Atom Optics – From de Broglie Wavesto Heisenberg Ferromagnets

Han Pu, Chris Search, Weiping Zhang, and Pierre Meystre

Optical Sciences Center, The University of Arizona, Tucson, AZ 85721

Abstract

We review some of the key developments that lead to the field of atom optics, and discuss how it hasrecently began to make connections with Werner Heisenberg’s trailblazing work on magnetism.

1. Introduction

At the September 10, 1923 session of the French Academy of Sciences, Louis Perrinpresented a note by Prince Louis de Broglie entitled ‘‘Radiations – Ondes et Quanta”[1]. This was a remarkable paper drawing on the analogy between ‘‘atoms of light“ ––les atomes de lumiere –– and electrons. De Broglie believed that an ‘‘atom of light”should be considered as a moving object with a very small mass (<10�50 g) and with aspeed nearly equal to c, although slightly less.” By analogy, he found that it was almostnecessary to suppose (‘‘il est presque necessaire de supposer”) that there is, associatedwith the electron motion, a fictitious wave at what is now known as the de Brogliewavelength. Amazingly, this somewhat flawed argumentation lead, together with otherdevelopments in atomic and optical physics, to one of the greatest scientific revolutions,the invention of quantum mechanics, carried out independently by Heisenberg [2] and bySchrodinger [3] in 1925.The optics of atomic de Broglie waves, or atom optics, is now thriving and rapidly ma-

turing [4]. Linear, nonlinear, quantum and integrated atom optics are witnessing excitingdevelopments and may soon lead to practical applications, such as e.g. sensors of unprece-dented accuracy and sensitivity.Experimental atom optics can be traced back to the seminal experiments by O. Stern [5],

who demonstrated the reflection and diffraction of atoms from metallic and crystalline sur-faces, and to R. Frisch [6], who measured the deflection of atoms as a result of the absorp-tion and spontaneous emission of light. These pioneering experiments were truly heroic,due to the difficulties associated with the extremely short room-temperature atomic de Bro-glie wavelength Lth, of the order of a fraction of an angstrom. The situation changed drasti-cally following the invention of laser cooling [7], which allows one to readily bring atomicsamples down to temperatures of the order of 10�6 kelvin. Since Lth / 1=

ffiffiffiffiT

p, it is easily

seen that such low temperatures lead to very large atomic de Broglie wavelengths, of theorder of microns or longer.Even lower temperatures can be achieved, most notably using forced evaporative cooling

[8, 9], which can lead to temperatures of the order of a few nanokelvin. At this point, evenatomic samples of modest densities (1013�1014 cm�3) suffer the effects of quantum statis-tics and must be described as undistinguishable particles. In the case of bosonic atoms, thiscan lead under appropriate conditions to the Bose-Einstein condensation of the sample, aspredicted by Einstein [10] in 1924 following the work of Bose [11], and first experimen-tally verified in an atomic vapor in 1995 [12, 13].

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# WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, 2002 0015-8208/02/5-705-0664 $ 17.50þ.50/0

Atom optics bridges many subfields of physics, from atomic and optical physics to statis-tical mechanics and to condensed matter physics. It also bears on information science,metrology, nonlinear science, and even relativity. It borrows from and contributes to thesefields in numerous ways. In the context of this meeting celebrating Werner Heisenberg, wefind it appropriate to discuss one specific aspect of atom optics, the close analogy betweenthe dynamics of Bose-Einstein condensates on optical lattices [14––18], and models of mag-netism in condensed matter physics [20––23], since Heisenberg was of course the father ofthe quantum theory of magnetism.

A particularly interesting geometry in the context of the present paper involves quantum-degenerate Bose gases trapped in optical lattices. They were first used to demonstrate‘‘mode-locked” atom lasers [14]. It was also predicted [15] and demonstrated [16] that theyundergo a Mott insulator phase transition as the depth of the lattice wells is increased. Wealso mention their application in studies of quantum chaos [17––19].

In a parallel development, recent experiments have established that the ground state ofoptically trapped 87Rb spinor Bose condensates must be ferromagnetic at T ¼ 0 [24, 25].What this means is that when located in optical lattices deep enough for the individual sitesto be independent, the ‘‘mini-condensates” at each lattice site behave as individual ferro-magnets. In the absence of external magnetic fields and of site-to-site interactions, thesemesoscopic magnets point in random directions. However, magnetic and optical dipole-di-pole interactions can couple neighboring sites, and as a result it can be expected that underappropriate conditions the mini-condensates will arrange themselves in ferromagnetic orantiferromagnetic configurations, that spin waves can be launched in the lattice, etc.

We recall that ferromagnetism remained mysterious until Heisenberg identified the crucialrole played by the exchange interaction. In particular, the magnetic dipole-dipole interactionwas too weak to explain the observed properties of ferromagnets. The traditional startingpoint for the study of magnetism in solids is now the Heisenberg Hamiltonian [23]Hspin ¼ �

PJij Si � Sj; where Si is the spin operator for ith electron, the Jij are exchange

coupling constants. They arise from the combined effects of the direct Coulomb interactionamong electrons and the Pauli exclusion principle.

