Probing the edge between integrability and quantum chaos in interacting few-atom systems - Quantum Journal
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Probing the edge between integrability and quantum chaos
in interacting few-atom systems
Thomás Fogarty1 , Miguel Ángel Garcı́a-March2 , Lea F. Santos3 , and N.L. Harshman4
1
Quantum Systems Unit, Okinawa Institute of Science and Technology Graduate University, Onna, Okinawa 904-0495, Japan
2
Instituto Universitario de Matemática Pura y Aplicada, Universitat Politècnica de València, E-46022 València, Spain
3
Department of Physics, Yeshiva University, New York, New York 10016, USA
4
Department of Physics, American University, 4400 Massachusetts Ave. NW, Washington, DC 20016, USA
Interacting quantum systems in the the eigenstate thermalization hypothesis (ETH)
chaotic domain are at the core of vari- [17, 28, 114], hinders localization [74, 79, 98],
arXiv:2104.12934v2 [quant-ph] 24 Jun 2021
ous ongoing studies of many-body physics, facilitates the phenomenon of many-body quan-
ranging from the scrambling of quantum tum scarring [8, 107], leads to diffusive trans-
information to the onset of thermaliza- port [12, 55] and causes the fast spread of quan-
tion. We propose a minimum model tum information [75, 91].
for chaos that can be experimentally re- Quantum chaos refers to properties of the spec-
alized with cold atoms trapped in one- trum and eigenstates that appear in the quan-
dimensional multi-well potentials. We ex- tum domain when the classical counterpart of
plore the emergence of chaos as the num- the system is chaotic in the sense of mixing and
ber of particles is increased, starting with positive Lyapunov exponent. The features are
as few as two, and as the number of wells is similar to what we find in random matrix the-
increased, ranging from a double well to a ory [76], namely the eigenvalues are strongly cor-
multi-well Kronig-Penney-like system. In related [44] and the eigenstates in the mean-
this way, we illuminate the narrow bound- field basis are close to random vectors [114].
ary between integrability and chaos in a This quantum-classical correspondence is well es-
highly tunable few-body system. We show tablished for systems with few degrees of free-
that the competition between the particle dom [103], such as billiards, the kicked rotor, and
interactions and the periodic structure of the Dicke model, where the source of chaos is re-
the confining potential reveals subtle in- spectively the shape of the billiard, the strength
dications of quantum chaos for 3 parti- of the kicks, and the collective interaction be-
cles, while for 4 particles stronger signa- tween light and matter. In the case of systems
tures are seen. The analysis is performed with many interacting particles, the semiclassi-
for bosonic particles and could also be ex- cal analysis is challenging and sometimes not well
tended to distinguishable fermions. defined, so the common approach has been to re-
fer to many-body quantum systems that present
the above mentioned properties of spectrum and
1 Introduction
eigenstates as chaotic, even when the classical
The interest in quantum chaos, especially when limit is not analyzed.
caused by the interactions between particles, The purpose of this work is to identify a mini-
has grown significantly in the last few years mum model of interacting particles that is chaotic
due to its relationship with several questions and that can be experimentally studied with cold
of current experimental and theoretical research atoms. There are theoretical examples in the
that arise in atomic, molecular, optical, con- literature of quantum systems with only 3 or 4
densed matter, and high energy physics, as well interacting particles that already exhibit chaotic
as in quantum information science. In inter- properties. They include the cesium atom, which
acting many-body quantum systems, quantum has 4 valence electrons [33]; systems composed
chaos ensures thermalization and the validity of of 4 particles of unequal masses in a harmonic
Accepted in Quantum 2021-06-21, click title to verify. Published under CC-BY 4.0. 1trap [49] and 3 particles with unequal masses ture that consists of ultra-narrow barriers [109].
on a ring [54]; 3 or 4 excitations in spin-1/2 This technique can create sharp-box potentials
chains with short-range [94] or long-range cou- and the heights of individual barriers can be var-
plings [115]; and even spin-1/2 chains with only ied, which adds a further tunable parameter to
3 sites [77]. In the context of thermalization due the system.
to chaos, we also find works that obtained the Our results apply to few-body atomic sys-
Fermi-Dirac distribution in systems with only 4 tems with identical particles that possess spa-
particles [31, 32, 34, 57, 97]. tial wave functions symmetric under particle ex-
We consider a one-dimensional system with N change. Wave functions with this spatial sym-
identical particles that is split into wells sepa- metry are clearly relevant for the description of
rated by delta-function barriers. A single bar- bosons, but our results are equally valid for dis-
rier defines a double-well system and many bar- tinguishable fermions, i.e. identical fermions with
riers results in the finite Kronig–Penney model internal degrees of freedom like spin. For a sys-
[65, 89]. We focus on the sector of states sym- tem of distinguishable fermions, the antisymme-
metric under particle exchange and spatial parity, try required by particle statistics can be carried
and that the particles interact via contact inter- by the spin or internal wave function. All permit-
actions which are modeled with a delta function ted symmetries for three fermions or bosons (dis-
as in the Lieb-Liniger Hamiltonian [72, 81]. The tinguishable or indistinguishable) are discussed
system is integrable when there are finite barri- in [36, 46, 50]; for more particles in [47]; for few
ers or finite interactions (not both). It is also particles in double or few wells in [37, 48].
solvable in the limiting cases of infinite barrier
Due to losses via three-body re-combination,
strength and infinite interaction strength. How-
experiments with a few bosons trapped in the ul-
ever, we provide numerical evidence that when
tracold regime have an additional difficulty when
the interaction strength and the barrier strength
compared with those with a few fermions. How-
are simultaneously finite, integrability is bro-
ever, for the order of tens of bosons, it was shown
ken. In this case, strong signatures of quantum
in [18] that one can successfully load dipole traps
chaos emerge for N = 4 particles in the pres-
by means of evaporative cooling. Smaller number
ence of just one barrier (double-well system). We
of atoms can be loaded in optical lattices [112], in
also demonstrate that the signatures of quantum
arrays of double wells [25], or in a two-site optical
chaos get enhanced as we increase the number of
ring, which can be appropriately reshaped into a
barriers, in which case strong level repulsion is
Gaussian trap [52]. An alternative experimen-
verified for as few as N = 3 particles. Our analy-
tal route leading to cooled atoms trapped in sev-
sis is done for bosons, but can also be extended to
eral wells is that of few atoms in optical tweezers,
systems with a small number of distinguishable
which for a single atom in the ground state was
fermions.
