Zhen-Lai Wangand Xiang-Song Chen

1 School of Mathematics and Physics,Hubei Polytechnic University,Huangshi 435003,China

2 School of Physics and MOE Key Laboratory of Fundamental Quantities Measurement,Huazhong University of Science and Technology,Wuhan 430074,China

Abstract We discuss a new gravitational effect that the wave packet of a free-fall quantum particle undergoes a spin-dependent transverse shift in Earth’s gravitational field.This effect is similar to the geometric spin Hall effect(GSHE)(Aiello 2009 et al Phys.Rev.Lett.103 100401),and can be called gravity-induced GSHE.This effect suggests that the free-fall wave packets of opposite spin-polarized quantum particles can be split in the direction perpendicular to spin and gravity.

Keywords:gravity-induced geometric spin Hall effect,quantum particle,universality of free fall

1.Introduction

The universality of free fall(UFF)is tested as a weak form of the Einstein equivalence principle,which is the most important guide when establishing Einstein’s general relativity.The classical tests of UFF with macroscopic masses have achieved a high precision quantified by the E?tv?s parameter η of about 10-13[1,2],and no violations were observed so far.To extend the domain of the test body,verifications of UFF based on microscopic particles in the quantum regime have been studied theoretically and experimentally since 1960s[3].Recently,WEP-test experiments using atom interferometers were proposed to reach the level of η ～10-15[4,5].Quantum systems are advantageous in testing WEP with regard to fundamental properties such as charge[6],matter/anti-matter[7–9],spin[10–12]and internal structures.Possible violations of equivalence principle were discussed extensively,such as by spin-gravity coupling[13–16],by spin–torsion coupling[17–19],in extended or modified theories of gravity,and in almost all tentative theories to unify general relativity and the standard model of particle physics[20,21].

Theoretical investigations have offered a wide variety of approaches to the WEP in the quantum domain[22–30].However,quantum particles differ critically from classical point-like particles in many respects,because of their wave-like features and inherent spacial extension.Even the notion of WEP for quantum systems is not very clear and it may be different from the conventional WEP for classical systems[31–38].In this paper,considering wave-like features and inherent spacial extension of quantum particles,we reveal an interesting phenomenon that the space-averaged free-fall point of quantum particles allows a spindependent transverse split in the gravitational field.Since such an effect is similar to the geometric spin Hall effect(GSHE)discussed in[39,40],we call it gravity-induced GSHE.

For a light beam,the GSHE states that a spin-dependent transverse displacement of the light intensity centroid is observed in a plane tilted with respect to the propagation direction.Unlike the conventional spin Hall effect of light as a result of light-matter interaction[41–43],the GSHE of light is of purely geometric nature.Besides,it is distinct from the Relativistic Hall effect[44]and the Wigner translation of electromagnetic beams[45]both as an effect of Lorentzboost-induced sideways shift of the energy-flux and energydensity centrioids.Analogously,the gravity-induced GSHE reported here also differs from the so-called gravitational Hall effect presented in the literature[46–49]which describes a helicity-dependent geodesic deviation correction.

In this paper,we discuss the gravity-induced GSHE for spin-polarized Dirac particle beams.The paper is organized as follows.First,we derive an approximate wave-packet solution of the covariant Dirac equation in the Newtonian limit.With this solution,we demonstrate the gravity-induced GSHE of freely-falling Dirac particles.Next,we present an alternative derivation of the gravity-induced GSHE in a simple method without employing detailed knowledge of the wave-function of the Dirac particle beam.Then,we go further to discuss another interesting configuration displaying gravity-induced GSHE.Finally,we give our conclusions.

2.Gravity-induced GSHE

2.1.Heuristic result from GSHE

To calculate the space-averaged free-fall point of the Dirac particle,what we need to know is the spatial distribution of the particle beam’s intensity in the detection plane.Following the method of Aiello et al[39],we choose the energy flux of the particle beam to represent its intensity and so consider the energy–momentum(E–M)tensor Tμνof the particle beam.Thus,the space-averaged free-fall point of the particle can be calculated as the barycenter of the energy fluxT z0′ across the tilted detection plane:

Note that the energy fluxT z0′ is defined in the detection frame and not in the beam frame.

