The CPT-violating effects on neutron's gravitational bound state

TL;DR

This paper derives analytical solutions for neutron gravitational bound states considering CPT-violating spin--gravity effects, revealing helicity-dependent phase shifts and setting bounds on CPT violation using experimental data.

Contribution

It provides the first analytical solutions for CPT-violating effects on neutron gravitational states and derives experimental bounds from transition frequency measurements.

Findings

Bound on CPT-violating parameter | ilde{b}|<6.9×10^{-21} GeV

Helicity-dependent phase evolution causes spin precession and transition shifts

Potential for tighter constraints with systematic error analysis

Abstract

Analytical solutions with effective CPT-violating spin--gravity corrections to the neutron's gravitational bound states are obtained. The helicity-dependent phase evolution due to $\vec{\sigma}\cdot\vec{\tilde{b}}$ and $\vec{\sigma}\cdot\hat{\vec{p}}$ couplings not only leads to spin precession, %shown by the tip of the spin on the Bloch sphere,but also to transition-frequency shifts between different gravitational bound states. Utilizing transition frequencies measured in the qBounce experiment,\cite{GRS} we obtain the rough bound $|\vec{\tilde{b}}|<6.9\times10^{-21}$GeV. Incorporating known systematic errors may lead to more robust and tighter constraints.

Equations

$$\hat{H}=\frac{{\hat{p}}_z^2}{2m_I}+m_{G}gz-\vec{\sigma }\cdot \overrightarrow{\stackrel{~}{b}}(1+{\Phi }_0),$$

$$\displaystyle\Psi_{\perp}(t,z)=\frac{1}{\sqrt{2}}\left(\begin{array}[]{c}\left[\cos(\frac{\Omega}{2}t)+ir_{+}\sin(\frac{\Omega}{2}t)\right]e^{-\frac{i\omega}{2}t}\\ \left[\cos(\frac{\Omega}{2}t)-ir_{-}\sin(\frac{\Omega}{2}t)\right]e^{+\frac{i\omega}{2}t}\\ \end{array}\right)\phi_{n}(z)e^{-iE_{n}t},$$

$$\displaystyle\Psi(t,z)=\left(\begin{array}[]{c}c_{1}~{}e^{i[\bar{b}_{\mathrm{eff}}(1+\Phi_{0})]z}\\ c_{2}~{}e^{-i[\bar{b}_{\mathrm{eff}}(1+\Phi_{0})]z}\\ \end{array}\right)\mathrm{Ai}[\frac{z}{L_{c}}-x_{n+1}]e^{-i\{E_{n}-\frac{[\bar{b}_{\mathrm{eff}}(1+\Phi_{0})]^{2}}{2m_{I}}\}t},$$

$$\displaystyle\left(\begin{array}[]{cc}\langle{n+}|\hat{H}_{LV}|n,+\rangle&\langle{n+}|\hat{H}_{LV}|n,-\rangle\\ \langle{n-}|\hat{H}_{LV}|n,+\rangle&\langle{n-}|\hat{H}_{LV}|n,-\rangle\\ \end{array}\right)$$

$$\displaystyle~{}~{}~{}~{}~{}~{}=-B_{0}\left[(1+\Phi_{0})+\frac{2}{3}gL_{c}x_{n+1}\right]\left(\begin{array}[]{cc}\cos\theta&\sin\theta e^{-i\phi}\\ \sin\theta e^{i\phi}&-\cos\theta\\ \end{array}\right)$$

$$\delta E_n=\mp B_0\left[(1+{\Phi }_0)+\frac{2}{3}g{L}_c{x}_{n+1}\right],$$

Full text

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1 Introduction

Lorentz-violating matter–gravity couplings[2] open a broad and interesting avenue for testing Lorentz symmetry. Recently, spin-independent Lorentz-violating neutron–gravity couplings have been thoroughly studied[3] in an attempt to analyze the GRANIT experiment.[4] However, to our best knowledge, an extensive study of spin-dependent fermion–gravity couplings is still under development.[5] Here, we provide a first glimpse at CPT-violating spin-dependent neutron–gravity couplings. A more detailed and complete analysis can be found in Ref. \refciteZXCPTV.

