Lattice scars: Surviving in an open discrete billiard

We study quantum systems on a discrete bounded lattice (lattice billiards). The statistical properties of their spectra show universal features related to the regular or chaotic character of their classical continuum counterparts. However, the decay dynamics of the open systems appear very different from the continuum case, their properties being dominated by the states in the band center. We identify a class of states ("lattice scars") that survive for infinite times in dissipative systems and that are degenerate at the center of the band. We provide analytical arguments for their existence in any bipartite lattice, and give a formula to determine their number. These states should be relevant to quantum transport in discrete systems, and we discuss how to observe them using photonic waveguides, cold atoms in optical lattices, and quantum circuits.

Introduction.-One of the most remarkable achievements in classical mechanics in the last century has been the establishment that the time evolution of certain dynamical systems is chaotic, i.e., it features an extreme sensitivity to initial conditions, usually portrayed by their Lyapunov exponents, which is a measure of an exponential divergence of trajectories in phase space. Even though the concept of trajectory no longer holds in quantum physics, quantum chaos, the quantummechanical study of classically chaotic systems, has also flourished [1]. Results of quantum chaos have been particularly remarkable in the study of billiards: domains wherein a particle moves ballistically except for elastic collisions with the boundary. One of the most surprising results in this field was the discovery by Heller [2] that the probability amplitude of certain wavefunctions-called "scarred wavefunctions" or, simply, "scars"-in a chaotic two-dimensional billiards is not uniform but concentrated along the trajectory of classical periodic orbits. This effect due to wave interference has now been observed in a number of systems, from microwaves in cavities [3], to electrons in quantum dots [4], to optical fibers [5].
The chaotic or regular properties of the dynamics in the closed system have important consequences when the system is opened. In particular, the decay properties of the particles inside a leaking billiard depend strongly on the system being regular or chaotic and on the presence of marginally stable periodic orbits (bouncing balls) [6]. With the development of atomic cooling and trapping techniques, beautiful experiments could be performed exploring different issues of quantum chaos [7]. The group of Nir Davidson confined rubidium atoms to a billiard realized by rapidly scanning a blue-detuned laser beam following the shape of the desired domain [8,9]. Opening a hole in the billiard, the number of atoms trapped as a function of time followed an exponential decay for chaotic domains, and a power-law decay for domains supporting stable trajectories [8]. They also showed the ap-pearance of islands of stability when the walls of chaotic billiards are softened [10] in agreement with theoretical arguments [11].
In this work, we study regular and chaotic billiards where the particle motion is restricted to a square lattice of discrete points. This model is adequate to describe several systems, including ultra-cold atoms trapped in optical lattices [12], cf. Fig. 1a, as well as the propagation of light along photonic waveguides [13,14], Fig. 1b. We consider billiards that in a continuum description feature regular and chaotic properties. By studying the statistics of their energy levels, we show that these behaviors are also present for the discrete case. Furthermore, we study the quantum dynamics in billiards with a leak localized on the border, and show that the population in both kinds of systems follows a similar trend: an initial exponential decay, followed by a power-law decay, until on occasions a final non-zero population is trapped in the system. We explain this unexpected behavior in terms of "lattice scars": scarred wavefunctions that are supported on the lattice structure and whose energy is at the band center, E = 0. We support our numerical findings with analytical arguments, and finally discuss their observability in several different atomic and optical setups.
Energy statistics.-We start by computing the eigenvalues E n and eigenfunctions ψ n of a lattice Hamiltonian where J lm (l, m = 1, . . . , N ) is the hopping amplitude from site m to site l, c m (c † m ) destroys (creates) a particle at site m, and the sum runs over all pairs of nearest neighbors of the N -points lattice. The shape of the domain is hence encoded in the hopping amplitudes or, equivalenty, on the set of neighbors of a given site.
Following the standard procedure, we unfold the set of eigenenergies into s n = (E n+1 − E n )/ E n+1 − E n , where the brackets · denote a local average. The Bosons are allowed to hop between sites with probability J, and there is a sink of particles (decay rate Γ) at a corner of the lattice. This model can be implemented using optical lattices (a), coupled waveguides (b) or coupled superconducting microwave resonators (c). In the first case, the sink can be implemented with a focused, resonant laser. For coupled waveguides, it is a guide with losses, while for (c) the loss comes from a resistor or a semi-infinite transmission line coupled to a few resonators.
