Nov 14, 2001 - in which the constituent atoms are characterized by an electric dipole, gives rise to a magnetic monopole distribution, which correctly preserves ...
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PHYSICAL REVIEW A, VOLUME 64, 061602共R兲
Electric and magnetic monopoles in rotating Bose-Einstein condensates C. R. Bennett, L. G. Boussiakou, and M. Babiker Department of Physics, University of York, Heslington, York YO10 5DD, England 共Received 7 February 2001; published 14 November 2001兲 We show that by virtue of the Ro¨ntgen interaction, a vortex of order n in a Bose-Einstein condensate 共BEC兲, in which the constituent atoms are characterized by an electric dipole, gives rise to a magnetic monopole distribution, which correctly preserves magnetic charge neutrality. If, on the other hand, the BEC atoms are characterized by a magnetic dipole, we show that the Aharonov-Casher interaction term leads to the prediction of an electric charge distribution associated with the vortex state, which too, exhibits electric charge neutrality. Our theory displays an exact symmetry between the electric and magnetic aspects of the problem and can thus be presented in a unified fashion. Illustrations are given for finite-size atomic gas BECs, superfluid helium, and spin-polarized hydrogen BECs in the n⫽1 vortex state. DOI: 10.1103/PhysRevA.64.061602
PACS number共s兲: 03.75.Fi, 03.65.Vf, 14.80.Hv, 41.20.⫺q
Since the initial generation of atomic gas Bose-Einstein condensates 共BECs兲 关1兴, it has now become routine to create them and trap them 关2兴 in magnetic and light traps 关3兴. In parallel with these developments, a number of theoretical studies 关4兴 have been carried out, with some emphasis shifting recently to the manifestation of vortex states 关4 – 8兴 and their electromagnetic properties. In particular, Leonhardt and Piwnicki 关6兴 recently predicted that the Ro¨ntgen interaction 关9兴, associated with the convective motion of electric dipoles of the BEC atoms, should give rise to a magnetic monopole located at the core of an n⫽1 vortex. What if the atoms forming the BEC are characterized by a magnetic dipole moment 共magnetic BEC兲 rather than an electric dipole moment 共electric BEC兲? This question is timely since now BECs can be maintained in light traps, allowing the magnetic properties of the alkali-metal BECs to be investigated 关8兴, and spinpolarized hydrogen BECs have recently been generated 关10,11兴. Will the analogue of the situation considered in 关6兴 be realized, leading to an electric monopole distribution from a vortex BEC state via the Aharonov-Casher effect 关12,13兴? Here we wish to show not just that the answer to this last question is in the affirmative, but that it is possible to formulate a theory in which both effects arise in a unified manner. Furthermore, we point out that the recent pioneering work by Leonhardt and Piwnicki 关6兴 violates overall magnetic monopole charge neutrality, while in our theory charge neutrality is preserved by the electric and magnetic monopole distributions associated with the vortex states of the two types of BEC. The first goal is to formulate a Lagrangian appropriate for the system. This leads to the constitutive relations involving electric polarization 共magnetization兲 of a system comprising a cylindrical BEC that is in an order n vortex state. We then derive expressions for the magnetic 共electric兲 monopole charge distributions accompanying the vortex state by consideration of the modified Gross-Pitaevskii equation. It suffices to model the constituent BEC particles in terms of a neutral two-particle atom of mass M ⫽m 1 ⫹m 2 involving two charged particles of charges e 1 ⫽⫺e 2 , masses m 1 and m 2 , position vectors qi and velocities q˙i , where i⫽1,2. The Lagrangian of the atomic system coupled to the electromagnetic scalar potential and vector potential A is written in the following form which is nonrelativistic in the motion 1050-2947/2001/64共6兲/061602共4兲/$20.00