accomplished in [60]. In subsequent papers it was
experimentally demonstrated the trapping of two
87 Rb bosonic atoms in two wells [61] or in uni-
2 Experimental realization
formly filled arrays of traps [68]. Recent advances
The experimental realization of our model can be allow the laser cooling of atoms in optical tweez-
done with a controllable number of interacting ers [88]; the trapping of individual atoms in op-
atoms trapped in one or several one-dimensional tical tweezer arrays [93]; and even the loading of
traps. Like experiments based on atoms in opti- atoms one by one [4, 30, 99] in a one-dimensional
cal lattices [14, 70], such systems allow precision array [30].
control and find potential applications in quan- Another experimental context for the results
tum engineering and quantum technologies [102]. in this paper is the ground breaking experiments
They are also suitable test beds for addressing that showed the accessibility and versatility of
the question of the transition from few- to many- systems with a very small number of interacting
body systems [15, 16, 102]. Our choice of the fermions. Reference [100] demonstrated that a
Kronig-Penney potential is also motivated by re- deterministic number of ultracold fermions could
cent experiments that have achieved coherent op- be extracted from a larger ensemble by apply-
tical lattices with a sub-wavelength spatial struc- ing a tightly confined one-dimensional dimple po-
Accepted in Quantum 2021-06-21, click title to verify. Published under CC-BY 4.0. 2tential. In this experiment, a few 6 Li atoms interactions, as it is the most common scenario
in the two lowest-energy Zeeman substates were for ultracold bosons.
trapped. The strength of the interactions be- The goal of our analysis of the Hamiltonian
tween atoms in different spin states could be H(N, W, τ, γ) is to understand how the signa-
controlled via Feshbach resonance. In a subse- tures of quantum chaos scale with the number
quent experiment by the same group, they con- of particles N , the number of wells W , the bar-
sidered one atom of one species (an impurity) rier strength τ and the interaction strength γ.
interacting with an increasing number of identi- The particularly simple form (1) of the Hamilto-
cal fermions, being able to build a small Fermi nian means that it is amenable to analytic and
sea adding fermions one by one in a controllable numeric calculations.
manner [110]. One can have more than two com-
ponents in a few fermion system, as in the exper- 3.1 Solvable and integrable limiting cases
iment reported in [84], where a one dimensional
system with a tunable number of spin compo- For a fixed N and W , consider the parameter
nents was realized. space window τ ∈ [0, ∞) and γ ∈ [0, ∞), de-
In addition to the controllable number of picted in Fig. 1. An interesting feature of the
atoms, another ingredient required for the re- Hamiltonian H(N, W, τ, γ) is that the model has
alization of our model is the ability to change exact solutions for four special limiting cases:
the trapping potential, creating double, triple or 1. H(N, W, 0, 0) = T N : N non-interacting
generally multiwell potentials. The same group identical particles in a one-dimensional in-
that realized few trapped fermions in [100] was finite square well with width L.
also able to trap few fermions in one dimensional
double-well [7, 78] and multi-well systems [6]. 2. H(N, W, 0, ∞): N hard-core identical parti-
cles in a one-dimensional infinite square well
with width L.
3 Model
3. H(N, W, ∞, 0): N non-interacting identi-
In the simplest case of equally-spaced barriers, cal particles distributed in W identical one-
the Hamiltonian describing our system takes the dimensional infinite square wells with width
form L/W .
1 4. H(N, W, ∞, ∞): N hard-core identical
H(N, W, τ, γ) = T N + τ V N,W + γU N (1a)
1 particles distributed in W identical one-
dimensional infinite square wells with width
where
L/W .
N
X ∂2 In all of these four cases, the configuration space
TN = − (1b)
i=1
∂x2i is sectioned into one or more N -dimensional poly-
N W
X X −1 topes with high symmetry. Solving for the spec-
V N,W = δ(xi − πk/W ) (1c) trum is equivalent to solving the Schrödinger
i=1 k=1 equation with Dirichlet boundary conditions.
X
UN = δ(xi − xj ). (1d) Exact solutions for the Schrödinger equation in
hiji these polytopes can be constructed from sym-
metrized combinations of one-particle states us-
This model realizes a system with N identical
ing methods of Refs. [59, 64, 82, 108].
interacting particles of mass m trapped in a one-
Beyond these four exactly solvable special
dimensional box of length L that is disrupted
cases, the Hamiltonian H(N, W, τ, γ) is also in-
by W − 1 delta-barriers. In the equation above,
tegrable along the four edges of the (τ, γ) pa-
xi ∈ [0, π] are the positions of the particles scaled
rameter space window (cf. Fig. 1). Referring to
by the length L/π, the energy scale is provided
the four ‘corner’ models denoted above, the inte-
by 1 = ~2 π 2 /(2mL2 ) (henceforth set to unity),
grable limits are:
and τ ≥ 0 and γ ≥ 0 are unitless parameters
describing the barrier strength, and interaction 1 ↔ 2 H(N, W, 0, γ): Without barriers, the Hamil-
strength, respectively. Here we consider repulsive tonian (1) for N identical particles in an
Accepted in Quantum 2021-06-21, click title to verify. Published under CC-BY 4.0. 3be different partitions of interacting parti-
2 integrable 4 cles and the size of the well is L/W .
(0,∞) (∞,∞)
Note that these four integrable models allow us
integrable
integrable CHAOS? to establish an adiabatic map between the en-
(t,g) ergy levels for the solvable (and superintegrable)
models at the corners of parameter space.
1 3
(0,0) integrable (∞,0) 3.2 Symmetries and degeneracies
To analyze signatures of chaos, the Hilbert space
Figure 1: Depiction of the (compactified) (τ, γ) parame- H must first be decomposed into subspaces with
ter space window τ ∈ [0, ∞) and γ ∈ [0, ∞). At the four fixed symmetry. The Hamiltonian (1) has two
corners of the window, the Hamiltonian H(N, W, τ, γ) symmetries for any interaction strength and bar-
has exact solutions and is superintegrable. The four rier strength in the (τ, γ) parameter window.
edges connecting these corners are integrable models First, it is symmetric under particle permuta-
where exact solutions can be found by the solution of tions SN , i.e. any permutation p ∈ SN is repre-
coupled transcendental equations. Except for possibly
sented as a linear transformation in configuration
the case of N = 2, the H(N, W, τ, γ) does not appear
integrable or solvable for arbitrary (τ, γ) in this parame- space (x1 , x2 , . . . , xN ) → (xp1 , xp2 , . . . , xpN ) that
ter space leaves the Hamiltonian (1) invariant. The sec-
ond symmetry is total parity inversion Π about
the center of the well, implemented as the affine
infinite square well with delta-function in- linear transformation xi → π − xi for all i.
