Manifold approach for a many-body Wannier-Stark system: localization and chaos in energy space

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by  C. A. Parra-Murillo

Manifold approach for a many-body Wannier-Stark system: localization and chaos in
energy space
C. A. Parra-Murillo and S. Wimberger
Institut fu¨r Theoretische Physik and Center for Quantum Dynamics,
Universita¨t Heidelberg, 69120 Heidelberg, Germany
We study the resonant tunneling effect in a many-body Wannier-Stark system, realized by ul-
tracold bosonic atoms in an optical lattice subjected to an external Stark force. The properties of
themany-bodysystemareeffectivelydescribedintermsofupper-bandexcitationmanifolds,which
allow for the study of the transition between regular and quantum chaotic spectral statistics. We
show that our system makes it possible to control the spectral statistics locally in energy space by
4
the competition of the force and the interparticle interaction. By a time-dependent sweep of the
1
Stark force the dynamics is reduced to a Landau-Zener problem in the single-particle setting.
0
2
PACSnumbers: 03.65.Xp,05.45.Mt

n
a
J I. INTRODUCTION ing regime [19, 24]. We show that the main spectral

7 properties can be captured in an effective theory based
on upper-band manifold excitations, that can, in prin-
] Oneofthemostremarkablefeaturesofaquantumsys-
s ciple, be measured in experimental realizations such as
tem is the ability of transporting electrons and atoms
a reported in [25]. Our approach allows us to characterize:
g across classically forbidden regions. This process is al-
the onset of quantum chaos, to distiguish the condition
- lowedbymeansofawellknowneffectreferredtoasquan-
t fortheemergenceoflocalizationinenergyspace[26,27],
n tum tunneling. In semiconductor physics [1], the use of
and to design the type of driving dynamics that can be
a a superlattice permits the enhancement of the transport
implemented in analogy with the well-known Landau-
u
along a specific spatial direction, supported, in a first
q Zener process [8, 9, 12–14, 28–30].
approximation, by the resonant coupling between elec-
.
t troniclevels. Nowadaysmanyofthesesolidstatephysics
a
m paradigmsareamplyinvestigatedinacleanermannerus- II. THE MANY-BODY WANNIER-STARK
ingultracoldatomsandopticalpotentials. Therapidad-
- SYSTEM
d vancesintheexperimentaltechniqueshaveopenedahuge
n field of research, in which the most common approaches
A. The System and the Two-band Model
o to the description of many-particle physics are based on
c
Bose-Hubbard-type Hamiltonians [2–4]. A basic feature
[ Our system consists of ultracold bosonic atoms in a
of these Hamiltonian models is that the kinetic energy
optical lattice [3, 4] subjected to an external Stark force.
1 isdescribedbymeansofdynamicalhoppings(tunneling)
v betweendifferentpotentialwells. Here,weareextending The force stimulates (a) the atomic transport along the
6 lattice, for instance, atomic Bloch oscillations [15–17],
such a model to higher energy bands coupled by means
4 and (b) between the Bloch bands, e.g. Landau-Zener
of an external Stark force.
4 transitions [10–14]. This latter process is characterized
1 The study of many-body effects at resonant tunnel-
by the exchange of particles between the bands and
. ing conditions is not straightforward since, in the pres-
1 is enhanced at specific values of force, F ≈ ∆ /2πr,
r g
0 ence of strong correlations, the complexity is overwhem- where ∆ is the energetic gap between the two bands.
g
4 ingly increased [5–7]. For the single-particle and mean At those values, resonantly enhanced tunneling takes
1 field limits, there are up to date experimental realiza-
place between Wannier-Stark levels distancing r wells
: tions of the Wannier-Stark system (see for instance [8–
v [9, 19, 23, 24]. The integer r is from now on called the
i 18]). Additionally, by investigating certain parameter order of the resonance. In the following, we restrict to
X
regimes, for example, in the case of strongly interact-
two coupled energy bands. Such a situation is realized,
r ing atoms (hard core bosons), an effective description is
a e.g., in a double-periodic lattice, see ref. [22].
