Einstein-de Haas Effect in a Dipolar Fermi Gas

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by  Ulrich Ebling

Einstein-de Haas Effect in a Dipolar Fermi Gas
Ulrich Ebling1,∗ and Masahito Ueda1,2
1Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
2RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan
(Dated: January 31, 2017)
WeshowthatananaloguetotheclassicalEinstein-deHaaseffectcanappearinultracolddipolar
Fermi gases. The anisotropic nature of dipole-dipole interactions can lead to a transfer of magne-
tization into orbital angular momentum. Remarkably, distinct from a Bose-Einstein condensate,
this transfer is accompanied by twisting motion, where individual spin components rotate in oppo-
7 site directions with larger orbital angular momenta than the full system, possibly leading to easier
1
experimental observation of the effect. This feature is induced by the deformation of the Fermi
0
surface and the direction of the twisting motion can be controlled by an s-wave scattering length
2
or external magnetic field, possibly providing a method of measuring scattering lengths of strongly

n dipolar atomic species.
a
J

8 Over the last decade, experimental progress in optical components can make the EdH effect easier to observe
2 cooling and trapping techniques has led to the achieve- in a Fermi gas. In addition, manipulation of either an
ment of quantum degeneracy of strongly dipolar atoms, s-wave scattering length or an external magnetic field
]
s from the pioneering work on Chromium [1, 2] to more can reverse this twisting motion and change the sign of
a recent experiments with Dysprosium and Erbium [3–6]. L −L , while leaving the orbital angular momentum of
g ↑ ↓
Interactions in these systems are not only long-ranged, the full system |L +L | unchanged. The sensitivity to
- ↑ ↓
t butalsoanisotropic;theydependontherelativeposition the scattering length could help resolve the outstanding
n
and orientation of atoms [7, 8]. Therefore, they intrinsi- issue of measuring scattering lengths in Dy [24] and Er.
a
u cally combine spin and orbital degrees of freedom. For
We consider a dipolar Fermi gas of mass m with mag-
q instance a dipolar Fermi gas deforms its Fermi surface
netic dipole moment µ, confined in a 2D geometry. The
.
t along the spin polarization axis [9, 10]. On the orbital
a system is confined in the z-direction by a strong har-
side, due to being both attractive and repulsive, dipo-
m monic confinement ω such that all atoms occupy the
z
lar Bose-Einstein condensates (BEC) are known to have
- harmonic oscillator ground state with respect to the z-
d complex stability properties [11, 12], observable recently axis. Inthex−yplane,weassumeanisotropicharmonic
n intheformofdropletsstabilizedbyquantumfluctuations
trap ω =ω =ω (cid:28)ω so that the gas will form a radi-
o [13–17]. On the spinor side, dipole-dipole interactions, x y z
ally symmetric disk. We study the case of zero or weak
c
unlike s-wave scattering, do not conserve total magneti-
[ magnetic field B perpendicular to the 2D disk. There-
zation [18] and are expected to lead to intricate ground
fore, the dipoles are not necessarily fully polarized and
2 state phase diagrams [19, 20] as well as the ability to
v can vary in space and time. In this Letter, for the sake
transfer magnetization into orbital angular momentum,
6 ofsimplicityofourtheoryandnumericalsimulations,we
analogous to the Einstein-de Haas (EdH) effect [21–23].
4 assume a 2-component Fermi gas with spin states {↑,↓},
4 The experimental vindication of the genuine EdH ef- which in addition to dipole forces may also interact via
5
fect in dipolar BECs has been elusive, despite substan- the s-wave scattering length a, where the system is not
0
tialexperimentalprogresses. InthisLetter,weproposea fully polarized.
.