teractions is solvable by coordinate Bethe These two symmetries allow eigenstates to be
ansatz [39, 40, 80]. The energies are the so- classified by irreducible representations of SN
lution of coupled transcendental equations and of parity Π and reduce the total Hilbert
that depend on N and γ. space H into subspaces with a given symmetry
(cf. [47]). In this work, we focus on the sector of
1 ↔ 3 H(N, W, τ, 0): N non-interacting particles
Hilbert space H[N ]+ ⊂ H containing states with
in an infinite square well with W −1 barriers.
bosonic symmetry under particle exchange and
The Hamiltonian separates into N identi-
positive parity. Note that a special property of
cal one-dimensional sub-Hamiltonians. The
delta-function operators V N,W and U N is that
spectrum for each sub-Hamiltonian is ob-
they vanish on certain subspaces of H, because
tained via solution to transcendental equa-
the support of the delta-functions coincides with
tions that depend on τ [89].
nodal lines of symmetrized eigenstates of T N . In
2 ↔ 4 H(N, W, τ, ∞): In this limit of finite wells particular, there is a subspace of H[N ]+ upon
with hard-core interactions, the solutions which V N,W vanishes when N is even or both N
are Tonks-Girardeau constructions derived and W are odd. Also, in the limits τ → ∞ and
from the symmetrized Slater determinants γ → ∞, both V N,W and U N must vanish on any
of the solutions of the non-interacting case states with finite energy; in this limit the wave
H(N, W, τ, 0) [41]. functions must have nodal surfaces that coincide
with the support of these operators.
3 ↔ 4 H(N, W, ∞, γ): With infinite barriers, the Degeneracies in the spectrum either originate
configuration space fractures into W N con- with the symmetries of the Hamiltonian or they
figurations, i.e. each particle i ∈ {1, . . . , N } are designated ‘accidental’. Because the only
is in a well j ∈ {1, . . . , W } with width L/W . symmetries for generic (τ, γ) are particle ex-
Each of these particle-well configurations is changes SN and parity Π, there should be no
an independent system, since with τ → ∞ degeneracies originating in symmetry in the in-
there is no tunneling among wells. Within terior of parameter space. This is because the
each configuration, the solutions are similar irreducible representations [N ]+ for totally sym-
to the coordinate Bethe ansatz solutions of metric bosonic states are one-dimensional. How-
H(N, W, 0, γ), except that now there may ever, additional symmetries arise in the limiting
Accepted in Quantum 2021-06-21, click title to verify. Published under CC-BY 4.0. 4cases of zero and infinite strengths τ or γ, includ-
ing the symmetries of separability and system
decoupling due to infinite barriers [48]. These
symmetries explain the integrability of the edge
models and the superintegrability of the corner
models. When τ and γ are both finite, then these
additional symmetries are broken and we expect
these energy levels that cross at certain parame-
ters along the edges of the model space to repel.
Level repulsion, which is a signature of chaos, is
thus expected in the interior of this model space
(see Fig. 1), which is indeed what we verify for
N ≥ 3.
Besides degeneracies originating in symme-
tries, several other kinds of accidental degen- Figure 2: Density of states for N = 2 (blue points),
eracies should be possible in our system. The N = 3 (red points) and N = 4 (yellow points). Solid
variation of two parameters is sufficient to allow lines show the leading term in Eq. (2) for the respective
for ‘diabolic points,’ topologically-stable conic particle numbers. Symbols denote different system pa-
rameters: γ = 10 and τ = 0 (circles), γ = τ = 10 and
degeneracies between two energy levels [9, 11].
W = 2 (triangles) and γ = τ = 10 and W = 10 (stars).
These degeneracies can be distinguished from
‘near misses’ by looking at how the wave func-
tion transforms when small loops in control space where we have used the notation O [N, W, τ, γ]
are taken and in subsequent work we will con- to indicate that generally, the coefficient in front
sider such loops. There are also degeneracies of the subleading term proportional to E N/2−3/2
for the solvable corner models arising from the will depend on all the parameters of the Hamil-
number theory of decomposing integers into sums tonian.
of squares of integers. Also called Pythagorean
We derive Eq. (2) in the Appendix for the four
degeneracies [48, 101], the density of such de-
solvable ‘corner’ models, and infer that it holds
generacies grows slowly with energy but even-
for the entire parameter window. More gener-
tually comes to dominate the spectrum [9]. Fi-
ally, adding one-dimensional delta-functions to
nally, there are other degeneracies characteristic
an otherwise free problem should not change the
of Bethe-ansatz solvable systems [53] that should
density of states. By Weyl’s Law [56], the vol-
be relevant for some of the integrable edge mod-
ume of phase space can be related to the den-
els. It is not clear how these additional ‘acciden-
sity of states, and boundary conditions (whether
tal’ degeneracies at the corners and edges affect
Dirichlet, Neumann or hybrid) do not change the
the spectrum of the interior of parameter space,
phase space volume to leading order.
but they inform our interpretation of the level
statistics presented below for the N = 2 case. In Fig. 2 we show that the leading term in
Eq. (2) agrees with the scaling of the density of
states regardless of number of wells W , barrier
3.3 Density of states
strength τ or interaction strength γ. Because
The density of states of the Hamiltonian the leading term in Eq. (2) is the same for the
H(N, W, τ, γ) is used to interpret some of the nu- entire parameter window for a given N , at least
merical results below. Perhaps surprisingly, it is heuristically we expect that the variation in level
independent of τ and γ to leading order in the en- statistics with the parameters (τ, γ) is governed
ergy E. The density of states in the sector H[N ]+ by the properties of the subleading term (or even
is lower order terms). In the Appendix, we support
this hypothesis by looking at how energy levels
change as the model parameters are adiabatically
1 π N/2 tuned along integrable edge models to connect
ρ[N ]+ (E) = E N/2−1
2N +2 (N + 1)! Γ(N/2 + 1) the spectra of the solvable corner models.
+O [N, W, τ, γ] (E N/2−3/2 ), (2) Note that for N = 2, the density of states is
Accepted in Quantum 2021-06-21, click title to verify. Published under CC-BY 4.0. 5a constant and the subleading correction is actu-
ally decreasing with energy as E −1/2 . In Fig. 2,
all data points for N = 2 lie on top of the line
given by the leading term in Eq. (2). For N = 3,
the density of states increases as E 1/2 and the
subleading term is constant. Only for N = 4 are
both the leading and the subleading term of the
density of states growing with energy. If the sub-
leading term is important for understanding the
density of level crossings as we propose, then this
helps to explain why N = 4 is the threshold when
the level statistics conform to the expectations of
random matrix theory across such a wide range
of parameter space, whereas the N = 3 only in a Figure 3: Energy level spacing distributions for (a,d,g)
limited range. N = 2 particles, (b,e,h) N = 3 particles and (c,f,i)
N = 4 particles with different interaction strengths γ.
