Single- to many-body crossover of a quantum carpet

Strongly interacting many-body system of bosons exhibiting the quantum carpet pattern is investigated exactly by using Gaudin solutions. We show that this highly coherent design usually present in noninteracting, single-body scenarios gets destroyed by weak-to-moderate interatomic interactions in an ultracold bosonic gas trapped in a box potential. However, it becomes revived in a very strongly interacting regime, when the system undergoes fermionization. We track the whole single- to many-body crossover, providing an analysis of de- and recoherence present in the system.

Introduction-Periodic self-replication of physical systems have been known since at least nineteenth century when Henry Fox Talbot discovered spatially repeating patterns in his experiments with diffraction gratings [1,2]. Such recurrences are most often associated with underlying wave nature of the system with light being the most straightforward example. As such, it comes as a no surprise that also quantum mechanical objects like ultracold atoms exhibit similar self-repeating behavior, however also in a time domain, giving rise to a rich family of phenomena, among others, quantum fractals [3,4], quantum echoes [5], quantum Talbot effect [6], and quantum scars [7,8], collectively addressed as quantum revivals [9,10].
However, recent study showed that even in the simplest many-body system of ideal fermions interesting phenomena arise [33]. The quantum carpet design gets much more pronounced, with canals and ridges becoming solitonlike-narrower and more distinct (see Fig. 1(d)). Such a fermionic quantum carpet appears when an ultracold fermionic gas is initially trapped in a box and then instantaneously released to a bigger one. Such a scenario is necessary to sustain a quantum carpet design in the large particle number limit, as other initial trappings cause destruction of a regular pattern. This strategy of preparing the nonequilibrium state might strike as troublesome, however experimental techniques involving atomic diffraction gratings or phase imprinting may reproduce similar dynamics [34][35][36][37][38][39][40].
In our work, we show that at the level of a singleparticle density, a gas of strongly interacting bosons in the limit of fermionization retrieves the same fermionic carpet behavior. We stress the distinction between this strongly correlated case and prototypical one of ideal fermions, as the differences appear already for a reduced single-particle density matrix, uncovering underlying structures of correlations and coherence. Moreover, as a noninteracting bosonic cloud exhibits a typical single-particle quantum carpet pattern, for a finite interaction strength between atoms there is some nontrivial crossover between a single-and many-body carpet of fermionized particles.
We perform the analysis of this crossover basing on Gaudin solutions from the Bethe Ansatz [41][42][43][44][45][46][47], which allows us to have access to exact dynamics after arbitrary long evolution time, contrary to previous approaches based on approximate methods. We utilize this advantage to investigate short and long time intrinsic decoherence in the system caused by introduction of interaction and eventual recoherence that happens in the nearly fermionized regime. Along the lines of previous studies [17], we provide an explicit physical mechanism in which quantum carpets give access to coherence and show how the de-and recoherence can be probed for highly nonequilibrium many-body states.
Fermionic quantum carpet of fermionized bosons-Throughout this work, we will consider a very particular scenario of exciting the many-body quantum system. Let the quantum gas of N atoms be initially confined inside a box potential with the length of D < L and stay in its ground state (throughout the work we consider a particular value D = 0.21L). Then, it gets released into a bigger box with the length of L that shares one of its walls with the initial one. When looking at the single particle density of this quantum gas in a spatiotemporal plot, arXiv:2011.04582v1 [cond-mat.quant-gas] 9 Nov 2020 such an excitation scheme produces a very thoroughly studied single-body quantum carpet pattern if the gas consists of ideal bosons ( Fig. 1(a)). On the other hand, if ideal fermions are considered, the pattern changes, getting more pronounced, yielding so-called fermionic quantum carpet ( Fig. 1(d)). It is necessary to note that such a behavior in the fermionic and fermionized cases is specific for this type of excitation-should the gas be initially confined in e.g. harmonic trap, the regularity would be lost [33].
