Ultra-Cold Neutrons in qBounce Experiments as Laboratory for Test of Chameleon Field Theories and Cosmic Acceleration

Abstract

The study of scalar field theories like the chameleon field model is of increasing interest due to the Universe’s accelerated expansion, which is believed to be caused in part by dark energy. These fields can elude experimental bounds set on them in high-density environments since they interact with matter in a density-dependent way. This paper analyzes the effect of chameleon fields on the quantum gravitational states of ultra-cold neutrons (UCNs) in qBounce experiments with mirrors. We discuss the deformation of the neutron wave function due to chameleon interactions and quantum systems in potential wells from gravitational forces and chameleon fields. Unlike other works that aim to put bounds on the chameleon field parameters, this work focuses on the quantum mechanics of the chameleonic neutron. The results deepen our understanding of the interplay between quantum states and modified gravity, as well as fundamental physics experiments carried out in the laboratory.

1. Introduction

In spite of the confirmation of the existence of the cosmic acceleration and a cosmological constant $\Lambda$ [1] and the simplest explanation of the cosmological constant [2,3,4,5], there remain some questions [6], which may find the answers within the single scalar field theory—the chameleon field theory [7,8,9,10]. A chameleon field has been suggested to drive the current phase of cosmic acceleration for a large class of scalar potentials. A chameleon field coupled to an ambient matter acquires an effective mass, depending on the matter density, and becomes heavier in the more dense matter. Thus, the properties of a chameleon field depend on the density of matter to which it is immersed. Because of this sensitivity to the environment, the scalar field was called chameleon, and it can couple directly to baryons and leptons with gravitational strength on Earth but it would be essentially massless on solar system scales [7,8,9,10,11].

Dark energy, interpreted as negative pressure causing the Universe’s acceleration, may originate from a chameleon field, which is responsible for inflating the Universe [12,13,14,15]. A competing possibility is that the acceleration is due to a modification of gravity, i.e., the left-hand side of Einstein’s equation rather than the right. Observations, which can distinguish these two possibilities are desirable, since measurements of expansion kinematics alone are not able to.

As has been pointed out in [16,17,18], ultra-cold neutrons, bouncing in the gravitational field of the Earth above a mirror [16] and between two mirrors [17,18], can be a good laboratory for testing the existence of a chameleon field. As has been shown theoretically [19,20,21] and observed experimentally [22,23,24,25], ultra-cold neutrons in the gravitational field of the Earth possess quantum gravitational bound states with binding energies of order ${10}^{-12}\mathrm{eV}$. The transitions between quantum gravitational states of ultra-cold neutrons, bouncing in the gravitational field of the Earth above a mirror and between two mirrors, were measured in [18,26,27,28], respectively. It has been found [17,18] that for the dependence of the sensitivity of the experimental data on the transition frequencies of the quantum gravitational states of ultra-cold neutrons for the chameleon–matter coupling constant, which is usually called $\beta$, the strong and weak coupling regime may be realised. The former is in agreement with the estimates obtained in [29,30] from the experimental data on the gravitational torsion balance.

The problem of applications of a chameleon field to the analysis of the observable phenomena in the universe and in the terrestrial laboratories depends strongly on the definition of the self-interaction of a chameleon field. Such a self-interaction plays an important role in the mechanism of the spontaneous creation of the effective mass of a chameleon field. According to [7,31,32,33,34,35,36], for the analysis of the properties of a massless chameleon field, one may use the potentials of the self-interaction of a chameleon, which we denote as $V\left(\varphi \right)\sim {\varphi }^{-n}$ with positive powers n and $V\left(\varphi \right)\sim {\varphi }^4$, respectively.

Below, we analyse a chameleon field within the ${\varphi }^{-n}$-theory with the potential of the self-interaction of a chameleon field $V\left(\varphi \right)={\Lambda }^4(1+{\Lambda }^n/{\varphi }^n)$ with $\Lambda =2.4\times {10}^{-3}\mathrm{eV}$ [7,17,31,32,33,34,35]. In turn, from the experimental data on the transition frequencies of quantum gravitational states [18], one could obtain the estimate on the chameleon–matter coupling constant $\beta <5\times {10}^9$.

Experimental studies have placed stringent constraints on chameleon field theories. The qBOUNCE collaboration has reported an upper bound on the chameleon–matter coupling constant, restricting it to ($\beta <6.9\times {10}^6$) [37]. Additionally, independent experiments by [38] have set strong upper limits on the chameleon field, while experiments conducted by Eric Adelberger et al. [29] have constrained it from below. The combination of these results suggests that the chameleon field, in its original formulation, is largely ruled out as a viable explanation for cosmic acceleration. Given this, our study does not aim to refine these existing bounds but instead focuses on a detailed quantum mechanical analysis of how a chameleon field influences the gravitationally bound quantum states of ultra-cold neutrons. By investigating neutron wave function deformations in a confined environment, we provide insights into the interplay between modified gravity models and quantum mechanics, which remains relevant for broader tests of fundamental physics.

In this paper, we investigate the contribution of a chameleon field to the wave function of ultra-cold neutrons moving above a mirror. This results in a qualitative phenomenon, related to the deformation of the wave function of ultra-cold neutrons, which may be observed experimentally in the Quantum Bouncing Ball (QBB) experiments [39,40,41], i.e., a measurement of the time evolution of the Schrödinger wave function of ultra-cold neutrons, bouncing above a mirror.

