[\left[\left[\frac{\partial \stackrel{^}{L}}{\partial t},\stackrel{^}{L}\right],\stackrel{^}{L}\right],\stackrel{^}{L}\right]+\cdots ,$ (16)

$A+B+\frac{\beta \xi }{|\xi |}\left[\mathrm{sin}\left(2|\xi |\right)+2|\xi |-1\right]+\frac{2\xi \left(\xi {\stackrel{˙}{\xi }}^{*}-\stackrel{˙}{\xi }{\xi }^{*}\right)}{{\left(2|\xi |\right)}^{3}}\left[\mathrm{sin}\left(2|\xi |\right)-2{|\xi |}^{3}\right]=0,$ (17)

${\stackrel{^}{H}}_{V}\left(t\right)\equiv {\stackrel{^}{V}}^{†}\left(t\right)\stackrel{^}{I}\left(t\right)\stackrel{^}{V}\left(t\right)-i{\stackrel{^}{V}}^{†}\left(t\right)\frac{\partial \stackrel{^}{V}\left(t\right)}{\partial t}=\lambda {\sigma }_{z},$ (18)

$\lambda =i\left[1-\mathrm{cos}\left(2|\xi |\right)\right]\left(\xi {\stackrel{˙}{\xi }}^{*}-\stackrel{˙}{\xi }{\xi }^{*}\right)+A\left[\mathrm{sin}\sqrt{2|\xi |}+\frac{1}{2\sqrt{2|\xi |}}-1\right]\sqrt{2|\xi |}\left[\alpha \xi +{\alpha }^{*}{\xi }^{*}\right]$ (19)

${\stackrel{˙}{\delta }}^{g}\left(t\right)=i\left[1-\mathrm{cos}\left(2|\xi |\right)\right]\left(\xi \stackrel{˙}{\xi }-\stackrel{˙}{\xi }{\xi }^{*}\right).$ (20)

4. 结论

Geometric Phase in a Quantum Coherent Tunable Coupling of Superconducting Qubits[J]. 凝聚态物理学进展, 2018, 07(01): 22-26. http://dx.doi.org/10.12677/CMP.2018.71004

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