The wave function of ideal gas of fermions is given by the single Slater determinant: where φ i (x), i = 1, ..., N denote different, orthonormal orbitals. When released into a box with the length of L, each of the orbitals starts to evolve . . , are mode solutions of a large box, where θ(x) is a Heaviside step function and eigenenergies read E k = k 2 π 2 2 /2mL 2 .
Λ n,k ≡ ϕ k |φ n are overlaps between initial orbitals φ n and box trap eigenfunctions ϕ k . Now, let us consider a case of fermionization, when bosons interact repulsively with an infinite strength.
Then, the wave function is given by a Tonks-Girardeau form: [48,49]. The Slater determinant is preceded by an unit antisymmetric function guaranteeing correct bosonic symmetry of the whole wave function. The orbitals and their evo-lution are the same in both cases and so is single-particle density, ρ(x, t) = N n=1 |φ n (x, t)| 2 , producing fermionic quantum carpet as seen in Fig. 1(d).
The difference appears at the level of a reduced singleparticle density matrix, (1) which can be analytically evaluated in the ideal fermionic case (see Supplemental Materials (SM) for derivation), where v 0 = π /2mL is the characteristic velocity that is connected to time of the system's revival T rev = 2L/v 0 = 4L 2 m/π . The density matrix can be expressed as a sum of travelling contributions that move with the velocities that are multiples of v 0 , where p denotes each of these terms. Each travelling contribution is associated with rectangular-shaped breathing structure in the density matrix, whose shape is roughly described by Fig. 2(e)). These structures are connected to the solitonlike objects from the singleparticle density and their presence is a signal of a coherence in the system. The spatial coherence is present all across the system in the ideal bosonic and ideal fermionic cases, however the structure of coherence is visibly different-in the latter case rectangular-shaped, narrow structures are present signifying correlations between solitonlike canals and ridges from the fermionic quantum carpet. Fermionized bosons do not recreate this type of coherence, with off-diagonal parts decaying as one goes away from the diagonal.
The time evolved density matrix in case of fermionized bosons cannot be cast into a simple form of Eq. (2). However, its fast numerical evaluation is possible, which we take advantage of [50]. The rectangular-shaped structures are suppressed as the off-diagonal terms decay the faster the further they are from the diagonal, which is a characteristic feature of a Tonks-Girardeau gas. However, temporal coherence is preserved as the system revives exactly at the multiples of the revival time, T rev . Note the revival time is the same for ideal bosons, ideal fermions and fermionized bosons.
The system with a finite interaction-Dynamics of highly nonequilibrium and strongly interacting systems, such as quantum carpets, is demanding to treat exactly [51][52][53]. Even numerically exact methods, such as a Hamiltonian diagonalization or cluster expansions suffer from large computational costs close to infinite interaction and proved to be too inefficient to provide reliable results. One of the examples of the system that is integrable and allows for an exact treatment is the Lieb-Liniger model of N bosons interacting by a contact delta potential: where γ is a dimensionless parameter quantifying interaction strength. Here we consider less explored case of atoms contained within a box with the length of L under open boundary conditions. Such a system can be solved exactly by means of the Bethe Ansatz in the form of Gaudin solutions, that take analytical, however complicated form [43,46] (see SM for details). It allows us to analyze dynamics generating quantum carpet pattern exactly for an arbitrary time and an arbitrary interaction strength. We investigate up to 3 atoms as computational expenses grow exponentially with the number of atoms [54], however we argue that behavior of much larger system can be tracked already from such a few body approach.