possiblebymappingthemany-bodylatticesystemtoan-
The many-body physics can then be described by the
alytically solvable effective Hamiltonians [19–21]. In this
celebrated Bose-Hubbard model extended to a two-band
paper, we investigate the resonant tunneling effect in a
scenario [22, 23]. The corresponding Hamiltonian reads
Bose-HubbardmodelextendedtoasecondexcitedBloch
bexapnedr.imTehnitsss(ysseteemre,f.th[2a2t,c2a3n] fboeridmetmaields)ia,thealys vreearyliziendteirn- Hˆ = (cid:88)L (cid:88)(cid:20)−J2β (cid:16)βˆl†+1βˆl+h.c.(cid:17)+ W2ββˆl†2βˆl2(cid:21)
esting spectral properties. These allow for the study of l=1 β
many-body effects as, for example, interaction-induced L
quantum chaos [5, 7], diffusion and relaxation in Hilbert + (cid:88)(cid:88)ω C (aˆ† ˆb +h.c.)+(cid:88)εβnˆβ
B µ l+µ l l l
space [22] and coherent dynamics in the weak interact-
l=1 µ β
2
+ (cid:88)L 2W nˆanˆb+ Wx (cid:16)ˆb†2aˆ2+h.c.(cid:17) . (1) left). We now rescale the Hamiltonian as Hˆ → Hˆ/∆g,
x l l 2 l l where∆g istypicallythelargestparameter;thuswehave
l=1 |C |/∆ (cid:28) 1, which means that only working with
|µ|>0 g
the largest dipole strength, C , is enough to capture the
The annihilation (creation) operators are defined by 0
essential features of the system. The effective Hamilto-
β (β†), and the number operators are nβ = β†β , with
l l l l l nian around the resonance of order r reads
band index defined as β = {a,b}. The hopping ampli-
tudesareJβ=a,b,andtheon-siteinterparticleinteraction Hˆ = (cid:88)L (cid:88)(cid:20)−Jβ (cid:16)βˆ† βˆ +h.c.(cid:17)+(∆ −ω r)nˆβ(cid:21)
per band has a strength Wβ. The interband coupling r 2 l+1 l β B l
is generated by dipole-like couplings (C ) and by the re- l=1 β
µ
ppaurlsaimveetinertesrpaacctieomna(Wkexs)i.tTimhepolasrsgibeleditmoeannsaiolintiaclaitlylyosfotlhvee + (cid:88)L ωBC0(cid:16)aˆ†lˆbl+h.c.(cid:17), (3)
ourproblem. Thereforeanumericalproceduretofindthe l=1
eigensystem is required. We quickly summarize the pro- where the energy separation between the Bloch bands is
cedure in the following, and refer the reader to ref. [23] ∆ = {0,∆ }. We then have a Hamiltonian consist-
a,b g
for details. ing of two tilted lattices. The eigenenergies of the inde-
pendent lattices are given by the Wannier-Stark ladder
formula [28]:
B. Numerical Treatment
εβ =ω l +∆ , with l ∈Z. (4)
l B β β β
In order to study the eigenenergy spectrum of the
InthecaseF =0,thatis,withoutinter-bandcoupling
Hamiltonian (1), we first implement a transformation
and no tilt, the eigenstates of Eq. (4) can be classified
into the interaction picture with respect to the exter-
accordingtotheirnumberofparticlesintheupperBloch
nal force. In this procedure the term (cid:80)l,βωBlnˆβl is band, defined as:
removed, and the hopping and dipole-like terms trans-
form as: βˆl†+1βˆl → βˆl†+1βˆlexp(−iωBt) and aˆ†l+µˆbl → Mi =(cid:104)εi|Mˆ|εi(cid:105), with Mˆ =(cid:88)lnˆbl. (5)
aˆ† ˆb exp(−iω µFt). The big advantage of the trans-
l+µ l B BywrittingtheHamiltonianmatrixrepresentation(4)in
formationintotheinteractionpictureisthatnowthenew
the basis |γ (cid:105), ordered by increasing upper-band occupa-
i
Hamiltonian is invariant under translation and periodic
tion number M, the Hamiltonian is reduced to the block
in time. Its period is given by T = 2π/ω , which is
B B matrix
known as the Bloch period. We now set periodic bound-
ary conditions in space, i.e. βˆ† = βˆ†. In order to di- H0,0 H†0,1 
aRthgeeofsnt.ara[l5inz,se7lat],thiewonittiahmllyed-iidmnevepanersniiadonnetngtFiovHLce+aknm1sbtiylattoens1ia{n|γ,iH(cid:105)ˆ}(dt)e,fiwneedusine Hˆr =H0,1 H..1.,1 ...... H†0,M−1 .