of the atomic system 关14 –17兴, L⫽m 1 q˙ 21 /2⫹m 2 q˙ 22 /2⫹
冕
d 3 r 兵 ⑀ 0 关 E2 共 r兲 ⫺c 2 B2 共 r兲兴 /2
⫹J共 r兲 •A共 r兲 ⫺ 共 r兲 共 r兲 其 ,
共1兲
˙ ⫺“ and B⫽“⫻A. The charge and where E⫽⫺A current densities are (r)⫽ 兺 i⫽1,2e i ␦ (r⫺qi ) and J(r) ⫽ 兺 i⫽1,2e i q˙i ␦ (r⫺qi ). An important feature is to express the theory in terms of the center of mass and relative coordinates R⫽(m 1 q1 ⫹m 2 q2 )/M and q⫽q1 ⫺q2 . This is because the electric and magnetic polarization fields P(r) and M(r) are defined as multipolar expansions, albeit in closed forms, about the the center of mass coordinate R 关14,15兴. A PowerZienau-Woolley 共PZW兲 gauge transformation 关15–17兴 is carried out next. After recasting the Lagrangian in center of mass and relative coordinates we obtain ¯ q˙ 2 /2⫹ L⫽M R˙ 2 /2⫹m
冕
d 3 r 兵 ⑀ 0 关 E2 共 r兲 ⫺c 2 B2 共 r兲兴 /2
˙ • 关 P共 r兲 ⫻B共 r兲兴 其 , ⫹P共 r兲 •E共 r兲 ⫹M共 r兲 •B共 r兲 ⫺R
共2兲
¯ is the reduced mass and M is the multipolar magwhere m netization of the two-particle system involving here orbital magnetic multipoles, rather than the spin magnetic moment of the particles, but it is possible to generalize the theory to incorporate spin. The interpretation of the various terms in the Lagrangian in Eq. 共2兲 is clear, including the last term that we identify as the Ro¨ntgen interaction term 关6,9,18兴. The Lagrangian in Eq. 共2兲 is clearly not symmetric between the electric and magnetic properties, specifically as far as the center of mass motion is concerned. This lack of symmetry suggests that an interaction term is missing, which like the ˙ and that couples the Ro¨ntgen term, should be first order in R magnetization to the electric field via the center of mass motion. The missing term is identified as the Aharonov-Casher term 关12兴 and is obtainable by adding a relativistic correction to the Lagrangian in Eq. 共1兲 so that → ⬘ where
64 061602-1
˙ •„“⫻M共 r兲 …/c 2 . ⬘ 共 r兲 ⫽ 共 r兲 ⫹R
共3兲
©2001 The American Physical Society
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Including this correction in the original Lagrangian, and following the same procedures as descibed above, leads to another term added to the Lagrangian in Eq. 共2兲 in the form of ˙ •„M(r)⫻E(r)…. Once this is the Aharonov-Casher term, R done, the Lagrangian becomes symmetric with respect to the magnetic and electric interactions. The corresponding Hamiltonian follows straightforwardly from the alternative Lagrangian and this can then be generalized to the case of a many-atom system appropriate for BEC. Before we come to this, however, it is instructive to examine the constitutive relations between the matter fields and the electromagnetic fields of the system that follow from the Euler-Lagrange equations for and A. The constitutive relations are found to be given by ˙ ⫻P共 r兲 , H共 r兲 ⫽B共 r兲 / 0 ⫺M共 r兲 ⫹R ˙ ⫻M共 r兲 /c 2 , D共 r兲 ⫽ ⑀ 0 E共 r兲 ⫹P共 r兲 ⫹R
共4兲
where D is the electric displacement and H is the magneticfield intensity. It is important to note that, without the Aharonov-Casher term, the last term in the second equation in Eq. 共4兲 would not have emerged. From the constitutive relations one concludes that if the motion of the condensate is arranged to be orthogonal to the polarization 共magnetization兲 field vector, then a magnetic 共electric兲 field will be generated that is proportional to the magnitude of the velocity ˙ 兩 and to the strength of the electric polarization 共magneti兩R zation兲 of the constituent atoms. The situation where this can be realized is that of a BEC in a vortex state. However, the polarization and magnetization of the condensate will depend on its wave function, so it is necessary to solve the associated Gross-Pitaevskii equation, generalized to include the Ro¨ntgen and Aharonov-Casher interaction terms, and which arises from the corresponding Hamiltonian in a many-body treatment. We obtain 关4,15兴