The barrier height is fixed to τ = 10 and the number
4 Indicators of chaos of wells is W = 2. The red solid line is the Poissonian
distribution and the blue solid line is the Wigner-Dyson
In the parameter window of interest, finite inter- distribution.
actions and finite barrier heights, the Hamilto-
nian (1) is diagonalized numerically using a ba-
sis consisting of N -particle non-interacting eigen- 4 particles for different interactions and at a fixed
+
states of T N in the Hilbert space H[N ] [46]. In barrier height of τ = 10.
the following we discuss the different indicators For weak interactions, γ = 1, a picket
of chaos that are used in this work along with the fence pattern is noticeable for N = 2 particles
numerical results. [Fig. 3 (a)], which indicates non-generic correla-
tions in the energy spectrum [10]. Additionally,
the peak at s = 0 signals excessive degeneracies
4.1 Energy spacing distribution typical for solvable systems [53, 113]. For N = 3,
To study the degree of short-range correlations the picket fence structure vanishes and the distri-
between the eigenvalues, we use the distribution bution is closer to Poissonian with some remain-
P (s) of the spacings between neighboring lev- ing evidence of additional degeneracies at s = 0
els obtained after unfolding the spectrum. In [Fig. 3 (b)]. For N = 4, the distribution is also
generic integrable models, the level spacing dis- close to Poissonian, except for a slight decrease
tribution is Poissonian, PP (s) = e−s , when the at s = 0 that provides evidence for level repulsion
energy levels are uncorrelated and not prohibited already at weak interactions [Fig. 3 (c)].
from crossing, although different shapes emerge At stronger interactions (γ ≥ 10), the distri-
for “picket-fence”-kind of spectra [10, 38, 85] butions for N = 2 particles in Fig. 3 (d) and
and systems with an excessive number of de- Fig. 3 (g) become close to Poissonian and level
generacies [113]. For chaotic systems with real repulsion is not evident. This Poissonian distri-
and symmetric Hamiltonian matrices, as the one bution suggests that for the N = 2 and W = 2
considered here, the level spacing distribution the Hamiltonian is integrable (or effectively so)
follows the Wigner-Dyson distribution [44, 76], for all values of (τ, γ); see the conclusion for a fur-
2
PWD (s) = (πs/2) exp −πs /4 , which indicates ther discussion on this point. In contrast, some
that the eigenvalues are correlated and repel each degree of level repulsion is already noticeable for
other. N = 3 [Fig. 3 (e) and Fig. 3 (h)] and the Wigner-
First we investigate the minimal case for chaos Dyson distribution is visible for N = 4 particles
in our system, where only one barrier is inserted [Fig. 3 (f) and Fig. 3 (i)].
centrally in the square well, thereby creating a To explore the crossover of the energy spec-
double well potential (W = 2). In Fig. 3 we trum from Poissonian to Wigner-Dyson and the
show the level spacing distributions for 2, 3 and role of interactions and the potential barrier, we
Accepted in Quantum 2021-06-21, click title to verify. Published under CC-BY 4.0. 6emerging for N = 3 in the region 15 . γ . 45
and 5 . τ . 35 where β ∼ 0.5 [Fig. 4 (b)]. For
N = 4 the fitting parameter β in almost the
entire parameter space is larger than 0.5, with
the areas of integrability confined to the edges of
the (τ, γ) parameter space [Fig. 4 (c)]. In fact,
we find the maximum to be around β ∼ 0.83 at
γ = 20 and τ = 12.5, indicating that the energy
spacing distribution gets close to Wigner-Dyson.
In Figs. 4 (d)-(f) we also consider multiple
wells, with W = 10. For more wells, the density
of states does not change to leading order in τ ,
but a greater proportion of the eigenstates of T N
feel the effect of the barriers more acutely. For
example, for N = 2 and W = 2, the barrier po-
Figure 4: Brody distribution parameter β as a function
of τ and γ for (a,d) N = 2, (b,e) N = 3, and (c,f)
tential V N,W vanishes on half of the eigenstates
N = 4 particles. The number of wells are (a-c) W = 2 of T N in H[N ]+ , whereas for N = 2 and W = 10,
and (d-f) W = 10. For W = 2 wells the maximum the barrier potential V N,W only vanishes on one
βmax = 0.8269 is found for N = 4 particles at γ = 20 tenth of the eigenstates of T N in H[N ]+ .
and τ = 12.5, while for W = 10, we find βmax = 0.9512 Comparing the case of W = 10 to W = 2
for N = 4 particles at γ = 45 and τ = 12.5. Black areas in Fig. 4, we see that larger values of β are in-
show β < 0 indicating degeneracies and picket-fence deed found for W = 10 and the region of the
spectra.
parameter space where β is large has increased
significantly for both N = 3 and N = 4. This
fit our numerical results to the Brody distribu- suggests that in systems with a large number of
tion [22] (see an alternative in [58]), wells, the energy level repulsion is enhanced and
chaos could be observed for systems with as few
Pβ (s) = (β + 1)bsβ exp(−bsβ+1 ), (3) as N = 3 particles. However, for N = 2, we find
β+2
β+1 a maximum of β ≈ 0.5 and this value does not
b= Γ . approach the chaotic limit by increasing the num-
β+1
ber of wells. As before, the case of only two par-
For the Wigner-Dyson distribution, β ∼ 1, while ticles resists the transition to chaos. In Sect. 4.4
the Poissonian distribution leads to β ∼ 0. below, we discuss in more detail how the signa-
Close to the integrability limits, vanishing in- tures of chaos change as the number of barriers
teractions (γ ≈ 0) with finite barriers (τ 6= 0) or is increased.
