Localized and extended states in a disordered trap

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by  Luca Pezzé

Localized and Extended States in a Disordered Trap
Luca Pezz´e and Laurent Sanchez-Palencia
Laboratoire Charles Fabry de l’Institut d’Optique, CNRS and Univ. Paris-Sud,
Campus Polytechnique, RD 128, F-91127 Palaiseau cedex, France
(Dated: January 27, 2011)
WestudyAndersonlocalization inadisorderedpotentialcombinedwithaninhomogeneoustrap.
Weshowthatthespectrumdisplaysbothlocalizedandextendedstates,whichcoexistatintermedi-
ateenergies. Intheregion of coexistence, we findthat theextendedstates result from confinement
1
by the trap and are weakly affected by the disorder. Conversely, thelocalized states correspond to
1
eigenstates of the disordered potential, which are only affected by the trap via an inhomogeneous
0
energy shift. These results are relevant to disordered quantum gases and we propose a realistic
2
schemeto observe thecoexistence of localized and extended states in these systems.
n
a PACSnumbers: 03.75.-b,03.75.Ss,72.15.Rn
J

6
2 Disorderunderliesmanyfields inphysics,suchaselec- correspondsto states which spreadoverthe full (energy-

tronics, superfluid helium and optics [1–3]. It poses dependent) classically allowed region of the bare trap,
]
r challenging questions, regarding quantum transport [4] andwhichwethuscall“extended”. Thesecondclasscor-
e
and the interplay of disorder and interactions [5]. In responds to states of width much smaller than the trap
h
this respect, ultracold gases offer exceptionally well con- size, which are localized by the disorder, and which we
t
o
trolled simulators for condensed-matter physics [6] and thus call “localized”. We give numerical evidence of the
.
t are particularly promising for disordered systems [7]. coexistence of extended and localized states for different
a
m They recently allowed for the direct observation of kinds of traps. We show that while the extended states
one-dimensional (1D) Anderson localization of matter areconfinedbythetrapandweaklyaffectedbythedisor-
-
d waves[8–11]. Itshouldbenoticedhoweverthatultracold der, the localized states correspond to eigenstates of the
n gases do not only mimic standard models of condensed- disorderedpotential, which are only affected by the trap
o
matter physics, but also raise new issues which require via an inhomogeneous energy shift. Finally, we propose
c
special analysis in its own right. For instance, they are an experimentally-feasible scheme using energy-selective
[
mostoftenconfinedinspatialtraps,whichhassignificant time-of-flight (TOF) techniques to observe this coexis-

2
consequences. On the one hand, retrieving information tence with ultracold Fermi gases.
v
about bulk properties requires specific algorithms [12]. Let us consider a d-dimensional gas of noninteracting
9
4 On the other hand, trapping induces novel effects, such particles of mass m, confined into a spatial trap VT(r)
0 as the existence of Bose-Einstein condensates in low di- and subjected to a homogeneous disordered potential
4 mensions [13], and suppression of quantum tunneling in V(r)ofzeroaverage,amplitudeV andcorrelationlength
R
.
6 periodic lattices [14]. σR. Hereafter, we use “red-detuned” speckle potentials
0 (V < 0), which are relevant to quantum gases [7, 17].
Consider Anderson localization [15]. In homogeneous R
0 For the trap, we take V (r)=(~2/2ma2)r/aα, being a
disorder,linearwavescanlocalizeowingtocoherentmul- T
1 | |
the trap length scale. For instance, α= and a=L/2
: tiplescattering,withpropertiesdependingonthesystem ∞
v for a homogeneous box of length L, while α = 2 and
dimension and the disorder strength [1]. A paradigm
Xi of Anderson localization is that localized and extended a = p~/mω for a harmonic trap of angular frequency
ω. We numerically compute the eigenstates ψ and
r states generally do not coexist in energy. This relies on | ni
a eigenenergies E of the Hamiltonian
Mott’s reductio ad absurdum [1]: Shouldthereexistalo- n
calized state and an extended state with infinitely close Hˆ = ~2∇2/2m+V(r)+V (r). (1)
T
energiesforagivenconfigurationofdisorder,aninfinites- −
imal change of the configuration would hybridize them, Theeigenstatesarecharacterizedbytheircenterofmass,
forming two extended states. Hence, for a given energy, r ψ rˆ ψ , and spatial extension (rms size), ∆r
n n n n
almost all states should be either localized or extended. ψ≡hrˆ2 ψ| | ri2 1/2. The quantity ∆r quantifies loca≡l-
Exceptions only appear for peculiar models of disorder (cid:0)h n| | ni− n(cid:1) n
ization: the smaller, the more localized.