冋
冉
冊
P2 d2 ⫹V trap 共 R兲 ⫹ •B共 R兲 ⫺d•E共 R兲 ⫹N 兩 共 R兲 兩 2 V 0 ⫹ 2M ⑀0
册
P ⫹„d⫻B共 R兲 ⫺ ⫻E共 R兲 …• 共 R兲 ⫽E 共 R兲 , M
共5兲
where N is the number of atoms, V trap is the trapping potential, V 0 ⫽4 ប 2 a/M is the scattering term with a the s-wave scattering length, and we have restricted the full multipolar forms of the electric polarization field P and magnetization field M to the 共leading兲 dipole approximation P共 R兲 ⫽Nd兩 共 R兲 兩 2 , M共 R兲 ⫽N 兩 共 R兲 兩 2 ,
共6兲
where d is the electric dipole moment and the permanent magnetic dipole moment. The term proportional to d 2 in Eq. 共5兲 arises from dipole-dipole interactions. To solve Eq. 共5兲 we assume that the BEC is confined within a cylinder of radius R 0 and height 2z 0 , so that it occupies the region 兩 z 兩 ⭐z 0 of the z axis. Whether z 0 is larger or smaller than R 0
FIG. 1. Variations of the derivative of the modulus squared of the BEC wave function in the n⫽1 vortex state when n 1d a⫽0.01 共solid curve兲, n 1d a⫽10 共dashed curve兲, and n 1d a⫽100 共dotted curve兲. The inset shows the modulus squared of the normalized wave function for the same values of n 1d a.
depends on the trap used, but below we discuss the cases where z 0 ⬇R 0 and z 0 is infinite. We perform a Madelung transformation 关5兴, and assuming that the wave function is independent of z, write (R)⫽ 兩 (r) 兩 e in / 冑4 z 0 R 20 , where (r, ,z) are the cylindrical coordinates of R. When using Eq. 共5兲 we will ignore the electromagnetic field interactions, which introduce small second-order corrections 关6兴, and the dipole-dipole interactions that are negligible compared to the s-wave scattering. We thus obtain the velocity profile of the BEC as ˆ បn/M r, ˙ ⫽ R
共7兲
and also the radial wave function satisfies the equation 关5兴
冋
d2 d
⫹ 2
册
1 d n 2 2M E ⫺ 2 ⫹ 2 ⫺4n 1d a 兩 共 兲 兩 2 共 兲 ⫽0, d ប
共8兲
with ⫽r/R 0 , n 1d ⫽N/2z 0 as the one-dimensional atomic density parameter and we set (R 0 )⫽0. The inset to Fig. 1 displays the numerical solution of Eq. 共8兲 for various values of n 1d a. Consider first the case in which the atoms forming the BEC are characterized by electric dipole moments that have been polarized in the z direction. We concentrate on the static case and take the divergence of the first equation in Eq. 共4兲. The divergence of the magnetic-flux density B is zero due to the Maxwell equation and the magnetization is assumed to be zero. Hence,
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˙ ⫻P兲 ⫽P• 共 “⫻R ˙ 兲 ⫺R ˙ • 共 “⫻P兲 “•H⫽“• 共 R ⫽
冋
册
m共 r 兲 បn d n 1d d 2 ␦共 r 兲⫹ 兩 r 兩 ⫽ , 兲 共 Mr dr 2 R 20 0
共9兲
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where we have identified the right-hand side of Eq. 共9兲 as a magnetic monopole charge density produced by the rotating BEC. Thus, we may write H⫽⫺“⌽. This is similar to the result obtained in 关6兴. However, in 关6兴 it was argued that, although the wave function is zero at the center of the cylinder, some atoms will leak into the core of the vortex and this gives rise to the monopole charge. The spatial variation of the wave function across the cylinder was ignored. In solving Eq. 