The time-dependent observables we consider include a square of an autocorrelation function [10]: that quantifies the coherence over time, manifesting how close the initial wave function is reproduced near the revival time, single-particle density ρ, and entanglement entropy, where λ j are eigenvalues of a single-particle density matrix. The last quantity allows us to analyze how the coherence behaves in the system as it is connected to the number of natural orbitals with a significant contribution. First, we take a look at the reduced single-particle density matrix (see Fig. 2) and the quantum carpet design coming from its diagonal (see Fig. 1). Not surprisingly, the weak interaction destroys the ideal single-particle quantum carpet, blurring the plot after a long evolution time. However, the regularity becomes revived as the system becomes strongly interacting, eventually arriving at the fermionic quantum carpet in the case of fermionization. At the level of the square of the autocorrelation function, one can also see that for a finite interaction, revivals become imperfect. As for the density matrix, situation is different. It is explicitly spatially coherent for a noninteracting system and its coherence length becomes visibly shorter for growing interaction, achieving its fermionized value for an infinite repulsion. It shows that while the time coherence is restored, the spatial one is lost, submitting to the general feature of a fermionized system. . Left inset shows evolution of entropy divided by its initial value for different interaction strengths. We observe the rapid growth of that quantity for intermediate interactions for which time coherence is lost over time. One can see that interaction introduces temporal incoherence in the system, which is later suppressed while entering coherent, fermionized regime.
Secondly, we analyze the behavior of a revival time as a function of interaction strength (see Fig. 3(a)). By plotting A(t) close to the revival time T rev (insets in Fig. 3(a)) for different interaction strengths, one can see that for a weak interaction perfect revival is destroyed as the maximum value of A decreases with the interaction strength. Nonetheless, the revival time is not shifted and remains at T rev . At the fermionization, system is again perfectly revived at T rev , however when the interaction strength becomes smaller, imperfect revival occurs at times slightly larger than T rev . Additionally, for intermediate interactions, another imperfect revival occurs, however its maximal value is smaller for three atoms than for two of them, suggesting that it may disappear in the limit of large number of atoms.
Thirdly, we look at the entanglement entropy (see Fig. 3(b)). For a noninteracting system it remains constant during the time evolution. In a weak-tomoderate interaction regime, the entanglement entropy rises rapidly in comparison to its initial value (left inset in Fig. 3(b)) and for strong enough interaction, it saturates at some value at times larger than the revival time (right inset in Fig. 3(b)). The situation is different for weak interactions, where we have checked that entropy does not thermalize even at very long times. Its value, averaged over one revival time, is not a monotonic function of the interaction strength-it has a distinctive maximum at some intermediate interaction, γ ≈ 15, which corresponds to disappearance of the revival visible at the level of the square of autocorrelation function. For a very strong interaction, system gets closer to the fermionization regime in which the entanglement entropy exhibit stable, oscillatory behavior, characteristic for the Tonks-Girardeau wave function.
Each of these signatures-regularity of a quantum carpet pattern, presence of the quantum revival and nonmonotonic behavior of the time-averaged entanglement entropy-heralds the loss of temporal coherence in the system due to interaction and then its reappearance in the strongly interacting regime. In previous studies, it has been shown that a quantum carpet design is a purely interference effect and as such it can serve as an alternative probe for decoherence effects that does not rely on the reconstruction of the Wigner function [17]. In such an approach, the decoherence present in the system is probed via access to the single-particle density, which is well suited to e.g. experiments involving diffracton gratings that can reproduce similar revival dynamics.
Here, we present a model in which temporal coherence of the system is affected by the interparticle interaction and a quantum carpet pattern is accordingly smeared out. Such a scenario could be realized not only in manybody atomic systems, but also in e.g. effectively interacting photons, allowing one to probe effect of interaction on coherence through diffraction and then free evolution. Coherent diffraction has been achieved also in other systems involving atoms [55], large molecules [56,57], light [58] and electrons [59]. Quantum carpets provide a natural tool to investigate different decoherence scenarios in these systems, among others due to interaction as presented in here.
Conclusions and outlook -We have analyzed exactly a single-to many-body crossover of a highly nonequilibrium state by investigating a spatiotemporal design known as a quantum carpet. The latter can be realized in plethora of systems, both for matter and light waves and might be utilized as a decoherence probe that provides a direct access to the Wigner function of the considered state. We have shown that interparticle interaction not only destroys the coherence in the system, but, if strong enough, can make the system coherent again. Such a behavior is possible due to a particular excitation scheme that allows a quantum carpet of fermionized bosons to preserve regularity. We have provided a thorough analysis of the crossover between bosonic and fermionic quantum carpets that emerge under such a scenario.