with strong local symmetries [16]. Then, a question
Numerical results for the 1D (d=1) case are reported
arises: Can inhomogeneous trapping modify this picture
in Fig. 1. In infinite, homogeneous disorder (α = ,
so that localized and extended states coexist in energy? ∞
L = ), all states ψ are localized, uniformly dis-
n
∞ | i
In this Letter, we study localization in a disordered tributed in space, and, for most models of disorder,
potential combined with an inhomogeneous trap. The theirextension∆z monotonicallyincreaseswiththeen-
n
centralresult of this work is the coexistence, at interme- ergy [18]. As Figs. 1(a),(f) show, using a finite flat box
diateenergies,oftwoclassesofeigenstates. Thefirstclass (L< ) only induces a trivialfinite-size effect: For low-
∞
2
Figure1: Extension(a)-(e),centerofmass(f)-(j)anddensityofstates[DOS](k)-(o)oftheeigenstatesversusenergyinvarious
kinds of 1D disordered traps. The plots result from accumulation of numerical data over 5000 realizations of a red-detuned
scporercekslpeopnodtsentotiaalnwinitfihnimteσsR2y|sVtRem|/~[228=]0a.n2d56t.heThhoerifizrosnttacollluinmenisr∆efezr0s=toL/a2√fla3t.bTohxeoofthleenrgctohluLm=ns50r0efσeRr.toTihnehocmurovgeednelionuesitnra(pas)
witha=12.5σR andvarioustrappowersα. Thesolidlinescorrespondtothenondisorderedcase,i.e.∆z0 inpanels(b)-(e)and
zcl(E) in panels (g)-(j). The last row shows the full DOS ρ(E) (solid black line), as well as the DOS restricted to localized
±
(ρ<,solid red line) and extended(ρ>, dashed blue line) states [19]. The dot-dashed green lines are the nondisordered limits.
enough energy E, we find ∆z L and the states are nature of the localized and extended states is even more
n
≪
notsignificantlyaffectedbythefinitesizeofthebox. For striking when one studies the wavefunctions. Let us fo-
larger energies however, boundary effects come into the cus for instance on the harmonic trap (α = 2) and on a
picture. The states are centered close to the box center narrow slice of the spectrum around E 4V , where
R
∼ | |
and their extension saturates to the value obtained for a ρ /ρ 14% of the states are localized [20]. Figure 2(a)
<
planewave,i.e.∆z0=L/2√3. Acentraloutcomeofthese shows≃the spatial density ψ (z)2 of all states found for
n
| |
results is that the curve giving ∆z versus E displays a asinglerealization ofthedisorder. Wecanclearlydistin-
n
single branch. In particular, there is no energy window guish localized (thick red lines) and extended (thin blue
where localized and extended states coexist. This find- lines) states, which shows that they coexist in the same
ing holds independently of the finite box size and is in energy window for each realization of the disorder. The
agreement with Mott’s argument [1]. localizedstatesareverynarrowandpresentnonode(e.g.
For inhomogeneous traps (α < ), we find a com- statesAandE)orafewnodes(e.g.statesCandH).They
pletely different behavior. The cur∞ves giving ∆z and may be identified as bound states of the local deep wells
n
z versus E now display two clearly separated branches ofthedisorderedpotential,similarlyasthelowest-energy
n
[seeFigs.1(b)-(e)and(g)-(j)]. Forlowenergy,the states states creating the Lifshits tail in bare disorder [18]. To
are strongly localized and, for E > 0, they are roughly confirmthis,letusdecomposetheeigenstates ψn ofthe
| i
uniformlydistributedinaregionboundedbythe(energy- disordered trap onto the basis of the eigenstates χp of
| i
dependent)classicalturningpoints,zcl(E),definedasthe the bare disordered potential [i.e. Hamiltonian (1) with
ssioolnutoiofntshoefsVtaTt(ezsclc)o=rreEsp.onFdorinhgigtohetrheenueprgpye,rtbhreanexchtenin- VizTed≡st0a],teasψsonci,atweedfitnodtheχepigψennen2ergi1esfoǫrp.aFsionrgalelsotcaatle-
| i |h | i| ∼
Figs. 1(b)-(e) grows and eventually saturates to that of |χpi such that ǫ′p ≃En, where ǫ′p =ǫp+hχp|VT(z)|χpi is
the eigenstates of the nondisordered trap, ∆z0(E). The the eigenenergy of χp shifted by the trapping potential
| i
centers of mass of these states approach the trap center [see Fig. 2(b)]. Conversely, the same decomposition for
and form the horizontal branch in Figs. 1(g)-(j). This an extended state shows a broad distribution of ampli-
branch corresponds to extended states. It is easily in- tude much smaller than unity. A localized state ψn of
| i
terpreted in terms of finite-size effects, similarly as for the disorderedtrap thus corresponds to a strongly local-
a finite flat box. The lower branch in Figs. 1(b)-(e) is ized state χp in the bare disorder, which is affected by
| i
more surprising. It identifies strongly localized states the trap by just the energy shift χp VT(z)χp . We gen-
h | | i
of relatively large energy. It has no equivalent in the erally find that ǫp χp VT(z)χp ψn VT(z)ψn ,
| | ≪ h | | i ≃ h | | i
flat box and cannot be interpreted as a finite-size effect. and, due to the reduced spatial extension of ψn , we
| i
The corresponding states are located close to the classi- get En VT(zn) [21]. This explains that the localized
≃
calturningpointszcl(E)andgeneratetheouterbranches statesarelocatedcloseto the classicalturning points,as
in Figs. 1(g)-(j). As Fig. 1 shows, this holds for all in- observed in Figs. 1 and 2(a).