共9兲 we note that the term involving the ␦ function cancels with a term arising from integrating the derivative that is proportional to 兩 (0) 兩 2 . This important feature, guaranteeing preservation of ‘‘charge neutrality,’’ was not exhibited by the treatment in 关6兴. Figure 1 shows the variation of the derivative of the modulus squared of the wave function. It is easy to see that this is proportional to the areal charge density in a symmetry plane containing the cylinder axis and so we can write 2 r m (r)⫽បnn 1d 0 d/M R 30 d 兩 ( ) 兩 2 /d . It can be seen that as the interaction becomes stronger or the number of atoms increases the wave function broadens and the associated charges move closer to the center and edge of the cylinder. Note that the overall integral of the density over the cylinder volume reduces to zero, as it should. We may think of this magnetic monopole charge distribution as akin to that found in a cylindrical capacitor, with one set of charges on an inner thin cylinder and another set of charges of opposite sign on an outer cylinder. A useful measure of the effect is the cumulative positive charge, Q m , in the inner region of the cylinder, which can be estimated by integrating the charge density up to r⫽R 0 /2 to give Q m ⫽បnN 0 d/M R 20 兩 ( ⫽1/2) 兩 2 . For an infinite cylinder, we may solve Eq. 共9兲 using Gauss’s law to obtain
冕 冋 ␦ ⬘
បnn 1d d H共 兲 ⫽rˆ 2 M R 30
共
0
兲⫹
d d⬘
册
兩 共 ⬘ 兲兩 2d ⬘
បnn 1d d ⫽rˆ 兩 共 兲兩 2. 2 M R 30
共10兲
We see that H is proportional to the modulus of the wave function squared, and since this is zero outside the cylinder, there is a magnetic field only inside the cylinder. This is a consequence of the conservation of ‘‘magnetic monopole charge.’’ If we wish to detect the magnetic field, it may be necessary to look for fields outside the BEC. To this end, we return to investigate a cylinder of a finite height 2z 0 . The solution for the potential can be found using Green functions 关20兴. We obtain ⌽ 共 ,z 兲 ⫽⌽ m 0 ⫻
⫹
冋
冕 ⬘ 冕 1
2
d
0
0
d⬘
⬘ ⫺ cos共 ⬘ 兲 h2
1⫺z 共 h f ⫹ 共 1⫺z 兲 2 兲 1/2 2 2
1⫹z 共 h f ⫹ 共 1⫹z 兲 2 兲 1/2 2 2
册
,
共11兲
PHYSICAL REVIEW A 64 061602共R兲
FIG. 2. A contour plot within a symmetry plane of the cylinder showing the variations of the electric 共magnetic兲 potential of a magnetic 共electric兲 dipole active BEC in the n⫽1 vortex state that occupies the shaded region. The potential exhibits angular symmetry around R⫽0 and is in units of ⌽ e0 for the electric potential and ⌽ m 0 for the magnetic potential 共see text兲. The condensate is such that z 0 /R 0 ⫽1 and n 1d a⫽100.
where z is in units of z 0 , h 2 ⫽ 2 ⫹ ⬘ 2 ⫺2 ⬘ cos(⬘), f 2 2 ⫽R 0 /z 0 , and ⌽ m 0 ⫽បnn 1d d/8 M R 0 . Figure 2 shows a contour plot of the potential on a symmetry plane through the cylinder axis. Clearly due to the finite height of the cylinder, the potential is not confined to the inside of the cylinder but leaks outside it. Inside the cylinder the potential exhibits its largest variations where the wave function changes most rapidly. It is instructive to carry out an order of magnitude analysis of the effects for the cases of a typical atomic gas BEC and for superfluid helium. In the case of an atomic gas BEC, we choose typical parameters appropriate for 87Rb 关4兴 in the n⫽1 vortex state confined in a cylinder with dimensions R 0 ⫽z 0 ⫽2 m, each atom carrying an electric dipole moment d⫽ea B , where a B ⫽0.53 Å is the Bohr radius, and the s-wave scattering length is taken to be a⫽59 Å. Setting n 1d a⫽100 gives a linear density of n 1d ⫽1.7⫻1010 m⫺1 and a total number of ⫺19 A and atoms N⫽7⫻104 . Thus, we obtain ⌽ m 0 ⫽3.3⫻10 ⫺28 2 Q m ⫽1.3⫻10 兩 ( ⫽1/2) 兩 V s, which is equivalent to a magnetic field of order 10⫺19 T. In the case of superfluid helium, we should rewrite the polarization as P( ) ⫽ E兩 ( ) 兩 2 , where E is an applied