1 dipolarFermigasasabettercandidate,becauseofaspe-
0 The Hamiltonian of our system is
cial feature of the Fermi system: the deformation of the
7
Fermi surface. We demonstrate that this effect leads to
1
v: anadditionaltwistingmotionoftheFermisystemontop Hˆ =(cid:90) d2r (cid:88) ψˆ†((cid:126)r)(cid:20)−(cid:126)2∇2 + mω2(cid:126)r2 +Pk(cid:21)ψˆ ((cid:126)r)
i of the center-of-mass orbital angular momentum that is k 2m 2 k
X converted from initial magnetization by the EdH effect. k=↑,↓
(cid:90)
r Using numerical simulations of the dynamics of a sim- +g d2rψˆ†((cid:126)r)ψˆ†((cid:126)r)ψˆ ((cid:126)r)ψˆ ((cid:126)r)
a ple two-component, two-dimensional (2D) Fermi gas, we 2D ↑ ↓ ↓ ↑
show that both spin components {↑,↓} acquire angular +1(cid:90) d2rd2r(cid:48) (cid:88) ψˆ†((cid:126)r)ψˆ†((cid:126)r(cid:48))V ((cid:126)r−(cid:126)r(cid:48))ψˆ ((cid:126)r(cid:48))ψˆ ((cid:126)r),
momenta in opposite directions, with individual angular 2 k l klmn m n
klmn
momenta considerably larger than that induced by the
(1)
EdH effect |L − L | (cid:29) |L + L |. Our results show
↑ ↓ ↑ ↓
that this twisting motion arises from Fermi surface de-
formation during the time evolution, which explains the where P = gµ B is the Zeeman splitting, g =
√ B 2D
absence of this effect in dipolar BEC [21]. Therefore, 2 2π(cid:126)2a/l m describes s-wave scattering in the 2D case
z
this excess of the angular momentum of individual spin with l = (cid:112)(cid:126)/mω characterizing the width of the sys-
z z
2
Boltzmann-Vlasov equation (BVE)
1.2
1 (cid:18) (cid:19)
d p(cid:126)
W((cid:126)r,p(cid:126))= − ·∇ +mω2(cid:126)r·∇ W((cid:126)r,p(cid:126))
0.8 dt m x p
1
n 1 (cid:16)(cid:104) (cid:105)(cid:17)
etizatio 00..46 0.5 + i1(cid:126)(cid:16)(cid:110)W((cid:126)r,p(cid:126)),i(cid:126)pσz+(cid:16)Ua((cid:126)r)+U((cid:126)r)−U˜((cid:126)r,p(cid:126))(cid:17)(cid:111)
agn 0.2 − 2 ∇pW((cid:126)r,p(cid:126)),·∇r Ua((cid:126)r)+U((cid:126)r)−U˜((cid:126)r,p(cid:126))
M 0
(cid:110) (cid:111)(cid:17)
0 0 4 8 + ∇ W((cid:126)r,p(cid:126)),·∇ U˜((cid:126)r,p(cid:126)) . (6)
r p
-0.2
Equation (6) involves single-particle contributions from
-0.4
0 1 2 3 4 5 6 7 8 the kinetic, trap and Zeeman energy, as well as contact
interactionsandtwocontributionsfromdipole-dipolein-
Time (ms)
teractions, the direct term that operates in real space
FIG.1. Einstein-deHaaseffect: Conversionofmagnetization and the exchange term in momentum space [30]. The
N −N (blue)intoorbitaltheangularmomentum(magenta) elementsofthemean-fieldpotentialsinEq.(6)aregiven
↑ ↓
duetotheDDIforB =0anda=0. Theinsetshowsthetime by
evolutionofthespinpopulationsofN (red)andN (green).