The calculation of the chameleon field is adapted to the experimental setup used in [39,40,41]; see Figure 1. Ultra-cold neutrons move with a along the x-axis with a velocity $v_0$ between a scatter (upper yellow) mirror (lower red) system in the spatial region, which we define as $z^2\le \frac{d^2}{4}$ [17]. Then, at the edge, which we localise at $x=0$ [39,40,41], ultra-cold neutrons fall down from the height $h=D-\frac{d}{2}+h_1$ above a mirror, localised at $z=-D$. The height $h_1=-{\ell }_0{\xi }_1=13.71$ μm is related to the ground quantum gravitational state with the binding energy ${\mathcal{E}}_1=mg{h}_1=-mg{\ell }_0{\xi }_1=0.602{\xi }_1\mathrm{peV}=1.412\mathrm{peV}$, where ${\ell }_0={\left(2m^{2}g\right)}^{-1/3}=5.87$ μm (see Appendix A), is the scale of the quantum gravitational states of ultra-cold neutrons in the gravitational field of the Earth [19,20,21] and ${\xi }_1=-2.34497$ (see Appendix A and [17]). For the numerical analysis, we take $v_0=6\mathrm{m}/\mathrm{s}$ [41], $d=25.5$ μm [28] and $D-\frac{d}{2}=47$ μm. Since the binding energy of the first excited gravitational state of ultra-cold neutrons is equal to ${\mathcal{E}}_2=mg{h}_2=-mg{\ell }_0{\xi }_2=2.614\mathrm{peV}$, with ${\xi }_2=-4.34177$ corresponding to the height $h_2=-{\ell }_0{\xi }_2=25.5$ μm, the distance $d=25.5$ μm allows us to select, between two mirrors, ultra-cold neutrons only in the ground gravitational state.

This paper is organised as follows. In Section 2, we construct the wave function of ultra-cold neutrons in the ground quantum gravitational state, moving between two mirrors in the spatial region $z^2\le \frac{d^2}{4}$, and define the wave function of ultra-cold neutrons in the spatial region $z\ge -D$ above a mirror. We show that the main contributions come from the excited states with a principal quantum number $k=6$, $k=7$, and $k=8$. In Section 3, we calculate a first-order correction to the wave functions of ultra-cold neutrons, induced by a chameleon field. From a unitarity condition, we obtain a new estimate on the chameleon–matter coupling constant $\beta \le 6.5\times {10}^8$. We discuss the obtained results in Section 4 and conclude in Section 6. In Appendix A, we define the wave function of the ground quantum gravitational state of ultra-cold neutrons between two mirrors in the spatial region $z^2\le \frac{d^2}{4}$ and above a mirror in the spatial region $z\ge -D$, respectively.

2. Wave Function of Ultra-Cold Neutrons Above a Mirror

For the analysis of the evolution of the wave function of ultra-cold neutrons in the spatial region above a mirror, we assume that (1) ultra-cold neutrons start to fall at $t=0$ and (2) before the fall the time evolution and the x–degree of freedom of ultra-cold neutrons are described by the Gaussian wave packet with a width $\delta =1$ μm [39,40,41]. At time t, the wave function of ultra-cold neutrons takes the form

$$\begin{array}{ccc}\hfill & & {\psi }_1(z,x,t)={\psi }_1\left(z\right){\left(4\pi {\delta }^2\right)}^{1/4}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{dp}{2\pi }}{e}^{-\frac{{\delta }^2{(p-p_0)}^2}{2}}\hfill \\ \hfill & & \times e^{-i({\mathcal{E}}_1+\frac{p^2}{2m})t+ipx}={\psi }_1\left(z\right){\left({\displaystyle \frac{{\delta }^2}{\pi {\delta }_t^4}}\right)}^{1/4}{e}^{-\frac{{(x-v_{0}t)}^2}{2{\delta }_t^2}}\hfill \\ \hfill & & \times e^{-i({\mathcal{E}}_1+\frac{p_0^2}{2m})t+i{p}_{0}x},\hfill \end{array}$$

(1)

where the wave function ${\psi }_1\left(z\right)$ is given in Appendix A and ${\delta }_t=\delta \sqrt{1+it/m{\delta }^2}$. The squared absolute value $|{\psi }_1{(z,x,t)|}^2$ is

$$\begin{array}{ccc}\hfill & & |{\psi }_1{(z,x,t)|}^2={\left|{\psi }_1\left(z\right)\right|}^2{\displaystyle \frac{1}{\sqrt{\pi {\Delta }_t^2}}}{e}^{-\frac{{(x-v_{0}t)}^2}{2{\Delta }_t^2}},\hfill \end{array}$$

(2)

where ${\Delta }_t=\delta \sqrt{1+t^2/m^2{\delta }^4}$.

Since in the spatial region between two mirrors $z^2\le \frac{d^2}{4}$, ultra-cold neutrons are only in the ground quantum gravitational state with the wave function, and the excited quantum gravitational states do not exist, the first-order correction to the wave function of the ground quantum gravitational state, induced by a chameleon field, appears only as a phase shift of the wave function [42].

