Quantum-tunneling dynamics of a spin-polarized Fermi gas in a double-well potential

0.8 MB·English

by  L. Salasnich

Quantum-tunneling dynamics of a spin-polarized Fermi gas in a double-well potential
L. Salasnich1, G. Mazzarella1, M. Salerno2, and F. Toigo1
1Dipartimento di Fisica “Galileo Galilei” and CNISM,
Universita` di Padova, Via Marzolo 8, 35122 Padua, Italy
2Dipartimento di Fisica “E.R. Caianiello”, CNISM and INFN - Gruppo Collegato di Salerno,
Universita` di Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy
Westudytheexactdynamicsofaone-dimensionalspin-polarizedgasoffermionsinadouble-well
0 potential at zero and finite temperature. Despite the system is made of non-interacting fermions,
1 its dynamics can be quite complex, showing strongly aperiodic spatio-temporal patterns during
0 the tunneling. The extension of these results to the case of mixtures of spin-polarized fermions in
2
interaction with self-trapped Bose-Einstein condensates (BECs) at zero temperature is considered

n as well. In this case we show that the fermionic dynamics remains qualitatively similar to the one
a observed in absence of BEC but with the Rabi frequencies of fermionic excited states explicitly
J dependingon the numberof bosons and on the boson-fermion interaction strength. From this, the

6 possibility to control quantum fermionic dynamicsby means of Feshbach resonances is suggested.
2
PACSnumbers: 03.70.Lm;03.75.+k;03.05.Ss

]
s
a I. INTRODUCTION out that even in absence of BEC and of interactions be-
g
tweenthefermions,thiskindofsystemexhibitsaninter-
-
t esting behavior.
n The control and manipulation of confined ultracold
a atomic gases make possible the study of the dynamics In the first part of the paper we focus on a spin-
u
ofmany-bodyquantumstateseitherforbosons[1–5]and polarized quantum gas in absence of BEC. We suppose
q
forfermions[6,7],aswellasfortheirmixtures. Ultracold that at the initial time the system is prepared in such
.
at atoms give the opportunity to investigate the most im- a way that all the fermions are localized at a given side
m portantquantumeffects witha veryhigh levelof control of the barrier and that a number of fermionic states can
and precision [8–11]. Among these effects, the quantum be excited by means of external resonantfields (e.g. res-
-
d tunneling is a topic ofwide interest. In this sense,oneof onant with the Rabi frequencies of the doublets). The
n the most important examples is provided by Bose gases dynamics sustained by the above Hamiltonian allows to
o confined in double-well shaped potentials. In these sys- study the transfer of matter which takes place through
c
temsthetunnelingofbosonsthroughthecentralbarrier, the barrier by tunneling. This effect was studied in
[
which separates the two potential wells, is responsible Ref. [18] for few bosons in a 1D double-well at zero-

1 for the atomic counterpart of the Josephson effect (see, temperature. Indeed, the non-interacting pure fermionic
v for instance, Refs. [11–14]) observed in superconductive system can be viewed as a realization in the infinitely
5
junctions [15]. The macroscopic quantum tunneling was strongrepulsiveinteractionlimitof1Dhard-corebosonic
1
6 studied also with two weakly-linked superfluids made of systems [22], where the Pauli exclusion principle mimics
4 interacting fermionic atoms for which it is possible to the hard-core interaction. By following Ref. [18], we fix
. obtain atomic Josephson junction equations [16]. thenumberoffermionsandanalyzethespatialvariations
1
The opticallattices anddouble-welltrapsareparadig- of the density profile of the fermionic cloud at different
0
0 matic external potentials to understand the tunneling in times, and the total fermionic fractional imbalance ZF
1 atomicsystemscharacterizedbyasmallnumberofparti- between the two wells as a function of time t. We carry
: cles [1, 4, 17]. The dynamics of smallsystems of spinless out this analysis both with one fermion and with more
v
bosons [18], of finite Fermi-Hubbard and Bose-Hubbard fermionsbypointingoutthe strikingdifferencesbetween
i
X systems[19], andofspin1/2fermions-especially within the two cases. In the former case the temporal behavior
r the quantum information community [20, 21] - were of Z (t) is fully periodic and its period (Rabi period) is
F
a
widely studied. given by the inverse Rabi frequency of the lowest dou-
Inthisworkweconsideraone-dimensional(1D)dilute blet. In the latter case the periodicity occurs on much
and ultracold gas of spin-polarized fermionic atoms in a longer times, equal to the minimum commonmultiple of
double-well potential in the absence and in the presence the Rabi periods of the populated doublets. Z (t) may
F
of a Bose-Einstein condensate (BEC) which is intrinsi- evenbe completely aperiodic ifsome ofthe Rabi periods
cally localized in one of the two potential wells. In both of the populated doublets are incommensurate. In any
caseswehavethatincorrespondenceto certainvaluesof case, the shape of the density profile generally exhibits
theparametersofthedouble-wellpotential,itseigenspec- spatio temporal patterns which do not replicate the ini-
trumisorganizedindoubletsspacedbetweenthemselves tial situation for rather long times. In Refs. [18, 19],
and made of quasi-degenerate states. Each doublet is the atomic systems therein considered are dealt with at
characterizedby its ownRabi frequency, proportionalto zero-temperature. Here we perform our analysis also at
the gap between the two corresponding states. It turns nonzero temperature. By keeping fixed the number of
2
fermions, we study the influence of the temperature on whereφ (x)arethecompletesetoforthonormaleigen-
α,j
the total fractional imbalance: there is a spatial broad- functions andǫ the correspondingeigenenergies. Here
α,j
enings of the density profile that become more evident α = 1,2,3,... gives the ordering number of the doublets
by increasing the temperature. of quasi-degenerate states, and j = S,A gives the sym-
In the second part of the paper we discuss the case of metry of the eigenfunctions (S means symmetric and A
a spin-polarized quantum Fermi gas interacting with a means anti-symmetric). It follows that the eigenvalues
quasi-stationary BEC at zero temperature. In particu- are ordered in the following way: ǫ1,S < ǫ1,A < ǫ2,S <
lar, we show that for a self-trapped BEC (e.g. a BEC ǫ2,A < ǫ3,S < ǫ3,A < ... . Without loss of generality, we
intrinsically localized by the nonlinearity in one of the consider real eigenfunctions φα,i(x).