↑ ↓
Note that the system not only fully demagnetizes, but tem- (cid:90) d2q
porarily reaches negative magnetization around 2.6ms. Ua((cid:126)r)=g (δ TrW((cid:126)r,(cid:126)q)−W ((cid:126)r,(cid:126)q)),
ij 2D (2π(cid:126))2 ij ij
(7)
(cid:90) d2qd2r(cid:48) (cid:88)
tem in the z-direction. The DDI potential is given by U ((cid:126)r)= V ((cid:126)r−(cid:126)r(cid:48))W ((cid:126)r(cid:48),(cid:126)q), (8)
ij (2π(cid:126))2 jikl kl
kl
V ((cid:126)r)=c√d√2(cid:0)V (r)(cid:2)σz σz − 1σ+σ− U˜ij((cid:126)r,p(cid:126))=(cid:90) (2dπ2(cid:126)q)2 (cid:88)V˜jlki((cid:126)q−p(cid:126))Wkl((cid:126)r,(cid:126)q), (9)
klmn πl5 1 kl mn 4 kl mn kl
z
−14σk−lσm+n(cid:3)− 14(x−iy)2V2(r)σk+lσm+n where V˜ denotes the momentum representation of the
−1(x+iy)2V (r)σ−σ− (cid:1). (2) dipole-dipole interaction potential. We numerically inte-
4 2 kl mn
grate Eq. (6) using the MacCormack method, where we
work in Fourier space to calculate the mean-field poten-
Its strength is determined by c = µ µ2/4π (µ is the
d 0 0 tialsinEqs.(8)and(9)andcopewithdifficultiesarising
permeability of vacuum), while its long-range behavior
fromthesingularityofV (r)atr =0byusingtechniques
1
in 2D is given by
applied to a dipolar BEC in Ref. [31].
We assume our system to be prepared initially in a
strong magnetic field perpendicular to the x−y plane,
V1(r)=e4rl22z (cid:104)(2lz2+r2)K0(4rl22)−r2K1(4rl22)(cid:105), (3) suchthatitisfullypolarizedalongthez-axis,withN↓ =
z z (cid:82) d2rd2pW ((cid:126)r,p(cid:126),t = 0) = 0. In a 2D geometry, this
V2(r)=e4rl22z (cid:104)K0(4rl22)+(cid:16)2rl2z2 −1(cid:17)K1(4rl22)(cid:105), (4) means that↓↓the DDI is isotropic and repulsive, and we
z z
use the corresponding equilibrium distribution derived
in Ref. [29]. Next, the magnetic field is ramped down
where K and K denote modified Bessel functions [25]. to very low values, such that the dipoles can evolve in
0 1
time. In our numerical simulations, we assume the trap
We treat the dynamics of the system in the Hartree-
parameters to be ω = 2π×84Hz and ω = 2π×47kHz
Fock approximation, by following the time evolution of z
with N = 500 and T = 0.2T . We choose 161Dy with
the one-body Wigner function F
m = 160.93u, µ = 9.93µ and g = 1.243 as atomic
B
species,whichforsimplicitywetreatasatwo-component
(cid:90) (cid:68) (cid:69) Fermi gas.
W ((cid:126)r,p(cid:126))= d2r(cid:48)ei(cid:126)r(cid:48)p(cid:126)/(cid:126) ψˆ† ((cid:126)r−(cid:126)r(cid:48)/2)ψˆ ((cid:126)r+(cid:126)r(cid:48)/2) ,
mn m n For the simplest case of B = 0 and a = 0, the time-
(5) evolution of the two spin components is depicted in the
which is the phase-space representation of the single- inset of Fig. 1. As shown there, from an initially fully
particle density matrix [9, 26–29]. In the semi-classical polarized spin state with N =N, the Fermi gas evolves
↑
approximation, in the collisionless regime, the dynam- into an unpolarized state with N = N . This magneti-
↓ ↑
ics of the system can be described by a collisionless zation M = N −N is fully converted into the orbital
↑ ↓
3
Magneticµfieldµ(µG)
5
100 0 -100 -200 -300 -400
4 1.5
3 1
m
omentu 12 6µms) 0.05
ngular m -10 L(tµ=µ2.z -0-.15
A
-2
-1.5
-3
-2
-4 -300 -200 -100 0 100 200 300 400
0 1 2 3 4 5 6 7 8
Scatteringµlengthµ(a )
0
Time (ms)
FIG. 3. Reversal of the twisting motion: Beyond certain val-
uesfora(blue)andB(magenta),thespincomponentsreverse
theirsenseofrotation. Shownistherelativeangularmomen-
tum L at t = 2.6ms. The turning points L = 0 (dashed
z z
line) correspond to the values at which E +E remains
kin trap
approximately constant.