In the spatial region $z\ge -D$, the pure ground gravitational state ${\psi }_1(z,x,t)$ should be treated as a mixed quantum gravitational state with the wave function of the z-degree of freedom given by

$$\begin{array}{c}\hfill {\psi }_1(z,t)=\sum _k{C}_k{\Psi }_k(z,t),\end{array}$$

(3)

where ${\psi }_1(z,t)={\psi }_1\left(z\right)e^{-i{\mathcal{E}}_{1}t}$, ${\Psi }_k(z,t)={\Psi }_k\left(z\right)e^{-i{E}_{k}t}$ with the wave functions ${\Psi }_k\left(z\right)$, given in Appendix A, and the coefficients $C_k$ are given by

$$\begin{array}{c}\hfill C_k={\int }_{-d/2}^{+d/2}dz{\Psi }_k^{*}\left(z\right){\psi }_1\left(z\right).\end{array}$$

(4)

The integrals are calculated over the spatial region $z^2\le \frac{d^2}{4}$, since the wave function ${\psi }_1\left(z\right)$ vanishes in the spatial region $z^2\ge \frac{d^2}{4}$. The binding energies of the first quantum gravitational states of ultra-cold neutrons and the values of the coefficients $C_k$ for $k=1,2,\dots ,15$ are adduced in

Table A1. The binding energies of the first 15 quantum gravitational states of ultra-cold neutrons in the gravitational field of the Earth above a mirror and the coefficients $C_k$ for $k=1,2,\dots ,15$ and different heights h = 31 μm, 39 μm, 47 μm, 55 μm, 63 μm. The last two rows show the probabilities $P_{total}={\sum }_k|C_k|$ for $k\in [1,2,\dots ,15]$ and $P_{considered}$ for the $C_k$ s plotted in bold, which are taken into account for the calculation of transition coefficients $a_{k^{\prime }k}\left(n\right)$.

k	${\mathit{\zeta }}_{\mathit{k}}$	${\mathit{E}}_{\mathit{k}}$	${\mathit{C}}_{\mathit{k}}$(h = 31 μm)	${\mathit{C}}_{\mathit{k}}$(h = 39 μm)	${\mathit{C}}_{\mathit{k}}$(h = 47 μm)	${\mathit{C}}_{\mathit{k}}$(h = 55 μm)	${\mathit{C}}_{\mathit{k}}$(h = 63 μm)
1	−2.33811	1.407	$2\times {10}^{-3}$	$1.08\times {10}^{-4}$	$+4.03\times {10}^{-6}$	$1.05\times {10}^{-7}$	$1.99\times {10}^{-9}$
2	−4.08795	2.461	$3.79\times {10}^{-2}$	$3.66\times {10}^{-3}$	$+2.23\times {10}^{-4}$	$9.27\times {10}^{-6}$	$2.67\times {10}^{-7}$
3	−5.52056	3.324	$\mathbf{0.23}$	$0.04$	$+3.89\times {10}^{-3}$	$2.44\times {10}^{-4}$	$1.03\times {10}^{-5}$
4	−6.78671	4.086	$\mathbf{0.61}$	$\mathbf{0.20}$	$+3.22\times {10}^{-2}$	$3.08\times {10}^{-3}$	$1.8\times {10}^{-4}$
5	−7.94413	4.782	$\mathbf{0.70}$	$\mathbf{0.52}$	$+\mathbf{0.15}$	$0.02$	$1.9\times {10}^{-3}$
6	−9.02265	5.432	$\mathbf{0.17}$	$\mathbf{0.71}$	$+\mathbf{0.42}$	$0.10$	$0.01$
7	−10.04020	6.044	$-\mathbf{0.18}$	$\mathbf{0.36}$	$+\mathbf{0.66}$	$\mathbf{0.30}$	$0.06$
8	−11.00850	6.627	$0.03$	$-\mathbf{0.13}$	$+\mathbf{0.55}$	$\mathbf{0.58}$	$\mathbf{0.20}$
9	−11.93600	7.186	$0.02$	$-0.08$	$+0.06$	$\mathbf{0.65}$	$\mathbf{0.44}$
10	−12.82880	7.723	$-0.04$	$0.08$	$-\mathbf{0.17}$	$\mathbf{0.31}$	$\mathbf{0.64}$
11	−13.69150	8.242	$0.05$	$-0.04$	$+0.02$	$-\mathbf{0.11}$	$\mathbf{0.53}$
12	−16.13270	9.712	$0.02$	$-0.04$	$+0.03$	$4.69\times {10}^{-3}$	$-0.05$
13	−16.90560	10.177	$-0.03$	$0.03$	$+0.01$	$-0.04$	$\mathbf{0.07}$
14	−19.12640	11.514	$0.01$	$0.03$	$-0.01$	$-0.03$	$0.05$
15	−19.83810	11.943	$-1.38\times {10}^{-3}$	$-0.02$	$-0.01$	$0.04$	$-0.02$
$P_{total}={\sum }_k{|C_k|}^2$			98.67%	98.68%	98.01%	96.80%	94.52%
$P_{considered}={\sum }_i{|{\mathbf{C}}_{\mathbf{i}}|}^2$			97.8%	96.72%	97.46%	95.29%	93.62%

in Appendix A. One may see that the main contribution to the mixed state comes from the quantum gravitational states with $k=6$, $k=7$, and $k=8$ with coefficients $C_6=0.42$, $C_7=0.66$, and $C_8=0.55$, and probabilities $P_6={\left|C_6\right|}^2=0.18$, $P_7={\left|C_7\right|}^2=0.44$, and $P_8={\left|C_8\right|}^2=0.30$, respectively. For the analysis of the first-order corrections to the wave functions of ultra-cold neutrons, caused by a chameleon field, we will use the following approximation of the wave function ${\psi }_1(z,t)$

$$\begin{array}{c}\hfill {\psi }_1(z,t)=C_6{\Psi }_6(z,t)+C_7{\Psi }_7(z,t)+C_8{\Psi }_8(z,t),\end{array}$$