That it has now become possible to ‘‘turn back the clock” and study such effects insystems where a ‘‘classical” interaction between sites is dominant, is an immediate conse-quence of the collective behavior of the atoms in each of the mini-condensates: because ofthe Bose enhancement resulting from the presence of coherent ensembles of Ni atoms ––typically several thousands in one-dimensional lattices –– in the ‘‘mini-condensate” at eachlattice site i; each of the mini-condensates acts as a mesoscopic ferromagnet Ni times stron-ger than that associated with an individual atom. The magnetic dipole-dipole interaction cantherefore become significant despite the large distance between sites. Hence it becomespossible to carry out detailed static and dynamical studies of magnetism in one- to three-dimensional condensate lattices [26, 27]. A particularly attractive feature is that since Bose-condensed atomic systems are weakly interacting, a detailed microscopic description ofthese systems is possible, in contrast to typical condensed-matter situations.

2. Ferromagnetism on a Bosonic Lattice

Our starting point is the Hamiltonian H describing an F ¼ 1 spinor condensate at zero tem-perature trapped in an optical lattice along the z-axis, subject to a magnetic dipole-dipoleinteraction Hdd. More precisely, we include the long-range magnetic dipole-dipole interactionbetween different lattice sites, but neglect it within each site, assuming that it is much weakerthan the s-wave ground-state collisions. We further assume that the optical lattice potential isdeep enough that there is no spatial overlap between the condensates at different lattice sites.Finally, the atoms are coupled to an external magnetic field Bext [28––30].

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For a one-dimensional lattice of mini-condensates with equal number of atoms, and as-suming that all Zeeman sublevels of the atoms share the same spatial wave function, thetotal Hamiltonian of the system takes the form [30]

H ¼Pi

l0aS2i þ

Pj 6¼i

lijSi � Sj � 3Pj 6¼i

lijSzi S

zj � gBSi � Bext

� �: ð1Þ

In this expression, the coefficient l0a account for spin-changing collisions within the indivi-dual mini-condensates, lij accounts for the magnetic dipole-dipole interaction between con-densates, and gB is the gyromagnetic ratio. mB being the Bohr magneton and gF the Landefactor. In the case of 87Rb the individual spinor condensates at the lattice sites are ferromag-netic, l0a < 0.In contrast to the usual situation in ferromagnetism, where the temperature plays an

essential role and the ferromagnetic transition occurs at the Curie temperature TC, the pre-sent analysis is at zero temperature, T ¼ 0. Nonetheless, effects similar to those of a finitetemperature analysis can be simulated by an appropriate choice of the applied magneticfield BextðrÞ. Specifically, we consider a situation where BextðrÞ consists of two contribu-tions, Bext ¼ Bzzzþ Bqqq, where q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ y2

pis the radial coordinate. Here, Bz is a con-

trolled, external applied field that we take along the z-axis, and Bq is an effective ‘‘stray”field, that we take along the transverse direction. It accounts for uncontrolled aspects of theexperimental environment, such as e.g. environmental magnetic fluctuations. We will seeshortly that this transverse field plays a role analogous to temperature in conventional ferro-magnetism.In the case of an infinite lattice, the Hamiltonian of a generic site i reads

hi ¼ l0aS2 � gBS � Bz þ 2

Pj 6¼i

lijSzj

� �zzþ Bq �

Pj 6¼i

lijSqj

� �qq

" #: ð2Þ

We determine its ground state in the Weiss molecular potential approximation [21, 22],which approximates this Hamiltonian by

hmf ¼ l0aS2 � gBS � Beff ; ð3Þ

where we have introduced the effective magnetic field

Beff ¼ ðBz þ 2LmzÞ zzþ ðBq �LmqÞ qq ð4Þ

with L ¼ NP

j 6¼i lij and ma ¼ hSaj i=N.For ferromagnetic mini-condensates, the ground state corresponds to the situation where

the condensate at site i is aligned along Beff and S takes its maximum possible value N.That is, the ground state of the mean-field Hamiltonian (3) is simply

jGSi ¼ jN; NiBeff; ð5Þ

where the first number denotes the total angular momentum and the second its componentalong the direction of Beff .Equation (5) illustrates clearly the essential element brought about by the presence of

mini-condensates at the individual lattice sites: in contrast to the situation for an incoher-ent atomic sample hwere the enhancement factor would be

ffiffiffiffiN

p, the coherent behavior of

all N atoms within one site results in a ground-state magnetic dipole moment equal to Ntimes that of an individual atom. It is this property that leads to a significant magnetic

Han Pu et al., Atom Optics666

dipole-dipole interaction even for lattice points separated by hundreds of nanometers. Assuch, the situation at hand is in stark contrast with usual ferromagnetism, where themagnetic interaction is negligible compared to exchange and the use of fermions is essen-tial [22].