H H
0,M−1 M,M
(N +2L−1)!
Ns = LN!(2L−1)!, (2) The diag(Hˆr)=⊕NM=0(Hˆr)M is a diagonal block matrix
constructed through the Hamiltonian terms preserving
whereListhenumberoflatticesitesandN thetotalpar- numberM,i.e.,thehoppingandenergytermsinEq.(1).
ticle number. We study the eigensystem of the Floquet The blocks on the diagonal are matrices with dimension
Hamiltonian [31] Hˆf = Hˆ(t)−i∂t, whose eigenenergies dM ×dM (see Fig. 1(a)), where dM is given by
εi aredefinedasthesetofeigenvaluesofHˆf lyingwithin 1(cid:18)M +L−1(cid:19)(cid:18)N −M +L−1(cid:19)
d = . (6)
the Floquet zone εi ∈ [−ωB/2,ωB/2] (see details of the M L L−1 L−1
diagonalization procedure in [23]). In the following we
introduce an effective, analytical description of the spec- The Hamiltonian only contains non-zero coupling be-
tral properties based on the upper-band excitation and tween those inter-manifold states with excess of parti-
its respective comparison with the exact numerical re- cles ∆M = ±1, with ”hopping” strength ωBC0 (see
sults. Fig. 1(b)). In addition, the Hamiltonian can be reduced
toatight-binding-typeonefortheupper-bandexcitation
manifolds (from now on labeled by M) when describing
C. Manifold Approach the averaged one-particle exchange processes in the res-
onant system. To do this, we use the closure relation
(cid:80)
|γ (cid:105)(cid:104)γ | = 1, which can be rewritten in terms of the
The non-interacting limit is described by the Hamil- i i i
tonian Eq. (1) with Wa,b,x = 0. Here it is possible to manifold projectors PˆM as follows
construct a local Hamiltonian around a single resonance
r. This implies that the site la may be connected to (cid:88)N PˆM =1ˆ, with PˆM =(cid:88)dM |γi;M(cid:105)(cid:104)γi;M|. (7)
the upper-band lattice site l = l − r (r sites to the
b a M=0 i=1
3
Interestingly, from the new Hamiltonian we can easily
recognizeanemerginglocalizationofitsrespectiveeigen-
(cid:80)
functions |φ(cid:105). To see this, we use |φ(cid:105) = Q |ψ (cid:105),
M M M
which together with the Schr¨odinger equation, Hˆ(cid:48)|φ(cid:105) =
r
E|φ(cid:105), yields the coefficient equation:
(E−(cid:15)M)Q =ω˜ (Q +Q ), (11)
M B M+1 M−1
with (cid:15) = ∆ − ω r + J − J . This equation can
g B a b
be solved using the ansatz Q = AJ (x ), where
M M(cid:48)−M B
x ≡2ω C /(cid:15) and J (x ) is the Bessel function of
FIG. 1: (Color online): Hamiltonian matrix written in the B B 0 M(cid:48)−M B
thefirstkind. Therefore,byusingtheidentity2kJ (x)=
|γ (cid:105)basis. (a)BlockstructureoftheHamiltonianinthenon- k
i
x(J (x)+J (x))wefindthatthesolutionofEq.(11)