(5)

which describes the exact wave function with an accuracy of about $8\%$. In this analysis, we treat the neutron wavefunction within the mirror gap as unperturbed by the chameleon field. This assumption is based on the fact that the region between the two mirrors is spatially narrow and bounded by dense materials, which significantly screen the chameleon field. In such confined environments, the scalar field tends toward a constant value, and its contribution to the potential becomes effectively uniform. As a result, it induces only a global phase shift in the wavefunction, which has no measurable impact on observables such as transition probabilities or spatial distributions. Therefore, within this region, we adopt the standard approximation of neglecting chameleon-induced modifications to the wavefunction. Perturbative corrections due to the chameleon field are introduced only after the neutron exits the confined region, in the spatial domain $z\ge -D$, where the field is no longer screened and exhibits spatial variation. This treatment is consistent with previous work, such as Ivanov et al. [17], and ensures a physically justified separation between the unperturbed and perturbed regimes.

3. Chameleon Field Correction in the ${\mathit{\varphi }}^{\mathbf{-}\mathit{n}}$-Theory

The potential of a chameleon field, coupled to ultra-cold neutrons, takes the form [16,17]

$$\begin{array}{c}\hfill \Phi \left(z\right)=\beta {\displaystyle \frac{m}{M_{\mathrm{Pl}}}}\varphi \left(z\right),\end{array}$$

(6)

where $M_{\mathrm{Pl}}=2.435\times {10}^{27}\mathrm{eV}$ is the Planck mass, $\beta$ is a chameleon–matter coupling constant, and $\varphi \left(z\right)$ is the profile of a chameleon field in the spatial region $z\ge -D$. As has been shown in [18], the chameleon–matter coupling constant obeys the constraint $\beta <5\times {10}^9$. As has been shown in [16,17], in the spatial region $z\ge -D$, the profile of a chameleon field, obtained as a solution of the non-linear equations of motion in the ${\varphi }^{-n}$-theory and in the strong coupling limit $\beta \ge {10}^5$, takes the form

$$\begin{array}{c}\hfill \varphi \left(z\right)=\Lambda {\left({\displaystyle \frac{n+2}{\sqrt{2}}}\right)}^{\frac{2}{n+2}}{\left(1+{\displaystyle \frac{z}{D}}\right)}^{\frac{2}{n+2}},\end{array}$$

(7)

where $\Lambda =2.4\times {10}^{-3}\mathrm{eV}$ [7,16,17,31,32,33,34]. For $D=59.75$ μm we may define the chameleon field potential as

$$\begin{array}{c}\hfill \Phi \left(z\right)=mg{\ell }_0{C}_{\varphi }{\displaystyle \frac{\varphi \left(z\right)}{\Lambda }},\end{array}$$

(8)

where $C_{\varphi }=\beta m\Lambda /M_{\mathrm{pl}}mg{\ell }_0=1.54\times {10}^{-9}\beta$ and

$$\begin{array}{c}\hfill {\displaystyle \frac{\varphi \left(z\right)}{\Lambda }}={\left({\displaystyle \frac{n+2}{2}}\right)}^{\frac{2}{n+2}}{\left(1+{\displaystyle \frac{z}{D}}\right)}^{\frac{2}{n+2}}.\end{array}$$

(9)

For the estimate $\beta <5\times {10}^9$ [18], the coupling constant $C_{\varphi }$ is $C_{\varphi }<7.7$. The power n of the potential of a self-interaction of a chameleon field is a free parameter of the ${\varphi }^{-n}$-theory of a chameleon field. For $n\ge 2$ the accuracy of the approximation Equation (8) is better than $1.5\%$.

The wave function ${\Psi }_k(z,t)$ of ultra-cold neutrons, including the first-order corrections induced by a chameleon field, takes the form

$$\begin{array}{ccc}\hfill & & {\Psi }_k(z,t)\to {\Psi }_k(z,t)+C_{\varphi }\sum _{k^{\prime }\ne k}{a}_{k^{\prime }k}\left(n\right){\Psi }_{k^{\prime }}(z,t),\hfill \end{array}$$

(10)

where we have denoted

$$\begin{array}{ccc}\hfill a_{k^{\prime }k}\left(n\right)& =\hfill & \hfill {\left({\displaystyle \frac{{\ell }_0}{D}}{\displaystyle \frac{n+2}{2}}\right)}^{\frac{2}{n+2}}{\displaystyle \frac{\langle k^{\prime }|{\xi }^{\frac{2}{n+2}}|k\rangle }{{\zeta }_{k^{\prime }}-{\zeta }_k}},\\ \hfill \langle k^{\prime }|{\xi }^{\frac{2}{n+2}}|k\rangle & =\hfill & \hfill {\int }_0^{+\infty }d\xi {\Psi }_{k^{\prime }}\left(\xi \right){\xi }^{\frac{2}{n+2}}{\Psi }_k\left(\xi \right)\end{array}$$

