Introduction 038

In the present paper, we investigate the interaction between the inertial force field and the intrinsic spin of a particle. According to the equivalence principle, the nature of the inertial force is gravitational force, and consequently both expressions of these two gravitomagnetic forces (namely, the gravitational Lorentz force and Coriolis force) can be derived from the equation of gravitational field. It is known that Mashhoon$^{,}$s approach to deriving the intrinsic spin -rotation coupling is suggested by analyzing the Doppler$^{,}$s effect of wavelight in the rotating frame with respect to the fixing frame \QCITE{cite}{}{Mashhoon1,Mashhoon2}. In this paper, however, the Hamiltonian of the coupling of the intrinsic spin of a particle with the rotating frequency of a rotating reference frame is obtained by comparing the inertial centrifugal force and Coriolis force with the electric force and magnetic force in electrodynamics\QCITE{cite}{}{Shen}. The reason why the coupling of spin (or gravitomagnetic moment) with noninertial frame is of great importance lies in that, with the development of laser technology and their applications to the gravitational interferometry experiment\QCITE{cite}{}{Ahmedov,Ciufolini,Hayasaka}, it becomes possible for us to investigate quantum mechanics in weak-gravity field. The utilization of these relativistic quantum gravitational effects enables physicists to test the fundamental principles of general relativity in microscopic areas. Although the equivalence principle still holds in the relativistic quantum gravitational effect\QCITE{cite}{}{Mashhoon2}, there are some physically interesting phenomena such as the violation of the principle of free falling body for the spinning particle\QCITE{cite}{}{Mashhoon2,Mashhoon3} moving in, for instance, the Kerr spacetime. 038

Since the analogy can be drawn between gravity and electromagnetic force in some aspects, Aharonov and Carmi proposed the geometric effect of the vector potential of gravity, and Anandan, Dresden and Sakurai et al. proposed the quantum-interferometry effect associated with gravity\QCITE{cite}{}{Anandan,Dresden}. In the rotating reference frame, a particle was acted on by the inertial centrifugal force and Coriolis force, which are respectively analogous to the electric force and magnetic force in electrodynamics\QCITE{cite}{}{Shen}. The matter wave in the rotating frame propagating along a closed path will thus possess a nonintegral phase factor (geometric phase factor), which has been called the Aharonov-Carmi effect, or the gravitational Aharonov-Bohm effect. Overhauser, Colella\QCITE{cite}{}{Overhauser}, Werner and Standenmann et al.\QCITE{cite}{}{Werner} have proved the existence of the Aharonov-Carmi effect by means of the neutron-gravity interferometry experiment. Note, Aharonov-Carmi effect results from the interaction between the momentum of a particle and the rotating frame. Although the interaction of a spinning particle such as neutron with the rotating frame has the same origin of the Aharonov-Carmi effect, i.e., both arise from the presence of the Coriolis force, the Aharonov-Carmi effect mentioned above does not contain the spin-rotation coupling. In the following we further investigate another geometric effect that a spinning particle possesses a geometric phase in the time-dependent rotating frame. 038

Berry$^{,}$s theory of the geometric phase proposed in 1984 is applicable only to the case of adiabatic approximation \QCITE{cite}{}{Berry}. In 1991, on the basis of the Lewis-Riesenfeld invariant theory \QCITE{cite}{}{Lewis}, Gao et al. proposed the invariant-related unitary transformation formulation that is appropriate to treat the cases of non-adiabatic and non-cyclic process\QCITE{cite}{}{Gao1}. Hence, the Lewis-Riesenfeld invariant theory is developed into a generalized invariant theory which is a powerful tool to investigate the geometric phase factor\QCITE{cite}{}{Gao3,Gao4}. In Sec.2, the time-dependent spin-rotation coupling is taken into consideration by using these invariant theories, and then we obtain exact solutions of the time-dependent Schr\"{o}dinger equation which governs the interaction between a spinning particle and the time-dependent rotating reference frame. 036

