Effects of ultra-light dark matter on the gravitational quantum well
EEffects of ultra-light dark matter on the gravitational quantum well
Paolo Castorina ∗ Dipartimento di Fisica, Universit`a di Catania,and INFN Sezione di Catania, Via Santa Sofia 64 I-95123 Catania, Italia
Alfredo Iorio † and Michal Malinsk´y ‡ Faculty of Mathematics and Physics, Charles University,V Holeˇsoviˇck´ach 2, 18000 Prague 8, Czech Republic (Dated: June 30th 2017)
Abstract
We study the influence of a periodic perturbation of the effective masses of the nucleons, due to theassumed semi-classical ultra-light dark matter background, on the motion of neutrons in a gravitationalquantum well. Our focus is on the transition probability between the lowest two energy states, with theRabi frequency in the kHz region corresponding to the series of “sweet spot” dark matter masses in the10 − eV ballpark. The relevant probability is written in terms of the specific mass and of the effectivecoupling to the ordinary matter. These parameters can be constrained by the non-observation of anysignificant deviations of the measured transition probabilities from the dark-matter-free picture. PACS numbers: 95.35.+d, 03.65.w, 28.20.v
I. INTRODUCTION
It is well known that most of the direct de-tection searches [1] for weakly interacting mas-sive particles (WIMPs), perhaps the most pop-ular [2] hypothetical form of the Dark matter(DM), may in a not-so-distant future hit theirconceptual sensitivity floor corresponding to theirreducible background due to neutrino interac-tions inside the detectors. Remarkably enough,the room is narrowing down quickly not only forthe “classical” WIMPs with masses at the levelof tens of GeV or above [3] but, with facilitieslike SuperCDMS [4] on the horizon, the same islikely to happen also for much lighter candidateswith sub-GeV masses. In view of that a lot ofattention has recently been paid to alternativesto WIMPs, corresponding to very light or evenultra-light DM (ULDM) candidates such as, e.g.,axions [5] or other types of scalar fields [6] withmasses reaching deep inside the sub-eV region.However, the interactions of such a substancewith the ordinary matter is likely to be very dif- ∗ Electronic address: [email protected] † Electronic address: [email protected]ff.cuni.cz ‡ Electronic address: [email protected]ff.cuni.cz ferent from the usual particle scattering picturefor WIMPs. Indeed, the observed energy densityin DM yields occupation numbers so high thatthe system acts coherently resembling a classi-cal wave rather than a set of individual quanta.Hence, in looking for the effects of such a formof matter in laboratory-based experiments onemay take the advantage of simple quantum me-chanical systems and consider their response tothe quasi-classical DM background.Recently the sensitivity of the atomic inter-ferometry to oscillating scalar ULDM has beenanalyzed in [7] for scalar fields φ of masses inthe range 10 − eV ≤ m φ ≤ f φ = m φ c /h , in therange of 10 − Hz ≤ f φ ≤ Hz. Indeed, thelinear and quadratic couplings [7] of the Stan-dard Model (SM) fields with the DM fields givea modulation of the fermion masses and of thefundamental constants. This affects the mass ofparticles and of the Earth, the former taken intoaccount by m → m + δm , the latter by a mod-ification of the local gravitational acceleration g → g + δg .In this letter we study how a GravitationalQuantum Well (GQW) experiment can eitherdetect or constraint these effects due to ULDM,1 a r X i v : . [ h e p - ph ] J u l hrough a resonance mechanism between theULDM oscillations frequencies and the frequen-cies associated to the neutron bouncing phe-nomenon in the GQW [8]. High precison GQWexperiments have already been shown to givestrong constraints on new physics, such as space-time noncommutativity [9], violation of theequivalence principle [10], and more. II. THE GRAVITATIONAL QUANTUMWELL