interacting case, for the system N/L = 7/4, with parameter k+1 k−1
(cid:80)
Fr=1 =0.25. SinceN =7thematrixcontains8blockscorre- is: EM = (cid:15)M and |φM(cid:105) = A M(cid:48)JM(cid:48)−M(xB)|ψM(cid:48)(cid:105),
spondingtotheN+1manifolds. Theintra-manifoldoffdiag- with A being a normalization constant. In energy space
onalcouplingsaresetbythehoppingtermsthatdonotcouple the eigenfunction can be written as
γ states with different manifold number. (b) Tight-binding-
i (cid:88)
type many-body Hamiltonian matrix for the same parame- φ (ε)≡(cid:104)ε|φ (cid:105)=A J (x )ψ (ε) (12)
M M M(cid:48)−M B M(cid:48)
ter from panel (a). The other parameters are: ∆ = 1.61, M
g
J = 0.082, J = −0.13, C = −0.094, C = 0.037, and
a b 0 ±1 where ψ (ε), the eigenfunctions of the Hamiltonian
M(cid:48)
C =−0.0022.
±2 Hˆ(cid:48)(ω˜ = 0), is a well-localized function around the en-
r B
ergy εr . These functions clearly satisfy the relation
M
ψ (ε − εr ) = ψ (ε). These are Wannier-like func-
Here |γ ;M(cid:105) ≡ |na,na,...(cid:105)⊗|nb,nb,...(cid:105) is a Fock state M M(cid:48) M(cid:48)
i 1 2 1 2 tions in energy space. The probability density is then
with M particles in the upper band. The projectors
Pˆ allow one to rewrite the Hamiltonian as 1ˆHˆ 1ˆ = now given by
M r
(cid:80)paMnd,Med(cid:48)PaˆsMHˆrPˆM(cid:48) with the eigenstate |ψ(cid:105) of Hˆr ex- |φM|A(|ε2)|2 =|J0(xB)ψM(ε)+ (cid:88) JM(cid:48)−M(xB)ψM(ε)|2,
M(cid:48)(cid:54)=M
|ψ (cid:105)=Pˆ |ψ(cid:105)=(cid:88)|γ ;M(cid:105)(cid:104)γ ;M|ψ(cid:105), (8) (13)
M M i i
whichmeansthat,forx (cid:28)1,theprobabilitymaximizes
i B
aroundthemanifoldenergyεr . Theconditionforthisto
M
and (cid:104)ψ |ψ (cid:105)=δ . The off-diagonal blocks are not happen is to be far from the resonance where for typical
M(cid:48) M M,M(cid:48)
squarematricesbuttheirdimensionisd ×d . They systemparameters: 2ω C (cid:28)∆ −ω r+J −J . Atthe
M M+1 B 0 g B a b
are computed from the single-particle exchange term resonance we have ∆ ≈ω r and x ≈1, which implies
g B B
anoverlappingofneighbormanifolds,sincetheexpansion
HˆM,M(cid:48) ≡(cid:104)ψM(cid:48)|(cid:88)L ωBC0(cid:16)aˆ†lˆbl+h.c.(cid:17)|ψM(cid:105). (9) ncoeegffiligciibelnet.sTinhiEsqin.t(r1o3d)uwceistha|kMin(cid:48)d−oMf h|y>bri0dibzeactoiomneenffoenct-
l=1 between the manifolds responsible for the destruction of
√ the strong localization of the eigenfunctions φM(ε) (see
Bychoosing(cid:104)γi;M|ψ(cid:105)=1/ dM weobtainasimplified refs. [26, 27] for other contexts of localization in energy
tight-binding-type Hamiltonian for the manifolds space).