Spin-rotation coupling 038

Here we suggest a simple method differing from Mashhoon$^{,}$s formulation to derive the Hamiltonian of spin-rotation coupling. Consider a particle that is moving with velocity $\vec{v}$ with respect to the rotating reference frame, the forces acting on the particle observed from the observer in the rotating frame are the inertial centrifugal force $\vec{F}_{cen}$ and Coriolis force $\vec{F}_{cor}$ that can be written respectively 038

\EQN{0}{1}{}{0,}{\RD{\CELL{\vec{F}_{cen}=-m\vec{\omega}\times (\vec{\omega}\times \vec{r}),\quad \vec{F}_{cor}=2m\vec{v}\times \vec{\omega},}}{1}{}{}{}}where the following relations are satisfied 038

\EQN{0}{1}{}{0,}{\RD{\CELL{\nabla \times \vec{F}_{cen}=0,\quad \nabla \cdot \vec{F}_{cor}=0.}}{1}{}{eq2}{}}Eq. (\QTSN{ref}{eq2}) shows $\vec{F}_{cen}$ is a conservative force, and one can introduce a vector potential $\vec{a}$ and a scalar potential $a_{0}$, where 038

\EQN{0}{1}{}{0,}{\RD{\CELL{\nabla \times \vec{a}=\vec{\omega},\quad \nabla a_{0}=\vec{\omega}\times (\vec{\omega}\times \vec{r}).}}{1}{}{eq3}{}}It follows from Eq. (\QTSN{ref}{eq1}) and Eq. (\QTSN{ref}{eq3}) that the inertial centrifugal force $\vec{F}_{cen}$ and Coriolis force $\vec{F}_{cor}$ are in analogy with the electric force and magnetic force in electrodynamics. 038

The Lagrangian of this particle in the rotating frame is 038

\EQN{0}{1}{}{0,}{\RD{\CELL{L=\frac{1}{2}mv^{2}-ma_{0}+2m\vec{v}\cdot \vec{a}.}}{1}{}{eq4}{}}It follows that the matter wave of neutron acquires an integral phase when it propagates along a closed path 038

\EQN{0}{1}{}{0,}{\RD{\CELL{\Delta \phi =2m\oint \vec{a}\cdot d\vec{l}=2m\vec{\omega}\cdot \vec{A}}}{1}{}{eq5}{}} where $\vec{A}$ denotes the area vector surrounded by the closed path. As is known to all, this phenomenon is called Aharonov-Carmi effect. 038

From Eq. (\QTSN{ref}{eq4}) we can obtain the following Hamiltonian 038

\EQN{0}{1}{}{0,}{\RD{\CELL{H=\frac{1}{2m}p^{2}+ma_{0}+\vec{\omega}\cdot \vec{L}+\frac{1}{2}m(\vec{\omega}\times \vec{r})^{2}}}{1}{}{eq6}{}}with $\vec{L}=\vec{r}\times \vec{p}$ being the orbital angular momentum. For a spinning particle such as neutron, there exists the interaction of gravitomagnetic moment with gravitomagnetic field that is similar to the interaction between magnetic moment and magnetic field in electrodynamics. In view of above discussions, we can give the Hamiltonian describing the coupling of neutron spin and gravitomagnetic field $\vec{\omega}$ as follows 038

\EQN{0}{1}{}{0,}{\RD{\CELL{H_{s-g}=\zeta (\nabla \times \vec{a})\cdot \vec{S}=\zeta \vec{\omega}\cdot \vec{S}}}{1}{}{eq8}{}}with $\zeta $ being the constant parameter which is determined from many other approaches such as from the Dirac equation with spin connections\QCITE{cite}{}{Hehl} and the result is $\zeta =1.$ We thus obtain the Hamiltonian of spin-rotation coupling 038