finite interactions (γ 6= 0) with vanishing barrier
height (τ ≈ 0), the energy spacing distributions 4.2 Off-diagonal ETH
can display a picket-fence or quasi-Poissonian dis-
tribution as discussed in Fig. 3 for N = 2, which Quantum chaos ensures the validity of the ETH,
results in β < 0. We highlight these non-generic so we can also use the indicators of ETH to detect
regions as black in Figs. 4 (a)-(f), showing that the transition to chaos. Two conditions need to
they are more prevalent at smaller particle num- be satisfied for a few-body observable O, evolving
bers, while their footprint in the parameter space according to
almost vanishes for N = 4. As we increase ∗
Cn e−i(En −Em )t Omn
X
the interactions (γ 0) and the barrier heights O(t) = Cm
m6=n
(τ 0), the degeneracies of the N = 2 system X
are destroyed and the level spacing distributions + |Cm |2 Omm , (4)
m
become closer to Poissonian for a large region
of the parameter space with β remaining below to reach thermal equilibrium. In the equation
0.35. above, Cm = hm|Ψ(0)i is the overlap between
For more particles, the distributions are dif- the eigenstate |mi of the Hamiltonian that de-
ferent, with indications of energy level repulsion scribes the system and the initial state |Ψ(0)i,
Accepted in Quantum 2021-06-21, click title to verify. Published under CC-BY 4.0. 7viation of the distribution. For Gaussian distri-
butions the kurtosis is given by KÔ = 3.
In our calculations of KÔ , we consider the ki-
netic energy operator T N (in the following we
drop the superscript to simplify the notation).
To ensure no effects from the bottom edge of the
spectrum, we choose a energy window far from
the ground state and include only states within
Em ∈ [Emid − ∆E, Emid + ∆E], where the cen-
ter of the energy window is high in the spectrum
at Emid = 700 and its width is ∆E = 100. In
Figs. 5 (a)-(f) we show the inverse of the kurto-
sis, 1/KT , with the maximal value 1/3 indicating
a Gaussian distribution and therefore the pres-
Figure 5: Inverse of the kurtosis KT−1 as a function of τ ence of chaos.
and γ for (a,d) N = 2 (b,e) N = 3 and (c,f) N = 4
particles. The number of wells are (a-c) W = 2 and (d- For N = 4 particles and W = 2 [Fig. 5 (c)],
f) W = 10. For W = 2 the minimum kurtosis is 4.3592 the distribution is close to Gaussian for barrier
at γ = 30 and τ = 15, while for W = 10 the minimum heights 1 . τ . 30 and interactions 20 . γ . 60,
kurtosis is 3.3274 at γ = 20 and τ = 7.5. which coincides with the region of β > 0.8 in
Fig. 4 (c). We choose the minimum value of
and Omn = hm|Ô|ni. The first condition is the kurtosis in Fig. 5 (c) and show the Gaus-
that of equilibration, which depends on the first sian probability distribution of Tmn in Fig. 6 (c).
term on the right-hand side of Eq. (4). At long In contrast, for lower particle numbers, N = 2, 3,
times, due to the lack of degeneracies of chaotic the distribution of the off-diagonal elements Tmn
Hamiltonians, the small values of the coefficients is sharply peaked, as seen in Fig. 6 (a) and
Cm ’s obtained when the systems is quenched far Fig. 6 (b), which is consistent with the kurtosis
from equilibrium, and the small values of the off- having values KT 3 in Fig. 5 (a) and Fig. 5 (b).
diagonal elements Omn ’s caused by the chaotic The kurtosis shows a similar enhancement due
eigenstates, the first term in Eq. (4) leads to to the presence of more barriers, attaining a min-
small fluctuations that decrease with system size imum of KT ≈ 3.3 for N = 4 particles in W = 10
and cancel out on average. So apart from small wells [Fig. 5 (f)]. Indeed, for 1 . τ < 20 and a
fluctuations, the observable reaches its infinite- broad range of interactions, the kurtosis is close
time average m |Cm |2 Omm . The second condi-
P
to 3. Increasing the number of barriers also
tion is that this infinite-time average approaches moves the band of minimal kurtosis to lower val-
the thermodynamic average as the system size in- ues of τ , as lower barrier heights are necessary
creases, confirming that the equilibrium is indeed to retain the competition with the inter-particle
thermal. These two steps are usually referred to interactions. In a similar region of the parame-
as off-diagonal- and diagonal-ETH, respectively. ter space, there is also a visible minimum of the
Here, we consider the off-diagonal-ETH to de- kurtosis for N = 3 particles [Fig. 5 (e)]. In fact,
tect the transition to chaos. The distribution of when taking the interaction and barrier height
Omn in chaotic (thermalizing) systems is Gaus- which give the minimum value of kurtosis [γ = 10
sian [13, 19–21, 66, 92], reflecting the chaotic and τ = 7.5 for N = 4], there are distinct Gaus-
structure of the eigenstates, while other forms sian probability distributions for both N = 4 and
emerge for integrable models [66]. To quantify N = 3 particles [see Figs. 6 (f,g)]. However, for
how close the distribution of Omn is to a Gaus- N = 2 the off-diagonal elements of the kinetic en-
sian, we use the kurtosis, ergy operator Tmn do not indicate the presence
h(Omn − hOmn i)4 i of chaos in either the kurtosis [Fig. 5 (d)] or the
KÔ = , (5) probability distribution [Fig. 6 (e)].
σ4
where h.i indicates the average over all pairs of To quantify how the off-diagonal elements Tmn
eigenstates |mi =
6 |ni and σ is the standard de- behave as a function of the energy difference ω =
Accepted in Quantum 2021-06-21, click title to verify. Published under CC-BY 4.0. 8Figure 6: Off-diagonal elements of the kinetic energy operator Tmn for (a-d) W = 2 wells and (e-h) W = 10 wells.
The probability distributions in the respective potentials are shown for (a,e) N = 2 (b,f) N = 3 and (c,g) N = 4.
(d,h) The ratio ΓT as a function of ω = |Em − En | in an energy window at Emid = 700 and width ∆E = 100. The
interaction strengths and barrier heights are chosen at the point of minimum kurtosis for 4 particles in Fig.5(c,f),
namely γ = 30 and τ = 15 for W = 2, and γ = 20 and τ = 7.5 for W = 10.
|Em − En | we also show in Fig. 6 the ratio “ramp” [27]. The correlation hole corresponds
to a dip below hS∞ i = h n |Cn |4 i, which is
P
|Tmn |2 the infinite-time average (saturation value) of
ΓT = 2, (6)
|Tmn | hSP (t)i. This dip emerges also in the spectral
form factor h n,m e−i(En −Em )t i, but contrary to
P
which is equal to π/2 for a Gaussian distribution. this one, the survival probability is a true dynam-
In Fig. 6 (d) and Fig. 6 (h) this ratio is shown ical quantity. In the equation above, h.i indicates
for W = 2 and W = 10 wells, respectively, for averages. The survival probability is non-self-
the same Hamiltonian parameters that gave the averaging [87, 96], so the correlation hole is not
lowest values of the kurtosis discussed in the pre- visible unless averages are performed. They can
ceding paragraph. For W = 2 wells, the values be done over initial states, disorder realizations,
of ΓT for N = 4 sit very close to π/2 over a large or, as in our case, they correspond to moving time
range of ω, confirming that the Gaussianity of the averages. The correlation hole detects short- and
distribution is preserved at different energy spac- long-range correlations in the spectrum, and in
ings. The ratio for N = 2, 3 has a large variance addition, it does not require unfolding the spec-
with the majority of the points being far from trum or separating it by symmetries [29, 92].