two wells of the potential) and for small boson-fermion Due to the symmetry of the problem, it is useful to
interactions, the fermionic imbalance dynamics remains introduce a complete set of orthonormalfunctions
qualitatively similar to the one of the pure fermionic
1
case with the only difference that the spacings of the ξ (x)= φ (x)+φ (x) , (4)
α,R α,S α,A
√2
fermionic levels (and therefore the Rabi frequencies) be- (cid:0) (cid:1)
come dependent on both the number of bosons and the and
boson-fermion interaction. This suggests the possibility
1
to controlthe fermionicquantumdynamicsby changing, ξ (x)= φ (x) φ (x) . (5)
α,L α,S α,A
√2 −
forexample,the inter-speciesscatteringlengthsbyusing (cid:0) (cid:1)
Feshbachresonances,afactwhichcouldbeofinterestfor If we fix the phase of φ and φ , such that
α,S α,A
applications to quantum computing. φ′ (+ ) < 0 for both j = S and j = A then ξ
α,j ∞ α,L
and ξ are mainly localized in the left and right well
α,R
respectively, at least for the low lying states.
II. THE SYSTEM The field operator ψˆ(x) can be written as
∞
We consider a confined dilute and ultracold spin- ψˆ(x)= ξ (x) cˆ , (6)
α,i α,i
polarized gas of N fermionic atoms of mass m in a
F αX=1i=XL,R
double-wellpotential. Fordiluteandultracoldatomsthe
onlyactivechannelofthe inter-atomicpotentialis the s- in terms of the single-particle fermi operators cˆ (cˆ† )
α,i α,i
wave scattering. For spin-polarized fermions the s-wave satisfying the usual anticommutation rules, which de-
channel is inhibited by statistics and consequently the stroy(create)onefermioninthestateξ (x)orξ (x).
α,L α,R
system is practically an ideal Fermi gas. The trapping By inserting Eq. (6) into (2), the Hamiltonian takes
potential of the system is given by the form
∞ ∞
V(r)=U(x)+ 12mω⊥2(y2+z2), (1) Hˆ = ǫ¯α cˆ†α,icˆα,i− Jα(cˆ†α,Lcˆα,R+cˆ†α,Rcˆα,L),
αX=1i=XL,R αX=1
(7)
We suppose that the system is one-dimensional (1D),
where nˆ = cˆ† cˆ is the fermionic number operator
due to a strong radial transverse harmonic confinement F,α,i α,i α,i
[23,24],withasymmetricdouble-welltrappingpotential of the α-th state in the i well (i = L,R: L means left
inthelongitudinalaxialdirection,thatwewilldenoteby andRmeansright). Noticethatthediagonalpartofthe
U(x). We are thus assuming that the transverse energy Hamiltonian (7), given by
¯hω of the confinement is much larger than the charac-
⊥ ∞
teristic energy of fermions in the axial direction [23, 24]. Hˆ = ǫ¯ cˆ† cˆ , (8)
0 α α,i α,i
On the basis of the dimensional reduction, the 1D αX=1i=XL,R
Hamiltonian of the dilute spin-polarized Fermi gas con-
fined by the potential U(x) is describesadouble-wellFermisystemwithaninfinitebar-
rier, i.e. with zero hopping terms. The energy
Hˆ = ψˆ†(x) ¯h2 ∂2 +U(x) ψˆ(x) dx, (2) ǫ¯ = ǫα,S +ǫα,A (9)
Z (cid:20)−2m∂x2 (cid:21) α 2
given by
where ψˆ(x) and ψˆ†(x) are the usual field operators an-
nihilating or creating a fermion at position x, therefore ¯h2 ∂2
ξ (x) +U(x) ξ (x) dx, (10)
obeying anticommutation rules. The single-particle sta- Z α,i (cid:20)−2m∂x2 (cid:21) α,i
tionary Schr¨odinger equation associated to the Hamilto-
nian (2) can be written as is the same for the left and right states described by
ξ (x) and ξ (x) The energy
α,L α,R
¯h2 ∂2 ǫ ǫ
(cid:20)−2m∂x2 +U(x)(cid:21)φα,j(x)=ǫα,jφα,j(x), (3) Jα = α,A−2 α,S (11)
3
is the energy of hopping from the L to the R well within with 0 f 1 and 0 f 1 parameters fixed by
α,L α,R
≤ ≤ ≤ ≤
the same doublet and gives directly half of the Rabi fre- the choice ofthe statisticaldensity operator. We remark
quency of the α-th doublet as thatf andf canbeanydistribution. Inparticular
α,L α,R
they could be the initial distributions corresponding to
J
Ω = α , (12) thermal equilibrium in the two fully separate wells with
α
¯h
different number of particles in each well. In this case
i.e. theRabiangularfrequencyΩRabi isgivenbyΩRabi = the statistical density operator is
α α
2Ω =2J /¯h [25].