Before going into the detailed calculation,we first explain a heuristic way of understanding the gravity-induced GSHE,by making a close connection to the ordinary GSHE,which originates from a non-zero spin projection in the detection plane and has no relevance to gravity.To deal with the gravitational interactions,we follow[50–52]and adopt two reasonable approximations.First,the spin-precession effect can be safely neglected in our case from the result of gravity probe B(GP-B)[53]experiment.Second,the motion of the particle wave-packet’s center can be approximately replaced by its classical trajectory.With these two approximations,the gravity-induced GSHE can be converted to a GSHE:gravity just induces a kinematic configuration that the particle can hit the detector with non-zero spin projection in the detection plane,and then the ordinary GSHE occurs.Considering the particles as set up in figure 1,we can directly quote the expression for ordinary GSHE as derived in[39],and write down our result for the gravity-induced GSHE:

2.2.Deriving Gravity-induced GSHE with wave-function

We turn now to the calculation of energy flux density.For the Dirac field,the familiar symmetric E–M tensor3It should be noted that the E–M tensors have various versions,e.g.the canonical one and symmetric one.It may be tricky to pick out a proper one in actual application.Fortunately,the use of different E–M tensors does not change qualitatively the key features of the GSHE.[61]is

Here+h.c.indicates the addition of the Hermitian conjugate of

A few parenthetical remarks are in order.First,this effect is purely the result of matter-wave effect of quantum particle because it would be vanished as ? →0.However,even in flat spacetime g →0,it just gives rise to the ordinary GSHE of the particle and of course does not completely disappear.Second,although this effect originates from the wave feature of quantum particle,the split is not due to the geodesic deviation out of the inherent spacial extension of wave,because the above derivation is acted in a uniform gravitational field and the tidal effect does not exist.Thus,this effect differs radically from the gravitational Hall effect[46–49]out of a helicity-dependent geodesic deviation correction and as an genuine gravitational effect.

2.3.Deriving Gravity-induced GSHE without wave-function

The simple spin-dependent result of equation(22)obtained by a lengthy calculation is not accidental.In fact,assuming the wave-function with cylindrical symmetry,we can derive the previous result by a clever method without more detailed knowledge of the particle wave-function.From equation(21),equation(1)can be re-expressed by

In the last step,expressing the area element fromK′ frame to K frame does not change the main result.It might be more convenient to deduce the previous result again in K frame.

One use of the symmetric E–M tensor is to construct a conserved angular momentum tensor:

In K frame,Tx0can be ignored compared to Tz0because the beam mainly carries energy along the propagation direction.Thus,the Tx0term can be ignored for a small tilted angle θ in the denominator of equation(23),and the remaining Tz0term can be computed via the sum rule of energy:

2.4.Another configuration of Gravity-induced GSHE

There is another possible configuration of gravity-induced GSHE.Figure 2 depicts the configuration that the particles with horizontal spin polarization as well as momentum along the longitudinal direction are released into free fall.Interestingly,the detection planex′-y′ inK′ frame is not tilted with respect to thex′-y′ plane in K frame in this configuration.Indeed,the ‘geometric’ factor here originates from the configuration of transverse spin polarization that the spin orientation is initially perpendicular to the momentum.Repeating the above analysis and computation,we can get the result of the gravity-induced GSHE for figure 2:

Figure 1.A schematic set-up to display gravity-induced GSHE.Dirac particles carrying spin and initial momentum along z axis are released to fall freely and hit a detection plane x′-y′tilted by an angle θ with respect to the horizontal plane.The detection and beam frames are denoted by K′and K,respectively.Just as an example,the spin orientation marked here is along the positive direction of z axis and σ = 1/2.

Figure 2.A sketch of particles initially carrying spin along the horizontal direction(x-axis)and momentum along the longitudinal direction(z-axis),and falling freely to a horizontal detection plane x′-y′.Just as an example,the spin orientation marked here is along the positive direction of x axis and σ = 1/2.

3.Discussion and summary

In conclusion,we revealed a nontrivial phenomenon called gravity-induced GSHE containing simultaneously quantum and gravitational effects.Though we considered an ideal metric describing a uniform gravitational field in a sufficiently small region,it does recover the Newtonian limit with neglecting terms of orderor higher.Such a new effect suggests that the ‘free-fall points’ of quantum particles(or matter waves)vary with their spin polarization.In comparison,the atom in the tests of UFF using atomic interferometers is also quantum matter,but treated as a classical point particle in the interaction with the gravitational field and only as matter wave in the interaction with the probing light pulse.Such special treatment is valid when the de Broglie wave-length of the atom is sufficiently small.This effect can be mainly interpreted as the result of the matter-wave effect of quantum particles in a gravitational field,seen from the following two respects:One can predict the ordinary GSHE of quatum particles or matter waves in the absence of gravitational field(g →0);and it is treated as an effect of quantum mechanics because it would vanish as ? →0.

The measurement of this effect will be of great interest and importance.As mentioned above,the displacement is an order of magnitude smaller than the de Broglie wavelength.Thus,this effect is extremely small for the spatial resolution of conventional detectors,considering the common particle beams such as electron,neutron and atom beams.However,the effect would perhaps be observed if the particle is extremely slow and the detector realizes a higher spatial resolution.To test the gravityinduced GSHE might be as a new probe of UFF of quantum particles,so as to clarify the notion of quantum WEP.

Acknowledgments

This work is supported by the National Natural Science Foundation of China(Grant Nos.11 535 005 and 11 275 077).Wang also gratefully acknowledges financial support from the Scientific Research Project of Hubei Polytechnic University(Project No.20xjz02R).

Appendix :first-order approximation of exponential operator

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