The energy gap between two neighboring manifolds
N
Hˆ(cid:48) (cid:39) (cid:88) εr |ψ (cid:105)(cid:104)ψ |+ω C (|ψ (cid:105)(cid:104)ψ |+h.c.),(10) characterizes the one-particle exchange process (see
r M M M B 0 M M+1 Fig. 3(b)) and can be estimated by straightforward di-
M=0 agonalization of the two-level Hamiltonian matrix
ωwBheCr0e√εMrM +=1.(∆Hger−e wωBeru)sMed+th(eJrael−atiJobn)MN,=anNdaω˜+BN≡b, H2r×2 =(cid:18)ωεrMC+1 ωεBrC0 (cid:19) , (14)
B 0 M
with M ≡N , and the order of the resonance is approx-
b
imately given by r ≈ ∆ /ω . To obtain the Hamilto- from which we obtain
g B
nian (10) we have assumed that there is no additional
(cid:115)
relevant subclass of Fock states in any M-subspace, and (cid:18) ω r J −J (cid:19)2 (cid:18)ω C (cid:19)2
∆ =∆ 1− B + a b +4 B 0 . (15)
all possible single-particle processes are equally proba- r g ∆ ∆ ∆
g g g
ble. Underthiscondition,theHamiltonian(10)averages
overhoppinganddipole-liketransitionprocesses,andits We notice that the minimal energy range of the many-
final dimension is just N +1. A different choice of the level spectrum is thus given by ∆E = N∆min, with
r
distribution of the coefficients (cid:104)γ ;M|ψ(cid:105) would lead to ∆min ≈ 2ωr|C | and ωr ≡ 2πF (for typical parame-
i r B 0 B r
a similar effective Hamiltonian, but restricted to fewer ters, ∆ ,ω (cid:29)|J −J |). This energy scale is shown in
g B b a
participating states. Fig. 2(a) for the single-particle case (N/L=1/10).
4
D. Effects of the Interparticle Interaction
(a)
Amoreprecisedescriptionofthemany-bodyspectrum
is obtained when considering the effects of the interac-
tions, i.e., for W (cid:54)= 0. This induces a splitting of
a,b,x
the internal manifold levels and couplings between the
|γ (cid:105) states. The coupling is strong especially at reso-
i
nant tunneling condition (e.g. F ≈ F ). In this region,
r
the manifold levels come closest (see Fig. 2(b)) and a
natural mixing of the manifold states occurs, that is,
the eigenstates of (1) become hybridized as explained in
(b)
Sec. IIC. This latter effect is associated with the occur-
rence of avoided crossings (ACs) around F , i.e., with
r
the lack of symmetries in the system [23]. We can now
easily estimate the largest manifold splittings generated
by interparticle interaction. This is done by consider-
ing the basis states {|γ (cid:105)} with M particles sitting in
i
a single-particle level, in one lattice site, for example
|γ (cid:105) ∼ |N −M,0,..(cid:105) ⊗|M,0,...(cid:105) . The energy cost due
i a b
to the interaction strengths W is thus given by
a,b,x
W
(UM) = a(N −M)(N −M −1),
a max 2
W FIG. 2: (Color online): (a) Numerically computed energy
(UM) = bM(M −1),
b max 2 spectrum versus the Stark force in the single-particle case
around r = 1, with N/L = 1/10. The inter-band cou-
(UM) = 2W (N −M)M, (16)
ab max x plingmanifestsitselfaroundtheresonancepositionF byan
r
avoided crossing. (b) Many-body spectrum for N/L = 4/5,
which allow one to compute the maximal intra-manifold around r = 1. The different color lines correspond to eigen-
splitting as U(M) ≡ max{(UM ) ,(UM) }. We states of the type: |ψ (cid:105) (black), |ψ (cid:105) (red), and the
β=a,b max ab max M=0 M=1
can also rewrite the width ∆E as follows mixed-likestates,i.e.,stateswith0<M <N arerepresented
bythegreenlines. Inpanel(a)|φ (cid:105)representthehybridized
±
∆E =N∆min+U(N). (17) states at F = Fr. The parameters are those of Fig. 1 with
r additional interaction strengths: W = 0.021, W = 0.026
a b
and W =0.023.
