Quantum gravitational states
of ultracold neutrons as a tool for probing of beyond-Riemann gravity

Nowadays the effective field theories (EFT) are a powerful tool for the analysis of the Nature Weinberg2021 ; Weinberg2016 ; Weinberg2009 . The general EFT by Kostelecký Kostelecky2004 , based on the General Gravity (GR) coupled to the Standard Model (SM), has been extended by Kostelecký and Li Kostelecky2021a ; Kostelecky2021b by the contributions of interactions, caused by beyond-Riemann gravity (BRG). These contributions are closely overlapped with the contributions of interactions violating Lorentz-invariance Kostelecky1997a ; Kostelecky1998a ; Kostelecky2002a ; Kostelecky2002b ; Kostelecky2003 ; Kostelecky2009 ; Kostelecky2016 . In Kostelecky2021b Kostelecký and Li have proposed to investigate the BRG as well as Lorentz-invariance violation (LV) contributions to the energy spectrum and transition frequencies of the quantum gravitational states of ultracold neutrons (UCNs).

This paper is addressed to the analysis of the BRG and LV contributions to the energy spectrum and transition frequencies of the quantum gravitational states of ultracold neutrons (UCNs). For the analysis of these problems we follow Ivanov2019 . In Ivanov2019 we have calculated the LV contributions to the energy spectrum and transition frequencies of the quantum gravitational states of UCNs, caused by the effective low-energy potential, derived by Kostelecký and Lane Kostelecky1999 in the frame work of the Standard Model Extension (SME) Kostelecky1997a ; Kostelecky1998a (see also Kostelecky2004 ) by using the Foldy-Wouthuysen transformations Foldy1950 ; Itzykson1980 .

The paper is organized as follows. In section II we discuss the effective low-energy potential, derived by Kostelecký and Li Kostelecky2021b for the analysis of BRG interactions in the terrestrial laboratories. We define such a potential in the standard coordinate frame related to the laboratory at the Institut Laue Langevin (ILL) in Grenoble. We specify the BRG and LV contributions to the phenomenological coupling constants of this potential. We adduce the wave functions of the quantum gravitational states of polarized and unpolarized UCNs. In section III we calculate the BRG and LV contributions to the energy spectrum and transition frequencies of the quantum gravitational states of polarized and unpolarized UCNs. Using the current experimental sensitivity of the qBOUNCE experiments we give some estimates of the phenomenological constants of the BRG and LV interactions. In section IV we discuss the obtained results and perspectives of further investigations of the BRG and LV interactions by using the quantum gravitational states of UCNs.

For the experimental analysis of the BRG and LV interactions in the terrestrial laboratories by using the quantum gravitational states of UCNs Kostelecký and Li propose to use the following Hamilton operator Kostelecky2021b

where the first two terms are the operators of the UCN energy and the Newtonian gravitational potential of the gravitational field of the Earth, respectively, with the gravitational acceleration $\vec{g}$ such as $\vec{g}\cdot\vec{z}=-gz$ Kostelecky2021b . Then, $\Phi_{\rm nRG}$ is the effective low-energy potential of the neutron-gravity interaction, calculated to next-to-leading order in the large neutron mass $m$ expansion and related to the contribution of Riemann gravity. It is equal to Kostelecky2021b

In turn, the potential $\Phi_{\rm nBRG}$ describes the BRG and LV contributions to neutron-gravity interactions

where the operators $H_{j}$ for $j=\phi,\sigma\phi,g$ and $\sigma g$ are equal to Kostelecky2021b

The non-relativistic Hamilton operator Eq.(1) is written in the coordinate system shown in Fig. 1, where $m$ is the neutron mass, $\vec{z}$ is a radius-vector of a position of an UCN on the $z$-axis, $\vec{p}=-i\nabla$ is a 3-momentum of an UCN and $\vec{\sigma}$ is the Pauli $2\times 2$ matrix of the UCN spin Itzykson1980 . The coefficients $(k^{\rm NR}_{\phi})_{n}$, $(k^{\rm NR}_{\ldots})^{j}_{n}$, $(k^{\rm NR}_{\ldots})^{jk}_{n}$, $(k^{\rm NR}_{\ldots})^{jk\ell}_{n}$, and $(k^{\rm NR}_{\ldots})^{jk\ell m}_{n}$ define the BRG and LV contributions, which can be tested in experiments with neutrons Kostelecky2021b in the following way.