(11)

with $\xi =(D+z)/{\ell }_0$. For the excited states ${\Psi }_k(z,t)$ with the principal quantum number $k=6$, 7, and 8, we obtain the following corrections

$$\begin{array}{ccc}\hfill & \hfill & \hfill {\Psi }_6(z,t)+C_{\varphi }{a}_{76}\left(n\right){\Psi }_7(z,t)+C_{\varphi }{a}_{86}\left(n\right){\Psi }_8(z,t),\\ \hfill & \hfill & \hfill {\Psi }_7(z,t)+C_{\varphi }{a}_{67}\left(n\right){\Psi }_6(z,t)+C_{\varphi }{a}_{87}\left(n\right){\Psi }_8(z,t),\\ \hfill & \hfill & \hfill {\Psi }_8(z,t)+C_{\varphi }{a}_{68}\left(n\right){\Psi }_6(z,t)+C_{\varphi }{a}_{78}\left(n\right){\Psi }_7(z,t).\end{array}$$

(12)

The wave function of the mixed state take the form

$$\begin{array}{c}\hfill {\psi }_1(z,t)\to {\overline{C}}_6{\Psi }_6(z,t)+{\overline{C}}_7{\Psi }_7(z,t)+{\overline{C}}_8{\Psi }_8(z,t),\end{array}$$

(13)

where the coefficients ${\overline{C}}_k$ for $k=6$, 7, and 8 are equal to

$$\begin{array}{ccc}\hfill {\overline{C}}_6& =\hfill & \hfill C_6+C_{\varphi }(C_7{a}_{67}\left(n\right)+C_8{a}_{68}\left(n\right)),\\ \hfill {\overline{C}}_7& =\hfill & \hfill C_7+C_{\varphi }(C_6{a}_{76}\left(n\right)+C_8{a}_{78}\left(n\right)),\\ \hfill {\overline{C}}_8& =\hfill & \hfill C_8+C_{\varphi }(C_6{a}_{86}\left(n\right)+C_7{a}_{87}\left(n\right)).\end{array}$$

(14)

The coefficients ${\overline{C}}_k$ must obey the unitarity conditions

$$\begin{array}{ccc}\hfill & \hfill & \hfill |{\overline{C}}_k|<1\mathrm{for}k=6,7,8;\\ \hfill & \hfill & \hfill |{\overline{C}}_6{|}^2+|{\overline{C}}_7{|}^2+{|{\overline{C}}_8|}^2<1.\end{array}$$

(15)

Such a constraint may be used for the estimate of the chameleon–matter coupling constant $\beta$. Figure 2 illustrates the dependence of the coefficients $C_k\left(D\right)$ on the distance $D=h+d/2$ for quantum states $k=1\dots 20$.

A numerical analysis shows that the contribution of $a_{86}\left(n\right)$ can be neglected in comparison with the contributions of $a_{67}\left(n\right)$ and $a_{87}\left(n\right)$. Then, $a_{67}\left(n\right)=-a_{76}\left(n\right)$ and $a_{87}\left(n\right)=-a_{78}\left(n\right)$. The behaviour of $a_{67}\left(n\right)$ and $a_{87}\left(n\right)$ as functions of n for $n\in [2,10]$ is shown in Figure 3. One may see that $a_{67}\left(n\right)\simeq -a_{87}\left(n\right)$. Using the obtained relations, the coefficients ${\overline{C}}_k$ are

$$\begin{array}{ccc}\hfill {\overline{C}}_6& =& C_6+C_{\varphi }{C}_7{a}_{67}\left(n\right),\hfill \\ \hfill {\overline{C}}_7& =& C_7+C_{\varphi }(C_8-C_6)a_{67}\left(n\right),\hfill \\ \hfill {\overline{C}}_8& =& C_8-C_{\varphi }{C}_7{a}_{67}\left(n\right)\hfill \end{array}$$

(16)

or

$$\begin{array}{ccc}\hfill {\overline{C}}_6& =& C_6+1.54\times {10}^{-9}\beta C_7{a}_{67}\left(n\right),\hfill \\ \hfill {\overline{C}}_7& =& C_7+1.54\times {10}^{-9}\beta (C_8-C_6)a_{67}\left(n\right),\hfill \\ \hfill {\overline{C}}_8& =& C_8-1.54\times {10}^{-9}\beta C_7{a}_{67}\left(n\right)\hfill \end{array}$$

(17)

One may see that the first-order perturbation theory, with respect to the coupling constant $C_{\varphi }=1.54\times {10}^{-9}\beta \ll 1$, the constraint

$$\begin{array}{c}\hfill |{\overline{C}}_6{|}^2+|{\overline{C}}_7{|}^2+|{\overline{C}}_8{|}^2=|C_6{|}^2+|C_7{|}^2+{|C_8|}^2<1\end{array}$$