FIG. 2. Top: Twisting motion of a dipolar Fermi gas. Both
imagingindividualspincomponents, ratherthanthefull
spin components (red, blue) acquire large angular momenta
system, thanks to the additional twisting motion of the
in opposite directions. The angular momentum of the entire
spin components we find in Fermi systems.
system(magenta,sameasinFig.1)ismuchsmallerthanthe
individual angular momenta. Bottom: Density plots at t = Tounderstandthephysicaloriginofthispeculiartwist-
2.6ms (dashed line) of N (x,y) = (cid:82) d2pW ((cid:126)r,p(cid:126)) for the ing motion, we take a look at the time evolution of
mn mn
spin components ↑, ↓, the sum and their difference. Spatial the expectation value of orbital angular momentum L.
patterns associated with the generated angular momentum
We split the Wigner function and other single-particle
areexpectedtobemorevisibleinanexperimentforindividual
quantities into a scalar and a vector part W ((cid:126)r,p(cid:126)) =
spin components or their differences. mn
W ((cid:126)r,p(cid:126))δ +W(cid:126) ((cid:126)r,p(cid:126))·(cid:126)σ. The scalar part of Eq. (10),
0 mn
L =L +L , corresponds to the total angular momen-
0 ↑ ↓
angular momentum L=L +L , where tum induced by the EdH effect. We find, that its time
↑ ↓
evolution is described by the subleading term (i.e. the
(cid:90) d2rd2p anti-commutator)intheBoltzmann-Vlasovequation(6).
L = (xp −yp )W ((cid:126)x,p(cid:126)). (10)
m (2π(cid:126))2 y x mm However, thetwistingmotion, capturedbythetimeevo-
lutionofL =L −L , isdominatedbythecommutator
z ↑ ↓
Wenotethat,comparedtothecaseofadipolarBEC[21],
term in Eq. (6), which explains its greater magnitude.
demagnetization is more complete in our Fermi system,
We find that for short times, L evolves according to
z
which quickly reaches zero magnetization [32].
√
A surprising effect reveals itself when we decompose
d 2c M M
thetotalorbitalangularmomentuminducedbytheEdH dtLz ≈ √dπ4lx5 y
effect into the the two spin components. We show the z
(cid:90) d2rd2r(cid:48)d2p
tsiemrveeetvhoaltuttihoenaonfgLu↑la,rLm↓oamndenLtu↑m+iLs↓noint oFnigly. 2c.reWateedobin- × (2π(cid:126))2 (xpy−ypx)V2((cid:126)r−(cid:126)r(cid:48))
the ↓-component as expected, but in both components ×(cid:2)(x−x(cid:48))2−(y−y(cid:48))2(cid:3)N ((cid:126)r(cid:48))W ((cid:126)r,p(cid:126)), (11)
0 0
such that the ↓-component exhibits an excess of angular
momentum, while the ↑-component rotates in the oppo- whereM denotesthetransversemagnetization. So, in
x,y
site direction. Even more remarkable is the fact that the order for a twisting motion with |L | > L to occur, a
z ↓
magnitudes of both L and L are considerably larger non-zero transverse magnetization must be present, and
↑ ↓
than that of the total angular momentum as shown in the total phase-space distribution of the system must
the upper panel of Fig. 2. In the lower panels of Fig. 2, be anisotropic. Both of these effects are related, since
we plot density profiles of both spin components, their the transverse magnetization in turn is the source of
sum and difference. It is evident from these plots that in the Fermi surface deformation [9]. Thus, small amounts