\EQN{0}{1}{}{0,}{\RD{\CELL{H_{s-r}=\vec{\omega}\cdot \vec{S}}}{1}{}{eq14}{}}which is consistent with Mashhoon$^{,}$s result\QCITE{cite}{}{Mashhoon1}. It follows from Eq. (\QTSN{ref}{eq8}) that the interaction of gravitomagnetic moment with gravitomagnetic field embodies the spin-rotation coupling in the noninertial reference frame. Once the gravitomagnetic field $\vec{\omega}$ is time-dependent, the matter wave of neutron possesses another integral phase factor due to spin-rotation coupling. It is believed that this geometric effect will be tested by means of neutron-gravity interferometry experiment in the near future. 036

Exact solutions of time-dependent spin-rotation coupling 038

The variation of the Earth$^{,}$s rotating frequency may be caused by the motion of interior matter, tidal force, and the motion of atmosphere as well. Once we have information concerning the Earth$^{,}$s rotating frequency, it is possible to investigate the motion of matter on the Earth. For the sake of detecting the fluctuation of the Earth$^{,}$s time-dependent rotation conveniently, we suggest a potential approach to measuring the geometric phase factor arising from the interaction of neutron spin with the Earth$^{,}$s rotation by using the neutron interferometry experiment. First we should exactly solve the time-dependent Schr\"{o}dinger equation of a spinning particle in the rotating system. 038

The Schr\"{o}dinger equation which governs the interaction of neutron spin with Earth$^{,}$s rotation is 038

\EQN{0}{1}{}{}{\RD{\CELL{i\frac{\partial }{\partial t}\left| \Psi (t)\right\rangle _{s}=H_{s-r}(t)\left| \Psi (t)\right\rangle _{s}.}}{1}{}{eq33}{}}Set $\vec{\omega}(t)=\omega _{0}(t)[\sin \theta (t)\cos \varphi (t),\sin \theta (t)\sin \varphi (t),\cos \theta (t)],$ and $\sigma _{\pm }=\sigma _{1}\pm i\sigma _{2}$ with $\sigma _{1},\sigma _{2}$ being Pauli matrices$,$ then the expression (\QTSN{ref}{eq14}) for $H_{s-r}(t)$ can be rewritten as 038

\EQN{1}{1}{}{}{\RD{\CELL{H_{s-r}(t) &=&\omega _{0}(t)\{\frac{1}{4}\sin \theta (t)\exp [-i\varphi (t)]\sigma _{+}+\frac{1}{4}\sin \theta (t)\exp [i\varphi (t)]\sigma _{-}}}{0}{}{}{}\RD{\CELL{&&+\frac{1}{2}\cos \theta (t)\sigma _{3}\}.}}{1}{}{eq341}{}} 038

In accordance with the invariant theory, an invariant which satisfies the following invariant equation\QCITE{cite}{}{Lewis} 038

\EQN{0}{1}{}{}{\RD{\CELL{\frac{\partial I(t)}{\partial t}+\frac{1}{i}[I(t),H_{s-r}(t)]=0}}{1}{}{eq340}{}}should be constructed often in terms of the generators of Hamiltonian (\QTSN{ref}{eq341}). Then it follows from Eq. (\QTSN{ref}{eq340}) that the invariant may be written in terms of \ Pauli matrices as follows 038

\EQN{0}{1}{}{}{\RD{\CELL{I(t)=\frac{1}{4}\sin \lambda (t)\exp [-i\gamma (t)]\sigma _{+}+\frac{1}{4}\sin \lambda (t)\exp [i\gamma (t)]\sigma _{-}+\frac{1}{2}\cos \lambda (t)\sigma _{3},}}{1}{}{}{}}where the time-dependent parameters $\lambda (t)$ and $\gamma (t)$ satisfy the following two auxiliary equations 038

\EQN{0}{1}{}{}{\RD{\CELL{\dot{\lambda}(t)=\omega _{0}(t)\sin \theta \sin (\varphi -\gamma ),\quad \dot{\gamma}(t)=\omega _{0}(t)[\cos \theta -\sin \theta \cot \lambda \cos (\varphi -\gamma )]}}{1}{}{eq342}{}}with dot denoting the time derivative. It is readily verified by using Eq. (\QTSN{ref}{eq342}) that the invariant $I(t)$ has time-independent eigenvalue $\sigma =\pm \frac{1}{2}$ and its eigenvalue equation is 038

