Tracking quantum state evolution by the Berry curvature with a two-level system

Abstract

We investigate two kinds of topological structures (sphere and torus) spanned by the controlled parameters of a driven two-level system’s Hamiltonian and consider the connection between the structures and the system’s dynamics. We discuss the Berry curvature obtained through the dynamical response method, show the certain physical and observable manifolds including the gapped region probed by integrating the Berry curvature and demonstrate the system’s state evolution can be tracked and manipulated by extracting the Berry curvature.

Appendix A: A concise comparison between two approaches for calculating the Berry curvature

In general case, the first Chern number of the ground state manifold is encoded in the gauge-invariant curvature 2-form which can be written as [3]

$$\begin{aligned} {\mathcal {B}=\textit{B}_{{\theta \phi }}\mathrm{d}\theta \wedge \mathrm{d}\phi ,} \end{aligned}$$

(A1)

where $\textit{B}_{{\theta \phi }}$ is the conventional Berry curvature. For our two-level Hamiltonian, the conventional Berry curvature of the ground state in the gapped region can be expressed as

$$\begin{aligned} \textit{B}_{{\theta \phi }}=\frac{1}{2}\hat{x}\cdot \left( \frac{\partial \hat{x}}{\partial \theta }\times \frac{\partial \hat{x}}{\partial \phi }\right) \end{aligned}$$

(A2)

via the normalized vector component $\hat{x}={\pmb {x}}/\Vert \pmb {x}\Vert$, which in the sphere and the torus cases are, respectively,

$$\begin{aligned} \pmb {x}_{\text {sph}}= & {} \left( {x}^{1},{x}^{2},{x}^{3}\right) _{\text {sph}} =\left\{ \hat{\sigma }^{\mu }\right\} _{\text {sph}} =\left\{ \hat{\sigma }^{x},\hat{\sigma }^{y},\hat{\sigma }^{z}\right\} _{\text {sph}}=\left( \Omega \cos \phi ,~\Omega \sin \phi ,~\Delta \right) \end{aligned}$$

(A3)

and

$$\begin{aligned} \pmb {x}_{\text {tor}}= & {} ({x}^{1},{x}^{2},{x}^{3})_{\text {tor}} =\left\{ \hat{\sigma }^{\mu }\right\} _{\text {tor}} =\left\{ \hat{\sigma }^{x},\hat{\sigma }^{y},\hat{\sigma }^{z}\right\} _{\text {tor}} =\left( \Delta \cos \phi ,~\Delta \sin \phi ,~\Omega \right) . \end{aligned}$$

(A4)

We can then calculate the conventional Berry curvatures of the two families via Eq. (A2) to be

$$\begin{aligned} \textit{B}_{{\theta \phi }}^\mathrm{{S}^2} =\frac{\Omega _1^2\sin \theta \left( \Delta _2\cos \theta +\Delta _1\right) }{2\left[ \left( \Delta _1\cos \theta +\Delta _2\right) ^2+\Omega _1^2\sin ^2\theta \right] ^{3/2}} \end{aligned}$$

(A5)

and

$$\begin{aligned} \textit{B}_{{\theta \phi }}^\mathrm{{T}^2}=-\frac{\Omega _1\left( 2\left( \Delta _1^2+\Delta _2^2\right) \cos \theta +\Delta _2 \Delta _1(\cos 2\theta +3)\right) }{4\left[ \left( \Delta _1\cos \theta +\Delta _2\right) ^2+\Omega _1^2\sin ^2\theta \right] ^{3/2}}. \end{aligned}$$

(A6)

The corresponding first Chern number of them can be calculated as

$$\begin{aligned} \mathbb {C}_1\left( \text {S}^2\right) =\frac{1}{2\pi }\int _{0}^{\pi }\mathrm{d}\theta \int _{0}^{2\pi }\mathrm{d}\phi ~\textit{B}_{\theta \phi }^\mathrm{{S}^2} =\frac{1}{2}\left( \frac{\Delta _1-\Delta _2}{\sqrt{\left( \Delta _1-\Delta _2\right) {}^2}}+\frac{\Delta _1+\Delta _2}{\sqrt{\left( \Delta _1+\Delta _2\right) {}^2}}\right) \end{aligned}$$

(A7)

and

$$\begin{aligned} \mathbb {C}_1\left( \text {T}^2\right) =\frac{1}{2\pi } \int _{0}^{2\pi }\mathrm{d}\theta \int _{0}^{2\pi }\mathrm{d}\phi ~\textit{B}_{\theta \phi }^\mathrm{{T}^2}=0. \end{aligned}$$

(A8)

From Eq. (A7), we notice that the first Chern number is not well defined for $\Delta _1=\pm \Delta _2$.