π/2, which is indicative of the peaked distribu- In cold atom systems the survival probability is
tions in Fig. 6 (a) and Fig. 6 (b). For W = 10 commonly used to probe the non-equilibrium dy-
wells, we find that not only ΓT ≈ π/2 for N = 4 namics of few- [23, 62] and many-body systems
particles, but also the N = 3 system approaches [26, 35, 42, 63], and can be experimentally mea-
this result. This suggests that the N = 3 system sured using interferometric techniques [24].
can be tuned between integrability and chaos by We take an initial state that has a homoge-
changing the trapping potential. neous probability distribution centered at Emid
inside an energy window of width ∆E
4.3 Survival Probability (
1
2∆E for E ∈ [Emid − ∆E, Emid + ∆E]
ρ(E) =
Spectral correlations get manifested also in the 0 otherwise.
evolution of the survival probability,
In [67, 95, 106] a general analytic solution was
2 derived for the survival probability for chaotic
hSP (t)i = h|hΨ(0)|Ψ(t)i| i
=h
X
|Cn |2 |Cm |2 e−i(En −Em )t i, (7) systems. For the square distribution used here,
n,m this solution is given by
" #
1 − S∞ sin2 (∆E t) ∆E t
in the form of what is known as the correla-
SP (t) = η − b2 +S∞ .
tion hole [2, 29, 43, 51, 67, 69, 71, 73, 86, 95, η−1 (∆E t)2 πη
104–106, 111], recently referred to also as the (8)
Accepted in Quantum 2021-06-21, click title to verify. Published under CC-BY 4.0. 9Figure 7: Survival probability of (a,d) 2 particles (b,e), 3 particles, and (c,f) 4 particles. The number of wells is (a-c)
W = 2 and (d-f) W = 10. The initial state is a square distribution centered at Emid = 700 and of width ∆E = 100.
Black lines are the moving time averages on the logarithmic timescale with window [log10 t − ∆t, log10 t + ∆t] and
∆t = 0.02. Green horizontal dashed line indicates the infinite time average of the survival probability S∞ . For
N = 3 and 4, red lines represent the analytic solution. For N = 4, the grey points represent the unaveraged value
for comparison. The parameters used for each row correspond to those that give the minimum kurtosis for 4 particles
in Fig.5, namely γ = 30 and τ = 15 for W = 2, and γ = 20 and τ = 7.5 for W = 10.
Here, η is the number of energy eigenvectors in cay and subsequent ramp of SP (t) matching pre-
the energy window ∆E. The initial decay of the cisely the analytics. For N = 3 and W = 10
survival probability is captured by the first term [Fig. 7 (e)], partial revivals still obscure the min-
in the brackets in Eq. (8), after which the dy- imum of the correlation hole, but the ramp to-
namics is described by the two-level form factor wards saturation can be seen to follow the ana-
lytic results.
b2 (t̄) = 1 − 2t̄ + t̄ ln(2t̄ + 1) Θ(1 − t̄)
" ! #
2t̄ + 1
+ t̄ ln − 1 Θ(t̄ − 1), 4.4 Dependence on the number of wells
2t̄ − 1
For a more systematic analysis of the onset of
with Θ(t̄) the Heaviside step function. chaos as the number of wells in the system is
In Fig. 7 the survival probability is shown for increased, we show in Fig. 8 (a) the minimum
different number of particles and number of wells of the kurtosis Kmin = min[KT ] in the range
(grey points). The moving time average (black γ, τ ∈ [0, 100]. Here we fix the size of the box
lines) smooths the data and allows us to identify L and focus on N = 3 and N = 4 particles. For
the correlation hole with its ramp towards S∞ . both cases the number of wells dictates the emer-
For W = 2 and N = 2, 3 [Figs. 7 (a,b)] there gence of chaos, but to different degrees. For the
is no obvious indication of a dip below S∞ that trivial case of W = 1 (no barriers) the kurtosis is
could be described by the two-level form factor. large and both N = 3 and N = 4 are integrable.
For N = 4 [Fig. 7 (c)], a noticeable dip manifests For W > 1 the kurtosis of N = 4 takes low val-
below the saturation point and follows closely the ues, Kmin ∼ 3, indicating chaos, essentially irre-
analytic solution in Eq. (8) (red lines). This is in- spective of the number of wells when W . 13.
dicative of chaotic behavior. In the W = 10 well However, for N = 3 the inclusion of more barri-
system the correlation hole for N = 4 is even ers causes a more subtle change to the kurtosis,
more pronounced [Fig. 7 (f)] with the initial de- which decreases slowly as more barriers are intro-
Accepted in Quantum 2021-06-21, click title to verify. Published under CC-BY 4.0. 10duced, attaining a minimum of Kmin ∼ 4 in the
region of 6 . W . 13. It is in this region that
the N = 3 system displays chaotic signatures as
discussed in the previous sections. Interestingly,
increasing the number of wells further (W > 13)
results in an increase of the kurtosis and indica-
tions of chaos are diminished. A similar tendency
is seen for N = 4, albeit in a less drastic manner,
as the kurtosis increases at a lower rate. For both
N = 3 and N = 4 we expect that as the limit of
N/W → 0 is approached the contest between the
ordering of the particles in the wells and their in- Figure 8: (a) Minimum of the kurtosis as a function
teractions is reduced and that the system slowly of the number of wells for N = 3 (red triangles) and
returns to what we would see in the continuum: N = 4 (yellow circles). Kurtosis is calculated in an
particles in a box [90]. energy window of width ∆E = 100 around Emid = 700.