an experiment, finding the signature spatial patterns of present at t=0 or later during the demagnetization can
the EdH effect is more likely to be successful by in-situ lead to a twisting motion of the dipolar Fermi gas. The
4
25.5 25.5 tion[9,33]andobtainthenormalizedenergyfunctionals
(a) (c)
25 25
(cid:90) d2rd2p p2
ω)24.5 24.5 E [W]= W ((cid:126)r,p(cid:126)), (12)
ħy ( 1 1 kin (2π(cid:126))2N 2m 0
erg (cid:90) d2rd2p mω2(cid:126)r2
En 0.5 0.5 Etrap[W]= (2π(cid:126))2N 2 W0((cid:126)r,p(cid:126)), (13)
0 0 (cid:90) d2rd2p
E [W]= PW ((cid:126)r,p(cid:126)), (14)
-0.5 -0.5 mag (2π(cid:126))2N z
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
Time (ms) Time (ms) (cid:90) d2rd2p
E [W]= Ua((cid:126)r)W((cid:126)r,p(cid:126)), (15)
25.5 25.5 a (2π(cid:126))22N
(b) (d)
25 25 1(cid:90) d2rd2p
ħω)24.5 24.5 Edir[W]= 2 2N(2π(cid:126))2U((cid:126)r)W((cid:126)r,p(cid:126)), (16)
Energy ( 0.51 0.501 Eex[W]=−12(cid:90) 2Nd2(r2dπ2(cid:126)p)2U˜((cid:126)r,p(cid:126))W((cid:126)r,p(cid:126)). (17)
-0.5
0 -1
These denote respectively the kinetic (12), trap (13) and
-1.5
-0.5 -2 Zeeman (14) energies as well as the short-range interac-
0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
tion energy (15) and the direct (16) and exchange (17)
Time (ms) Time (ms)
parts of the DDI. In Fig. 4, we show the time evolution
FIG. 4. Time evolution of different energy contributions. In of these energy contributions for values of a and B on
the left column, B =0 and a=50a0 (a) and a=300a0 (b), both sides of the sign change of the twisting motion (see
on both sides of the sign change in Fig. 3. Likewise, in the
Fig. 3). The DDI is initially repulsive, hence E [W(t=
dir
right column, a=0 and B =−20µG (c) and B =−200µG
0)]>0. For a=B =0, total energy conservation means
(d). Thereversalofthetwistingmotionthataccompaniesthe
that E [W]+E [W] must increase. However, a and
EdH effect coincides with a decrease/increase of the sum of kin trap
kinetic and trap energies (black line). B can be tuned such that Ekin[W]+Etrap[W] must de-
crease, either if E [W(t = 0)] < E [W(t → ∞)] or
dir a
E [W(t = 0)]+E [W(t = 0)] < 0, where we denote
dir mag
by W(t → ∞) the demagnetized state after the short-
termEdHdynamics. Thiscoincideswiththesignchange
of L depicted in Fig. 3.
z
ThereasonforthisistheFermisurfacedeformationef-
dependence of Eq. (11) on an anisotropic real and mo-
fect[9,10]. Sogoetal. showed,thatanincrease/decrease
mentum space distribution explains why such an effect
ofthekineticenergydetermineswhetheraFermigasex-
has not been predicted for a dipolar BEC.
pands or contracts along the transverse spin direction
when undergoing the Fermi surface deformation [33]. A
Let us now turn our attention to another intriguing decrease in kinetic energy leads to a contraction along
featureofthetwistingmotionthataccompaniestheEdH the spin polarization axis, while an increase leads to an
effect in dipolar Fermi gases. So far, we have considered elongation. In turn, this behavior determines the sign of
a system without s-wave scattering and magnetic field therelativerotationofbothspincomponentsinEq.(11).