α α
ρˆ =e−β(Hˆ0−µF,LNˆF,L−µF,RNˆF,R) , (22)
0
III. TUNNELING DYNAMICS OF FREE where Hˆ is given by Eq. (8), Nˆ = ∞ nˆ and
FERMIONS IN DOUBLE WELL POTENTIAL 0 F,i α=1 F,α,i
µF,i the number operator and the chemPical potential of
fermions in the i-th well, respectively. In this way we
To study the dynamics sustained by the Hamiltonian find
(7), we analyze the problem within the Heisenberg pic-
1
ture. TheHeisenbergequationofmotionfortheoperator f = , (23)
ψˆ(x,t) is α,L eβ(ǫ¯α−µF,L)+1
1
f = . (24)
i¯h∂ ψˆ(x,t)=[ψˆ(x,t),Hˆ]= ¯h2 ∂2 +U(x) ψˆ(x,t), α,R eβ(ǫ¯α−µF,R)+1
∂t (cid:20)−2m∂x2 (cid:21)
The chemical potentials µ and µ are fixed by the
(13) F,L F,R
number of particles in the left and right wells at time
and the time-dependent density operator is
zero:
nˆ (x,t)=ψˆ†(x,t)ψˆ(x,t). (14)
F ∞
1
N (0)= , (25)
Tnˆhe Heainsednnˆberg eqrueaadtions of motion for the operators F,L αX=1eβ(ǫ¯α−µF,L)+1
F,α,L F,α,R ∞
1
i¯hddtnˆF,α,L = −Jα(hˆα,L−hˆα,R), (15) NR,L(0)=αX=1eβ(ǫ¯α−µF,R)+1 . (26)
d
i¯h nˆ = J (hˆ hˆ ), (16) Itiseasytoshowthatthetime-dependentdensitypro-
F,α,R α α,L α,R
dt − file n (x,t) of the Fermi system,
F
where hˆ =cˆ† cˆ , and hˆ =cˆ† cˆ . The equa-
α,L α,L α,R α,R α,R α,L n (x,t)= ψˆ†(x,t)ψˆ(x,t) , (27)
tions of motion for the operators hˆ and hˆ are F h i
α,L α,R
can be written as
d
i¯h hˆ = J (nˆ nˆ ), (17)
α,L α F,α,L F,α,R
dt − − ∞
d n (x,t)= n (t) ξ (x)2 , (28)
i¯h hˆ = J (nˆ nˆ ). (18) F F,α,i α,i
dt α,R α F,α,L− F,α,R αX=1i=XL,R
We want to analyze the spatio-temporal evolution of
where n (t) = nˆ (t) . In addition, from Eqs.
F,α,i F,α,i
our system both at zero and finite temperature T. To h i
(15)-(18) it is straightforward to get the following cou-
this end, we evaluate the thermal average of both sides
pled ODEs
of the Heisenberg equations of motion obtained so far.
The thermal average of an operator Aˆ(t) is given by n¨ +2Ω2 n =2Ω2 n , (29)
F,α,L α F,α,L α F,α,R
Tr Aˆ(t)ρˆ n¨F,α,R+2Ω2α nF,α,R =2Ω2α nF,α,L (30)
0
Aˆ(t) = h i =A(t), (19)
h i with Ω given by Eq. (12).
Tr ρˆ α
0
h i Notice that the total number nF,α = nF,α,L(t) +
where ρˆ0 is the chosen statistical density operator. It is nF,α,R(t) of fermions in the α-th doublet and the
clear that a thermal averageevaluatedby using the den- fermionic hopping number hα(t)=hα,L(t)+hα,R(t) are
sity operator ρˆ = e−β(Hˆ−µNˆ) (with β = 1/(k T)) does both constants of motion.
B
Itis notdifficult to show thatthe ODEs(29)and (30)
not give rise to dynamics (time-dependent observables)
with the initial conditions (20) and (21) have the follow-
[26]. We want a statistical operator ρˆ which produces
0
ing solutions
the initial conditions
nF,α,L(0)= nˆF,α,L(0) =fα,L , (20) nF,α,L(t)=fα,L cos2(Ωαt)+fα,R sin2(Ωαt), (31)
h i
n (0)= nˆ (0) =f , (21) n (t)=f cos2(Ω t)+f sin2(Ω t). (32)
F,α,R F,α,R α,R F,α,R α,R α α,L α
h i
4
0.1 0.1
0 F
n 0.05 0.05
0 0
) -40 -20 0 20 40 -40 -20 0 20 40
x 0.06
(
U 0.1
-0.5
0.04
F
n 0.05
0.02
0 0
-40 -20 0 20 40 -40 -20 0 20 40
x x
-1
-40 -20 0 20 40 1
x
0.5
FIG.1: Thedoublewellpotential(35)forA=5·10−7,B =1,
and C = 5. For these values of the double-well potential
parameters,thereare30energeticlevels-correspondingto15 ZF 0
doublets-with energysmaller thantheheightofthebarrier.