The system of a Schrödinger quantum particle with mass $m$ bouncing in a linear gravitational field is known as the quantum bouncer Gibbs1975 ; Gea-Banacloche1999 ; Rosu2001 ; Robinett2004 . Above a horizontal mirror, the linear gravity potential leads to discrete energy eigenstates of a bouncing quantum particle. An UCN, bound on a reflecting mirror in the gravity potential of the earth, can be found in a superposition of quantum gravitational energy eigen-states. The quantum gravitational states of UCNs have been verified and investigated Nesvizhevsky2002 ; Nesvizhevsky2003 ; Nesvizhevsky2005 ; Abele2007 at the UCN beamline PF2 at the Institute Laue-Langevin (ILL), where the highest UCN flux is available worldwide. The qBOUNCE collaboration develops a gravitational resonant spectroscopy (GRS) method Abele2010 , which allows to measure the energy difference between quantum gravitational states with increasing accuracy. Recent activities Sedmik2019 , and a summary can be found in Jenke2019 . The energy difference can be related to the frequency of a mechanical modulator, in analogy to the Nuclear Magnetic Resonance technique, where the Zeeman energy splitting of a magnetic moment in an outer magnetic field is connected to the frequency of a radio-frequency field. The frequency range in GRS used so far is in the acoustic frequency range between 100 and 1000 ${\rm Hz}$. The quantum gravitational states of UCNs have peV energy, on a much lower energy scale compared to other bound quantum systems. Any gravity-like potential or a deviation from Riemann gravity would shift these energy levels Jenke2021 ; Jenke2011 ; Jenke2014a ; Cronenberg2018 and an observation would point to new physical understanding.

Our choice of the laboratory frame is related to the following. Indeed, the qBOUNCE experiments are being performed in the laboratory at Institut Laue Langevin (ILL) in Grenoble. The ILL laboratory is fixed to the surface of the Earth in the northern hemisphere. Following Kostelecky2002a ; Kostelecky2002b ; Kostelecky2003 ; Kostelecky2009 ; Kostelecky2016 (see also Kostelecky2021b ; Ivanov2019 ) we choose the ILL laboratory or the standard laboratory frame with coordinates $(t,x,y,z)$, where the $x$, $y$ and $z$ axes point south, east and vertically upwards, respectively, with northern and southern poles on the axis of the Earth’s rotation with the Earth’s sidereal frequency $\Omega_{\oplus}=2\pi/(23\,{\rm hr}\,56\,{\rm min}\,4.09\,{\rm s}=7.2921159\times 10^{-5}\,{\rm rad/s}$. The position of the ILL laboratory on the surface of the Earth is determined by the angles $\chi$ and $\phi$, where $\chi=44.83333^{0}$ N is the colatitude of the laboratory and $\phi$ is the longitude of the laboratory measured to east with the value $\phi=5.71667^{0}$ E Grenoble . The beam of UCNs moves from south to north antiparallel to the $x$–direction and with energies of UCNs quantized in the $z$–direction. The gravitational acceleration in Grenoble is $g=9.80507\,{\rm m/s^{2}}$ Ivanov2019 ; Grenoble . Following Ivanov2019 we may neglect the Earth’s rotation assuming that the ILL laboratory frame is an inertial one.

The energy spectrum of the quantum gravitational states of polarized UCNs with the RG, BRG and LV corrections are defined by the integrals

Using the table of integrals in Albright1977 ; AiryF2004 we obtain the RG, BRG and LV contributions to the energy spectrum of the quantum gravitation states of unpolarized UCNs. We get

Since the binding energies of the quantum gravitational states of UCNs are of a few parts of $10^{-12}\,{\rm eV}$, the RG contribution is of order of a few parts of $10^{-33}\,{\rm eV}$ and can be neglected. This concerns also the contributions proportional to $\frac{2}{3}\,mgE^{(0)}_{k}\leq 10^{-25}\,{\rm eV^{3}}$ for $k\leq 10$ Jenke2019 ; Sedmik2019 .

As a result, the energy spectrum of the quantum gravitational states of UCNs together with the BRG and LV contributions is equal to

where $g=2.15\times 10^{-23}\,{\rm eV}$ Ivanov2019 . The LV contribution, proportional to $\big{(}k^{(\rm NR)}_{\sigma g}\big{)}^{\prime}_{n}$, is the same for all energy level. It depends only on the neutron spin-polarization.

According to the energy spectrum Eq.(13), for the non-spin-flip $|k\sigma\rangle\to|k^{\prime}\sigma\rangle$ and spin-flip $|k\sigma\rangle\to|k^{\prime}\sigma^{\prime}\rangle$ transitions we get Ivanov2019

for $(\sigma=\uparrow,\sigma^{\prime}=\downarrow)$ or $(\sigma=\downarrow,\sigma^{\prime}=\uparrow)$, respectively.

For current sensitivity $\Delta E=2\times 10^{-15}\,{\rm eV}$ Jenke2019 (see also Ivanov2019 ) and for the $|1\rangle\to|4\rangle$ transition Ivanov2019 we are able to obtain the upper bound on the BRG contribution $\big{|}\big{(}k^{(\rm NR)}_{\phi})_{n}\big{|}$ and an estimate for $\big{(}k^{(\rm NR)}_{\sigma g}\big{)}^{\prime}_{n}$, i.e.,

The upper bound $\big{|}\big{(}k^{(\rm NR)}_{\phi})_{n}\big{|}<10^{-3}\,{\rm GeV}$ is one order of magnitude better in comparison with the result $\big{|}\big{(}k^{(\rm NR)}_{\phi})_{n}\big{|}<1.3\times 10^{-2}\,{\rm GeV}$, obtained in Kostelecky2021b . Then, our result $\big{(}k^{(\rm NR)}_{\sigma g}\big{)}^{\prime}_{n}=0$ agrees well with that by Kostelecký and Li Kostelecky2021b . The spin-flip transitions may also admit an upper bound $\big{|}\big{(}k^{(\rm NR)}_{\sigma g}\big{)}^{\prime}_{n}\big{|}<10^{8}$. However, it seems unrealistic, since the main contribution to $\big{(}k^{(\rm NR)}_{\sigma g}\big{)}^{\prime}_{n}$ is caused by LV interactions Kostelecky2011b .