This expression is obtained by considering the maximal a
splitting of the highest manifold M = N, which occurs
for those eigenstates, whose maximal projection in the
Computing the following commutator
|γ (cid:105) space is given by the state with N particles in the
i
s|Nam00e,u..p.(cid:105)pe.r-band level, say, the state |γi(cid:105) ∼ |000,...(cid:105)a⊗ (cid:34) Hˆ ,Mˆ(cid:35)=(cid:88) ωBCµaˆ† ˆb + Wx aˆ†2ˆb2−h.c. (19)
In genebral, far from the resonance, any eigenstate of ∆g l,µ ∆g l+µ l 2∆g l l
Eq. (1) can be rewritten in the dressed-like state ba-
shows that the transition from weak to strong mixing is
sis {|M,θ ,θ ,θ (cid:105)} characterized by the integers θ =
a b x β determined by the competition between one- and two-
(cid:104)ε |(cid:80) nˆβ(nˆβ−1)/2|ε (cid:105) and θ =2(cid:104)ε |(cid:80) nˆanˆb|ε (cid:105). The
i l l l i x i l l l i particle exchange between the bands. This competition
eigenenergies can then be approximated by
is the stronger the closer we are in resonance, at which
we have that ω /∆ ≈1/r. Furthermore, it is expected
εi(M,θ(cid:126))≈Mi∆r+Waθa,i+Wbθb,i+Wxθx,i. (18) thatforafillingBfactgorN/L∼1, single-andtwo-particle
transitions have the same occurrence probability, which
The mixing between the different M-manifolds in the makes the system strongly mixed. In the case N/L(cid:28)1,
presenceofinter-particleinteractionisnowalsotriggered the dominant effect is a one-particle exchange character-
by the coupling between inter-manifold states with ex- ized by the energy scale ∆ , and similarly for N/L(cid:29)1,
r
cess ∆M = ±2. This fact implies that the Hamiltonian thetwo-particleexchangeprocessdominateswithenergy
Eq. (10) contains a new term, which is just a second scale proportional to W . These two latter cases favor
x
neighbor transition from the manifold M to the mani- the weak manifold mixing. Therefore, the eigenstates of
foldM±2. Thisisnothingbutanextendedtight-binding (1)areexpectedtobelocalizedinenergyspaceaccording
model, with increasing on-site energies ε (M,θ(cid:126)). There- to the effective Hamiltonian (10).
i
fore,eveninthepresenceofweakinteractions,theeigen- So far, we have studied the properties of the Hamil-
states (cid:104)ε|φ (cid:105) preserves localization features. tonian (1) by using rather simple approximations based
M
5
on the concept of manifold excitation. The results have time
showntobeinagoodagreementwiththenumericalones.
We now show two direct consequences of the spectral
properties;firstwebrieflydescribetheimplementationof
Landau-Zener-type dynamics by making the Stark force
time-dependent. In this way, driving individual eigen-
states |ε (cid:105) across an avoided crossing may be straightfor-
i
wardly implemented. Secondly, we study the connection
between strong manifold mixing and quantum chaos in
the resonant tunneling regime.
FIG. 3: (Color online) Driving over the single particle spec-
trum in Fig. 2(a): Parametric-time evolution (from left to
III. NUMERICS: LANDAU-ZENER DYNAMICS right) of a state initially prepared in the manifold M = 0
(bottom left of the panels) across the resonance r = 1, with
Around a local resonance, our system provides a nat- thelinearpulseF(t)=F0+αt. Dynamicalparameters: (left)
ural scenario for the study of Landau-Zener-type tran- adiabatic transit (λ = 0.1), (center) non-adiabatic transit
(λ = 1) and (right) diabatic transit (λ = 50). The color
sitions. This can be done by defining a pulse F(t) =
scale is from black (small) to white (large), and δ = 0.05 in
F +αt, with α=∆F/∆T. Here ∆F is the effective re-
0 Eq. (16). The parameters are the same as in the previous
gion of resonant tunneling and ∆T is the sweeping time.
figure.