normalized level spacing distribution P (s) for a continuum regular billiard follows a Poisson distribution, P P (s) = exp(−s) [15], while for chaotic billiards it follows the Wigner surmise, P W (s) = π 2 se −πs 2 /4 , from Random Matrix Theory (RMT) [16]. We show in Fig Similarly to the continuum case, we see that the former agrees well with a Poisson distribution, while the stadium presents a distribution in agreement with RMT. It is worth mentioning that we do not find any indication of the semi-Poisson behavior that was found for the very similar spin stadium billiard in Ref. [17]. Results for long range correlations calculated through the Power Spectrum P δ (k) of the δ n statistics (as defined in Ref. [18]) are shown in the inset of Fig. 2. The comparison with the theoretical results is very good. The decrease in the average value of the Power Spectrum for small values of the frequency k can be understood as the effect of bouncing ball orbits [19].
Dynamics in open systems.-Having established the static properties of the discrete rectangular and stadium billiards, we proceed now to analyze their dynamics in the presence of a leaking hole on the border of the billiard. We have studied the evolution of a localized wavepacket with initial momentum p 0 and width w, described by a pure state Here, H is given by Eq. (1) while γ k describes the loss rate: γ k = Γ within the leak-located on the billiard boundary and with radius σ = 2 in units of the lattice constant-while γ k = 0 otherwise [20]. This is equivalent to the evolution under an effective non-Hermitian Hamiltonian with imaginary on-site energies γ k . The number of particles remaining in the system after The average of N (t) over all possible positions of the hole, and over a range of initial momenta p 0 is shown in Fig. 3. For classical systems, one expects very different population dynamics for the two billiards [6,11,21]: a rapid exponential decay for the chaotic one, and a power-law decay for the regular one. For quantum systems, unless there is a large number of decay channels or holes, a purely algebraic decay is expected [11,22]. These predictions have been experimentally confirmed in previous experiments in a large variety of systems, from microwaves billiards [22] to cold atoms in optics billiards [9]. Here, we observe two features that strikingly contradict these expectations: (i) the population dynamics is similar for both discrete billiards, and (ii) there is a fraction of population that remains trapped for arbitrarily long times. Indeed, N (t) decays rapidly at short times tJ 1000, then it levels off, and finally saturates for tJ 10 4 [cf. Fig. 3, inset]. The numerical data is accurately fitted by the formula [23] For a given initial wavepacket ρ 0 and position of the hole, this can be rationalized in terms of the decomposition of ρ 0 over the eigenstates of the closed billiards with rapid (exponential) decay for short times, E, those with algebraic decay, A, and those that survive the presence of the leak for t 10 4 , S = N (t = 0) − E − A. The rapidly decaying eigenstates correspond to those that overlap the site where the leak opens, or to trajectories of the wavepacket that reach the hole after only a few bounces off the walls. Algebraic decay is associated with orbits that go through many bounces before leaking out [21]. Following this idea, we have performed a quantitative analysis of the eigenenergies, E open n = ε n + iΓ n , of the non-Hermitian Hamiltonian with imaginary on-site energies, for rectangular and stadium billiards. We find that the widths Γ n can be divided into three sets: (i) a very small number of states (2−8) with large imaginary parts, Γ n ≥ 10 −1 , which we expect to decay for times ≤ 10 1 . (ii) A large fraction ( 95%) of states with Γ n ≈ 10 −4 − 10 −1 which decay slowly and whose widths follow, in the chaotic case, a Porter-Thomas distribution as RMT predicts [1]. Finally, (iii) a small number of states with very small decay rates; among these, a few Γ n are numerically equivalent to zero.
Lattice scars.-The probability density of one of these non-decaying eigenstates, |ψ open n | 2 , in the rectangle (stadium) is shown in Fig. 4 (top, resp. bottom) when the leaking hole is at the top-right corner of the billiard. The long lifetime of these states is quickly understood by noting that their densities vanish at the position of the hole. Their spatial distribution on the billiard, however, is far from that of a "bouncing ball" orbit [11,21,22,25]. This is due to the restricted dynamics given by Eq. (1), that (2) at the indicated times. Note how the state on the rectangle is a superposition of the two scars. Lower density is indicated by blue (dark grey) and higher by lighter colors; maximum density is at red spots. Small white dots point the lattice sites, and the purple line is the circular edge of the stadium. See [24]. only allows to hop from one site to its nearest neighbor. This, together with the geometry of the square lattice, amounts to the system being bipartite on two disjoint sublattices, A, B. Application of a theorem by Inui et al. [26] implies then that in the closed system one can find r solutions to the Schrödinger equation at the band center, E = 0, which vanish on one of the sublattices, say ψ k∈B = 0. Here, r is the number of sites on the occupied lattice (here, N A ) minus the rank of the matrix J lm [26]. Once we open the system, a small portion of these degenerate states in the middle of the band acquires a large width with a purely imaginary eigenvalue, while the others remain at E = 0. The number of states at the band center that do not remain bound after opening the leak is given directly by the number of A sites overlapping the leak. These large-width states dominate the decay for short times. States remaining at (ε = 0, Γ = 0), on the other hand, will dwell in the billiard for very long times. We refer to them as lattice scars.