\EQN{0}{1}{}{}{\RD{\CELL{I(t)\left| \sigma ,t\right\rangle =\sigma \left| \sigma ,t\right\rangle .}}{1}{}{}{}} 038

According to the Lewis-Riesenfeld invariant theory, the particular solution $\left| \sigma ,t\right\rangle _{s}$ of Eq. (\QTSN{ref}{eq33}) is different from the eigenfunction $\left| \sigma ,t\right\rangle $ of the invariant $I(t)$ only by a phase factor $\exp [i\phi _{\sigma }(t)]$. Then the general solution of the Schr\"{o}dinger equation (\QTSN{ref}{eq33}) can be written as 038

\EQN{0}{1}{}{}{\RD{\CELL{\left| \Psi (t)\right\rangle _{s}=\tsum_{\sigma }C_{\sigma }\exp [i\phi _{\sigma }(t)]\left| \sigma ,t\right\rangle ,}}{1}{}{eq25}{}}where 038

\EQN{1}{1}{}{}{\RD{\CELL{\phi _{\sigma }(t) &=&\int_{0}^{t}\left\langle \sigma ,t^{^{\prime }}\right| i\frac{\partial }{\partial t^{^{\prime }}}-H_{s-r}(t^{^{\prime }})\left| \sigma ,t^{^{\prime }}\right\rangle dt^{^{\prime }},}}{0}{}{}{}\RD{\CELL{C_{\sigma } &=&\langle \sigma ,t=0\left| \Psi (0)\right\rangle _{s}.}}{1}{}{eq26}{}} 038

In order to obtain the analytic solution of the time-dependent Schr\"{o}dinger equation (\QTSN{ref}{eq33}), we introduce an invariant-related unitary transformation operator $V(t)$ 038

\EQN{0}{1}{}{}{\RD{\CELL{V(t)=\exp [\frac{\beta (t)}{2}\sigma _{+}-\frac{\beta ^{\ast }(t)}{2}\sigma _{-}],}}{1}{}{eq36}{}}where the time-dependent parameter 038

\EQN{0}{1}{}{}{\RD{\CELL{\beta (t)=-\frac{\lambda (t)}{2}\exp [-i\gamma (t)],\quad \beta ^{\ast }(t)=-\frac{\lambda (t)}{2}\exp [i\gamma (t)].}}{1}{}{eq37}{}}$V(t)$ can be easily shown to transform the time-dependent invariant $I(t)$ to $I_{V}(t)$ which is time-independent: 038

\EQN{0}{1}{}{}{\RD{\CELL{I_{V}\equiv V^{\dagger }(t)I(t)V(t)=\frac{1}{2}\sigma _{3}.}}{1}{}{eq38}{}}The eigenstate of the $I_{V}=\frac{1}{2}\sigma _{3}$ corresponding to the eigenvalue $\sigma $ is denoted by $\left| \sigma \right\rangle $ which is of the form 038

\EQN{0}{1}{}{}{\RD{\CELL{\left| \sigma \right\rangle =V^{\dagger }(t)\left| \sigma ,t\right\rangle .}}{1}{}{}{}}By making use of $V(t)$ in expression (\QTSN{ref}{eq36}) and the Baker-Campbell-Hausdorff formula\QCITE{cite}{}{Wei}, one can obtain $H_{V}(t)$ from $H_{s-r}(t)\QCITE{cite}{}{Gao1}$ 038

\EQN{7}{1}{}{}{\RD{\CELL{H_{V}(t) &=&V^{\dagger }(t)H_{s-r}(t)V(t)-V^{\dagger }(t)i\frac{\partial V(t)}{\partial t}}}{1}{}{}{}\RD{\CELL{&=&\frac{1}{2}\{[\cos \lambda \cos \theta +\sin \lambda \sin \theta \cos (\gamma -\varphi )]+\dot{\gamma}(1-\cos \lambda )\}\sigma _{3}.}}{1}{}{}{}}From Eqs.(\QTSN{ref}{eq342}), it is shown that 038

