Abstract
The decoupling of spin and density dynamics is a remarkable feature of quantum onedimensional manybody systems. In a fewbody regime, however, little is known about this phenomenon. To address this problem, we study the time evolution of a small system of strongly interacting fermions after a sudden change in the trapping geometry. We show that, even at the fewbody level, the excitation spectrum of this system presents separate signatures of spin and density dynamics. Moreover, we describe the effect of considering additional internal states with SU(N) symmetry, which ultimately leads to the vanishing of spin excitations in a completely balanced system.
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Introduction
Simplified quantum models of interacting particles in one dimension have been, for decades, a favorite starting point for theoretical physicists of different fields, such as particle physics and condensed matter^{1}. This is mainly due to the possibility of finding exact mathematical solutions for these models, which can provide insight into more complex problems. This is the case, for instance, of onedimensional models with contact interactions, for which the solutions can usually be found through the method known as the Bethe ansatz^{2}. As examples, we have the LiebLiniger model^{3,4} for bosons and the GaudinYang model for twocomponent fermions^{5,6}.
Additional interest has been drawn recently to this area due to the arrival of experimental techniques that allow for the construction of effectively onedimensional systems of cold atoms in optical lattices^{7}. These experiments feature a remarkable degree of control over different parameters, such as the confinement frequency, number of atoms and interactions between them^{8,9}. While theoretical models with exact solutions usually assume periodic boundary conditions, different approaches have made it possible to study onedimensional systems of atoms in confining geometries that range from the harmonic trap^{10,11,12} to doublewells^{13,14,15,16}, boxes^{17,18} and onedimensional lattices^{19,20}.
An effect of particular interest in this context is the phenomenon known as spincharge separation, where the degrees of freedom related to the density (“charge”, in analogy to electronic systems) and spin components can be described independently. In spinless systems, the quantization of phonon modes in manybody bosonic^{21,22} and fermionic^{23} ensembles has been wellestablished through the hydrodynamic approach. In onedimensional twocomponent systems, the separation of density and spin dynamics is captured by the celebrated TomonagaLuttinger Liquid (TLL) model^{24}. Theoretical proposals for the observation of spincharge separation usually employ the TLL description^{25} or assume weakly interacting manybody Fermi^{26} or Bose^{27} mixtures. Recently, the hydrodynamic approach has been also used to study these effects in the presence of temperature^{28}. The strongly interacting regime, however, is where this effect is expected to manifest itself more dramatically, with the freezing of the spin degrees of freedom as compared to the density dynamics^{29}. Additionally, many works have explored the possibility of describing a strongly interacting mixed system as an effective spin chain^{30,31,32,33,34,35,36}, where the separation of the spin and density sectors arises naturally in a dynamical context^{37,38}. Such a framework is therefore appropriate for studying this particular phenomenon.
Experimentally, ultracold atomic setups are good candidates for measuring spincharge separation in detail. Unlike experiments in condensed matter, ultracold atomic ensembles usually allow for a fine tuning of the interactions and the trapping geometries, as well as a precise control over the number of atoms^{39}. While spincharge separation has been observed in quantum wires^{40}, measurements with cold atoms are still in an early stage^{41,42}. Therefore, new proposals for the observation of this effect, which take into account the possibilities and limitations imposed by current experiments with cold atoms, are welcome. In this work, we provide such a proposal by presenting an analysis of the dynamics of a strongly interacting fewbody system of fermions with SU(N) symmetry after a sudden change  a quench  in the trapping potential. Recently, systems of cold atoms in optical lattices with SU(3) and SU(4) symmetries have been explored theoretically^{43,44}. The correlations of SU(N) impenetrable systems have also been calculated for an increasing number of internal components and display interesting properties^{45}.
We begin by presenting the formalism used to describe a system of strongly interacting atoms in a trap, where the Hamiltonian can be mapped into a spin chain with positiondependent exchange coefficients. We then describe the quench protocol, which essentially consists of changing the trap from a split well (where we assume a Gaussian barrier in the center of the system) to a simple harmonic well (see Fig. 1 for a sketch of this protocol). The groundstate configurations for these two systems are considerably different, and by changing the potential we can expect a nontrivial time evolution in the spin and density sectors. Initially, we describe the effect of the quench in the density sector and its consequences on the spin chain dynamics. By combining the dynamics of both sectors, we can extract the signatures of the separation between the density and spin oscillations in the system, showing how this effect can be observed in fewbody ensembles, even as the number of internal components is increased. Moreover, we show that for a completely balanced system (where each atom is in a different internal state) the spin signature vanishes, and the excitation spectrum is analogous to that of a gas of impenetrable bosons. Multicomponent cold atomic ensembles with strong interactions are currently within experimental reach^{46} and often exhibit exotic dynamical effects, such as edge states^{47,48}. In these systems, the internal states of the atoms can be manipulated with laser pulses, and the behavior of each component can be measured with precision. Studying cold atoms with different internal symmetries in a highly controllable environment can lead to insight on particle physics models and even shed light on the validity of unification schemes^{49}. Parts of the work presented here have been published, with modifications, in ref.^{50}.