From the ground state (5) and for Bz ¼ 0, the components ma ¼ 1N hGSj Sai jGSi ¼ cos qa

of the magnetization, where qa is the angle between Beff and the a-axis, are

mz ¼2Lmzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð2LmzÞ2 þ ðBq �LmqÞ2q ; mq ¼

Bq �Lmqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið2LmzÞ2 þ ðBq �LmqÞ2

q : ð6Þ

For Bq � 3L, the only solutions are mz ¼ 0 and mq ¼ 1. That is, the lattice of mini-conden-sates is magnetically polarized along the transverse magnetic field. For Bq < 3L, in con-trast, there are two coexisting sets of solutions: (i) mz ¼ 0 and mq ¼ 1; and (ii)

mz ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ðBq=3LÞ2

qand mq ¼ Bq=3L. However, it is easily seen that the state asso-

ciated with the latter solution has the lower energy. Hence it corresponds to the true groundstate, while solution (i) represents an unstable equilibrium.

We have, then, the following situation: As Bq is reduced below a critical value 3L, thecondensate lattice ceases to be polarized along its direction. A phase transition occurs,characterized by a spontaneous magnetization with a finite mz along the z-direction.

From the preceding discussion, one might expect that the site-to-site coupling provide bythe magnetic dipole-dipole interaction conceive that the magnetic dipole-dipole interactionwill lead to the excitation of spin waves.1Þ This, however, is not quite the case. Simpleorder-of-magnitude estimates indicate that this interaction alone is not sufficient to excitespin waves in a lattice of Bose-Einstein condensate under presently achievable experimentalconditions. One way out of this difficulty is to add an external laser field to couple themini-condensates on the various sites via the electric dipole-dipole interaction. With thisadded interaction, the Hamiltonian (1) becomes [32, 33]

H ¼Pi

l0aSS2i � gBSSi � B�

Pj 6¼i

JzijSSzi SS

zj �

Pj 6¼i

JijðSSð�Þi SS

ðþÞj þ SS

ðþÞi SS

ð�Þj Þ

� �: ð7Þ

The explicit form of the coupling coefficients for the case of linearly polarized opticalfields is given in Ref. [27].

From the Hamiltonian (7), we can derive the Heisenberg equations of motion for the spinexcitations as

i�h@SSð�Þ

q

@t¼ ðw0 þ DwqÞ SSð�Þ

q �Pj 6¼q

cqjSSð�Þj ; ð8Þ

where we invoked the mean-field approximation to replace the spin operator SSzq by itsground state expectation value. The frequencies w0 ¼ �gBB and Dwq ¼ 2

Pj 6¼q J

zqjNj�h de-

scribe the precessing of the qth spin caused by the external magnetic field and the staticmagnetic dipolar interaction. The site-to-site spin coupling coefficients cqj ¼ 2JqjNq�h deter-mine the propagation of the spin waves.

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1) We recall that in analogy with phonons, which are normal modes of motion of atoms displacedfrom their equilibrium position in a crystal, spin waves are normal modes of spin excitation in materi-als with an ordered magnetic structure. These waves and their quanta of excitation, the magnons, werefirst discussed by F. Bloch, a student of Heisenberg, in 1930 [31].

In the long-wavelength limit, Eq. (8) leads to an effective Schrodinger equation [27]

i@Sðy; tÞ

@t¼ � b1

2

@2

@y2� b0 þ wðyÞ

� �Sðy; tÞ ; ð9Þ

where we have introduced the continuous limit quantities SSð�Þq ! Sðy; tÞ, cqj ! cðy� y0Þ,

and w0 þ Dw1 ! wðyÞ, and we have defined bn ¼ ð2=lLÞÐdh cðhÞ h2n for n ¼ 0; 1.

Eq. (9) describes the motion of ‘‘waves” caused by spin excitations in the x-y plane. Themagnon dispersion relation is presented in Ref. [27], which also discusses possible techni-ques to experimentally establish the existence of spin waves in the lattice of mini-conden-sates.

3. Outlook

While the possibility of atom optics was of course obvious from de Broglie’s work, it tookthree quarters of a century before it became somewhat practical, and it is only now that webegin to get a glimpse at its future promise.While device applications such as rotation sensors and improved clocks are rather evi-

dent, the true technological potential of this emerging field remains unclear. In fundamentalscience, though, the situation is less murky. Atom optics now has a profound impact on ourunderstanding of problems ranging from statistical and manybody physics to atomic physicsand to condensed matter physics. In particular, we have shown in this note how it opens upthe way to the study of magnetism in situations under exquisite control, both theoreticallyand experimentally.Discussions with Brian Anderson on the experimental aspects of this problem are grate-

fully acknowledged. This work is supported in part by the US Office of Naval Researchunder Contract No. 14-91-J1205, by the National Science Foundation under Grant No.PHY98-01099, by the US Army Research Office, by NASA, and by the Joint ServicesOptics Program.

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