The GQW is a system conventionally madeof a quantum particle in a potential well real-ized by i) a homogeneous gravitational field withits gradient oriented in the (by definition) verti-cal direction, say x , and ii) a horizontal mirroralong, say, y (usually placed at x = 0), wherethe particle experiences perfect elastic reflection,see, e.g., [11], and also Fig. 1 here.The eigenvalue equation, ˆ H Ψ s = E s Ψ s , s = 1 , , ... , has well known separable formΨ s ( x, y ) = ψ s ( x ) χ ( y ). Here the eigenfunctionscorresponding to x are those of bound states (asfor any potential well) and, as well known [11],are given in terms of the Airy function ϕψ s ( x ) = A s ϕ ( x/x + α s ) , (1)where the α s = {− . , − . , − . , ... } identify the zeroes of ϕ , x s = − α s x , with x ≡ (¯ h / (2 m g )) / . One can introduce a di-mensionless coordinate z ≡ x/x + α s , in terms ofwhich the normalization coefficients are writtenas A s ≡ ( x (cid:82) + ∞ α s dzϕ ( z )) − / . The eigenvaluesare E s = − m g x α s . (2)For m (cid:39) . neutron ,and g (cid:39) . s , one has x (cid:39) . µ m, and m g x (cid:39) . y , χ ( y ) corresponds to apacket of plane waves of continuous energy spec-trum χ ( y ) = (cid:90) + ∞−∞ g ( k ) e iky dk , (3) where g ( k ) determines the shape of the packetin phase space.In GQW experiments with neutrons [12, 13],it has been possible to identify their quan-tum states, ψ s ( x ), by realizing a horizontal slitwith the upper boundary corresponding to ascatterer/absorber, above the horizontal mirror.When the absorber is at a height less than a crit-ical value, h < h crits , the neutrons shot into theslit with energy E s (and greater) do not make itout on the other side of the apparatus, as theyare absorbed by the scatterer/absorber. Thiscritical value corresponds to the classical turn-ing point for that given quantized energy , thatis, h crits ≡ E s /m g = − α s x = x s . Detailed de-scription of the experimental set-up can be foundin [12], and in the review [8] (see also Fig. 1 here),while the report of the first identification of thelowest quantum state is in [12].Recently, Nesvizhevsky et al. [13] were ableto measure the critical heights for the first twoquantum states, obtaining the following results x exp = 12 . ± . syst ) ± . stat ) ( µ m) ,x exp = 21 . ± . syst ) ± . stat ) ( µ m) . (4)The corresponding theoretical values can be de-termined from x n = − α n x for α = − . α = − . x = 5 . µ m yielding x = 13 . µ m and x = 24 . µ m, correspondingto the energy eigenvalues E = 1 .
407 peV and E = 2 .
461 peV. These values are contained inthe error bars, and allow for maximum absoluteshifts of the energy levels with respect to thepredicted values:∆ E exp = 6 . × − J = 0 .
41 peV , ∆ E exp = 8 . × − J = 0 .
54 peV . (5)In this experiment, neutrons exhibited a meanhorizontal velocity of (cid:104) v y (cid:105) (cid:39) . − . The extent to which the Equivalence Principle can besaid to hold in this experiment is discussed in [8]. Inthis letter we do not consider any such violations, i.e.,for us the inertial and gravitational masses are indis-tinguishable. bsorber m g xE x ( μ m ) E ( peV ) oscillates with n ω ϕ P mirror E classically forbiddenregion FIG. 1: A neutron with energy E n is absorbed whenthe slit aperture h is equal or less than the corre-sponding classical turning point x s = E s /m g = − α s x . In the picture, x = 13 . µ m, and x =24 . µ m, E = 1 .
407 peV, E = 2 .
