Could trapped quintessence account for the laser-detuning-dependent acceleration of cold atoms in varying-frequency time-of-flight experiments?

Hai-Chao Zhang, Xin-Ping Xu, Jing-Fang Zhang, and Chuan Wang

Phys. Rev. D 105, 102006 – Published 26 May 2022

Abstract

Using a trapped quintessence model, a series of time-of-flight (TOF) experiments with different frequencies of probe light are designed and performed. The varying-frequency TOF (VFTOF) experiments demonstrate that the free-fall acceleration of test atoms is dependent on the detuning of the probe light frequency with respect to the atomic transition frequency. In appropriately designed experiments, if the scalar field in the model accounts for the accelerated expansion of the Universe entirely, the field will result in an observable fifth force. Meanwhile, the trapped quintessence model still satisfies all experimental bounds on deviations from general relativity due to both the saturation effect and the short interaction range of the scalar field. The scalar saturates at a value corresponding to the cosmological constant when the microscopic nonrelativistic matter density is large enough. The interaction range of the scalar is inversely proportional to the square root of the microscopic nonrelativistic matter density. The interaction range has been estimated to be several $μ m$ in the current cosmic density $\sim 10^{- 27} kg / m^{3}$ . The Universe is assumed to be permeated with fuzzy dark matter, which means that the microscopic nonrelativistic matter density defined through the quantum wave functions of the ultralight particles can be used on the cosmic scale. In an almost completely empty space between atoms of a dilute atomic gas in an ultrahigh vacuum chamber, the interaction range of the scalar field may approach the order of $\sim 1 μ m$ in the presence of dark matter; then, the scalar field might be detected in laboratories. Since the trapped quintessence model hypothesizes that the scalar strongly couples to nonrelativistic matter but cannot couple to radiation, the source for generating the fifth force is experimentally set up by the laser-irradiated background atoms in the ultrahigh vacuum chamber. The mass density of the source is altered by detuning the frequency of the probe laser light from the atomic resonance transition. The test atoms are prepared by the laser cooling technique and located initially above the probe light. When the test atoms are released from their initial positions, they are able to pass through the region of the source that generates the fifth force to be measured. Thus, if the scalar field exists, the corresponding fifth force might be sensed by the test atoms even if the interaction range is extremely short. By measuring the free-fall acceleration of the test atoms with the TOF method step by step in the detuning frequency domain of the probe light, we derive the dispersion curves of the measured acceleration versus the frequency detuning of the probe light. When the nonrelativistic matter density of the source increases due to the energy gained from the laser light, the test atoms are pulled to the center of the source, and vice versa. If the trapped quintessence model is correct, the observed detuning-dependent acceleration in the VFTOF scheme suggests a closed Universe, i.e., a positive spatial curvature of the Universe.

Received 22 July 2021
Accepted 5 May 2022

DOI:https://doi.org/10.1103/PhysRevD.105.102006

Physics Subject Headings (PhySH)

Conformal field theory Cooling & trapping Dark energy Dark matter Experimental studies of gravity Light-matter interaction Quantum cosmology

Gravitation, Cosmology & AstrophysicsAtomic, Molecular & OpticalParticles & Fields

Authors & Affiliations

Hai-Chao Zhang ^*, Xin-Ping Xu, Jing-Fang Zhang, and Chuan Wang

Key Laboratory for Quantum Optics, Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, Shanghai 201800, China

