Table Of ContentUltra–cold gases and the detection of the Earth’s rotation: Bogoliubov space and
gravitomagnetism
A. Camacho∗
Departamento de F´ısica, Universidad Auto´noma Metropolitana–Iztapalapa
Apartado Postal 55–534, C.P. 09340, M´exico, D.F., M´exico.
E. Castellanos†
ZARM, Universita¨t Bremen, Am Fallturm, 28359 Bremen, Germany
(Dated: February 1, 2012)
The present work analyzes the consequences of the gravitomagnetic effect of the Earth upon a
bosonic gas in which the corresponding atoms have a non–vanishing orbital angular momentum.
Concerning the ground state of the Bogoliubov space of this system we deduce the consequences,
2 on the pressure and on the speed of sound, of the gravitomagnetic effect. We prove that the effect
1 on a single atom is very small, but we also show that for some thermodynamical properties the
0 consequencesscale as a non–trivial function of the numberof particles.
2
n
The deep analogies that can be found between elec- the experimental corroboration has severe hurdles. In-
a
J tromagnetismand gravitationhave a long history, a fact deed, this fact is related to the smallness of this effect
1 readily understood looking at Coulomb’s law of electric- (the ratio between the gravitomagnetic and gravitoelec-
3 ityandNewton’slawofgravitation[1]. Theanalogywas triceffectsoftheEarthis 10−7). Clearly,theuseofan
∼
taken further and the possible existence of a magnetic atomicsysteminthiscontextseemstobeaverybadidea,
] component in the gravitational interaction between the due to the smallness of the gravitationalpassive mass of
c
q Sun and the remaining celestial objects of our solar sys- anatom. This lastassertionis a correctone, thoughone
- tem was put forward in the nineteenth century [2, 3]. A mustaddthatthis statementis validfor oneatomic sys-
r
g more complete and profound relation between these two tem. Thepointhereconcernsthepossibilityofenhancing
[ interactions emerged with the formulation of Einstein’s the consequences of this effect, upon a quantal system,
theoryofgeneralrelativity(GR),inwhichagravitomag- resortingtoagas. Thepresentworkaddressesthisissue,
1
netic field appears as an inexorable consequence of the namely, we show that in a Bose–Einsteincondensate the
v
presence of a current of mass–energy [1], though several effectuponasingleatomappears,inthecontextofsome
2
7 theories of gravitation predict a gravitomagnetic contri- thermodynamicalpropertiesasthespeedofsound,multi-
6 bution [4]. This effect has several consequences, one of pliedbyanon–trivialfunctionofthenumberofparticles.
6 them is relatedto the appearance ofa gravitationalLar- This fact acts as an enhancer for this effect.
. mor theorem [5], i.e., the exterior gravity of a rotating Let us consider a rotating uncharged, idealized spher-
1
0 source couples to the angular momentum of a test body ical body with mass M and angular momentum J~. In
2 andgivesrisetoaLarmorprecession,inthesamewayas theweakfieldandslowmotionlimitthegravitomagnetic
1 a magnetic field couples with the angular momentum of field may be written, using the PPN parameters ∆1 and
v: an electrically charged particle. Other manifestations of ∆2, as
i this effect arethe so–calledframe–draggingandgeodetic
X
precession [6]
ar Thegravitomagneticfieldisoneofthemostimportant B~(~x)= 7∆1+∆2 GJ~−3(J~·xˆ)xˆ. (1)
predictions of GR and has no Newtonian counterpart. 4 c2 ~x3
(cid:16) (cid:17)
| |
CurrentlytheresultsassociatedtothemotionoftheLA-
GEOS and LAGEOS–II satellites provide observational The case of GR implies 7∆1+∆2 =1 [5], where c is the
4
evidenceforthiseffect[7]. Theextantspectrumofexper- speedoflight,~xisthepositionvector,GistheNewtonian
iments, or of astrophysical observations, have a weight, gravitational constant, and xˆ is the unit vector related
mainly, in the classical realm [6]. Of course, there are to ~x . We now assume that this gravitomagnetic field
alsoexperimentalproposalsinthequantalworld,namely, couples to the orbital angular momentum of an atom in
there is some evidence of a shift in the energy for some the same way as it does in the case of a classicalangular
fermions [8, 9]. The need for a more profound work in momentum [5].
