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036 Introduction
038 One can easily verify that the field equation of general relativity in low-motion weak-field approximation is analogous to Maxwell$^{,}$s equation of electromagnetic field. It is the most outstanding point that the former field (gravitational field) also possesses both the gravitoelectric potential written as $\frac{g_{00}-1}{2}$ and the gravitomagnetic potential as $\vec{A}=(g_{01},g_{02},g_{03}),$ and the corresponding gravitomagnetic field strength is of the form $\vec{B}=-\frac{1}{2}\nabla \times \vec{A}$. A particle with intrinsic spin possesses the gravitomagnetic moment of such magnitude that it equals the spin of this particle. The interaction between the gravitomagnetic moment and the gravitomagnetic field is thus also called the spin-gravity coupling\QCITE{cite}{}{Kleinert,Mashhoon1}, of which the Hamiltonian is given by
038 \EQN{0}{1}{}{}{\RD{\CELL{H=\frac{1}{2}\vec{B}\cdot \vec{S}.}}{1}{}{eq1}{}}
038 The strongest gravitomagnetic field that we can find on the Earth arises from the Earth$^{,}$s rotation. Since the Earth is a noninertial reference frame due to its rotation, a spinning particle is coupled to a more strong gravitomagnetic field, which represents the spin-noninertial frame coupling in addition to the interaction expressed by Eq. (\QTSN{ref}{eq1}). It is apparently seen that the interaction of angular momentum of a particle with noninertial frame is related to the Coriolis force\QCITE{cite}{}{Shen}. These two gravitomagnetic fields have different origins and properties: the gravitomagnetic field caused by mass current, expressed by $\vec{B}=-\frac{1}{2}\nabla \times \vec{A},$ is analogous to the magnetic field produced by electric current, and its strength is dependent on the Newtonian gravitational constant $G,$ while the gravitomagnetic field associated with the Coriolis force depends on the choice of the coordinates and in consequence its strength is independent of the Newtonian gravitational constant. That is, in accordance with Newton$^{,}$s law the coordinate transformation from the rotating frame to the fixing frame results in this inertial force observed by the observer fixed in the rotating reference frame. Apparently, due to the smallness of $G,$ the coupling of the latter gravitomagnetic field with intrinsic spin is much stronger than that of Eq. (\QTSN{ref}{eq1}).
038 In the present paper, we investigate the interaction between this 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 the expressions of these two gravitomagnetic forces can be derived from the equation of gravitational field. This work is given in what follows and we thus obtain the Hamiltonian of the spin-rotation coupling. 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 in the fixing frame\QCITE{cite}{}{Mashhoon1,Mashhoon2}. In this paper, however, the transformation of the gravitomagnetic potential is studied through the coordinate transformation, and as a result, the Hamiltonian of the coupling of the intrinsic spin of a particle with the rotating frequency of a rotating reference frame is then obtained.
038 The reason why the coupling of spin (or gravitomagnetic moment) with noninertial frame is of great importance is that, with the development of laser technology and their applications to the gravitational interferometry experiment\QCITE{cite}{}{Ahmedov,Ciufolini,Hayasaka}, it is possible 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. Although the equivalence principle is still accurate\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} in, for instance, the Kerr space-time.