461 peV. The prob-ability | ψ s | is maximal not at x s , but rather at asmaller value, that in the picture are x max = 7 . µ m,for | ψ | , and x max = 18 . µ m, for | ψ | . Therefore,setting h at about 15 µ m (green line) neutrons in thestate ψ should not be seen unless a Rabi transition, ψ → ψ induced by ULDM, with probability P ,takes place. The second energy level, E , and theprobability function, | ψ | , are drawn in dots. III. ULDM EFFECTS
The interactions of DM fields φ with theSM matter can be described by the effective la-grangian densities − L intn = (cid:32) √ ¯ hc Λ n,f (cid:33) n m f ¯ ψ f ψ f φ n , (6)where n indicates the order of the interaction, f stands for the type of the SM matter under con-sideration and m f and ψ f denote its masses andfield operators, respectively. All these structuresare weighted by the inverse of the relevant high-energy scales, Λ n,f , that also include the a-prioriunknown couplings.The main implication of (6) is the space-and time-dependent modulation of the fermionmasses in the ULDM background m efff m f = 1 + √ ¯ hcφ ( (cid:126)r, t ) n Λ nn,f , (7)and, in turn, the variation of the local gravita-tional acceleration (due to the modulation of the mass of the Earth) assuming the DM field per-meates through the Earth body, thus making itsmass change slightly in time [7].To evaluate how this affects the GQW weconsider the specific case proposed in [7] andassume that the local gravitational accelerationand the mass of the neutron both vary periodi-cally in time as g ( t ) = g + g cos ( ωt ) , (8) m ( t ) = m + m cos ( ωt ) , (9)where ω = nω φ , with ω φ = m φ c / ¯ h .The motion along the x axis is governed bythe Hamiltonian ˆ H = ˆ p x / (2 m ) + mg ˆ x which, by(8) and (9), becomesˆ H = ˆ p x m + m cos( ωt )) (10)+ ( m + m cos( ωt ))( g + g cos( ωt ))ˆ x . Since the corrections due to ULDM interactionare small, i.e. m (cid:28) m amd g (cid:28) g , we canwrite ˆ H (cid:39) ˆ H + ˆ V ( t ), with ˆ H ≡ ˆ p x / m + m g ˆ x , ˆ V ( t ) ≡ ˆ V cos( ωt ) andˆ V = − ˆ p x m m m + (cid:18) m m + g g (cid:19) m g ˆ x , (11)that are the expressions we shall consider in eval-uating the ULDM effects. A. Time independent corrections
Just for curiosity let us recall that, for n = 1,the oscillation frequency in terms of the ULDMmass m φ is given by f φ = 2 πω = 2 . × ( m φ [ eV ]) Hz (12)and that the time of flight of neutrons in theGQW studied in [8] is T (cid:39)
40 ms. Obviously,for ωT (cid:28)
1, i.e. m φ (cid:28) . × − eV, onecan neglect the time dependence in ˆ V ( t ) and themere effect of the ULDM consists in the shifts inthe energy levels of the unperturbed Hamilto-nian ˆ H , E and E . For m φ < − eV thesecan be readily evaluated by looking at (cid:104) s | ˆ V | s (cid:105) , s = 1 , , ... , with ˆ V in (11), and, in order to becompatible with the measurement, they should3e within the maximum allowed shifts ∆ E ,∆ E of Eq. (5). The numerical calculation, re-ported in the Appendices, gives E (cid:20) m m .
341 + g g . (cid:21) ≤ .
41 peV (13) E (cid:20) m m .
333 + g g . (cid:21) ≤ .
54 peV (14)where E , are the unperturbed eigenvalues.Clearly the inequalities (13) and (14) give nostrong bound on such ULDM couplings. B. Time dependent corrections
For longer transition times or for higherULDM Compton frequencies (i.e., larger m φ )the time dependence of ˆ H can not be neglected.Moreover, the time variation of ˆ V ( t ) may stim-ulate efficient transitions among different eigen-states leading, eventually, to much stronger lim-its. In particular, the 1 → P = Ω Ω + δω sin (cid:32) √ Ω + δω t (cid:33) (15)where δω ≡ ω − ω , with ω = 2 πf , f ≡ E − E h (cid:39)
254 Hz , (16)the characteristic frequency, t the time, andΩ = 1¯ h (cid:104) | ˆ V | (cid:105) . (17)Combining Eqs. (11) and (1) one gets (see Ap-pendix B)Ω = g m x ¯ h (cid:18) m m ( I + I ) + g g I (cid:19) (18)where I = x A A (cid:90) ∞ α dzϕ ( z − α + α )( z − α ) ϕ ( z ) ,I = x A A (cid:90) ∞ α dzϕ ( z − α + α ) zϕ ( z ) . (19) Since, in the case of our interest, m g x (cid:39) .