(18)

is fulfilled. However, for the upper bound $\beta <5\times {10}^9$ on the chameleon–matter coupling constant, we obtain that $C_{\varphi }<7.69$. Since this violates a perturbative analysis of the wave functions of ultra-cold neutrons, the upper bound on the coupling constant $\beta$ should be substantially decreased. Using the numerical values of the coefficients $C_k$ for $k=6,7,8$, we obtain

$$\begin{array}{ccc}\hfill {\overline{C}}_6& =& 0.42+1.02\times {10}^{-9}\beta a_{67}\left(n\right),\hfill \\ \hfill {\overline{C}}_7& =& 0.66+0.20\times {10}^{-9}\beta a_{67}\left(n\right),\hfill \\ \hfill {\overline{C}}_8& =& 0.55-1.02\times {10}^{-9}\beta a_{67}\left(n\right).\hfill \end{array}$$

(19)

and new formulas for n = 2 and h = 47

$$\begin{array}{ccc}\hfill {\overline{C}}_5& =\hfill & \hfill 0.15+0.37\times {10}^{-9}\beta \\ \hfill {\overline{C}}_6& =\hfill & \hfill 0.42+0.13\times {10}^{-9}\beta \\ \hfill {\overline{C}}_7& =\hfill & \hfill 0.66-1.80\times {10}^{-9}\beta \\ \hfill {\overline{C}}_8& =\hfill & \hfill 0.55+2.01\times {10}^{-9}\beta \\ \hfill {\overline{C}}_{10}& =\hfill & \hfill 0.17-0.06\times {10}^{-9}\beta \end{array}$$

(20)

A reasonable upper bound on the coupling constant $C_{\varphi }$ compatible with a perturbative analysis of contributions of a chameleon field is

$$\begin{array}{c}\hfill C_{\varphi }{a}_{67}\left(n\right)=1.54\times {10}^{-9}\beta a_{67}\left(n\right)\le 0.1.\end{array}$$

(21)

Equation (21) does not represent an experimentally derived constraint on the chameleon–matter coupling constant $\beta$. Rather, it defines a theoretical upper limit that ensures the validity of the first-order perturbative treatment employed in this analysis. Equation (21) gives a new upper bound on the chameleon–matter coupling constant $\beta \le 6.5\times {10}^8$, which is of an order of magnitude smaller compared with the upper bound $\beta <5\times {10}^9$, obtained in [18] from the transition frequencies of quantum gravitational states of an ultra-cold neutron, confined between two mirrors. For the upper bound Equation (21), the coefficients ${\overline{C}}_k$ with $k=6,7,8$ are

$$\begin{array}{ccc}\hfill 0.42\le & {\overline{C}}_6& \le 0.49,\hfill \\ \hfill 0.66\le & {\overline{C}}_7& \le 0.67,\hfill \\ \hfill 0.55\ge & {\overline{C}}_8& \ge 0.48\hfill \end{array}$$

(22)

with the sum of the squared absolute values equal to $|{\overline{C}}_6{|}^2+|{\overline{C}}_7{|}^2+{|{\overline{C}}_8|}^2=0.92$.

The estimates of the coefficients ${\overline{C}}_k$ with $k=6,7,8$ may be extracted from the experimental data [28] on a free fall of ultra-cold neutrons in the spatial region $z\ge -D$ above a mirror with the wave function

$$\begin{array}{ccc}\hfill {|\Psi (z,x,t)|}^2& \hfill =& \hfill |{\overline{C}}_6{\Psi }_6(z,t)+{\overline{C}}_7{\Psi }_7(z,t)+{\overline{C}}_8{\Psi }_8{(z,t)|}^2\\ \hfill & \hfill & \hfill \times {\displaystyle \frac{1}{\sqrt{\pi {\Delta }_t^2}}}{e}^{-\frac{{(x-v_{0}t)}^2}{2{\Delta }_t^2}}.\end{array}$$

(23)

The experimental data on a free fall of ultra-cold neutrons from [18] may be fitted by only one parameter $\kappa =0.20\times {10}^{-9}\beta a_{67}\left(n\right)$, defining the coefficients ${\overline{C}}_k$ as follows ${\overline{C}}_6=0.42+5\kappa$, ${\overline{C}}_7=0.66+\kappa$, and ${\overline{C}}_8=0.55-5\kappa$, respectively.