Energies are measured in units of ¯hω⊥, lengths in units of -0.5
a⊥ = ¯h/mω⊥.
p -1
0 1500 3000 4500 6000
t
The population imbalance z (t) within the α-th double
α
is FIG. 2: (Color online). Density profile nF vs. space x and
total fermionic imbalance ZF vs. time t with NF =1. Plots
zF,α(t)=nF,α,L(t)−nF,α,R(t)=(fα,L−fα,R) cos(2Ωαt), of nF(x) are for different values of t. We define t1 as 2π/Ω1.
(33) In this figure: t = 0 and t = 0.25t1 (the two upper panels
and the total fermionic imbalance ZF(t) is given by of nF from left to right); t = 0.88t1 and t = 1.5t1 (the two
lowerpanelsofnF fromlefttoright). ForboththenF(x)and
1 ∞ 1 ∞ ZF(t)plots,thecontinuouslinerepresentsdataforKBT =0,
ZF(t)= N zα(t)= N (fα,L−fα,R) cos(2Ωαt). the dashed line for KBT =0.038 (≃(ǫ¯2−ǫ¯1)), and the dot-
F αX=1 αX=1 dashedlinefor KBT =0.087 (≃(ǫ¯3−¯ǫ1)). Theplotsat zero
(34) temperature are obtained with thefermion initially localized
Wemodelthedouble-welltrappingpotentialU(x)(see inthegroundstateoftherightwell;theplotsatfinitetemper-
Fig. 1) in the form ature are obtained with NF,L(0) = 0 and NF,R(0) = 1 with
level occupation defined by Eq. (26). Times are measured in
U(x)=Ax4+B(e−Cx2 1). (35) unitsof(ω⊥)−1,energiesinunitsof¯hω⊥,andlengthsinunits
−
of a⊥ = ¯h/mω⊥.
We study the time-dependent density profile (27) and p
thetotalfermionicimbalance(34)bothatzeroandfinite
temperaturefordifferentnumberoffermions. Theresults
deformed by this thermal effect, as we can see from the
of this study are shown in Fig. 2 (N = 1), Fig. 3
F
Z panel of Fig. 2.
(N =2), Fig. 4 (N =6), and Fig. 5 (N =12). F
F F F
Inourcalculationswehavesupposedthatthefermions Whenthenumberoffermionsisgreaterthanone(Figs.
ofthesystemare40Katoms,andsettheharmonictrans- 3-5), doublets other than the lowest one are involved in
versefrequencyω to160kHz,sothata 0.1µm[24]. the dynamics. At T = 0 the temporal evolution of the
⊥ ⊥
≃
The absolute temperatures used in Figs. 2-5 are calcu- totalfractionalimbalance-thecontinuouslineoftheZ
F
lated according to the above assumptions. In each of plots-isnomorecharacterizedbyasinglefrequencyasin
these figures we show, for fixed values of T, the spatial the N =1 case; it is, in fact, characterizedby a mixing
F
evolution of n (x,t) at different times and the temporal of frequencies, each of them proportional to the gap of
F
evolutionofZ . WhenN =1andT =0- the continu- the corresponding filled doublet. As in the N =1 case,
F F F
ous line of Fig. 2 - only the lowestdoublet is involvedin at finite temperatures the fermions can partially occupy
thedynamics. Atzerotemperature,thetemporalbehav- doublets higher in energy than those filled at T =0 (see
ior of Z is harmonic and is characterized by only one thecaptionsofFigs. 3-5). Theeffectistoinduceadefor-
F
frequency: 2Ω . At finite temperatures - the dashed and mationintheoscillationsofthetotalfermionicfractional
1
the dot-dashed lines of Fig. 2 - the fermions are allowed imbalance - with respect to T =0 case - as shownin the
to partially occupy doublets which are empty at T = 0 plots of Z - the dashed and the dot-dashed lines - of
F
(seethecaptionsofFig. 2),andtheharmonicityofZ is Figs. 3-5. The thermal effects influence the fermionic
F
5
0.15 0.15
0.25 0.25
0.1 0.1 0.2 0.2
F F0.15 0.15
n n
0.05 0.05 0.1 0.1
0.05 0.05
0 0 0 0
-40 -20 0 20 40 -40 -20 0 20 40 -40 -20 0 20 40 -40 -20 0 20 40
0.1 0.2
0.15
0.08 0.25
0.15
0.06 0.1 0.2
F F 0.1 0.15
n 0.04 n
0.05 0.1
0.02 0.05 0.05
0 0 0 0
-40 -20 0 20 40 -40 -20 0 20 40 -40 -20 0 20 40 -40 -20 0 20 40
x x x x
1 0.9
0.6
0.5
0.3
ZF 0 ZF 0
-0.3
-0.5
-0.6
-0.9
-1
0 1500 3000 4500 6000 0 1500 3000 4500 6000
t t
FIG. 3: (Color online). Density profile nF vs. space x and FIG. 4: (Color online). Density profile nF vs. space x and
total fermionic imbalance ZF vs. time t with NF =2. Plots total fermionic imbalance ZF vs. time t with NF =6. Plots
of nF(x) are for different values of t. We define t1 as 2π/Ω1. of nF(x) are for different values of t. We define t1 as 2π/Ω1.