Results
System description
Our goal is to describe the dynamics of a strongly interacting fewbody system with internal (“pseudospin”) degrees of freedom. We focus our description on a fermionic system, but the formalism is equally valid for bosons with the correct adaptations to the Hamiltonian. We consider initially an SU(2) system, where the internal degrees of freedom are described by ↑〉 and ↓〉. Later we will generalize our approach to systems with higher symmetries. We denote the particle numbers in each species by N_{↑} and N_{↓}, the total number of particles thus being given by N = N_{↑} + N_{↓}. For simplicity, we adopt the notation N_{↑} + N_{↓} (e.g. 3 + 1 for a system with three particles of species ↑〉 and one of species ↓〉). Experimentally, twocomponent fermionic systems can be realized by preparing trapped ensembles of ^{6}Li atoms in the two lowest hyperfine states^{51,52}.
The Hamiltonian for the system under consideration is given by
where \({H}_{0}(x)=\frac{{\hslash }^{2}}{2m}\frac{\partial }{\partial x}+V(x)\) denotes the single particle Hamiltonian, where V(x) is a trapping potential. The remaining term accounts for the contact interactions, where atoms in different internal states interact with strength g (interactions between atoms in the same internal state are forbidden due to the Pauli principle). Since we are dealing with atoms of the same element in different internal states, we consider all masses equal. In our calculations, we use the length, energy and time units of the harmonic trap, that is \({l}_{0}=\sqrt{\hslash /m{\omega }_{0}}\), ε_{0} = \(\hslash \)ω_{0} and T_{0} = 1/ω_{0}, respectively. The interaction strength g is given in units of \({\hslash }^{2}\)/ml. We also assume, for simplicity, \(\hslash \) = m = 1.
A case of particular interest in onedimensional systems is the limit of strong interactions (\(g\gg 1\)), where the Hamiltonian can be mapped to that of a spin chain^{32,33,34,35,53,54}. In the case of fermions, Eq. (1) can be written, up to linear order in 1/g, as^{36}
where P_{i,i+1} is the permutation operator, which exchanges two neighboring atoms of different components. When all interactions between atoms in different internal states are the same, the system exhibits an SU(N) symmetry^{55}. In the following sections, we focus only on strong repulsive interactions, and consider a fixed value of g in our calculations. For the formalism considered here, it has been shown that the static properties and spatial distributions are welldescribed already for g ~ 10^{32}. In the strongly attractive limit, the model described here corresponds to the fermionic analog^{56} of the SuperTonksGirardeau gas^{57}.
The exchange coefficients α_{i} are solely determined by the geometry of the trapping potential through the wave function for a system of spinless fermions, which we label Φ(x_{1}, ..., x_{N}). Obtaining the eigenstates of Eq. (2), we can calculate the spatial distributions of each atomic component in the trap as
where \({\rho }_{\uparrow ,\downarrow }^{i}(x)={m}_{\uparrow ,\downarrow }^{i}{\rho }^{i}(x)\), \({m}_{\uparrow ,\downarrow }^{i}\) is the probability of finding (↑, ↓) spins at site i and ρ^{i}(x) represents the spatial distribution of each individual particle in the trap. In Section 3 we provide details on how to calculate the exchange coefficients and the spatial densities.