thisdirectionisalsoapointthathastobeunderlined,an Followingthis analogybetweengravitomagnetismand
issue already addressed in the context of interferometry. magnetism we may now write down the interaction
either neutronic or atomic [10]. The quantal aspect of Hamiltonian that describes the coupling between grav-
itomagnetism and the orbital angular momentum of an
atom, here denoted by L~
∗ [email protected]
† [email protected] W = L~ B~. (2)
− ·
2
an additional simplification, which does not restrict the
validityofourresults,weassumethatallthe particlesin
Forthe sakeofsimplicity we nowimpose some restric- thed–statehaveeigenvalueforLz equaltos=+2. Very
tions. Firstly,theEarthhasaperfectsphericalsymmetry close to the temperature T = 0, the second term in this
witharadiusR,massM,constantmassdensity,andro- Hamiltonian becomes [13]
tation frequency equal to ω. Under these conditions we
have that [5]
aˆ†aˆ†aˆ aˆ =N2+2N aˆ†aˆ
p~ q~ p~+~k q~−~k ~k ~k
2MR2 ~Xk=0Xp~=0Xq~=0 ~Xk6=0
J~= ω~ez. (3) +N aˆ†aˆ† +aˆ aˆ . (6)
5 ~Xk6=0(cid:16) ~k −~k ~k −~k(cid:17)
With these approximations the N–body Hamiltonian
Clearly, the Hamiltonian for an atom must include
has the following structure
this last term the one will be considered in the N–body
Hamiltonianoperator(assumingthatthe gasis sodilute
that only the two–body interaction potential is required Hˆ = U0N2 +mgzN +
[11]). The system under study will be a Bose–Einstein
2V
gcgaaatsseaednraectlaoatsoehmdeisignhwtaitzcho<pn<taasisRnivewerigothrfavrvoeisltupamteicoetnVtaol,tmphaaersEtsiacmlretsha’nosfdstulhore-- ~Xk6=0h~22mk2 +mgz+ 4g5ωcR2 ~ + UV0Niaˆ~†kaˆ~k
face. In addition, the interaction between two particles
wteimllpbeeraastusuremoedf tthoebseydsotemminiastevderbyylso–wsc(aktate<ri<ng,1,i.ew.,htehree +N2UV0haˆ~†kaˆ†−~k+aˆ~kaˆ−~ki. (7)
~k and a are the wave vector and the scattering length, This Hamiltonian can be diagonalizedintroducing the
respectively)[12]. ThisentailsthefollowingHamiltonian Bogoliubov transformations [12]
for the N–body system.
1
ˆb = aˆ +α aˆ† , (8)
Hˆ = ~22mk2aˆ~†kaˆ~k ~k p1−α2kh ~k k −~ki
~kX=0
+2UV0 ~Xk=0Xp~=0Xq~=0aˆp†~aˆq†~aˆp~+~kaˆq~−~k ˆb~†k = 11−α2khaˆ~†k+αkaˆ−~ki. (9)
p
+ mgzaˆ†aˆ + 2sgωR~aˆ† aˆ , (4) These two operators fulfill the same algebra related
~kX=0 ~k ~k ~kX=0s=X0,±2 5c2 ~k,s ~k,s to aˆ~k and aˆ~†k, i.e., they are also bosonic operators. In
this last expression the following definitions have been
introduced
4πa~2
U0 = . (5)
m ~2k2 4gωR~
ǫ = +mgz+ , (10)
The lastterminourHamiltonianis relatedtothe fact k 2m 5c2
that the coupling between gravitomagnetismand orbital
angular momentum is absent for the case of vanishing
l. In addition, since we assume that the gas has a very Vǫ Vǫ Vǫ
k k k
α =1+ 2+ . (11)
lowtemperature(thisphrasemeanssmallerthanthecon- k U0N −rU0Nr U0N
densationtemperature)thenalmostalltheparticleshave
l=0andafewoneswillhavenon–vanishingangularmo- The final form for our Hamiltonian is
mentum and, at this point, we assume that they are in
the d–state, i.e., l = 2, because the symmetry require-
mentsassociatedtothewavefunctiondiscardthecaseof Hˆ = U0N2 +mgzN
l =1 [11], i.e., the first case with non–vanishing angular 2V
momentum is l = 2 and not l = 1. The parameter s + ǫ (ǫ + 2U0N)ˆb†ˆb
dtheenootpeesrtahteorfiLvez,pnoassmibeillyitises=rela1t,e0d, to2.thTeheeisgeenovpaelruaetsoorsf ~kX6=0nr k k V ~k ~k
† ± ±
(aˆ~k andaˆ~k) are bosonic creationandannihilation opera- 1 U0N +ǫ ǫ (ǫ + 2U0N) . (12)
tors,andfulfilltheusualBosecommutationrelations. As −2h V k−r k k V io
3
The last summation diverges, a result already known
[14, 15], and this divergence disappears introducing the
so–called pseudo–potential method, which implies that 2U0N
E = ǫ ǫ + . (20)
we must perform the following substitution [15] k r k k V
(cid:16) (cid:17)