We now numerically calculate the Berry curvature using the dynamical method as follows [2]:

$$\begin{aligned} \textit{B}_{{\theta \phi }}^{\text {sph}}= & {} \frac{\langle \psi (t)|\partial _{\phi }\hat{\textit{H}}_{\text {sph}}|\psi (t)\rangle }{\theta _t^{\text {sph}}}\nonumber \\= & {} \frac{1}{2\theta _t^{\text {sph}}} \bigg \langle \psi (t)\bigg |\partial _{\phi }\left( \begin{array}{cc} \Delta &{} \Omega {\textit{e}}^{-\textit{i}\phi } \\ \Omega {\textit{e}}^{\textit{i}\phi } &{} -\Delta \\ \end{array} \right) \bigg |\psi (t)\bigg \rangle =\frac{\Omega _1\sin \theta }{2\theta _t^{\text {sph}}} \bigg \langle \psi (t)\bigg |\left( \begin{array}{cc} 0 &{} -i \\ i &{} 0\\ \end{array} \right) \bigg |\psi (t)\bigg \rangle \end{aligned}$$

(A9)

$$\begin{aligned}= & {} \frac{\Omega _1\sin \theta }{2\theta _t^{\text {sph}}} \langle \hat{\sigma }^y\rangle =\frac{\Omega _1\sin \theta }{2\theta _t^{\text {sph}}} \text {Tr}\left( \hat{\rho }\hat{\sigma }^y\right) \end{aligned}$$

(A10)

and

$$\begin{aligned} \textit{B}_{{\theta \phi }}^{\text {tor}}= & {} \frac{\langle \psi (t)|\partial _{\phi }\hat{\textit{H}}_{\text {tor}}|\psi (t)\rangle }{\theta _t^{\text {tor}}}\nonumber \\= & {} \frac{1}{2\theta _t^{\text {tor}}} \bigg \langle \psi (t)\bigg |\partial _{\phi }\left( \begin{array}{cc} \Omega &{} \Delta {\textit{e}}^{-\textit{i}\phi } \\ \Delta {\textit{e}}^{\textit{i}\phi } &{} -\Omega \\ \end{array} \right) \bigg |\psi (t)\bigg \rangle =\frac{\Delta _1\cos \theta +\Delta _2}{2\theta _t^{\text {tor}}} \bigg \langle \psi (t)\bigg |\left( \begin{array}{cc} 0 &{} -i \\ i &{} 0\\ \end{array} \right) \bigg |\psi (t)\bigg \rangle \nonumber \\ \end{aligned}$$

(A11)

$$\begin{aligned}= & {} \frac{\Delta _1\cos \theta +\Delta _2}{2\theta _t^{\text {tor}}}\langle \hat{\sigma }^y\rangle =\frac{\Delta _1\cos \theta +\Delta _2}{2\theta _t^{\text {tor}}} \text {Tr}\left( \hat{\rho }\hat{\sigma }^y\right) ,\nonumber \\ \end{aligned}$$

(A12)

where $\hat{\rho }$ is the density matrix of the lower dressed state $|\psi (t)\rangle =|\psi _g\rangle$ in Eq. (22). Equations (6), (A9) and (A11) show the approach to measure Berry curvature is a leading order approximation, good to the first order in the ramping rate $\theta _{t}$ (sphere case). However, we notice that the first Chern number in the torus case is no longer a constant number outside the gapped region. The various oscillations in Figs. 3 and 5 noted in the text could not happen in the case of the conventional Berry curvature. From Ref. [2], we know that the method is not that perfect when the ramp velocity is sufficiently large (large deviation from the linear response regime). Therefore, the dynamically non-adiabatic response method we adopt here actually is not equivalent to the conventional one under certain circumstances.