4. Discussion

We have analysed a quantum free fall of ultra-cold neutrons from the gap of a spatial region $z^2\le \frac{d^2}{4}$ between two mirrors with $d=25.5$ μm into the spatial region above a mirror, located at $z=-D$ below the lower mirror at $D=47$ μm. Between two mirrors, ultra-cold neutrons are prepared in the ground quantum gravitational state and move with a velocity $v_0=6\mathrm{m}/\mathrm{s}$ in the x-direction with the wave function, taken in the form of a Gaussian wave packet. For a depth $D-\frac{d}{2}=47$ μm of a spatial region of a quantum free fall, we have shown that the wave function of the z-degrees of freedom of ultra-cold neutrons can be approximated by a superposition of three excited quantum gravitational states with the principal quantum numbers $k=6,7$, and 8 with coefficients $C_6=0.42$, $C_7=0.66$, and $C_8=0.55$, respectively. The probability of an observation of ultra-cold neutrons in such a mixed quantum gravitational state is $P=|C_6{|}^2+|C_7{|}^2+{|C_8|}^2=92$. Using perturbation theory, we calculated the first-order corrections to the wave functions of ultra-cold neutrons in the spatial region of a quantum free fall. A numerical analysis shows that the main contributions to the wave function of the z-degrees of freedom of ultra-cold neutrons appear only to three excited quantum gravitational with the principal quantum numbers $k=6,7$, and 8 and from these excited states. The contributions of a chameleon field changes the coefficients $C_k$ with $k=6,7$, and 8, as follows $C_6\to {\overline{C}}_6+5\kappa$, $C_7\to {\overline{C}}_7\to 0.66+\kappa$, and $C_8\to {\overline{C}}_8=0.55-5\kappa$, where $\kappa =0.20\times {10}^{-9\beta a_{67}}\left(n\right)$ and $a_{67}\left(n\right)$ is a function of the power n of the potential of a self-interaction of a chameleon field $0.2\ge a_{67}\ge 0.1$ for $n\in [2,10]$. A perturbative analysis of the contributions of a chameleon field assumes that the effective coupling constant $C_{\varphi }=1.54\times {10}^{-9}\beta$ is much smaller compared with unity, i.e., $C_{\varphi }\ll 1$. For the estimate $\beta <5\times {10}^9$, we obtain $C_{\varphi }<7.7$. Since this value is not compatible with a perturbative analysis of a chameleon field and violates the unitarity condition on the coefficients ${\overline{C}}_6$, ${\overline{C}}_7$, and ${\overline{C}}_8$, i.e., $|{\overline{C}}_k{|}^2<1$ and $|{\overline{C}}_6{|}^2+|{\overline{C}}_7{|}^2+{|{\overline{C}}_8|}^2<1$, the estimate $\beta <5\times {10}^9$ should be improved in agreement with a perturbative analysis of chameleon field and a unitarity condition. A reasonable theoretical upper bound on the effective coupling constant $C_{\varphi }\le 0.1$ compatible with a unitarity condition of the coefficients ${\overline{C}}_6$, ${\overline{C}}_7$, and ${\overline{C}}_8$ leads to a new constraint on the chameleon–matter coupling constant $\beta$, i.e., $\beta \le 6.5\times {10}^8$, which is of order of magnitude smaller compared with the estimate $\beta <5\times {10}^9$ [18]. A new upper bound $\beta \le 6.5\times {10}^8$ is compatible with the experimental data on the transition frequencies of quantum gravitational states of ultra-cold neutrons [18] at the level of a sensitivity $S=\Delta \omega /\omega \le 0.6\%$. For the theoretical transitions $|6\rangle \to |7\rangle$ and $|7\rangle \to |8\rangle$, the upper bound $\beta \le 6.5\times {10}^8$ may be observed for the sensitivities $S_{67}\le 0.6$ and $S_{78}\le 1.5\%$, respectively.

With the stability of the new upper bound $\beta \le 6.5\times {10}^{-8}$, one may prove by varying the depth of a quantum free-fall region. Indeed, decreasing the depth of a quantum free-fall region to $D=30$ μm leads to the following approximation of the wave function ${\psi }_1\left(z\right)$, owing an approximation of the wave function ${\psi }_1(z,t)$

$$\begin{array}{c}\hfill {\psi }_1(z,t)=C_3{\Psi }_3(z,t)+C_4{\Psi }_4(z,t)+C_5{\Psi }_5(z,t)\end{array}$$

(24)

with the coefficients equal to $C_3=0.28$, $C_4=C_5=0.66$. In the region of a quantum free fall, ultra-cold neutrons are in the mixed state, described by the wave function Equation (24), with a probability $P=|C_3{|}^2+|C_4{|}^2+{|C_5|}^2=0.95$. The contributions of a chameleon field are given by ${\overline{C}}_3=0.28+0.99\kappa$, ${\overline{C}}_4=0.66-1.22\kappa$ and ${\overline{C}}_4=0.66+1.22\kappa$, where $\kappa =1.54\times {10}^{-9}\beta a_{54}\left(n\right)$. The function $a_{54}\left(n\right)$ varies over the region $0.06\le a_{54}\le 0.03$ for $n\in [2,10]$. One may show that the upper bound $\beta <5\times {10}^9$, obtained in [18], violates a unitarity condition, whereas a new upper bound $\beta \le 6.5\times {10}^8$, obtained above, does not violate a unitarity. It is also compatible with a perturbative analysis of the contributions of a chameleon field. At $\beta \le 6.5\times {10}^8$ the coefficients ${\overline{C}}_k$ are defined by the inequalities $0.28\le {\overline{C}}_3\le 0.33$, $0.66\ge {\overline{C}}_4\ge 0.59$, and $0.66\le {\overline{C}}_3\le 0.70$, and describe the mixed state Equation (24) of ultra-cold neutrons with the probability $P=|{\overline{C}}_3{|}^2+|{\overline{C}}_4{|}^2+{|{\overline{C}}_5|}^2=0.95$.