Dynamics
We now describe the procedure that induces the dynamical evolution of the system, which consists of a sudden change of the trapping potential. Our initial choice of V(x) is given by a harmonic trap with an additional Gaussian bump in the center, as shown in Fig. 1(a). For t > 0, the bump is suddenly turned off. The time evolution of the spinless fermion wave function can then be described in terms of the evolution of the single particle orbitals under the same quench protocol^{19,58}. We are thus able to construct Φ(x_{1}, ..., x_{N}, t) for all times t, which in turn determines the timedependence of the exchange coefficients in Eq. (2). As a general rule, we can assume that the exchange coefficients are proportional to the overlap between the single particle distributions; additionally, since the trapping potentials are symmetric at all times, we have α_{i}(t) = α_{N−i}(t). In Fig. 2(a) we show the time evolution of the spatial densities obtained from Φ(x_{1}, ..., x_{N}, t). At this point, we are not considering the spin sector, so the densities shown correspond to the total (“charge”) density
which is normalized to the total number of particles. Its dynamical behavior is what should be expected for the coherent density oscillations of a TonksGirardeau gas^{58}. In panel (b), we see the behavior of α_{1}(t)/g and α_{2}(t)/g after the sudden change in the potential. We observe that the periodicity of the dynamics in the motion of the infinitely repulsive gas is reflected in the dynamics of the exchange coefficients. Particularly, since the system is approximately split in two parts at t = 0, we have that α(t = 0) ~ 0 (there is nearly no overlap between the initial densities at the center of the system). When the densities in the central region become larger, we see that the numerical value of α_{2}(t) surpasses that of α_{1}(t). This indicates that the spin correlations of the system should change between these two points, as we will see next.
Given the established timeperiodicity of Hamiltonian (2), we can analyze the dynamics of the spin sector using Floquet theory (see Section 3 for details). While most studies performed in this context must deal with the issue of thermalization due to the external driving^{59}, in our case the timedependence of the spin chain originates directly from the dynamics of the density sector generated by the sudden change of the trapping potential. To describe the time evolution of the system in these terms, we first find an analytical fit of the exchange coefficients in terms of Fourier modes. In Fig. 2(b), this approximation is shown as the black dashed curves. The full time evolution of the spin sector is performed through the numerical integration of the Schrödinger equation with the CrankNicolson method^{60}. Our quantities of interest are the dynamical densities ρ_{c}(x, t), ρ_{↑}(x, t) and ρ_{↓}(x, t), as well as the squared width of the distribution for each spin density, defined as \(\langle {x}_{\uparrow ,\downarrow }^{2}(t)\rangle =\int \,dx\,{\rho }_{\uparrow ,\downarrow }(x,t){x}^{2}\). While we choose to focus on the time evolution of paritysymmetric operators, other quantities (like, for instance, 〈x_{↑,↓}(t)〉) could also be explored. Such cases, however, would require a different choice of quench protocol or initial spin state to break the symmetry across the system.
SU(2)
We choose initially a fermionic system with SU(2) symmetry. In this case, the permutation operator is given by \({P}_{i,i+1}=\frac{1}{2}(1+{{\boldsymbol{\sigma }}}_{i}\cdot {{\boldsymbol{\sigma }}}_{i+1})\), which allows us to write the Hamiltonian as
where E_{0} has the same meaning as in Eq. (2). We notice that this Hamiltonian reproduces the fermionic cases presented, for instance, in refs^{33,37}. We focus mostly on the 3 + 1 case, which can be interpreted as a fewbody Fermi background in the presence of an impurity^{61}. As we will show next, in this simple setup it is already possible to describe the decoupling of spin and density dynamics. At the end of this section, we include some results for a 4 + 2 combination, which points to the possibility of realizing the protocol described here in larger systems.
The ground state of Hamiltonian (5) with repulsive interactions (g > 0) has antiferromagnetic correlations and can be described (in the 3 + 1 case) by
aside from a normalization factor. Here, we have assumed a homogeneous potential, such that the exchange coefficients are identical and equal to 1. However, it can be shown that similar results hold for a harmonic trap. To observe how the ground state correlations change with the choice of α_{i}, we define the operator P_{edge} = 〈↓↑↑↑gs〉^{2} + 〈↑↑↑↓gs〉^{2}, which provides information regarding the position of the impurity (the ↓ atom) in the system. In Fig. 3 we show the values of P_{edge} for different choices of α_{1} and α_{2}, where again we assume a paritysymmetric potential.
We can readily see that, for α_{1} = α_{2}, we have a constant result of \({P}_{{\rm{edge}}}=\frac{1}{4}\mathrm{(2}\sqrt{2})\). Above the diagonal (α_{2} > α_{1}) we have the region that includes, for instance, the coefficients obtained from a harmonic trap. In this case, the antiferromagnetic correlations are even more prevalent. If, however, \({\alpha }_{2}\ll 1\), we obtain a larger probability of finding the impurity at the edges. These ground state correlations are obtained when considering a potential such as the doublewell assumed for t = 0. In fact, we can plot the trajectory of the exchange coefficients after the quench in the trapping potential described above. This is shown by the dashed curve in Fig. 3. The white dot denotes the values at t = 0; the curve is then traveled back and forth periodically as t increases. The fact that this trajectory crosses over the diagonal indicates that the sudden quench in the potential should induce major changes in the spin correlations of the system.