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 will thus possess a nonintegral 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 making use 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 a spinning particle such as neutron interacting 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 will propose another geometric effect that a spinning particle possesses a geometric phase in the time-dependent rotating frame. 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}{}{Gao2,Gao3}. 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 Gravitomagnetic field and spin-rotation coupling
038 The Kerr metric of the exterior gravitational field of the ratating spherically symmetric body is of the form
038 \EQN{1}{1}{}{}{\RD{\CELL{ds^{2} &=&(1-\frac{2GMr}{c^{2}(r^{2}+a^{2}\cos ^{2}\theta )})c^{2}dt^{2}-\frac{r^{2}+a^{2}\cos ^{2}\theta }{r^{2}+a^{2}-\frac{2GMr}{c^{2}}}dr^{2}}}{0}{}{}{}\RD{\CELL{&&-(r^{2}+a^{2}\cos ^{2}\theta )d\theta ^{2}-\sin ^{2}\theta (\frac{2a^{2}\sin ^{2}\theta }{r^{2}+a^{2}\cos ^{2}\theta }\frac{GMr}{c^{2}}}}{0}{}{}{}\RD{\CELL{&&+r^{2}+a^{2})d\varphi ^{2}+\frac{2a\sin ^{2}\theta }{r^{2}+a^{2}\cos ^{2}\theta }\frac{GMr}{c}dtd\varphi ,}}{1}{}{eq2}{}}where $r,\theta ,\varphi $ are the displacement of spherical coordinate, $a$ is so defined that $ac$ is the angular momentum of unit mass of the gravitational body, and $M$ denotes the mass of this gravitational body. Since the space-time coordinate of Kerr metric (\QTSN{ref}{eq2}) is in the fixing reference frame, we can transform it into that in the fixing reference frame. Due to the smallness of the Earth$^{,}$s rotating velocity, one can use the Galileo transformation
038 \EQN{0}{1}{}{}{\RD{\CELL{dr^{^{\prime }}=dr+vdt,\quad d\theta ^{^{\prime }}=d\theta ,\quad d\varphi ^{^{\prime }}=d\varphi +\omega dt,\quad dt^{^{\prime }}=dt}}{1}{}{eq3}{}}with $v$ being the radial velocity of a particle moving relative to the rotating reference frame, $($ $r^{^{\prime }},\theta ^{^{\prime }},\varphi ^{^{\prime }},t^{^{\prime }})$ and $(r,\theta ,\varphi ,t)$ the space-time coordinates of the rotating frame and fixing frame, respectively. $\omega $ denotes the rotating frequency of the rotating frame with respect to the fixing reference frame. Substitution of Eq. (\QTSN{ref}{eq3}) into Eq. (\QTSN{ref}{eq2}) yields
038 \EQN{1}{1}{}{}{\RD{\CELL{ds^{2} &=&[1-\frac{2GMr}{c^{2}(r^{2}+a^{2}\cos ^{2}\theta )}-\frac{(r^{2}+a^{2}\cos ^{2}\theta )}{r^{2}+a^{2}-\frac{2GMr}{c^{2}}}\frac{v^{2}}{c^{2}}}}{0}{}{}{}\RD{\CELL{&&-\sin ^{2}\theta (r^{2}+a^{2}+\frac{2a^{2}\sin ^{2}\theta }{r^{2}+a^{2}\cos ^{2}\theta }\frac{GMr}{c^{2}})\frac{\omega ^{2}}{c^{2}}}}{0}{}{}{}\RD{\CELL{&&-\frac{2a\sin ^{2}\theta }{r^{2}+a^{2}\cos ^{2}\theta }\frac{GMr}{c}\frac{\omega ^{2}}{c^{2}}]c^{2}dt^{^{\prime }2}-\frac{r^{2}+a^{2}\cos ^{2}\theta }{r^{2}+a^{2}-\frac{2GMr}{c^{2}}}dr^{^{\prime }2}}}{0}{}{}{}\RD{\CELL{&&-(r^{2}+a^{2}\cos ^{2}\theta )d\theta ^{^{\prime }2}-\sin ^{2}\theta (\frac{2a^{2}\sin ^{2}\theta }{r^{2}+a^{2}\cos ^{2}\theta }\frac{GMr}{c^{2}}}}{0}{}{}{}\RD{\CELL{&&+r^{2}+a^{2})d\varphi ^{^{\prime }2}+\frac{2(r^{2}+a^{2}\cos ^{2}\theta )}{r^{2}+a^{2}-\frac{2GMr}{c^{2}}}\frac{v}{c}dr^{^{\prime }}cdt^{^{\prime }}}}{0}{}{}{}\RD{\CELL{&&+[\frac{2a\sin ^{2}\theta }{r^{2}+a^{2}\cos ^{2}\theta }\frac{GMr}{c}+2\sin ^{2}\theta (r^{2}+a^{2}+}}{0}{}{}{}\RD{\CELL{&&\frac{2a^{2}\sin ^{2}\theta }{r^{2}+a^{2}\cos ^{2}\theta }\frac{GMr}{c^{2}})\omega ]dt^{^{\prime }}d\varphi ^{^{\prime }},}}{1}{}{eq4}{}}where $\frac{\omega ^{2}r^{2}}{c^{2}}\sin ^{2}\theta $ in $g_{tt}$ results in the inertial centrifugal force written as $\vec{F}=m\vec{\omega}\times (\vec{\omega}\times \vec{r}).$ Neglecting the terms associated with $\frac{a^{2}}{r^{2}}\ll 1$ in $g_{t\varphi ^{^{\prime }}},$ one can obtain
038 \EQN{1}{1}{}{}{\RD{\CELL{g_{t\varphi ^{^{\prime }}}d\varphi ^{^{\prime }}dt^{^{\prime }} &=&(\frac{2aGMr\sin ^{2}\theta }{cr^{2}}+2\omega r^{2}\sin ^{2}\theta )dt^{^{\prime }}d\varphi ^{^{\prime }}}}{0}{}{}{}\RD{\CELL{&=&(\frac{2aGM\sin \theta }{cr^{2}}+2\omega r\sin \theta )dt^{^{\prime }}r\sin \theta d\varphi ^{^{\prime }}.}}{1}{}{eq5}{}}Thus the gravitomagnetic potential can be written as \EQN{0}{1}{}{}{\RD{\CELL{A_{\varphi }=\frac{2aGM\sin \theta }{cr^{2}}+2\omega r\sin \theta ,\quad A_{r}=2v,\quad A_{\theta }=0.}}{1}{}{eq6}{}}It follows that the first term $\frac{2aGM\sin \theta }{cr^{2}}$ of $A_{\varphi }$ is exactly analogous to the magnetic potential $\frac{\mu _{0}}{4\pi }\frac{ea}{r^{2}}\sin \theta $ of the rotating charged sphrical shell in the electrodynamics. Then we can calculate the exterior gravitomagnetic strength of the rotating gravitational body, and the result is $\vec{B}_{g}=\frac{2G}{c}(\frac{\vec{a}}{r^{3}}-\frac{3(\vec{a}\cdot \vec{r})\vec{r}}{r^{5}})\QCITE{cite}{}{Ahmedov}.$
038 In accordance with the equation of geodesic line of a particle in the post-Newtonian approximation, the gravitomagnetic strength can be defined by $-\frac{1}{2}\nabla \times \vec{A}$ with $\vec{A}==(g_{01},g_{02},g_{03})$ as assumed above. Set \ $\beta _{\varphi }=2\omega r\sin \theta ,\beta _{r}=2v,\beta _{\theta }=0,$ then the gravitimagnetic strength that arises from the choice of the reference frame is given as follows:
038 \EQN{0}{1}{}{}{\RD{\CELL{-\frac{1}{2}\nabla \times \vec{\beta}=-2\omega \cos \theta e_{r}+2\omega \sin \theta e_{\theta }}}{1}{}{eq7}{}}with $e_{r},e_{\theta }$ being the unit vector. It follows from Eq.(\QTSN{ref}{eq7}) that this gravitimagnetic strength is related to the rotation of noninertial frame and independent of the Newtonian gravitational constant $G. $ From the point of view of Newtonian mechanics, it is the inertial force field in essence rather than the field that is produced by mass current. Since we have assumed that the velocity of a particle is parallel to $e_{r},$ i.e., $\vec{v}=ve_{r},$ the Lorentz force acting on the particle in the gravitomagnetic field is given by \EQN{0}{1}{}{}{\RD{\CELL{\vec{F}=m\vec{v}\times (-\frac{1}{2}\nabla \times \vec{\beta})=2v\omega \sin \theta e_{\varphi }=2m\vec{v}\times \vec{\omega},}}{1}{}{eq8}{}}We conclude from Eq. (\QTSN{ref}{eq8}) that the Lorentz force in rotating reference frame is the familiar Coriolis force and the rotating frequency $\vec{\omega}$ can be regarded as the gravitomagnetic field strength.