We have calculated the number of lattice scarred states for a wide range of system sizes; the results are presented in Fig. 5. We see that most rectangles feature no scars: they appear only when one side is close to an integer multiple of the other. In contrast, almost all stadia have at least one scar, with a larger number when M ≈ 2N , a trace itself of the embedded rectangle [27].
The spatial distribution of the bound E open = 0 states will hence reside in sites k ∈ A, which on a square lattice are linked by 45 • lines. This requirement, besides the boundary conditions appropriate to each billiard, which restrict the allowed "bounces" off the walls, results in trapped eigenfunctions such as those in Fig. 4. Indeed, we have independently checked that all states with Γ = 0 do have ε = 0. Moreover, we have also verified the prediction derived from the theorem in [26] that the number of states with ε = Γ = 0 equals r as defined above.
From the arguments above, we conclude that an initial state on an open discrete billiard will evolve until at long times t 1/J, all probability amplitude is concentrated on a (superposition of) scar state(s). This prediction is confirmed observing the probability density on the billiard of an initial state after evolving it according to Eq. (2), see Fig. 4: the resemblance of this with the eigenstate probability density is evident [24].
Physical implementations.-Our predictions above can be investigated using cold atoms trapped in optical lattices [28] (Fig. 1a), where single-site resolution for preparation and measurement has already been demonstrated in several labs [29]. A major challenge in these systems is to produce a lattice with a customized boundary, a task that can be achieved thanks to the improved optics in recent experiments, which allows projecting arbitrary optical potentials onto the trapping plane [29].
While these setups have the advantage of being dynamically tunable, our models can also be probed with all-optical systems. In the infrared or visible range, this can be done using photonic lattices [14,30], which are nothing but customized optical circuits with tens of waveguides imprinted on an appropriate substrate using a laser, cf. Fig. 1b. These waveguides can be arranged on parallel rows, forming a square lattice where light may tunnel between neighboring guides. Doping one or more of the waveguides, or coupling them to outgoing lines, would result in a sink. The paraxial propagation of light in this setup is described by Eq. (2), and the final population of bosons is the distribution of light at the substrate's end.
Finally, the same ideas can be studied using microwave quantum optics, Fig. 1c. Inspired by recent designs of coupled harmonic oscillators [31], we suggest creating a lattice of capacitively coupled microwave LC resonators. When the capacitive coupling is weaker than the on-site energy, the rotating-wave approximation will apply [32] and the resulting model will be once more a Bose-Hubbard model that describes the hopping of microwave photons in the array. The leak can be introduced using either resistive elements or outgoing cables that extract energy from a few sites. The distribution of energy can be measured using a probe antenna that is moved over the circuit to scan the electromagnetic field.
In conclusion, we have studied two quantum billiards on a square lattice: a rectangle and the Bunimovich stadium. We have shown that their level statistics agree with those of a regular and a chaotic billiard, respectively. However, the dynamics of a wavepacket on the open billiards turns out to be rather similar for both cases, and presents a number of, to the best of our knowledge, so-far unnoticed features. The most remarkable is the appearance of lattice scars: states found at the band center, and whose probability density collapses around spatially-concentrated orbits that live on only one of the two square sublattices. These allows them to survive for long times even in the presence of localized decay channels. We have also given an analytic formula to calculate the surviving population at long times. Finally, we have discussed three possible experimental setups, within current capabilities, to test our predictions.
Propagation through periodic lattices is a subject of interest in fields as diverse as biological molecules [33], optical waveguides [14,30], and solid-state [34] and coldatom systems [12], and we expect that this work will open new perspectives in the study of quantum dynamics in classically-chaotic regimes. In the near future, we plan to extend our analysis to interacting fields on the lattice, which will be relevant to new applications of the discrete non-linear Schrödinger equation [28] in the context of quantum nonlinear dynamics and quantum chaos.