In this figure: t = 0 and t = 0.74t1 (the two upper panels In this figure: t = 0 and t = 0.76t1 (the two upper panels
of nF from left to right); t = 0.82t1 and t = 1.5t1 (the two of nF from left to right); t=0.83t1 and t=1.52t1 (the two
lowerpanelsofnF fromlefttoright). ForboththenF(x)and lowerpanelsofnF fromlefttoright). ForboththenF(x)and
ZF(t)plots,thecontinuouslinerepresentsdataforKBT =0, ZF(t)plots,thecontinuouslinerepresentsdataforKBT =0,
the dashed line for KBT =0.047 (≃(ǫ¯3−ǫ¯2)), and the dot- the dashed line for KBT =0.068 (≃(ǫ¯7−ǫ¯6)), and the dot-
dashedlineforKBT =0.164 (≃(ǫ¯5−¯ǫ2)). Theplotsatzero dashedlineforKBT =0.219(≃(ǫ¯9−¯ǫ6)). Theplotsatzero
temperature are obtained with the 2 fermions initially in the temperatureare obtained with the6 fermions initially in the
2lowerstatesoftherightwell;theplotsatfinitetemperature 6lowerstatesoftherightwell;theplotsatfinitetemperature
are obtained with NF,L(0) = 0 and NR,L(0) = 2 with level are obtained with NF,L(0) = 0 and NR,L(0) = 6 with level
occupationdefinedbyEq. (26). UnitsarethesameasinFig. occupationdefinedbyEq. (26). UnitsarethesameasinFig.
2. 2.
density profile as well. In particular, when the temper- tial(35)atzero-temperature. Inparticular,weareinter-
ature is finite, nF(x) experiences a broadening with re- ested in the changes in the Rabi frequencies and in the
spect to the case T = 0. From the plots of nF shown in fermionicimbalancedynamicsinducedbythepresenceof
Figs. 2-5, we see that the higher is the temperature the theBEC.Tothisregardwerestrict,forsimplicity,tothe
wider is this broadening. caseofasmallnumberofexcitedfermionicstatespresent
Finally,itisworthobservingthatduringthedynamics inthefermionicdensityandtoweakboson-fermioninter-
ofourmodelthesystemisnotinthermalequilibriumand actions. Inthissituationthecondensatewillremainself-
it will never reach it since the particles do not interact trapped(practicallystationary)inthecourseoftimeand
between themselves. Each doublet preserves its energy the fermionic imbalance dynamics will be qualitatively
and its number of particles. similar to the one discussed in the previous section. The
fermionic spectrum and the Rabi frequencies, however,
will depend on the boson-fermion interaction due to the
IV. TUNNELING DYNAMICS OF presence of a bosonic effective potential in the fermionic
SPIN-POLARIZED FERMIONS INTERACTING Schr¨odinger equation of motion (see below).
WITH INTRINSICALLY LOCALIZED BEC
To describe the spin-polarized fermionic gas in inter-
action with the BEC we adopt a mean field description
In this sectionwe consider the case ofa spin-polarized for the condensate but treat the fermions still quantum
quantum Fermi gas in interaction with a BEC which is mechanically. In this case one can show [28, 29] that the
intrinsicallylocalizedinoneofthetwowellsofthepoten- dynamicsofthemixtureisdescribedbythefollowingset
6
0.4 0.3 macroscopicquantumself-trappingBose-Josephsonjunc-
tion with frozen fermions was done in [27]. For sim-
0.3
0.2
F plicity, we also assumed equal masses and the same
n 0.2
trapping potential for both bosons and fermions. The
0.1
0.1
bosonic cloud is self-trapped in one of the two wells
0 0 when the nonlinear strength g N exceeds a certain fi-
-40 -20 0 20 40 -40 -20 0 20 40 B B
0.3 nite critical value [12]. This condition can be written
0.25
0.25 as a N /a > (ǫ ǫ )/(h¯ω ) [12], and it is fully
0.2 0.2 sat|isfiBe|dBin ou⊥r num1e,Aric−al1e,xSperime⊥nts.
nF0.15 0.15 Note that the BEC wavefunction evolves according to
0.1 0.1
a nonlinear Gross-Pitaevskii equation (GPE) in which
0.05 0.05
the fermionic density enters as a potential, while the
0 0
-40 -20 0 20 40 -40 -20 0 20 40
x x fermionic dynamics is still linear with the condensate
density playing the role of an additional external poten-
0.4
tial. From this it is clear that the fermionic eigenstates
0.2 and eigenvalues can be well approximated by the linear
0 Schr¨odinger equation
ZF-0.2 ∂χ ¯h2 ∂2
j
-0.4 i¯h ∂t =(cid:20)−2m∂x2 +Ueff(x)(cid:21)χj , (38)
-0.6
-0.8 withthe effective potential Ueff(x)=U(x)+gBFn¯B(x),
with n¯ (x) denoting the stationary bosonic density. We
-1 B
0 1000 2000 3000 4000 5000 6000 have numerically verified that indeed the bosonic cloud
t
is practically stationary. From Eq. (38) the possibility
FIG. 5: (Color online). Density profile nF vs. space x and to manipulate the fermionic dynamics by changing NB
totalfermionicimbalanceZF vs. timetwithNF =12. Plots or the inter-species scattering length becomes evident.