An increase in the depth of the quantum free-fall region, for example to $D=70$ μm, leads to a mixed quantum gravitational state of ultra-cold neutrons, which should contain a complete set of pure quantum gravitational states ${\Psi }_k(z,t)$. A truncated wave function, containing the contributions of the state ${\Psi }_k(z,t)$ with the largest coefficients, takes the form

$$\begin{array}{ccc}\hfill & & {\psi }_1(z,t)=C_{10}{\Psi }_{10}(z,t)+C_{11}{\Psi }_4(z,t)+C_{13}{\Psi }_{13}(z,t),\hfill \end{array}$$

(25)

with $C_{10}=0.35$, $C_{11}=0.58$, and $C_{13}=-0.14$, respectively. The probability of an observation of ultra-cold neutrons in the state with the wave function Equation (25) is $P=|C_{10}{|}^2+|C_{11}{|}^2+{|C_{13}|}^2=0.48$. For the analysis of a contribution of a chameleon field, we may neglect the contribution of the wave function ${\Psi }_{13}(z,t)$. The probability of the state with the wave function ${\psi }_1(z,t)=C_{10}{\Psi }_{10}(z,t)+C_{11}{\Psi }_4(z,t)$ is $P=|C_{10}{|}^2+{|C_{11}|}^2=0.46$. The contribution of a chameleon field changes the coefficients $C_{10}$ and $C_{11}$, as follows ${\overline{C}}_{10}=C_{10}+C_{11}\kappa$ and ${\overline{C}}_{11}=C_{11}-C_{10}\kappa$, where $\kappa =1.54\times {10}^{-9}\beta a_{10,11}\left(n\right)$ with $0.28\ge a_{10,11}\left(n\right)\ge 0.10$ for $n\in [2,10]$. For $\beta \le 6.5\times {10}^8$ we obtain $0.35\le {\overline{C}}_{10}\le 0.51$ and $0.58\ge {\overline{C}}_{11}\ge 0.48$.

The proposed analysis of the dependence of the upper bound of the chameleon–matter coupling constant $\beta$ on the depth of a quantum free-fall region of ultra-cold neutrons confirms the stability of the estimate $\beta \le 6.5\times {10}^8$. Of course, such a theoretical constraint should be improved experimentally by fitting the experimental data from the QBB experiments by only one parameter $\kappa$.

5. Binding Energy of Ultra-Cold Neutrons to Second-Order Perturbation Theory

For the completeness of our analysis of the contributions of a chameleon field to the wave function of the mixed state of ultra-cold neutrons, bouncing above a mirror in the gravitational field of the Earth, we have to analyse the second-order corrections to the binding energies of ultra-cold neutrons. The binding energy of the quantum gravitational k-state, calculated to second-order perturbation theory, is equal to [42]

$$E_k=-mg{l}_0{\zeta }_k+mg{l}_0{C}_{\varphi }\langle k|{\displaystyle \frac{\varphi }{\Lambda }}|k\rangle +mg{l}_0{C}_{\varphi }^2\sum _{k^{\prime }\ne k}\langle k|{\displaystyle \frac{\varphi }{\Lambda }}|k^{\prime }\rangle a_{k,k^{\prime }}\left(n\right).$$

(26)

The numerical analysis we carry out for the height $h=47$ μm and the quantum gravitational states with a principle quantum number $k=6,7,8$, which give the main contributions to the mixed state $\Psi (z,t)$. At $\beta =7.7\times {10}^7$, the first- and second-order corrections $\Delta E_1$ and $\Delta E_2$ to the binding energies of the quantum gravitational states of ultra-cold neutrons under consideration are shown in Figure 4. The smallness of the second-order corrections with respect to the first-order ones testifies to the applicability of perturbation theory for the analysis of contributions of a chameleon field. However, a much stronger dependence of the second-order corrections to the binding energies on the power n makes the search for the experiments meaningful, where the second-order corrections to the binding energies of ultra-cold neutrons may be observed. In this case, one may make stronger constraints on the power n of the potential of a self-interaction of a chameleon field.

6. Conclusions

This study refines the upper bound on the chameleon–matter coupling constant ($\beta \le 6.5\times {10}^8$), advancing our understanding of scalar field theories and their role in cosmic acceleration. By leveraging the unique properties of ultra-cold neutrons in qBounce experiments, we demonstrate the capability of laboratory-based setups to probe fundamental physics with unprecedented precision.

The findings highlight the utility of ultra-cold neutron experiments as precise laboratories for testing fundamental theories in physics. By providing a significantly tighter upper bound on the chameleon–matter coupling constant, this research underscores the sensitivity of quantum gravitational systems to subtle scalar field interactions.

These results have profound implications for the role of chameleon fields as drivers of cosmic acceleration. The refined constraints on $\beta$ challenge the viability of certain parameter ranges within chameleon field theories, narrowing the scope of models compatible with observed cosmic acceleration. By demonstrating the sensitivity of terrestrial experiments to such scalar field interactions, this work strengthens the case for chameleon fields as potential contributors to dark energy while also placing stringent limits on their behaviour.

One critical implication is the potential of qBounce setups to distinguish between competing dark energy models. The refined constraints can aid in differentiating chameleon field effects from alternative mechanisms driving cosmic acceleration. Moreover, the methodology described herein can be adapted to investigate other scalar fields, broadening its applicability in particle physics and cosmology.

Future work should focus on enhancing experimental sensitivities, particularly in detecting higher-order perturbations induced by chameleon fields. Exploring variations in experimental parameters, such as mirror separation or neutron velocity, may yield deeper insights into scalar field dynamics. The integration of complementary observational data from astrophysical phenomena could further validate the theoretical models tested here.

The implications extend beyond chameleon fields, offering insights into scalar field dynamics, quantum mechanics, and gravitational phenomena. These findings lay the groundwork for future experimental and theoretical advancements, highlighting the interplay between cosmology and quantum physics in addressing some of the Universe’s most profound questions.