B = 0. In Fig. 3, we show the relative orbital angular Thisexplainswhyanincreaseofanappliedmagneticfield
momentum of both spin components at t = 2.6ms, ap- or an s-wave scattering length beyond a critical value re-
proximatelywhenthemagnetizationhasreachesitsmin- verses the twisting motion of the Fermi gas due to an
imum, for various values of B and a. While the total an- overall reduction of kinetic and trap energies.
gular momentum remains largely unchanged, apart from In conclusion, we have shownthat dipolarFermi gases
magnetic fields where the EdH is fully suppressed, the can exhibit transfer of spin into orbital angular momen-
counter-rotating spin components actually change their tum analogous to the Einstein-de Haas effect. We have
relative motion at non-zero values of both B and a. In further demonstrated that the EdH effect is more pro-
other words, the ↓ spin component starts rotating in op- nounced than in a dipolar BEC and occurs in combina-
posite direction as would be expected from the change tion with a twisting motion, where individual spin com-
of magnetization accompanying an interaction process ponents acquire excessive angular momenta of opposite
{↑,↑} → {↓,↓} that generates orbital angular momen- directionswhichcanbecomeconsiderablylargerthanthe
tum. To investigate this feature, we take a look at the total angular momentum induced by the EdH effect. We
time evolution of the different energy contributions. We find that this twisting motion is caused by the dynami-
calculate the expectation value of the individual parts of cal anisotropic deformation of the phase-space distribu-
the Hamiltonian Eq. (1) in the Hartree-Fock approxima- tion and therefore is an effect unique to a Fermi gas.
5
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Grant No. JP26287088 from the Japan Society for the
[25] See Supplemental Material for the derivation of the 2D
Promotion of Science, a Grant-in-Aid for Scientific Re- interaction terms.
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U. E. acknowledges support from a postdoctoral fel-
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[30] For short-range interactions, the direct and exchange
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6
SUPPLEMENTAL MATERIAL Due to the symmetry of χ(z), the two bottom rows of
Eq. (20) vanish. It remains to compute the integrals
Dipole-dipole interactions in two dimensions I (ρ)= 1 (cid:90) dzdz(cid:48)e−zlz22e−zl(cid:48)z22(x2+y2−2(z−z(cid:48))2)
1 πl2 [x2+y2+(z−z(cid:48))2]5/2
√z
In the usual three-dimensional case, the Hamiltonian = √ 2 eρ2/4lz2(cid:104)(cid:0)2l2+ρ2(cid:1)K (ρ2)−ρ2K (ρ2)(cid:105),
for dipole-dipole interaction is given by πlz5 z 0 4lz2 1 4lz2
(22)
HDDI =21(cid:90) d3rd3r(cid:48) (cid:88) ψˆk†((cid:126)r)ψˆl†((cid:126)r(cid:48)) I2(ρ)= 4π3lz2 (cid:90) dzdz(cid:48)[x2+e−yz22+/lz2(ez−−z(cid:48)2z/(cid:48)l)z22]5/2
klmn √ ρ2
×Vk3lDmn((cid:126)r−(cid:126)r(cid:48))ψˆm((cid:126)r(cid:48))ψˆn((cid:126)r). (18) = √2πe44llzz52 (cid:104)K0(4ρl2z2)+ 2lz2ρ−2ρ2K1(4ρl2z2)(cid:105), (23)
where ρ2 =x2+y2, and K and K denote the modified
0 1
Here, the interaction potential for the case of free mag-
Bessel functions of the zeroth order and the first order,
netization is
respectively. Weobtainasafinalresultthedipole-dipole
interaction potential for a 2D system
√
V3D ((cid:126)r)=c r2(cid:126)σkl·(cid:126)σmn−3((cid:126)r·(cid:126)σkl)((cid:126)r·(cid:126)σmn), (19) V ((cid:126)r)=c√d 2(cid:0)V (r)(cid:2)σz σz − 1σ+σ−
klmn d r5 klmn πl5 1 kl mn 4 kl mn
z
−1σ−σ+ (cid:3)− 1(x−iy)2V (r)σ+σ+
4 kl mn 4 2 kl mn
where (cid:126)σ denotes the Pauli matrices. With the identities −41(x+iy)2V2(r)σk−lσm−n(cid:1), (24)
σ± =σx±iσy, Eq. (19) becomes (cid:112) (cid:112)
where V (ρ)= π/2l5I (ρ) and V (ρ)=4 π/2l5I (ρ).