of nF(x) are for different values of t. We define t1 as 2π/Ω1. In order to check these predictions, we have numer-
In this figure: t = 0 and t = 0.77t1 (the two upper panels ically determined the stationary states of the mixture
of nF from left to right); t=0.83t1 and t=1.52t1 (the two by solving in a self-consistent manner the time indepen-
lowerpanelsofnF fromlefttoright). ForboththenF(x)and dent equations corresponding to Eqs. (36) and (37) [29]
ZF(t)plots,thecontinuouslinerepresentsdataforKBT =0, for the case of attractive interactions. In Fig. 6a) we
the dashed line for KBT = 0.142, ((ǫ¯13 −ǫ¯12) < KBT < show the stationary bosonic wavefunction localized in
(ǫ¯ǫ¯1124)−<ǫ¯K12B)),Ta<nd(tǫ¯h16e−do¯ǫ1t2-d))a.shTehdelpinloetfsoratKzBerTot=em0p.3e3r3a,tu((rǫ¯e15ar−e the left well of the potential and the first 30 fermionic
stationary levels obtained in presence of BEC with the
obtained with the 12 fermions initially in the 12 lower states
of theright well; theplotsat finitetemperatureareobtained self-consistent method. Notice that the lowest levels de-
with NF,L(0) = 0 and NR,L(0) = 12 with level occupation viatefromthecorrespondingonesobtainedinFig. 1and
defined byEq. (26). Units are thesame as in Fig. 2. the quasi degeneracy of the lowest levels is removed due
to the bosoniceffective potential(in the presentcasethe
Rabifrequencies oflowestlevels areincreaseddue to the
of coupled equations
levelsplitting). Inpanelb)ofFig. 6weshowthefirstten
∂Ψ ¯h2 ∂2 fermionic stationary wavefunctions corresponding to the
i¯h ∂t =(cid:20)−2m∂x2 +U(x)+gBNB|Ψ|2+gBFnF(cid:21)Ψ, lowest ten energy levels of panel a), while in panel c) we
depictthefirstthreeexcited(e.g. nonstationary)states,
(36)
ζ (x), constructed from the lowest stationary wavefunc-
i
∂χ ¯h2 ∂2 tions as: ζ (x)=(χ (x)+χ (x))/√2. Note that the
i¯h j = +U(x)+N g Ψ2 χ , (37) i 2i−1 2i
∂t (cid:20)−2m∂x2 B BF| | (cid:21) j fermioniclowestenergyeigenstate is asymmetric,having
thesamelocalizationofthecondensateduetotheattrac-
where nF(x,t) = Nj=F1|χj(x,t)|2 denotes the fermionic tive Bose-Fermi interaction, while the next level has the
density with χj(x,Pt) (j = 1,...,NF) the set of orthonor- opposite localization. This implies that the lowest ex-
mal wave functions which satisfy Eq. (37), Ψ(x,t) is the cited states are not localized in the same potential well,
bosonic wavefunction normalized to one and such that as for the pure fermionic case considered before, but are
n (x,t) = N Ψ(x,t)2 is the bosonic density with N extended between the two wells.
B B B
| |
thetotalnumberofbosons. The1Dinteractionstrengths Excited fermionic densities corresponding to ten
are g = 2h¯2a /(ma2) and g = 2h¯2a /(ma2), fermions and including the lowest excited states are
B B ⊥ BF BF ⊥
with a and a the boson-boson and boson-fermion shown in panel d) of Fig. 6 together with the stationary
B BF
s-wavescatteringlengthsanda = ¯h/(mω )thechar- density. Theseexciteddensitieswereused,togetherwith
⊥ ⊥
acteristic length of the strong transpverse harmonic con- the BEC wavefunction shown in the top part of panel
finement of frequency ω [23, 24]. A similar study for 6a), as initial conditions to calculate the total fermionic
⊥
7
0.4 a 2.5 c 0.04 ν1 a 0.3 ν2 b
ψ 0.2
0.0 2.0 Aν00..0002 Aν0.1 ν1
0.006.000 0.005 0ν.010 0.015 0.020 00.0.000 0.003 ν 0.006 0.009
ξi1.5 0.3
-0.4 0.04
ε-i0.8 1.0 zF0.02 zF0.0
0.00
0.5 -0.02 -0.3
-30 -15 0 15 30 -30 -20 -10 0 10 20 30 0 200 400 600 800 1000 0 1000 2000 3000 4000
x x t t
5 0.3 0.012
b d 0.3 ν3 c d
4 Aν0.2 ν2 0.009
0.1 ν
3 0.2 0.0 1 να
0.000 0.002 0.004ν 0.006 0.008 0.010
χi nF 0.8 0.006
2 0.4
0.1 zF0.0 0.003
1
-0.4
0.000
0-30 -20 -10 0 10 20 30 0.0-30 -20 -10 0 10 20 30 -0.80 4000 t 8000 12000 0.0000 -0.0005 aBF-0.0010 -0.0015
x x
FIG. 7: Panels a)-c): dynamics of the fermionic density im-
FIG. 6: Panel a): condensate localized wavefunction Ψ(x) balance ZF(t) (bottom) and corresponding Fourier spectrum
(top curve) corresponding to NB = 470 bosons in a dou- (top part of panels) as obtained from numerical integrations
ble well potential and in the presence of 30 spin polarized of Eqs. (36) and (37) using as initial conditions the station-
fermions with attractive boson-boson and boson-fermion in- ary localized bosonic wavefunction and the non stationary
teractions aB =−0.001, aBF =−0.001. Horizontal lines de- fermionicdensitieswithone(a),two(b)andthree(c)excited
notethefirst30fermionicenergylevelsinthepresenceofthe states depicted in Fig. 6. Panel d): dependence of the Rabi
condensate while the bottom curve represents the trapping linear frequency να on the boson-fermion interaction for the
potentialwithparametersfixedasinFig. 1. Panelb): lowest first fiveexcited levels indicated with dots, squares, plus,tri-
ten fermionic eigenfunctions χi(x) (from bottom to top) in angleandcrosses,respectively. Theverticallineindicatesthe
the presence (continuous curves) and in the absence (dotted value of aBF at which the other panels have been evaluated.