603 peV (see discussion after Eq.(2)) one ob-tains Ω[Hz] = 914 . × δ m (20)where δ m = m m ( I + I ) + g g I . (21)Therefore, one can think of preparing the sys-tem so that only the ground state of energy E is populated [12, 13]. Then, within a given timeof flight (which, for ultra-cold neutrons with (cid:104) v y (cid:105) (cid:39) . − , is typically of the order of atenth of a second but, in settings with reflectivevertical mirrors on the edges of the main horizon-tal one, it may be stretched significantly) thereis a finite probability for the system to jump tothe first excited state ψ . The correspondingprobability P is, as usual, maximalized at theresonance, δω = 0, i.e. at ω . = 1596Hz, cor-responding, for n = 1, to m φ (cid:39) × − eV(scaling properly for higher n ) and obeys P max12 = sin (cid:18) Ω t (cid:19) (cid:39) Ω × t (cid:39) × δ m (22)for t = 1 s and δ m (cid:28) − . Therefore, an exper-imental limit on P gives a bound on δ m .Concerning the apparent smallness of P max12 for more realistic values of δ m two comments arein order. First, with reference to Fig.1, one canthink of a detector that can distinguish betweenthe ψ and ψ states (with some efficiency),placed at the end of the apparatus. Then, al-though the transition probability P is small,even a handful of observed events of the ψ typemay constitute the desired signal. Second, aftera DM quantum φ has been absorbed by the neu-tron, inducing the transition 1 →
2, it may thenbe emitted again, inducing the transition 2 → P = P , see Appendix B ). On the otherhand, the probability of the absorbtion and thesubsequent emission is P P = P and, hence,it is strongly suppressed. IV. CONCLUSIONS
The ultra-light dark matter is an intrigu-ing hypothesis that has recently attracted a lot4f attention as one of the most interesting al-ternatives to the notorious WIMP paradigm.Different in many aspects (in particular, highlevel of coherence in its interactions with mat-ter), its laboratory-based searches may often beperformed in table-top experiments focusing onthe response of simple quantum-mechanical sys-tems on the presence of the corresponding quasi-classical background perturbations.In this study, we focus on one of such sys-tems, namely, the gravitational quantum wellwhich has been recently subject to an intensivelaboratory study aimed at measuring the ener-gies of the lowest-laying bound states of neu-trons bouncing off the horizontal mirror. Inparticular, we focus on the influence of a pe-riodic perturbation modelling the variations ofthe effective gravitational acceleration due tothe assumed semi-classical ULDM backgroundon the transition probabilities of the lowest en-ergy (quasi-)stationary neutron states passingthrough the apparatus. With the Rabi frequencycorresponding to the resonance in the groundto the first excited state transition amplitudein the kHz region (leading to a series of “sweetspot” DM masses m φ [eV] ∼ × − /n withinteger n ) we rewrite the relevant probability asa function of the DM mass and effective cou-pling to the ordinary matter. These parameterscan be, subsequently, constrained from the thenon-observation of any deviations of the mea-sured transition probabilities from their theoret-ical ULDM-free spectrum. Acknowledgments
A. I. and M.M. acknowledge financial supportfrom the Grant Agency of the Czech Republic(GA ˇCR), contracts 14-07983S and 17-04902S.
Appendix A: Time independent correc-tions
The time independent corrections require thecalculation of the matrix elements (cid:104) s | ˆ V | s (cid:105) with t =