electric field and ⫽0.052 is the susceptability, in effect replacing the volume density times the dipole moment with a susceptability times an externally applied electric field. Thus, for every 1 V m⫺1 of applied field, we obtain the charge Q m ⫽1.2⫻10⫺20兩 ( ⫽1/2) 兩 2 V s and a magnetic field of order 10⫺14 T 共equivalent to the values in 关6兴兲. This larger magnitude is due to the smaller helium mass and larger density 共approximately 1021 m⫺3 in the case of the atomic BEC above and 1028 m⫺3 for superfluid helium兲. The field predicted here for the Rb atomic BEC is clearly too small to be measured using present experimental techniques. However, one anticipates that in the future denser atomic gas BECs
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We can see that the above equation for E becomes identical to that for H when d is replaced by ⫺ 0 . Thus we can also evaluate the electric field for an infinitely long cylinder with the same substitution as above in Eq. 共10兲 and the equation for the electrostatic potential, E⫽⫺“⌽, is given by Eq. 共11兲 e 2 2 with ⌽ m 0 replaced by ⌽ 0 ⫽⫺បnn 1d 0 /8 M R 0 . Figure 1 now serves to display a function proportional to the areal electric charge density in a symmetry plane such that 2 r e (r)⫽⫺បnn 1d /M c 2 R 30 d 兩 ( ) 兩 2 /d , and Fig. 2 shows the electrostatic potential. The equivalent cumulative negative electric charge in the inner region of the cylinder is approximately Q e ⫽⫺បnN /M c 2 R 20 兩 ( ⫽1/2) 兩 2 . Again we should ask if this electric potential is measurable. The best candidate for observing the effect in this case is spin-polarized hydrogen 关10,11兴, which has recently been produced as a BEC 关10兴. The hydrogen is doubly spinpolarized, thus ⫽ b ⫽បe/2m e ⫽1.9⫻10⫺23 Am2 , a typi-
cal density is 5⫻1021 m⫺3 , 关10兴 and the s-wave scattering length is a⫽0.72 Å 关10,11兴. Again assuming that n 1d a ⫽100 and z 0 ⫽5 mm 关10兴, we obtain ⌽ e0 ⫽⫺2.7 ⫻10⫺16 V and Q e ⫽⫺1.9⫻10⫺27兩 ( ⫽1/2) 兩 2 C. Once more these quantities appear too small to be experimentally accessable at present. However, the effects become amenable to measurement if denser spin-polarized hydrogen BECs can be produced 共with densities of the order of those of superfluid helium兲. In conclusion, we have shown how a dipole-active BEC generates electric and magnetic monopole charge distributions and their associated electric and magnetic fields when the condensate is in a vortex state. We have also shown how, even if no condensate atoms occupy the vortex core, monopole charge distributions, both electric and magnetic, arise which are globally neutral. The problem becomes similar to that of finding the potential in a cylindrical capacitor. In the case of an atomic gas BEC of finite size of either kind, the vortex state generates fields outside it that are small when calculated on the basis of current estimates. Their experimental detection will have to await the advent of denser atomic gas BECs than are currently available. The predicted magnetic fields are much larger for a denser and more electrically polarizable system such as a superfluid. In the case of superfluid helium, we have shown that the magnetic distributions are sufficiently large to be amenable to experimental detection. It is envisaged that future denser BECs of spinpolarized hydrogen would give rise to electric distributions that could be measurable. We thank Edward Hinds, University of Sussex, Gerald Hechenblaikner, University of Oxford, and Lincoln Carr, University of Washington for useful discussions. C.R.B. would like to thank the University of York for financial support. This work was carried out under the EPSRC Grant No. GR/M16313.