lines)oftheBECwavefunctiondepictedinpanela)(anoffset Parameters are fixed as in Fig. 6. Time is measured in units
of 0.5 between curves has been added to avoid overlapping). of ω⊥−1 and aBF is measured in unitsof a⊥.
Panel c): first three excited states in the presence of BEC
constructedwiththelowesteigenstatesofpanelb)(seetext).
An offset of 0.7 between curves was added to avoid overlap-
panels b),c), with corresponding values of panel d)).
ping. Panel d): fermionic densities for 10 fermions including
Frompanelsa)-c)ofFig. 7itisclearthatthefermionic
one (dotted line), two (dash dotted line), and three (dashed
imbalance dynamics is given by a superposition of har-
line) excited states depicted in panel c. The continuous line
denote thestationary fermionic density in absence of excited monics in number equal to the number of excited states
states. Energiesaremeasuredinunitsof¯hω⊥,lengthsinunits andwithperiodsfixedbythelevelspacingRabifrequen-
ofa⊥ = ¯h/mω⊥. Thebosonicandfermionicwavefunctions cies. Also note from panel c) of Fig. 7 the presence of
are bothpnormalized to one. beatings of period 1/(ν2 ν3) generated by the the two
−
closefrequenciesν andν ,asexpectedforlinearsystems
2 3
(noticealsothe presenceofsmallpeaksclosetotheRabi
brequency ν probably due to the quasi-stationarity of
imbalance density, Z (t), from direct numerical integra- 1
F
the bosons).
tions of Eqs. (36) and (37). The results are depicted
in Fig. 7, panels a)-c). From panel a) of this figure we The dependence of the Rabi frequency on the Bose-
see that when only one excited state is present in the Fermi interaction is depicted in Fig.7d) for the first five
density,the imbalancedynamicsisperiodic withasingle excitedstates. Notethatthefrequencyofthefirstexcited
frequency in the spectrum. The Rabi linear frequency state has a large variationwith aBF due to the fact that
ν = ΩRabi/(2π) = Ω /π calculated from the dynamics the corresponding eigenstates are the ones most effected
α α α
of the fermionic imbalance density is found in very good by the presence of the condensate (see Fig. 6b). Similar
agreement (see panel d)) with the value obtained from dependences are also obtained by changing the number
the energy spectrum and calculated from the nonlinear of atoms in the condensate and keeping fixed the fixed
eigenvalue problem associated with Eqs. (36) and (37) boson-fermion interaction.
using the self-consistent approach [29]. The same good From this we conclude that in the BEC self-trapped
agreement is found in the case of two and three excited regimethe dynamics forthe fermionic density imbalance
states present in the fermionic density panels (compare remains qualitatively similar to the one observed in the
8
purefermioniccasebutwiththeRabifrequenciesexplic- ingdeformationsofthedensityprofilewithrespecttothe
itly dependent on the boson-fermion interaction. Since zero-temperature case. We also have discussed the pos-
this interaction can be easily changed by external mag- sibility to include in the system a bosonic component,
netic fields via Feshbach resonances, we have found an which, under given hypothesis, does not produce impor-
effective way to controlthe the dynamics of the spin po- tantchangesinthedynamicsofthesystematleastwithin
larizedfermionicgaswhichcanbe implemented inareal very low temperature regimes. In particular, we have
experiment. shown that the presence of a self trapped Bose-Einstein
condensate weakly interacting with the spin-polarized
fermigasallowstoachieveaquiteeffectivecontrolofthe
V. CONCLUSIONS Rabi frequencies of excited fermionic states by chang-
ing the boson-fermion interaction with external mag-
We have investigated a confined dilute and ultracold netic fields via Feshbach resonances. This Bose-Einstein
spin-polarized gas of fermionic atoms in a double-well condensate-inducedcontrolofthefermionicquantumdy-
potential. We have analyzed the quantum tunneling namicsandcanbeimplementedinarealexperimentand
through the central barrier by studying the density pro- can be of interest for applications to quantum comput-
fileandthetotalfermionicfractionalimbalance. Wehave ing.
pointedoutthatdespite the fermionsdonotinteractbe- This work has been partially supported by Fondazione
tween themselves, the dynamics is quite complex, and it CARIPARO through the Project 2006: ”Guided soli-
exhibits very interesting aperiodicity features. We have tons in matter wavesandoptical waveswith normaland
performed our analysis both at zero and finite tempera- anomalous dispersion”. MS acknowledges partial sup-
turebydiscussingthe roleofthetemperatureinproduc- port from MIUR through a PRIN-2007 initiative.