2 LV corrections due to spin-dependent interactions

The main vertical hamiltonian after averaging over the horizontal degrees of freedom is[3]

[TABLE]

where we have started with the hamiltonian in Ref. \refciteYuriEPI and performed a series of redefinitions and approximations. The stationary solution of (1) is

[TABLE]

where $\phi_{n}(z)\equiv\frac{\mathrm{Ai}[\frac{z}{L_{c}}-x_{n+1}]}{L_{c}^{1/2}|\mathrm{Ai}^{\prime}[-x_{n+1}]|}$, $\Omega\equiv\sqrt{\omega^{2}+4\omega B_{0}\cos\theta+4B_{0}^{2}}$, and $r_{\pm}\equiv\left[\omega+2B_{0}(\cos\theta\pm\sin\theta{e^{-i\phi}})\right]/{\Omega}$; the initial state is assumed to be an eigenstate of $\sigma_{x}$, $|X\uparrow\rangle$.

The probability profile for unrealistically large $B_{0}\equiv|\vec{\tilde{b}}|$ as well as the spin-precession on the Bloch sphere are shown in Fig. 1. The eigensolution of $\hat{H}=\hat{p}_{z}^{2}/(2m_{I})+m_{G}\,g\,z-\bar{b}_{\mathrm{eff}}\,(1+\Phi_{0})\,\sigma_{z}\hat{p}_{z}/m_{I}$ is

[TABLE]

where the parity-odd nature of $\vec{\sigma}\cdot\hat{\vec{p}}$ again dictates opposite phase evolutions for the two helicity components, and $c_{1}$ and $c_{2}$ are constants to be determined.

For $\hat{H}_{LV}=-\vec{\sigma}\cdot\vec{\tilde{b}}\,(1+\Phi_{0}+gz)-\bar{b}_{\mathrm{eff}}\,(1+\Phi_{0})\,\sigma_{z}\hat{p}_{z}/m_{I}$, we can use the matrix elements

[TABLE]

to calculate the shift in the eigenenergies

[TABLE]

where the upper or lower sign depends on whether the spin state is $|\hat{n}+\rangle=(e^{-i\phi}\cos\tfrac{1}{2}\theta,\,\sin\tfrac{1}{2}\theta)$ or $|\hat{n}-\rangle=(\sin\tfrac{1}{2}\theta,\,-e^{i\phi}\cos\tfrac{1}{2}\theta)$, respectively. From (14), the transition-frequency shift is given by $\delta\nu_{mn}^{\pi}=\mp\frac{2g}{3h}\,|\vec{\tilde{b}}|\,L_{c}(x_{m+1}-x_{n+1})$. Comparing $\delta\nu_{mn}^{\pi}$ with the precisely measured frequencies in the qBounce experiment,[1] we obtain the rough upper bound $|\vec{\tilde{b}}|<6.946\times 10^{-21}\,$GeV.

3 Summary

In this work, we have discussed CPT-violating spin–gravity corrections on the neutron’s gravitational bound states. With several analytical solutions, we have demonstrated that the phase evolution depends on the helicity of the wave-function components. The resulting phenomena are spin precession and $\theta$- and $\phi$-dependent probability variations. Using degenerate perturbation theory, we have also calculated the transition-frequency shift. Comarison with the measurements in Ref. \refciteGRS has yielded the rough bound $|\vec{\tilde{b}}|<6.9\times 10^{-21}\,$GeV, which can be improved further if systematic errors from known physics are taken into account, or if polarized neutrons are used in the future.

Acknowledgments

The author appreciates valuable encouragement and helpful discussions with M. Snow and A. Kostelecký as well as help from many others. This work is partially supported by the National Science Foundation of China under grant no. 11605056, no. 11875127, and no. 11575060.