Concerning(18), if we impose the condition of vanish-
ing gravitational constant, i.e. g = 0, then we recover
1 U0N +ǫ ǫ (ǫ + 2U0N) the usual Hamiltonian [15].
− 2h V k−q k k V i →
− 21hUV0N +ǫk−qǫk(ǫk+ 2UV0N)− 21ǫk(cid:0)UV0N(cid:1)2i. (13) ThVe2p∂rPes0s)uraess(oPc0ia=ted−t∂∂oEV0th)eangdrosupnededstoaftesooufntdhe(vsBo=-
q−Nm ∂V
Finally, this last summation will be approximated by goliubov space become, respectively
anintegral. Itisnoteworthytomentionthattheoriginal
expression has as lower limit the condition k =0, which
implies that the integraldoes nothaveaslowe6rlimit the 2πa~2N2 194 a3N
value 0. In other words, P0 = mV2 h1+ 15 r Vπ
15
1 α +O(α2), (21)
(cid:16) − 16√2 (cid:17)i
1 U0N 2U0N 1 U0N 2
+ǫ ǫ (ǫ + )
− 2~kX6=0h V k−r k k V − 2ǫk(cid:0) V (cid:1) i 4πa~2N 242 a3N
= ~2V 8πaN 5/2 ∞f(x)dx, (14) vs2 = m2V h1+ 15 r Vπ
−8mπ2 V Z 15
(cid:0) (cid:1) α 1 α +O(α2). (22)
(cid:16) − 16√2 (cid:17)i
In this last expression we have that
Notice that the possibility of detecting the term de-
pending upon the gravitomagneticeffect (δvgm) requires
4ωR~ mgzV s
α2 = 1+ . (15) that,if∆vs isthe experimentalerrorrelatedtothe mea-
h 5mc2zi U0 surement of the speed of sound, then ∆v < δvgm . In
s | s |
our case this entails
Additionally
3 aωgR
f(x)=x2 1+x2 x 2+x2 1 , (16) ∆(vs)< √2 c2 N. (23)
h − p − 2x2i
Thedetectionofthespeedofsoundincondensateshas
already a long history [16, 17]. The main difficulty, in
the experimental context, is related to the fact that (for
ǫ V
k
x= . (17) the case of an atom) aωgR 10−23. Nevertheless, this
rU0N c2 ∼
contributiontothespeedofsounddoesnotdependupon
With these conditions we deduce the final structure of thedensityofparticlesbutuponthenumberofparticles.
the N–body Hamiltonian Inotherwords,this tiny contributionis enhancedby the
number of particles (N) related to a bosonic gas. The
density in condensationexperiments rangesfrom1013 to
Hˆ =E0+~Xk6=0Ekˆb~†kˆb~k. (18) 1w0h1o5sepavrotliuclmesepiser1.c4ucbmic3cwme h[1a8v]e, thheantce for a condensate
In this last expression E0 denotes the energy of the 3 aωgR
groundstate ofthe correspondingBogoliubovspace[12]. N 10−2m/s. (24)
√2 c2 ∼
Undertheseconditionsanexperimentaluncertaintyof
2πa~2N2 128 a3N ∆v 10−3m/s would allow the detection of this field.
E0 = mV 1+ 15 r Vπ s ∼
h
1 15 α +α2U0N. (19) ACKNOWLEDGMENTS
(cid:16) − 16√2 (cid:17)i V
On the other hand, we have that the energy of the This researchwaspartiallysupportedby DAADgrant
Bogoliubov excitations (E ) is given by [12] A/09/77687
k
4
[1] I. Ciufolini and J. A. Wheeler, Gravitation and Iner- [13] C. J. Pethick and H. Smith, Bose–Einstein Condensa-
tia (Princeton University Press, Princeton, New Jersey, tion inDiluteGases (CambridgeUniversityPress,Cam-
1995). bridge, 2004).
[2] G. Holzmu¨ller, Z. Math. Phys.15, 69–73 (1870). [14] G. F. Gribakin and V. V. Flambaum, Phys. Rev. A48,
[3] F. Tisserand, Comp. Rend.75, 760–768 (1872). 546–563 (1993).
[4] W.-T. Ni, Phys.Rev.D7, 2880–2883 (1973). [15] M. Ueda, Fundamentals and New Frontiers of Bose–
[5] B. Mashhoon, arXiv–gr–qc:0311030v2 (2008). Einstein Condensation (World Scientic, Singapore,
[6] I.Ciufolini, arXiv–gr–qc:0809:3219v1 (2008). 2010).
[7] I.Ciufolini,Class.QuantumGrav.17,2369–2380(2000). [16] M. R. Andrews et al, Phys. Rev. Lett. 79, 553–556
[8] B.J.Venemaetal,Phys.Rev.Lett.68,135–138 (1992). (1997).
[9] B. Mashhoon, Phys.Lett. A198, 9–12 (1995). [17] M. R. Andrews et al, Phys. Rev. Lett. 80, 2967–556
[10] B. Mashhoon et al, Phys.Lett. A249, 161–166 (1998). (1998).
[11] R. K. Pathria, Statistical Mechanics (Butterworth– [18] A. Griffin, T. Nikuni,and E. Zaremba, Bose–Condensed
Heinemann,Oxford, 1996). Gases at Finite Temperatures (Cambridge University
[12] L. Pitaevski and S. Stringari, Bose–Einstein Condensa- Press, Cambridge, 2009).
tion (Clarendon Press, Oxford,2003).