10 st = ω = Ω × - × - × - × - δ m - - - P → FIG. 2: Transition probability vs δ m at the resonance δω = 0, i.e. for m φ = 4 . × − eV (provided n = 1; for higher n the m φ down-scales accordingly),for two different times of flight, t = 1 s and t = 10 s.For a specific choice of P the allowed region for theeffective dark matter coupling δ m stretches below thecorresponding curve. s = 1 , , ... and ˆ V in Eq.(11), i.e. (cid:104) s | ˆ V | s (cid:105) = (cid:90) ∞ (cid:90) ∞ dxdx (cid:48) (cid:104) s | x (cid:105)(cid:104) x | ˆ V | x (cid:48) (cid:105)(cid:104) x (cid:48) | s (cid:105) (A1)where the eigenfunctions are real, ψ ∗ s ( x ) = (cid:104) s | x (cid:105) = (cid:104) x | s (cid:105) = ψ s ( x ), and given by the Airyfunction, ψ s ( x ) = A s ϕ ( x/x + α s ). In particular (cid:104) s | ˆ x | s (cid:105) = (cid:90) ∞ dxψ s ( x ) xψ s ( x )= A s (cid:90) ∞ dx x ϕ ( x/x + α s )= A s x (cid:90) ∞ α s ϕ ( z )( z − α s ) dz = x ( R s − α s ) , (A2)where R s = (cid:18)(cid:90) ∞ α s ϕ ( z ) dz (cid:19) − (cid:90) ∞ α s ϕ ( z ) zdz (A3)Since the unperturbed eigenvalues are given by E s = − α s m g x , (A4)the first term of the correction turns out to be (cid:104) s | m g ˆ x (cid:18) m m + g g (cid:19) | s (cid:105) = E s (cid:18) m m + g g (cid:19) [1 − R s /α s ] (A5)5nalogously, by using the property of thederivatives of the Airy function, ϕ (cid:48)(cid:48) ( z ) = zϕ ( z ),the second term of the correction yields (cid:104) s | − ˆ p m m m | s (cid:105) = ¯ h m m m ( m g x ) R s (A6)Putting all together, the final result is (cid:104) s | ˆ V | s (cid:105) = E s (cid:20) m m (cid:18) − R s α s (cid:19) + g g (cid:18) − R s α s (cid:19) (cid:21) (A7)which, by numerical evaluation of R s , gives (13)and (14). Appendix B: Transition probability
For the evaluation of the probability of thetransition 1 → P , one needs the matrix el-ement Ω in eq.(17). To compute the correctiondue to the position dependent term one needs(see before) (cid:104) | ˆ x | (cid:105) = (cid:90) ∞ dxψ ( x ) xψ ( x ) (B1)= A A (cid:90) ∞ dx ϕ (cid:18) xx + α (cid:19) x ϕ (cid:18) xx + α (cid:19) = A A x (cid:90) ∞ α ( z − α ) ϕ ( z ) ϕ ( z − α + α ) . To compute the correction due to the momentumdependent term one needs (cid:104) | p | (cid:105) = − ¯ h (cid:90) ∞ dxψ ( x ) d dx ψ ( x ) (B2)= − ¯ h A A (cid:90) ∞ dx ϕ (cid:18) xx + α (cid:19) d dx ϕ (cid:18) xx + α (cid:19) = +¯ h A A (cid:90) ∞ dx ddx ϕ (cid:18) xx + α (cid:19) ddx ϕ (cid:18) xx + α (cid:19) = A A x (cid:90) ∞ α zϕ ( z ) ϕ ( z − α + α ) . where the third line comes from partial integra-tion, and we used ϕ (+ ∞ ) = 0 = ϕ ( α ).By combining the previous results one gets¯ h Ω = ( m g + m g ) x I + ¯ h m m (1 /x ) I , (B3)with I , I in Eqs.(19), which, after simple alge-bra, gives Eq.(18).Notice the explicit symmetry 1 ↔ P = P . [1] J. Liu, X. Chen and X. Ji, Nature Phys. , no.3, 212 (2017).[2] N. A. Bahcall, Proc. Nat. Acad. Science (2015) 12243[3] D. S. Akerib et al. [LUX Collaboration], Phys.Rev. Lett. (2017) 021303 [arXiv:1608.07648[astro-ph.CO]].[4] R. Agnese et al. [SuperCDMS Collabora-tion], Phys. Rev. D (2017) 082002[arXiv:1610.00006 [physics.ins-det]].[5] D. J. E. Marsh, Phys. Rept. (2016) 1[arXiv:1510.07633 [astro-ph.CO]].[6] L. Hui, J. P. Ostriker, S. Tremaine andE. Witten, Phys. Rev. D (2017) 043541[arXiv:1610.08297 [astro-ph.CO]].[7] A. A. Geraci, A. Derevianko, Phys. Rev. Lett. (2016) 261301 [8] G. Pignol, Int. J. Mod. Phys. A (2015)1530048[9] O. Bertolami, J. G. Rosa, C. M. L. de Arag˜ao,P. Castorina, D. Zappal`a, Phys.Rev. D (2005) 025010[10] P. W. Graham, D. E. Kaplan, J. Mardon, S.Rajendran, and W. A. Terrano, Phys. Rev. D (2016) 075029[11] S. Fl¨ugge, Practical Quantum Mechanics,Springer-Verlag (Berlin) 1999.[12] V. V. Nesvizhevsky et al. , Phys. Rev. D (2003) 102002[13] V. V. Nesvizhevsky et al. , Eur. Phys. J. C (2005) 479[14] J. J. Sakurai, Modern Quantum Mechanics,Addison-Wesley (Reading, MA) 1985.(2005) 479[14] J. J. Sakurai, Modern Quantum Mechanics,Addison-Wesley (Reading, MA) 1985.