We focus on the dynamical driving of a state from the
lowest manifold |ψ (cid:105) for the single-particle case. We
M=0
havethentwomanifoldsM ={0,1}andthespectrumof
this system is shown in Fig. 1(a). We define the sweep-
ingrateαusingtheHeisenbergrelation∆Td≈1, where
d=∆E/N =∆ is mean level spacing at F .
s r r Fig. 2(a)) while being transformed into the excited state
TheHamiltonianisnowtime-dependent,H(F(t)),and
withM =1. Thisresultisexpectedaccordingtotheex-
the temporal evolution is computed by using a fourth-
changeofcharactertypicalofanavoidedcrossing. Thisis
order Runge-Kutta method. In analogy with the LZ
a dynamical effect following from the adiabatic theorem,
problem, the evolution across an avoided crossing can
which was already experimentally probed in Arimondo’s
be diabatic, non-adiabatic and adiabatic. In the cur-
group at Pisa University [8, 13, 14, 30]. On the other
rentcasethedynamicalregimesaredeterminatedbythe
hand, the central panel shows the non-adiabatic transit
parameter λ ≡ α/d∆F. Thus for λ (cid:28) 1 we have slow
for which the probability is split in both energy path.
drivingthroughresonances,i.e.,anadiabaticpassage;for
This effect is similar to the action of a beam splitter on
λ(cid:29)1 a diabatic one, also referred to as sudden quench,
an incident light beam. Finally the right panel depicts
and for λ∼1 we have a non-adiabatic evolution.
thediabaticpassageforwhichthestatedoespreserveits
To follow the evolution of a state across the RET
manifold number M but not its energy path. This lat-
regime, we compute the detection probabilities p =
(cid:104) i (cid:105) ter is characterized by the fidelity of the initial state, i.e.
|(cid:104)εi|Uˆ(t)|ψ(0)(cid:105)|2, where Uˆ(t)=Tˆexp −i(cid:82)0tHˆ(F(t))dt . |(cid:104)ψ(0)|Uˆ(t)|ψ(0)(cid:105)|2 ≈1.
The distribution probability of the evolved wavefunction
in the local energy space can be represented by means of
the local density of states [32, 33]:
(cid:88)
P (ε,t)= p δ(ε−ε ), (20)
ψ i i
i
(a)
with the delta function defined as (b)
g=0.1
1 δ
δ(ε−ε )= lim . (21) g=1.0
i δ→0π(ε−εi)2+δ2
Infigure3weshowtheevolutionoftheLDOSP (ε,t)
ψ
in the single particle case N = 1, i.e. the Hamiltonian
Hˆ(cid:48) in Eq. 10 is a 2×2 matrix. The initial condition is
r FIG. 4: (Color online) Interaction effects: (a) The panel
|ψ(0)(cid:105)∼|100,...(cid:105)a⊗|000,...(cid:105)b, M =0. |ψ(0)(cid:105) is evolved shows the level spacing distribution P(s) for N/L = 6/5
from F0 <Fr=1 to Ff >Fr=1 (left to right in Fig. 2(a)) (Ns = 1001) for the interparticle control parameter g = 0.1
for the two-state system N/L=1/10, and we can easily (red/greyhistogram,η=0.97)andg=1.0(blackhistogram,
appreciate the different types of dynamics, that is: the η=0.035), andWx =Wa =Wb =0.025. (b)Parameterη as
left panel shows the adiabatic regime, for which |ψ(0)(cid:105) afunctionofg. Theblacklinecorrespondstoanexponential
nearly follows the energy path (see low-energy state in fit. The reminding parameters are those of Fig. 2.