In Fig. 8 (b) and Fig. 8 (c) we show the opti- (b) Interaction and (c) barrier height corresponding to
the points of minimum kurtosis in (a).
mal interactions and barrier heights for achieving
the minimum kurtosis shown in Fig. 8 (a). Fig-
ure 8 (c) shows that the optimal barrier strengths
of chaos becomes more robust, even for weaker
τ for N = 3 and N = 4 have close agreement
barriers and interactions.
for all W , and that these values decrease with
increasing number of wells (for W ≤ 20). This In contrast, for N = 2, the spectrum is numeri-
reduction in τ is necessary to preserve the compe- cally indistinguishable from an integrable system
tition between barriers and interactions, as when throughout the parameter window for W = 2,
the number of barriers is increased the impact and deviates only slightly from this as the num-
of τ is magnified. This effect can be seen in ber of wells is increased. Several possible origins
the shift of the chaotic region in Fig. 5 (c) and for the non-chaotic behavior of N = 2 can be
(f). Fig. 8 (b) shows that the optimal interaction hypothesized, including some undiscovered inte-
strengths γ found for both N = 3 and N = 4 con- gral of motion, some sort of partial integrability
verge to a similar value in the region 7 . W . 20, of a subset of eigenstates of the Hamiltonian, a
which encompasses the areas of low kurtosis in proliferation of accidental degeneracies from sev-
Fig. 8 (a). This suggests that both N = 3 and eral sources that create the appearance of inte-
N = 4 particles are chaotic in the same regions of grability, or a combination of all these. These
the parameter space [τ, γ]. Increasing the num- possibilities will be considered in more detail in
ber of wells beyond W = 20 breaks this trend and a subsequent work. For now, we note with some
the parameters for N = 3 and N = 4 diverge as irony that integrability is much less generic and
the indications of chaos are lost. more complicated than chaos.
The intermediate case of N = 3 stands at the
ragged edge where symmetry and integrability
5 Discussion and Conclusion dissolve into chaos and randomness. By tuning
the interaction and barrier parameters and in-
To summarize, between N = 2 and N = 4 our creasing the number of barriers, the full gamut
model makes a clear transition. For N = 4 par- of possibilities can be realized on the same small
ticles and just W = 2 wells, there are clear sig- atomic system. The N = 3 case has some com-
natures of chaos when both the barrier and the mon aspects with the N = 2 case and other fea-
interactions have finite strength. The density of tures similar to N = 4. For example, the density
states grows rapidly with energy, and numerical of states grows with the energy like the N = 4
analysis gives evidence of the highly-correlated and unlike the N = 2 case, although it grows
spectrum typical of random matrices, of the va- sublinearly while the N = 4 case grows linearly.
lidity of the off-diagonal ETH, and of a clear On the other hand, the probability distributions
ramp in the survival probability. As the num- for the off-diagonal elements of the kinetic energy
ber of wells is increased, evidence for the onset are closer to the N = 2 than to the N = 4 case
Accepted in Quantum 2021-06-21, click title to verify. Published under CC-BY 4.0. 11for only two wells, but when increasing the num- Acknowledgments
ber of wells, it gets closer to the Gaussian dis-
tribution characteristic of the N = 4 case. The The authors thank M. Olshanii, T. Busch, A.
variety of possible scenarios for N = 3 atoms Fabra and M. Boubakour for insights on inte-
is most clear in Fig. 8 (a), where the degree of grability and conversations about chaos. TF
chaoticity measured with the minimum kurtosis acknowledges support from JSPS KAKENHI-
Kmin makes sweeping changes as the number of 21K13856 and the Okinawa Institute of Science
wells is increased. and Technology Graduate University. We are
As we discuss above, current experiments with grateful for the help and support provided by
subwavelength lattice potentials and determinis- the Scientific Computing and Data Analysis sec-
tically prepared few-body states provide the ideal tion of Research Support Division at OIST. LFS
platform to probe this boundary between inte- was supported by the NSF grant No. DMR-
grability and chaos. Furthermore, these small 1936006. M.A.G.M. acknowledges funding from
atomic systems have been proposed as the work- the Spanish Ministry of Education and Voca-
ing units of larger quantum information process- tional Training (MEFP) through the Beatriz
ing devices and protocols. Therefore, under- Galindo program 2018 (BEAGAL18/00203) and
standing their information, control, and entan- Spanish Ministry MINECO (FIDEUA PID2019-
glement properties in these different regimes be- 106901GBI00/10.13039/501100011033).
comes important.
For example, this model lends itself naturally A Density of states derivation
to “digitization”. The number of particles in each
well becomes a useful observable in the infinite To derive the density of states, we first calculate
barrier limit; they are integrals of motion in fact. the total number of states (with any symmetry
Similarly, coherent superpositions of eigenstates or parity) and energy less than E:
of these or other integrals of motion in the limit-
ing edge models could be used for storing quan- 1 π N/2
N (E) = E N/2 (9)
tum information (c.f. [83]). A quench from an 2N Γ(N/2 + 1)
integrable limit to the chaotic parameter regime +O [N, W, τ, γ] (E (N −1)/2 ).
would break these integrals of motion and effec-
tively scramble the information held in the initial The density of states in Eq. (2) is the derivative
state. of this with respect to energy.
An important extension of this work that is To establish this result (9), first consider the
experimentally relevant is determining how sen- simplest limiting case H(N, W, 0, 0). There is a
sitive our results are to the idealizations of delta- solution of H(N, W, 0, 0) for every set of non-
barriers, precisely symmetric positions and uni- negative integers n = {n1 , . . . , nN } with energy
En = N 2
P
form barrier height. We expect that small de- i=1 ni . The space of solutions therefore
viations of the periodic Kronig-Penney lattice, is a (hyper)cubic lattice in the all-positive ‘quad-
such as finite width barriers, would not signif- rant’ (really 2N -rant) of RN . Since each state
icantly alter the chaotic regions of our system. takes up a unit volume in this quantum number
The versatility of our model also allows us to ex- space, to find the number of states NN (E) with
plore the possibility of emergent integrability [1] energy less than E, one √ takes the volume of an
when more disorder is introduced into the sys- N -ball with radius r = E and divides by 2N
tem via non-regularly spaced barriers or barriers to account for the all-positive condition, giving
of different heights. Both aspects are inspiring the leading term in Eq. (9). The first correction
and we leave them for future research. Another term comes from the N -sphere boundary of the
interesting scenario which we have not explored N -ball, which has one dimension lower.
is the case in which the interactions are attrac- The spectrum of H(2, 2, 0, 0) is depicted as
tive, whereby the system is now furnished with model 1 in Fig. 9. In this simple case, states
a bound state and whose limit at infinite inter- with energy less than E√ lie within the quarter
actions is the so-called super Tonks-Girardeau circle with radius r = E. That quarter disk
gas [3, 5, 45]. therefore has area πE/4, agreeing with (9).