homogeneous traps. When the trap power α increases, Let us now decompose the states ψ of the disor-
n
the branchof extended states gets denser at the expense dered trap onto the basis of the eigen|staites ψ0 of the
of that of localized states, and completely vanishes for bare trap [i.e. Hamiltonian (1) with V = 0],|aspsiociated
α= (homogeneous box). to the eigenenergy E0. For a localized state, the distri-
∞ p
Thecoexistenceoflocalizedandextendedstatesinthe bution is broad. Conversely, for an extended state, the
sameenergywindowfordisorderedtrapsisconfirmedon distribution is sharp and peaks at E0 E to a value
p ≃ n
more quantitative grounds in the last row of Fig. 1. It equal to a fraction of unity [see Fig. 2(c)]. An extended
shows the full density of states (solid black line), as well state may thus be seen as reminiscent of an eigenstate
as the density of localized (ρ , solid red line) and ex- of the bare trap, which is weakly affected by the disor-
<
tended (ρ , dashed blue line) states [19]. The different der. Still, the main peak in Fig. 2(c) is smaller than
>
3
coexist in the same energy window. The disordered po-
tential combined with a smooth trap permits localized
states to sit outside the classically allowed region occu-
pied by extended states [see Fig. 2(a)]. Then, the Mott
argument does not apply here because the spatial seg-
regation can be strong enough to suppress hybridization
for an infinitesimal change of the disorder configuration.
Let us now discuss a possible scheme to observe the
coexistence of localized and extended states in a disor-
dered trap. Consider a gas of noninteracting ultracold
fermions prepared in a given internal state, at temper-
ature T and chemical potential µ. A class of energies
E E . ∆ [see Fig. 3(a)] deep in the Fermi sea
n
| − |
(i.e. with µ E k T) can be selected by applying
B
− ≪
a spin-changing radio-frequency (rf) field of frequency
ν =E/h and duration τ h/∆ (with h the Planck con-
∼
stant) [14, 22, 23]. The rf field transfers the atoms of
correspondingenergies to an internalstate insensitive to
thedisorderedtrap. Thetransferedatomsexpandfreely,
Figure 2: Eigenstates for a single realization of a 1D disor- which provides their momentum distribution:
dered harmonic trap. (a) Non-normalized spatial densities,
r|ψedn(lzin)|e2s,cvoerrrteiscpaollnyddtisoplloaccaeldizteodtshtaetirese,igaenndentheirngyblEuen.linTehsictko DE,∆(k) ≃ X |ψˆn(k)|2, (2)
|En−E|.∆
extendedstates[19]. Notethatextendedandlocalized states
whereψˆ (k)istheFouriertransformofψ (z)[TOFtech-
may occupy almost-degenerate energy levels (e.g. H and I). n n
The states C and D are projected: (b) over the eigenstates nique]. In the coexistence region, E,∆(k) has two sig-
D
of the disordered potential, χp , and (c) over those of the nificantly different contributions: For localized states,
harmonic trap, ψ0 . The par|amieters are as in Fig. 1. ψˆ (k)2 is centered around k 0 with tails of width
| pi ∆| kn | ∆z−1. Conversely, for e≃xtended states, ψˆ (k)2
is pnea∼ked ant k √2mE/~ with long tails toward|snsma|ll
≃
unity. Onlyforsignificantlyhigherenergy,thestate ψ momenta. We however found that averaging over real-
n
results from weak perturbation of ψ0 , and ψ0 ψ| 2i izations of the disorder blurs the central peak associated
| ni |h p| ni|
displays a main peak of the order of unity as predicted to the localized states in E,∆(k). In turn, the quan-
D
by standard perturbation theory. tity E,∆(k) E,∆(k) E,∆(0)/ E,∆(0)2 displays
C ≡ D ×D D
Our resultscannow be easilyinterpreted. Inbare dis- two distinct peaks for a rf pulse of realistic durations
order,thetypicalsize∆z ofthelocalizedstatesincreases [see Fig. 3(b)]. The central one is more pronounced for
faster than the classicallyallowedregionzcl E1/α pro- narrower pulses. Selecting either the localized states or
∝
vided by the trap. For low energy, ∆z zcl so that the the extended states [19] confirms that the central peak
≪
states are strongly localized by the disorder and weakly corresponds to the localized states and the side peak to
affected by the trap. For higher energyhowever,the dis- the extended states [see Inset of Fig. 3(b)].