\EQN{0}{1}{}{}{\RD{\CELL{\cos \lambda \cos \theta +\sin \lambda \sin \theta \cos (\gamma -\varphi )=0,}}{1}{}{eq312}{}}thus, the expression (\QTSN{ref}{eq39}) can be rewritten as 038

\EQN{0}{1}{}{}{\RD{\CELL{H_{V}(t)=\frac{1}{2}\dot{\gamma}(t)[1-\cos \lambda (t)]\sigma _{3}.}}{1}{}{eq313}{}}Based on (\QTSN{ref}{eq26}) and (\QTSN{ref}{eq313}), the geometric phase of the neutron whose eigenvalue of spin is $\sigma $ can be expressed by 038

\EQN{0}{1}{}{}{\RD{\CELL{\phi _{\sigma }(t)=-\frac{1}{2}\{\tint_{0}^{t}\dot{\gamma}(t^{^{\prime }})[1-\cos \lambda (t^{^{\prime }})]dt^{^{\prime }}\}\left\langle \sigma \right| \sigma _{3}\left| \sigma \right\rangle .}}{1}{}{eq314}{}} 038

Since we know the eigenvalues and eigenstates of $I_{V}(t)=\frac{1}{2}\sigma _{3},$ with the help of (\QTSN{ref}{eq25}), (\QTSN{ref}{eq26}), and (\QTSN{ref}{eq314}), it is easy to get the general solution of the time-dependent Schr\"{o}dinger equation which governs the neutron spin-rotation coupling is given 038

\EQN{0}{1}{}{}{\RD{\CELL{\left| \Psi (t)\right\rangle _{s}=\tsum_{\sigma }C_{\sigma }\exp [i\phi _{\sigma }(t)]V(t)\left| \sigma \right\rangle}}{1}{}{eq315}{}}with the coefficients $C_{\sigma }=\langle \sigma ,t=0\left| \Psi (0)\right\rangle _{s}.$ 038

It follows from the expression (\QTSN{ref}{eq312}) that the dynamical phase of solutions of Eq. (\QTSN{ref}{eq33}) vanishes, and the geometric phase is expressed by Eq. (\QTSN{ref}{eq314}). Since geometric phase appears only in systems whose Hamiltonian is time-dependent or possessing some evolution parameters, this enables us to obtain the information concerning the variation of the Earth$^{,}$s rotation by measuring the geometric phase difference of spin polarized vertically down and up in the neutron-gravity interferometry experiment. 036

Concluding remarks 038

This paper obtains the expression for the Hamiltonian of spin-rotation coupling by coordinate transformation of Kerr metric from the fixing reference frame to the rotating reference frame. By making use of the Lewis-Riesenfeld invariant theory and the invariant-related unitary transformation formulation, we obtain exact solutions of the time-dependent Schr\"{o}dinger equation governing the interaction of neutron spin with Earth$^{,}$s rotation. We propose a potential method to investigate the time-varying rotating frequency of the Earth by measuring the phase difference between geometric phases of neutron spin down and up. In view of the above discussions, the invariant-related unitary transformation formulation is a useful tool for treating the geometric phase factor and the time-dependent Schr\"{o}dinger equation. This formulation replaces the eigenstates of the time-dependent invariants with those of the time-independent invariants through the unitary transformation. Additionally, it should be pointed out that the time-dependent Schr\"{o}dinger equation is often seen in the literature, whereas the exact solutions of time-dependent Klein-Gordon equation is paid less attention to. Work in this direction is under consideration and will be published elsewhere. 038

\TeXButton{Acknowledgment}{Acknowledgment} This project was supported by the National Natural Science Foundation of China under the project No.$30000034$. 059

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