038 In the following we will derive the Hamitonian of spin-rotation coupling by investigate the Dirac equation with spinor connection
038 \EQN{0}{1}{}{}{\RD{\CELL{\lbrack i\gamma ^{\mu }(\partial _{\mu }-\frac{i}{4}\sigma ^{\lambda \tau }\omega _{\lambda \tau \mu })-mc]\psi =0}}{1}{}{eq9}{}}with $\sigma ^{\lambda \tau }=\frac{i}{2}(\gamma ^{\lambda }\gamma ^{\tau }-\gamma ^{\tau }\gamma ^{\lambda }).$ In the rotating frame, we have the following form of the line element of space-time
038 \EQN{0}{1}{}{}{\RD{\CELL{ds^{2}=(1-\frac{\omega ^{2}}{c^{2}}\vec{x}\cdot \vec{x})c^{2}dt^{2}-d\vec{x}\cdot d\vec{x}-2(\vec{\omega}\times \vec{x})\cdot d\vec{x}dt}}{1}{}{eq11}{}}by neglecting the gravitatinal effect associated with the gravitational constant $G$ and utilizing the weak-field low-motion approximation. Then further calculation yields the following spinor connections\QCITE{cite}{}{Hehl}
038 \EQN{1}{1}{}{}{\RD{\CELL{\omega _{\lambda \tau 0} &=&-\epsilon _{\lambda \tau \eta }\frac{\omega ^{\eta }}{c},\quad \omega _{0\tau 0}=-\omega _{\tau 00}=0,\quad \quad}}{0}{}{}{}\RD{\CELL{\omega _{\lambda \tau \mu } &=&0(\mu =1,2,3)}}{1}{}{eq10}{}}with $\epsilon _{\lambda \tau \eta }$ being three-dimensional Levi-Civita tensor. By making use of Eq. (\QTSN{ref}{eq9}), Eq.(\QTSN{ref}{eq11}) and Eq. (\QTSN{ref}{eq10}), one can arrive at the following Dirac equation
038 \EQN{0}{1}{}{}{\RD{\CELL{i\frac{\partial }{\partial t}\psi =H\psi}}{1}{}{eq12}{}}with
038 \EQN{0}{1}{}{}{\RD{\CELL{H=\beta mc^{2}+c\vec{\alpha}\cdot \vec{p}+\vec{\omega}\cdot \vec{L}+\vec{\omega}\cdot \vec{S}.}}{1}{}{EQ13}{}}We thus obtain the Hamitonian of spin-rotation coupling
038 \EQN{0}{1}{}{}{\RD{\CELL{H_{s-r}=\vec{\omega}\cdot \vec{S}}}{1}{}{eq14}{}}which is consistent with Mashhoon$^{,}$s result\QCITE{cite}{}{Mashhoon1}.
036 Exact solutions of time-dependent spin-rotation coupling
038 The variation of the Earth$^{,}$s rotating frequency maybe 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 measure 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 such as neutron 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) &=&\frac{1}{4}\omega _{0}(t)\{\sin \theta (t)\exp [-i\varphi (t)]\sigma _{+}+\frac{1}{2}\sin \theta (t)\exp [i\varphi (t)]\sigma _{-}}}{0}{}{}{}\RD{\CELL{&&+\cos \theta (t)\sigma _{3}\}.}}{1}{}{}{}}
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 construced to solve the time-dependent Schr\"{o}dinger equation (\QTSN{ref}{eq33}).
038 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}{2}\sin \lambda (t)\exp [i\gamma (t)]\sigma _{-}+\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 L-R 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 $\left| \sigma \right\rangle .$ 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{1}{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}}}{0}{}{}{}\RD{\CELL{&=&\{[\cos \lambda \cos \theta +\sin \lambda \sin \theta \cos (\gamma -\varphi )]+\dot{\gamma}(1-\cos \lambda )\}\frac{1}{2}\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}{eq28}, the geometric phase of a photon whose eigenvalue of helicity 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 $I_{V}(t)=J_{3},$ 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 describes the motion of photon in the fiber experiment 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}}{2}{315}{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 possesses some evolution parameters, this enables us to obtain the information concerning the variation of the Earth$^{,}$s rotation by measuring the geometric phase 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 by coordinate transformation of Kerr metric from the fixing reference frame to the rotating 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 detecting the time-varing rotating frequency by mesuring the phase difference of geometric phase 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 investigated in the literature, whereas the 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.$30000034$.
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