order would localize the states on a scale exceeding zcl. Finally, we have performed similar calculations as
The states are then bounded by the trap and the effect above in a 2D harmonic trap. Figure 4(a) shows the
of disorder becomes small. This forms the branch of ex- centers of mass r of the eigenstates with E 4V ,
n n R
≃ | |
tended states in both the disordered box and traps. In the color scale giving ∆r . Figure 4(b) shows a density
n
addition,somestronglylocalizedstateswithverylowen- plot of ∆r versus r for the same data. Again, the
n n
| |
ergyinthebaredisorderandlocatedaroundpointz are eigenstatesclearlyseparateintotwoclasses: Somestates
n
shifted by the trap to approximately the energy V (z ). are extended (large ∆r ) and centered nearby the trap
T n n
This forms the branch of localized states only in disor- center (small r ). The other states are strongly local-
n
| |
dered traps (α < ) since, in the box, a state cannot ized (small ∆r ) and located nearby the line of classical
n
be placed at inter∞mediate energy due to the infinitely turning points (rn rcl(E)=p2E/mω2). Hence, the
| |≃
sharp edges. Quantitatively, since the localized states in two classes of states can coexist at intermediate energies
the bare disorder are uniformly distributed in space, the also in 2D disordered traps.
density of localized states can be estimated to roughly In conclusion, we have shown that, in a disordered in-
scale as ρ (1/α) E1/α−1, which is consistent with homogeneoustrap,localizedandextendedstates canco-
<
∝ ×
the disappearance of the branchof localized states when existina givenenergywindow. The localizedstates cor-
α grows and with its vanishing for α = (see Fig. 1). respond to eigenstates of the disordered potential which
∞
Still, itis strikingthat localizedandextended states can are only affected by the trap via an inhomogeneous en-
4
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munoimtse),ntfourm∆di=str2ib~uωt.ioInnsDetE:,∆C(Ek,∆)((dka)sohfedalglrseteantelsin(es,oalirdbibtrlaacryk [6] MI..BlLocehweentsatel.i,nReevt.Mal.o,d.APdhvy.s.P8h0y,s8.8556(,200284)3. (2007);
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| | 87(2010); L.Fallani, C.Fort,andM.Inguscio,Adv.At.
and the otherparameters are as in Fig. 1.
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| |
[19] We use the condition ∆z < 10σR to identify localized
states, and ∆z>(3/4)∆z0(E) for extended states.
ergy shift. Conversely, the extended states spread over [20] Weobtainedqualitatively-similarresultsfordifferenten-
ergies and different traps.
the classically allowed region of the trap and are weakly
affectedbythedisorder. Thiseffectisdirectlyrelevantto [21] For instance, we find En ≃ 3.97|VR|, ǫp ≃ 0.24|VR| and
χp VT χp 3.71VR for state C in Fig. 2.
presentday experiments with disordered quantum gases, [22] Mh .|Gre|ineir≃et al.,|Nat|ure (London) 426, 537 (2003).
which are most often created in harmonic traps [11, 24– [23] W. Guerin et al.,Phys. Rev.Lett. 97, 200402 (2006).
27]. We have proposed a realistic scheme to observe it [24] D. Cl´ement et al., Phys.Rev.A 77, 033631 (2008).
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kinds of inhomogeneous disordered systems. [27] B. Deissler et al.,NaturePhys. 6, 354 (2010).
We thank T. Giamarchi and B. van Tiggelen for dis- [28] Here, we use a box of length 3000σR much larger than
∆z for all states considered.
cussionsandANR(ContractNo.ANR-08-blan-0016-01),