[1] S.F¨olling et al.,Nature448, 1029 (2007). F. Toigo, J. Phys. B: At. Mol. Opt. Phys. 42, 125301
[2] I. B. Spielman, W. D. Phillips, and J. V. Porto, Phys. (2009).
Rev.Lett. 98, 080404 (2007). [15] A. Barone and G. Patern`o, Physics and Applications of
[3] T. Salger, C. Geckleler, S. Kling, and M. Weitz, Phys. the Josephson effect (Wiley, New York, 1982); H. Otha,
Rev.Lett. 99, 190405 (2007). in SQUID: Superconducting Quantum Devices and their
[4] S.Trotzky et al.,Science 319, 295 (2008). Applications, edited by H.D. Hahlbohm and H. Lubbig,
[5] P. Wu¨rtz, T. Gericke, A. Koglbauer, and H. Ott, Phys. (de Gruyter,Berlin, 1977).
Rev.Lett. 103, 080404 (2009). [16] L.Salasnich, N.Manini, andF.Toigo, Phys.Rev.A77,
[6] Y.Shin,M.W.Zwierlein, C. H.Schunck,A.Schirotzek, 043609 (2008).
and W. Ketterle, Phys. Rev. Lett. 97, 030401 (2006); [17] J. Est´eve et al.,Nature455, 1216 (2009).
G. B. Partridge, Wenhui Li, Y. A. Liao, R. G. Hulet, [18] S, Zo¨llner, H.-D. Meyer, and P. Schmelcher, Phys. Rev.
M. Haque, H. T. C. Stoof, Phys. Rev. Lett. 97, 190407 Lett. 100, 040401 (2008); S, Zo¨llner, H.-D. Meyer, and
(2006);N.Strohmaier,Y.Takasu,K.Gu¨nter,R.J¨ordens, P. Schmelcher, Phys.Rev. A 78, 013621 (2008).
M. K¨ohl, H. Moritz, and T. Esslinger, Phys. Rev. Lett. [19] K. Ziegler, e-print arXiv:0909.4932 (2009).
99, 220601 (2007). [20] M. Lewenstein et al.,Adv.Phys. 56, 243 (2007).
[7] Ultra-cold Fermi Gases, edited by M. Inguscio, W. Ket- [21] J. I. Cirac in Ultra-cold Fermi Gases, edited by M. In-
terle, and C. Salomon, (IOSPress, Amsterdam, 2007). guscio, W. Ketterle, and C. Salomon, (IOS Press, Ams-
[8] L. Pitaevskii and S. Stringari, Bose-Einstein Condensa- terdam, 2007).
tion (Oxford University Press, Oxford,2003). [22] M. Girardeau, J. Math. Phys. (N.Y.) 1, 516 (1960).
[9] F. Dalfovo, S. Giorgini, L. Pitaevskii, and S. Stringari, [23] L. Salasnich and F. Toigo, J. Low. Temp. Phys. 50, 643
Rev.Mod. Phys. 71, 463 (1999). (2008).
[10] C. J. Pethick and H. Smith, Bose-Einstein Condensa- [24] G. Mazzarella, L. Salasnich, and F. Toigo, Phys. Rev.A
tioninDiluteGases,(CambridgeUniversityPress,Cam- 79, 023615 (2009).
bridge, 2001). [25] J.J. Sakurai, Advanced Quantum Mechanics (Addison
[11] A.J.LeggettandF.Sols,Found.Phys.21,353(1991);I. Wesley Longman, New York,1967).
Zapata,F.Sols,andA.J.Leggett,Phys.Rev.A57,R28 [26] K.K.Das,M.D.Girardeau,andE.M.Wright,Phys.Rev.
(1998); A.J. Leggett, Rev.Mod. Phys. 73, 307 (2001). Lett. 89, 170404 (2002).
[12] A. Smerzi, S. Fantoni, S. Giovanazzi, and S. R. Shenoy, [27] S.F. Caballero Ben´ıtez, E.A. Ostrovskaya, M. Gul´acs´ı
Phys. Rev. Lett. 79, 4950 (1997); S. Raghavan, A. and Yu.S. Kivshar, J. Phys. B: At. Mol. Opt. Phys. 42,
Smerzi, S.Fantoni, and S. R. Shenoy, Phys. Rev. A 59, 215308 (2009).
620 (1999). [28] T. Karpiuk, M. Brewczyk, S. Ospelkaus-Schwarzer, K.
[13] M. Albiez, R. Gati, J. F¨olling, S. Hunsmann, M. Cris- Bongs, M. Gajda, and K. Rza¸´zewski, Phys. Rev. Lett.
tiani, M. K. Oberthaler, Phys. Rev. Lett. 95, 010402 93, 100401 (2004); T. Karpiuk, M. Brewczyk, and K.
(2005); R. Gati and M. K. Oberthaler, J. Phys. B: At. Rza¸´zewski, Phys. Rev.A 69, 043603 (2004).
Mol. Opt.Phys.40, 10, R61-R89(2007). [29] M. Salerno, Phys. Rev.A 72, 063602 (2005).
[14] G.Mazzarella,M.Moratti,L.Salasnich,M.Salerno,and