1 z 1 2 z 2
V3D ((cid:126)r)= cd (cid:2)σ+σ+ (cid:0)−3(x−iy)2(cid:1) Contact interaction in 2D
klmn r5 kl mn 4
+σ−σ− (cid:0)−3(x+iy)2(cid:1)
kl mn 4 A simpler calculation for the contact interaction term
+(σz σz −1σ+σ− − 1σ−σ+ (cid:1)(cid:0)x2+y2−2z2(cid:1)
kl mn 4 kl mn 4 kl mn 4π(cid:126)2a(cid:90)
+(cid:0)σkzlσm+n+σk+lσmzn(cid:1)32(−xz+iyz) Hc = m d3rψˆ↑†((cid:126)r)ψˆ↓†((cid:126)r)ψˆ↓((cid:126)r)ψˆ↑((cid:126)r)
+(cid:0)σkzlσm−n+σk−lσmzn(cid:1)32(−xz−iyz)(cid:3). (20) =4π(cid:126)2a(cid:90) dz|χ(z)|4
m
(cid:90)
To achieve a 2D setup, we assume a very strong har- × d2rφˆ†↑(x,y)φˆ†↓(x,y)φˆ↓(x,y)φˆ↑(x,y)
monic confinement in the z-direction, such that all par- √
2 2π(cid:126)2a(cid:90)
ticles occupy the harmonic oscillator ground state with = d2rφˆ†(x,y)φˆ†(x,y)φˆ (x,y)φˆ (x,y)
l m ↑ ↓ ↓ ↑
respect to the z-axis. We then split the field opera- z
(cid:90)
tors in Eq. (18) into a product of 2D field operators =g d2rφˆ†(x,y)φˆ†(x,y)φˆ (x,y)φˆ (x,y) (25)
and the harmonic oscillator ground state wave function 2D ↑ ↓ ↓ ↑
lψˆm=((cid:126)r(cid:112))=(cid:126)/φˆmmω(x.,yW)χe(zsu),bwsthiteurteeχt(hzi)s=int(oπlz2t1)h1e/4He−azm2/il2tloz2nainand 2p√ro2vπid(cid:126)e2sa/ulsmw.ith the effective coupling constant g2D =
z z z
Eq. (18) and carry out the integration over z and z(cid:48).
The 2D version of the dipole-dipole interaction potential
is therefore given by Time evolution of orbital angular momentum
For a 2D system, the angular momentum is only de-
(cid:90) fined for the z-component such that in the phase-space
V2D (x,y)= dzdz(cid:48)|χ(z)|2|χ(z(cid:48))|2V3D (x,y,z−z(cid:48))
klmn klmn picture the expectation value of the angular momentum
=(cid:90) dzdz(cid:48)π1lz2e−z2/lz2e−z(cid:48)2/lz2Vk3lDmn(x,y,z−z(cid:48)). operator Lˆ =(cid:90)xˆpˆyd2−rdyˆ2pˆpx for a spin component is given by
(21) Lmn = (2π(cid:126))2 (xpy−ypx)Wmn((cid:126)r,p(cid:126)). (26)
7
We calculate its time evolution by multiplying the colli-
sionless Boltzmann-Vlasov equation
d (cid:18) p(cid:126) (cid:19) 1 (cid:16)(cid:104) (cid:105)(cid:17)
W((cid:126)r,p(cid:126))= − ·∇ +mω2(cid:126)r·∇ W((cid:126)r,p(cid:126))+ W((cid:126)r,p(cid:126)),i(cid:126)pσ +Ua((cid:126)r)+U((cid:126)r)−U˜((cid:126)r,p(cid:126))
dt m x p i(cid:126) z
1(cid:16)(cid:110) (cid:16) (cid:17)(cid:111) (cid:110) (cid:111)(cid:17)
− ∇ W((cid:126)r,p(cid:126)),·∇ Ua((cid:126)r)+U((cid:126)r)−U˜((cid:126)r,p(cid:126)) + ∇ W((cid:126)r,p(cid:126)),·∇ U˜((cid:126)r,p(cid:126)) . (27)
2 p r r p
with xp − yp and subsequent integration over phase Eq. (27). Neglecting the anti-commutator, we find
y x
space. The single-particle part is straightforward:
d (cid:114)2 c (cid:90) d2rd2r(cid:48)d2p