关1兴 M.H. Anderson et al., Science 269, 198 共1995兲; C.C. Bradley et al., Phys. Rev. Lett. 75, 1687 共1995兲; K.B. Davies et al., ibid. 75, 3969 共1995兲. 关2兴 F. Dalfovo et al., Rev. Mod. Phys. 71, 463 共1999兲. 关3兴 D.M. Stamper-Kurn et al., Phys. Rev. Lett. 80, 2027 共1998兲; E.M. Wright et al., Phys. Rev. A 63, 013608 共2000兲. 关4兴 A.L. Fetter, e-print cond-mat/9811366; F. Dalfovo and S. Stringari, Phys. Rev. A 53, 2477 共1996兲; R.J. Donnelly, Quantized Vortices in Helium II 共Cambridge University Press, Cambridge, 1991兲, and references therein. 关5兴 S.I. Shevchenko, J. Low Temp. Phys. 121, 429 共2000兲. 关6兴 U. Leonardt and P. Piwnicki, Phys. Rev. Lett. 82, 2426 共1999兲. 关7兴 K.W. Madison et al., Phys. Rev. Lett. 84, 806 共2000兲; K.G. Petrosyan and L. You, Phys. Rev. A 59, 639 共1999兲; E.V. Goldstein et al., ibid. 58, 576 共1998兲. 关8兴 T.-L. Ho, Phys. Rev. Lett. 81, 742 共1998兲. 关9兴 W.C. Ro¨ntgen, Ann. Phys. Chem. 35, 264 共1888兲; M. Babiker, et al., Proc. R. Soc. London, Ser. A 332, 187 共1973兲; M. Wilkens, Phys. Rev. Lett. 72, 5 共1994兲. 关10兴 T.J. Greytak et al., Physica B 280, 20 共2000兲; T. Willmann,
Appl. Phys. B: Lasers Opt. 69, 357 共1999兲. 关11兴 T.J. Greytak in Bose-Einstein Condensation, edited by A. Griffin, D.W. Snoke, and S. Stringari 共CUP, Cambridge, 1996兲; I.F. Silvera, ibid. 关12兴 Y. Aharonov and A. Casher, Phys. Rev. Lett. 53, 319 共1984兲. 关13兴 K. Sangster et al., Phys. Rev. Lett. 71, 3641 共1993兲; K. Sangster et al., Phys. Rev. A 51, 1776 共1995兲. 关14兴 V.E. Lembessis et al., Phys. Rev. A 48, 1594 共1993兲. 关15兴 L.G. Boussiakou et al. 共unpublished兲. 关16兴 M. Babiker and R. Loudon, Proc. R. Soc. London, Ser. A 385, 439 共1983兲. 关17兴 C. Cohen-Tannoudji et al., Photons and Atoms, Introduction to Quantum Electrodynamics 共Wiley, New York, 1989兲. 关18兴 H. Wei, R. Han, and X. Wei, Phys. Rev. Lett. 75, 2071 共1995兲. 关19兴 L. Dragos, Magnetofluid Dynamics 共Abacus Press, Tunbridge Wells, 1975兲. 关20兴 G. Barton, Elements of Green’s Functions and Propagation 共Clarendon Press, Oxford, 1989兲; P.M. Morse and H. Feshbach, Methods of Theoretical Physics 共McGraw-Hill, New York, 1953兲.
will become available. The superfluid helium estimate indicates that the effects are amenable to experimental detection, as deduced in 关6兴. Next we turn our attention to the case of a BEC in which the constituent atoms are characterized by magnetic dipoles aligned along the z direction. Taking the divergence of the second equation in Eq. 共4兲, and following analogous steps to those used in the above case, we obtain
⑀ 0 “•E⫽⫺ ⫽⫺
⫽⫺
1 c2 1 c2
˙ ⫻M兲 “• 共 R ˙ 兲 ⫺R ˙ • 共 “⫻M兲兴 关 M• 共 “⫻R
បn 2
c Mr
冋
␦共 r 兲⫹
册
d n 1d 兩 共 r 兲兩 2⫽ e共 r 兲 . dr 2 R 20
共12兲
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