As a future line of work, quantum carpets can be investigated in other many-body scenarios involving different interactions encountered in atomic systems, such as attractive or dipolar potentials. Moreover, two-and three-dimensional geometries would also produce different type of decoherence and affect the quantum carpet design differently.

Supplemental Materials
Appendix A: Deriving ρ(x, x , t) We now proceed to evaluate reduced single-particle density matrix for noninteracting fermions. If φ i (x , t) is a time-dependent natural orbital (in ideal fermionic case natural orbitals are just initial single particle orbitals), then the reduced single-particle density matrix reads where µ i = 1/N are equal weights. We can define parts ρ i of the full density matrix, associated with a given initial orbital: After the release from the initial confinement, it can be then expanded in single-particle orbitals of the larger box: The real part of a given contribution can be written as Analogously for the imaginary part: with The function I kl can be evaluated as With the following identitiesĨ we can immediately writeĨ Then, we can reorganize terms to get By analogous procedure for J kl we get It allows to write the whole reduced single-particle density matrix in a concise form: The solutions in terms of the Bethe Ansatz functions were constructed by M. Gaudin [43,46] (21) By {x} we denote x 1 , . . . , x N , integers i , i = 1, . . . , N take two values ±1 and summing over { } means sum over all 2 N possibilities 1 , . . . , N . The sum over P runs through all permutations P ∈ S N and constants N {k} assure proper normalization. The states are parametrized by sets of N positive numbers {k} called quasi-momenta obtained by solving equations Set {k} directly corresponds to the set {n}, 1 ≤ n 1 ≤ . . . ≤ n N which can be understood as a set of quantum numbers for our system. Eigenenergies are equal to In the remaining part we will often encounter Ndimensional integrals and it is useful to note that for Bose-symmetric function F ({x}) integration over N dimensional hypercube L N can be transformed [52] into integration over the fundamental domain We use that property to calculate normalization constants where we have introduced The property (21) allowed us to get rid of sign functions appearing in wavefunctions. The integrals (25) can be calculated analytically.

Initial state
In our paper we consider scenario of geometric quench, therefore at the beginning of the evolution the whole gas is contained in a box of smaller length D < L. Our initial state correspond to the ground state of such a system, quasi-momenta are obtained from where we put n i = 1, i = 1, . . . , N . We then calculate the normalization constant of the state N k 0 similarly to (24). As we want to track the evolution of our state in bigger box of length L we need to represent the initial state with eigenstates of bigger box here sum over {k} in principle runs through all solutions of equations (22). Coefficients C {k} are calculated in the following way 1 + ic P i k 0 P i − P j k 0 P j I N P 1 k 0 P 1 − σ Q1 k Q1 , . . . , P N k 0 P N − σ QN k QN , D .
In practice it is sufficient to take finite number M of lowest energy states (for two particles M ≈ 250, for three M ≈ 7000 to get convergence of autocorrelation function in the fermionized regime). Such a representation allows us to evolve state easily Ψ({x}, t) = M u=1 e −iEut C u ψ u ({x}) (29) we have changed indices in order to emphasize that now we work with finite sums.

Time-dependent observables
The quantity that is particularly easy to calculate is the autocorrelation function Next, we go to single-particle density ρ(x, x , t) = We may use property (21) to simplify the integral Explicit form of time-evolved single-particle density reads The integrals involving sign functions can be calculated analytically upon dividing the integration region into parts where sign functions do not change its values. Thus, all results for single-particle density presented in our paper are obtained from evaluation of analytical expressions. Having full matrix (33) calculated for a finite number of points N grid = 101 we diagonalize it obtaining eigenvalues λ j (t) used to calculate entropy