L ≈ d (xp −yp )V ((cid:126)r−(cid:126)r(cid:48))
dt z π4l5 (2π(cid:126))2 y x 2
z
(cid:90) d2rd2p (cid:18) p(cid:126) (cid:19) ×(cid:2)(x−x(cid:48))2N ((cid:126)r(cid:48))W ((cid:126)r,p(cid:126))
(xp −yp ) − ·∇ W((cid:126)r,p(cid:126))=0, (28) x y
(2π(cid:126))2 y x m x −(x−x(cid:48)) (y−y(cid:48))(N ((cid:126)r(cid:48))W ((cid:126)r,p(cid:126))−N ((cid:126)r(cid:48))W ((cid:126)r,p(cid:126)))
y y x x
(cid:90) d2rd2p(xp −yp )m(cid:0)ω2x∂ +ω2y∂ (cid:1)W((cid:126)r,p(cid:126)) −(y−y(cid:48))2Ny((cid:126)r(cid:48))Wx((cid:126)r,p(cid:126))(cid:3), (31)
(2π(cid:126))2 y x x px y py
=m(cid:90) d(22πrd(cid:126)2)p2 (cid:0)ωx2−ωy2(cid:1)xyW((cid:126)r,p(cid:126)) wpehnedres Nonj((cid:126)rt)ra=nsv(cid:82)edrs2epWmja((cid:126)rg,np(cid:126)e)t.izaTthioins ecxopmrepsosinoenntosnlWyxd,ey-.
Further, for isotropic phase space distributions, Eq. (31)
=0. (29)
is zero.
Ifweassumetransversemagnetizationtobesmalland
for small times to have approximately the same phase-
It would only be non-zero in an anisotropic trap. space distribution, such that W ((cid:126)r,p(cid:126))=M W ((cid:126)r,p(cid:126)),
x,y x,y 0
Eq. (31) reduces to
Beforewecarryon,weseparateall2×2matricessuch
√
as W((cid:126)r,p(cid:126)) into a scalar and vector part by decomposing d 2c M M
L ≈ √d x y
themintermsoftheunitmatrixandthePaulimatrices. dt z π4l5
z
For a matrix A, this means (cid:90) d2rd2r(cid:48)d2p
× (xp −yp )V ((cid:126)r−(cid:126)r(cid:48))
(2π(cid:126))2 y x 2
×(cid:2)(x−x(cid:48))2−(y−y(cid:48))2(cid:3)N ((cid:126)r(cid:48))W ((cid:126)r,p(cid:126)), (32)
A=A 11+A(cid:126)·(cid:126)σ, (30) 0 0
0
where N ((cid:126)r(cid:48)) = (cid:82) d2p(cid:48)f((cid:126)r(cid:48),p(cid:126)(cid:48)), we can identify the two
0
ingredients for the appearance of counter-rotating spin
where A = 1(A +A ) and A(cid:126) =2Tr((cid:126)σA). components: Non-zero magnetization in the x-y-plane
0 2 ↑↑ ↓↓
(M ,M > 0) and an anisotropic deformation of the
x y
Forangularmomentum,L correspondstothetotalor- Fermi surface of the phase-space distributions for the
0
bital angular momentum of the system which is changed transverse spin components. While the initial distribu-
by the Einstein-de Haas effect. We find, that it is not tion fulfills neither condition, transverse magnetization
affected by the commutators in Eq. (27), but rather the does occur during the demagnetization dynamics. This
next-order gradient terms. However, to gain insight into in turn automatically leads to a Fermi surface deforma-
thetwistingmotion,wemustlookatL =L −L ,which tion along the axis of transverse magnetization due to
z ↑ ↓
in leading order is induced by the commutator term in the anisotropy of DDI.