Accepted in Quantum 2021-06-21, click title to verify. Published under CC-BY 4.0. 122 4 Parallel arguments give the same leading terms
(0, ) ( , ) for the other three solvable corner models. For
example, in the limit of no barriers and infinite
interactions H(N, W, 0, ∞), we follow the con-
struction of Girardeau and find that there are N !
states for every strictly increasing set of integers
{n|n1 < n2 < · · · nN }; c.f. [47]. Therefore, infi-
1 3 nite interactions exclude cases when two or more
(0, 0) (, 0) quantum numbers are the same. This is depicted
for H(2, 2, 0, ∞) in Fig. 9, where the spectrum is
missing the diagonal states with n1 = n2 . For
N = 2, the number of states missing from the
area estimation in Eq. (9) is therefore propor- √
tional to the length of this boundary r = E.
More generally for N particles, the geometri-
Figure 9: These four diagrams depict the spectrum for
the four solvable cases of N = 2 particle and W = 2
cal structures in the quantum number ‘quadrant’
wells; c.f. Fig. 1 and Sec. 3.1. Model 1: no interactions corresponding to all numbers different remains
or barriers; model 2: infinite interactions no barriers; N -dimensional even in the presence of infinite
model 3 no interactions, infinite barriers; model 4 infi- interactions. However, the structures with two
nite interactions and barriers. Each dot represents an quantum numbers equal correspond to interior
energy eigenstate with energy E = n21 + n22 given by the boundaries of the quadrant and have dimension
sum of the squares of the integer coordinates (n1 , n2 ) N − 1. Therefore corrections to account for the
of the point. States with n1 ≥ n2 represent the sym-
‘missing states’ appear at the subleading order
metrized states with positive parity in black and with
with negative parity in cyan. States with n1 < n2 rep- rN −1 = E (N −1)/2 . Further corrections to the
resented by magenta and yellow dots are antisymmetric number of states appear at next-to-subleading or-
and have negative and positive parity, respectively. In der E (N −2)/2 when either two pairs of quantum
models 1 and 2, all states are two-fold degenerate except numbers are the same or three quantum numbers
in model 1 (no barriers, no interactions) when n1 = n2 . are the same.
Additionally, models 3 and 4 have additional two-fold
degeneracies (in model 4, states along the diagonal rep- Note that in the example with N = 2 in Fig. 9,
resented by split dots with twice the area), four-fold (in the Tonks-Girardeau map shifts the symmetric
model 4, represented by quartered dots along the diag- states (n1 , n2 ) in model 1 to (n1 + 1, n2 ) in model
onal) and eight-fold (in models 3 and 4, represented by 2. Since an integrable model connects these two
pairs of quartered dots with exchanged integer quantum limiting cases, this establishes a one-to-one adi-
numbers). The dashed √ quarter-circle is the boundary abatic mapping between the two spectra. From
E = 150 with radius 150 and is included to aid visu-
alization of the spectral flow. this mapping the number of level crossings that
occur as γ is tuned from 0 to ∞ can be explicitly
calculated without actually solving for the spec-
trum on the integrable model that links these two
cases.
Similarly, for the case H(N, W, ∞, 0) of infi-
nite barriers and no interactions, there are W N
solutions for every set of non-negative integers n
where all integers ni are multiples of the number
of wells W . This increases the volume associ-
ated with each set of quantum numbers from 1
to W N and that factor of 1/W N exactly cancels
the degeneracy factor W N giving the same lead-
ing term. This is depicted for the simplest case
of N = 2 and W = 2 in model 3 of Fig. 9. As
before, we do expect the coefficient on the sub-
leading term to depend on the intricate combina-
Accepted in Quantum 2021-06-21, click title to verify. Published under CC-BY 4.0. 13torics of putting N identical particles in W wells. [3] G. E. Astrakharchik, J. Boronat,
A parallel argument holds for the fourth solvable J. Casulleras, and S. Giorgini. Be-
model H(N, W, ∞, ∞). yond the Tonks-Girardeau gas:
Note again that these four models with exact Strongly correlated regime in quasi-
solutions are connected by integral models and one-dimensional bose gases. Phys.
exact spectral maps can be constructed for all Rev. Lett., 95:190407, Nov 2005. DOI:
of these cases. Because these four extreme cases 10.1103/PhysRevLett.95.190407. URL
all have the same leading term in the density of https://link.aps.org/doi/10.1103/
states that depends only on N , we assume that PhysRevLett.95.190407.
all the models that lie in this region of param- [4] Daniel Barredo, Sylvain de Léséleuc,
eter space have the same property, and this is Vincent Lienhard, Thierry Lahaye, and
further supported by the phase space argument Antoine Browaeys. An atom-by-atom
presented in Sec. 4.1. assembler of defect-free arbitrary two-
When we elect to consider only one symmetry dimensional atomic arrays. Science, 354
sector, e.g. bosons with positive parity, that re- (6315):1021–1023, 2016. DOI: 10.1126/sci-
duces the number of states (9) by a factor of 1/2 ence.aah3778.
for parity and 1/N ! for symmetrization at leading [5] M T Batchelor, M Bortz, X W Guan,
order. At subleading order, the correction coef- and N Oelkers. Evidence for the super
ficient depends on the combinatorics of N par- Tonks-Girardeau gas. J. Stat. Mech.,
ticles in W wells and on the parameters (τ, γ). 2005(10):L10001–L10001, oct 2005. DOI:
For example, for the simplest case of N = 2 and 10.1088/1742-5468/2005/10/l10001. URL
W = 2 depicted, we see the importance of states https://doi.org/10.1088/1742-5468/
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in corrections of order E for the length of that
[6] J. H. Becher, E. Sindici, R. Klemt,
diagonal.
S. Jochim, A. J. Daley, and P. M.
As a final comment, if the leading term of the Preiss. Measurement of identical par-
density of states is independent of the parame- ticle entanglement and the influence
ters (τ, γ), then the subleading term contains in- of antisymmetrization. Phys. Rev.
formation about the density of level crossings for Lett., 125:180402, Oct 2020. DOI:
the integrable models which presumably becomes 10.1103/PhysRevLett.125.180402. URL
level repulsions in the non-integrable parameter https://link.aps.org/doi/10.1103/
region. Future work will investigate this connec- PhysRevLett.125.180402.
tion between spectral flow, level crossings, and
[7] Andrea Bergschneider, Vincent M
energy level statistics.
Klinkhamer, Jan Hendrik Becher, Ralf
Klemt, Lukas Palm, Gerhard Zürn, Selim
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