6
IV. NUMERICS: INTERACTION-INDUCED quantumnumbersM andθ(cid:126)(seeSec.IID),seeFig.2(b).
QUANTUM CHAOS We thus see that the mixing properties of our system
allow for a straightforward identification of the onset of
In this final section we study the static eigenspectrum quantum chaos, which is a local effect in the plane ε vs
i
at strong mixing conditions. Given the eigenenergies ε , F. Nevertheless, a globally chaotic spectrum can also be
i
thespectracanbeanalyzedbytheuseofrandommatrix reached by decreasing the energy bandgap ∆ such that
g
measures[34]. Arobusttestoftheuniversalpropertiesof gW /∆ ∼ 1. In this case, not only resonant tunneling
x g
the discrete eigenspectrum is the so-called level spacing between two neighboring wells due to the Stark force is
distribution P(s ), where the spacing are defined as s = possible,butalsoviainteraction[20,21]. Thesetwopro-
i i
(ε −ε )/(cid:104)s(cid:105). The spectrum must be unfolded in order cessesareequallyrelevantand,asimmediatesignatureof
i+1 i
to compare with the random matrix distributions, i.e., this, the local resonances are destroyed. There are then
such that (cid:104)s (cid:105)=1 (see ref. [23] for details). avoided crossings in the entire energy spectrum. There-
i
At the resonances F , the crossover between regu- fore, we can switch from global to local quantum chaos
r
lar (Poisson), P (s) = exp(−s), and quantum chaotic by varying the bandgap ∆ .
P g
(GOE)statistics, P (s)=πsexp(−πs2/4)/2, isreached V. CONCLUSIONS
W
by varying the inter-particle interaction. We define the
parameter g, that allows one to tune the interaction, We studied the spectral properties of a many-body
i.e., Wβ,x → gWβ,x. In the experiment this can be Wannier-Stark system, especially in the resonant tun-
done via Feschbach resonances [3]. As expected from neling regime. We showed that the main characteris-
the discussion in subsection IID, for an energy band tics of our model are well described in terms of upper-
gap ∆g (cid:46) 1 and g = 1, all systems with N/L ∼ 1 band excitation subspaces, i.e., M-manifolds. This ef-
exhibit chaotic features characterized by the GOE dis- fective description provides an intuitive understanding
tribution (see Fig. 4(a)-black). Nevertheless, this is not of interaction-induced quantum chaos. Furthermore, we
a general rule for every interaction strength, since for have shown that the knowledge of the spectral proper-
weaklyinteractingparticlesthemanifoldmixingbecomes tiesofthesystemallowsustocontrolthedynamicswhen
weaker. Hence quantum chaos can be tuned most easily drivingasimpletwo-statesystemacrossanavoidedcross-
by increasing the inter-particle interaction, for instance, ing,intheLandau-Zenerframework[29,30]. Oursystem
g = 0 → 1. To see this transition we compute the pa- canbeexperimentallyrealizedanditoffersaninteresting
rameter frameworkforfutureinvestigationsofmany-bodyphysics
(cid:82)s0(P(s)−P (s))ds [12, 19, 21, 22, 24].
η = 0 W , (22)
(cid:82)s0(P (s)−P (s))ds
0 P W
where s = 0.4729... is the intersection point between VI. ACKNOWLEDGMENTS
0
P (s) and P (s). We show the behavior of η (η =1 for
P W
a poissonian case and η =0 for GOE) as a function of g We acknowledge financial support from the DFG
in fig. 4(b). We see that our two-band model allows us (FOR760), the HGSFP (GSC 129/1) and the Institute
toconcentrateahighdensityofmany-bodyenergylevels for Theoretical Physics at Heidelberg University. It is
around a resonance, which induces the chaotic features our pleasure to warmly thank J. Madron˜ero for lively
of the system. This is not the case far from the resonant discussions. S.W. thanks furthermore the organizers of
regime, since here the system is just weakly mixed due theworkshop, L.SirkoandS.Bauch, andM.Ku´sforhis
to the presence of the M-manifolds. Then the spectrum kind hospitality at the Polnish Academy of Sciences in
is nearly regular [22, 23], and well characterized by the Warsaw.
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