It is important to emphasize that in the coefficient $(k^{\rm NR}_{\phi})_{n}$ the dominate role belongs to the BRG interactions. According to Kostelecky2021b , the coefficient $(k^{\rm NR}_{\phi})_{n}$ has the following structure (see Table III in Ref.Kostelecky2021b ):

where the phenomenological coupling constants in the right-hand-side of Eq.(16) are fully induced by the BRG interactions (see Table I in Ref.Kostelecky2021b ).

The energy spectrum of the quantum gravitational states of unpolarized UCNs, calculated by taking into account the 2-fold degeneracy of the energy levels Ivanov2019 (see also LL1965 ; Davydov1965 ), is equal to

Using Eq.(15) in Ref.Ivanov2019 we define the contributions to the transition frequencies of the quantum gravitational states of UCNs

and

One may see that the experimental analysis of the transition frequencies between the quantum gravitational states of unpolarized UCNs should lead to the estimates Eq.(15).

We have analyzed a possibility to test contributions of interactions, induced by non-Riemann geometry beyond the standard Riemann General Relativity and the Lorentz-invariance violation (LV), by Kostelecký and Li Kostelecky2021b . Using the effective low-energy potential, derived in Kostelecky2021b , we have calculated the contributions of beyond-Riemann gravity or the BRG contributions and the LV contributions to the energy spectrum and transition frequencies of the quantum gravitational states of polarized and unpolarized UCNs. Such UCNs are used as test-particles for probes of contributions of interactions beyond the Standard Model (SM) and Einstein’s gravity Jenke2019 ; Sedmik2019 ; Abele2021 ; Abele2003 ; Abele2009 ; Jenke2009 ; Abele2010 ; Jenke2011 ; Abele2011c ; Abele2012 ; Jenke2014a ; Jenke2014b ; Schmiedmayer2015 ; Cronenberg2015 ; Abele2016 ; Konrad2017 ; Cronenberg2018 ; Ivanov2020 . We have got the following constraints $\big{|}\big{(}k^{(\rm NR)}_{\phi})_{n}\big{|}<10^{-3}\,{\rm GeV}$ and $\big{(}k^{(\rm NR)}_{\sigma g}\big{)}^{\prime}_{n}=0$. The upper bound $\big{|}\big{(}k^{(\rm NR)}_{\phi})_{n}\big{|}<10^{-3}\,{\rm GeV}$ is one order of magnitude better in comparison with the constraint obtained in Kostelecky2021b . Then, from our analysis of the transition frequencies of the quantum gravitational states of UCNs follows that $\big{(}k^{(\rm NR)}_{\sigma g}\big{)}^{\prime}_{n}=0$, whereas in Kostelecky2021b such a value $\big{(}k^{(\rm NR)}_{\sigma g}\big{)}^{\prime}_{n}=0$ has been imposed by assumption. It is important to emphasize that for the experimental sensitivity $\Delta E=2\times 10^{-17}\,{\rm eV}$, which should be reached in the qBOUNCE experiments in a nearest future Abele2010 , we should expect the upper bound $\big{|}\big{(}k^{(\rm NR)}_{\phi})_{n}\big{|}<10^{-5}\,{\rm GeV}$.

As has been pointed out by Kostelecký and Li Kostelecky2021b , the coefficient $(k^{\rm NR}_{\phi})_{n}$ should appear in the nonrelativistic Hamilton operator in Minkowski spacetime Kostelecky1999a but it produces no measurable effects in that context because it amounts to an unobservable redefinition of the zero of energy or, equivalently, because it can be removed from the theory via field redefinitions Kostelecky1997a ; Kostelecky1998a . The observability of $(k^{\rm NR}_{\phi})_{n}$ is thus confirmed to be a consequence of the coupling to the gravitational potential, the presence of which restricts the applicability of field redefinitions Kostelecky2021a .

In the perspective of the further analysis of BRG and LV interactions by the quantum gravitation states of UCNs we see in the use of i) the total effective low-energy potential Eq.(3) for the calculation of the BRG and LV contributions to the energy spectrum and the transition frequencies of the quantum gravitational states of UCNs, and of ii) the quantum bouncing ball experiments with a free fall of UCNs in the gravitational field of the Earth Abele2011c ; Abele2012 .

We are grateful to Alan Kostelecký for fruitful discussions and comments. The work of A. N. Ivanov was supported by the Austrian “Fonds zur Förderung der Wissenschaftlichen Forschung” (FWF) under the contracts P31702-N27, P26636-N20 and P33279-N and “Deutsche Förderungsgemeinschaft” (DFG) AB 128/5-2. The work of M. Wellenzohn was supported by the MA 23.