^*zhanghc@siom.ac.cn

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Issue

Vol. 105, Iss. 10 — 15 May 2022

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Images

Figure 1
Vertical profile of the scalar field generated by a cylindrical source. The dotted curve is obtained by the definition of Eq. (10) together with the closed Universe parameters shown in Eq. (5), which is noisily in the inside of the source due to the numerical calculation. The solid curve is the smooth result of the dotted curve. The dashed straight line corresponds to the scalar field’s average value in the region of the source, which is calculated by Eq. (12). The interaction range $ƛ_{c}$ is strongly dependent on the EBD. The larger the EBD, the shorter the Compton wavelength. Thus, in order to clearly display the character of the scalar field, we draw the curves by choosing the EBD with an imaginary value of $ρ_{b} = 1 \times 10^{- 28} kg \cdot m^{- 3}$ that is even lower than the current cosmic density $\sim 10^{- 27} kg \cdot m^{- 3}$ . The source radius $R_{cyl}$ is selected as 0.75 mm.
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Figure 2
Schematic of experimental setup for testing the scalar fifth force. The number density of background rubidium-87 atoms in the vacuum cell ( $30 mm \times 30 mm \times 70 mm$ ) is adjusted to a desired pressure by running the electric current through a dispenser (not shown). The source for generating the scalar fifth force is part of the background atoms that is irradiated by a pair of probe laser beams. The microscopic matter density of the source is adjusted by tuning the frequency of the probe laser light. The test atomic cloud is formed by a standard laser cooling technique, which is located 8.5 mm above the axis center of the probe light. The free-fall acceleration of the cold atomic cloud is measured by the TOF method. If the fifth force of the source is large enough, the measured acceleration should reflect the scalar fifth force, besides the gravity of the Earth. The TOF signals are recorded with a photodiode (PD) by collecting the fluorescence of the test atoms when they encounter the probe light. The lens is used to collect a larger solid angle range of the fluorescence.
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Figure 3
Variation of the source’s mass density as a function of laser detuning. In order to highlight the detuning dependence of the energy density of the source, the average intensity $\bar{I} = P_{0} / (π σ_{p}^{2})$ rather than the spatial-position-dependent intensity $I$ shown in Eq. (21) is used to calculate the variation of the energy density with Eq. (19). The solid curve corresponds to the upper and lower limits of the integral on $v_{z}$ being $\pm 2 Γ / k = \pm 9.467 m / s$ for the $D_{2}$ transition of ${}^{87}{Rb}$ atom. To recognize the importance of $2 Γ / k$ , the dotted curve is drawn for comparison, in which the upper and lower limits of the integral on $v_{z}$ are chosen as $\pm 550 m / s$ being about twice the average velocity $270 m / s$ of the atomic gas at room temperature. In both curves, the EBD is chosen to be the same as $ρ_{b} = 1 \times 10^{- 25} kg \cdot m^{- 3}$ .
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Figure 4
Spatial profile of the laser-adjusted matter density of the source. The solid curve shows that the matter density varies along the $y$ axis, which is numerically calculated in Eq. (19) with the spatial-dependent intensity $I$ shown by Eq. (21) in a fixed value of the detuning. The parameters in the calculation are as follows: the per beam power $P_{0} = 250 μ W$ , the $1 / e^{2}$ Gaussian radius $σ_{p} = 0.75 mm$ of the laser beam, the value of detuning selected as $Δ = 1 \cdot Γ$ , and the EBD selected as $ρ_{b} = 1 \cdot 10^{- 25} kg \cdot m^{- 3}$ . The dotted curve shows the Gaussian profile of the laser intensity for comparing to the profile of the mass density of the source. The profile of the mass density of the source does not behave like the Gaussian profile of the laser intensity and indeed depends on the laser power.
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Figure 5
Scalar fifth force of the laser-adjusted source versus the light frequency detuning from the atomic resonance frequency. The magnitude of the fifth force depends on the spatial curvature of the Universe: (a) using the parameters of $M_{1}$ and $M_{2}$ in the almost flat Universe shown in Eq. (4) and (b) using the parameters of $M_{1}$ and $M_{2}$ in the closed Universe shown in Eq. (5). The mass density of the source is regarded as a cylindrical form of $ρ = (ρ_{cyl} - ρ_{b}) Θ (R_{cyl} - r) + ρ_{b}$ , where the radius of the cylinder is selected as the $1 / e^{2}$ Gaussian radius of the laser beam, i.e., $R_{cyl} = σ_{p} = 0.75 mm$ . The difference between the source’s mass density and the EBD is calculated using Eq. (19) with the average light intensity $\bar{I} = P_{0} / (π σ_{p}^{2})$ . The initial distance $L$ between the test atomic cloud and the source is selected to be 8.5 mm. In both curves (a) and (b), the EBD and the laser power per beam are the same as that in Fig. 3, i.e., $ρ_{b} = 1 \cdot 10^{- 25} kg \cdot m^{- 3}$ , $P_{0} = 30 μ W$ , respectively.
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Figure 6
Acceleration variations versus the detuning of the probe beams. The experimental data are marked by the squares. Every square corresponds to an average value over the four values of the acceleration that are fitted from the TOF signals repeated 4 times. The error bar for each square is the standard deviation based on the four measurements. (a) The probe power per beam is $P_{0} = 30 μ W$ , and the repumping light is locked to the $F_{g} = 1 \to F_{e} = 2$ resonant transition, i.e., $Δ_{repump} = 0$ . (b) The probe power per beam is $P_{0} = 150 μ W$ , and the repumping light detuning is $Δ_{repump} = 0$ . (c) The probe power per beam is $P_{0} = 150 μ W$ , and the repumping light detuning is $Δ_{repump} = + 2 π \cdot 10 MHz$ . (d) The probe power per beam is $P_{0} = 150 μ W$ , and the repumping light detuning is $Δ_{repump} = - 2 π \cdot 10 MHz$ . The repumping power per beam in the probe beams is fixed as $P_{repump} = 300 μ W$ for all of the cases (a)–(d). The calculated curves are derived by using Eq. (13) together with Eq. (19) based on the assumption that the experimental data $δ a$ can be explained as the scalar fifth force shown in Eq. (13). The solid curves correspond to the EBD $ρ_{b} = 1.3 \cdot 10^{- 26} kg \cdot m^{- 3}$ . The dotted curves correspond to the EBD $ρ_{b} = 1.9 \cdot 10^{- 26} kg \cdot m^{- 3}$ . In all of the calculated solid and dotted curves, the cosmic parameters are selected as the parameters for the closed Universe, i.e., $M_{2} = 4.40353 meV / c^{2}$ and $M_{1} = M_{2} / 3$ shown in Eq. (5). If we use the parameters for the almost flat Universe as shown in Eq. (4) as $M_{2} = 4.96168 meV / c^{2}$ and $M_{1} = M_{2} / 8$ , the EBD $ρ_{b}$ is fitted on the order of $\sim 5 \cdot 10^{- 27} kg \cdot m^{- 3}$ , being slightly larger than or near the current cosmic density. The two probe beams here are linearly and parallel polarized. Similar experimental dispersion curves can also be demonstrated in the case of circular polarized probe beams.
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