To quantify this effect we now calculate the time evolution of the spin density and of the squared width for each component. It is important to notice that, for t > 0, the external potential is a simple harmonic trap. In Fig. 1(b,c), we show the time evolution of ρ_{↑}(x, t) and ρ_{↓}(x, t), respectively. We notice that, while the underlying dynamics seen in Fig. 2 is still present, we now have an additional oscillation mode. Specifically, after the sudden change in the potential, we observe a tendency of the majority atoms to spread to the edges, while the impurity localizes towards the center.
In Fig. 4(a) we show the time evolution of the squared width for the density of each component, over a larger time interval. This can be interpreted as induced breathing modes for the background and the impurity. Additionally, we show the dynamical behavior of the total density (Eq. (4)). Besides corroborating the results found in Fig. 1(b,c), these curves show how the fluctuations in the density and spin sectors are captured as two oscillations modes in the dynamics of the spin densities for each individual component. In Fig. 4(b,c) we show the Fourier transform of the width oscillations, defined as \({\tilde{x}}^{2}(\omega )=\int \,dt{e}^{i\omega t}\langle {x}^{2}(t)\rangle \), where the contributions of the spin and density excitations appear as two separate peaks, the lower frequency corresponding to the spin dynamics.
Here, we can see that the dynamics of the minority component is strongly dominated by the spin excitations. On the other hand, the majority component has a more balanced distribution of oscillations in the density and spin sectors. In Fig. 4(b,c), we additionally include the theoretical predictions for the density and spin oscillations (as black and gray dashed lines, respectively). The first is obtained by calculating the oscillation frequency of the single particle spatial orbitals and corresponds to the expected result for the TonksGirardeau or the spinless fermion gas, namely ω = 2ω_{0}. The second is extracted by calculating the gaps in the Floquet quasienergy spectrum of the timeperiodic spin chain. We point out that the coherence in the oscillations of each component over long times is due to the assumption that the spatial sector is described by a spinless fermion wave function, which in a real system is only approximately true for very strong interactions. Depending on the real value of the parameter g, we could expect to find excitation peaks which are slightly different than the ones predicted here. In our formalism, modifying g means changing the energy gaps in Eq. (5), which translates into a frequency shift in the spin excitation peaks (with the frequency decreasing as the interaction strength is increased).
In Fig. 4(d–f), we extend our results to the 4 + 2 case. The timedependence of the exchange coefficients for N = 6 is obtained by calculating these coefficients for one or two periods and then fitting a function by expanding the result in Fourier modes, as done in Fig. 2 for the N = 4 case. While the general behavior (relative magnitude and position of the excitation peaks) are maintained with respect to the N = 4 system, we find additional peaks in the lowfrequency side of the spectrum, which originates from having a larger number of particles in each component. Nevertheless, these excitations can also be captured by analyzing the Floquet quasienergy gaps of the timedependent spin chain. These results indicate that the protocol we employ is also suited for larger systems, provided that the imbalance in spin the populations is respected. In the next sections, we show how increasing the number of internal components will affect the behavior of these quantities.
SU(3)
We now consider the case of a threecomponent strongly interacting fermionic gas with SU(3) symmetry. These systems are particularly interesting due to their connections to the quark model in the framework of quantum chromodynamics. We label the three internal states as ↑〉, →〉 and ↓〉. While the Hamiltonian can still be described by Eq. (2), the permutation operator is now given by
where λ is the vector composed by the eight generators of the SU(3) group, namely the GellMann matrices. A system described by Eq. (7) can be mapped into the LaiSutherland model^{62,63} through P_{i,i+1} = S_{i} ⋅ S_{i+1} + (S_{i} ⋅ S_{i+1})^{2}−1, which is a particular case of the spin1 bilinear biquadratic model^{64,65}. We keep the number of particles fixed as N = 4, with N_{↑} = 2, N_{→} = 1 and N_{↓} = 1. The quench protocol and the quantities considered are the same as in the previous section.
In Fig. 5(a), we show 〈x(t)^{2}〉 for each component and for the total density. Since we have two minority particles, each interacting with the remaining atoms with interaction strength g, the results for each of these components are identical. The excitation peaks seen in Fig. 5(b–d) reveal the contributions of the density and spin oscillations to the dynamics of each component (naturally, since we have N_{→} = N_{↓} = 1 plots (c) and (d) show identical results). Still, we can see that the majority component has a larger contribution to the total density excitations. The minority cases, however, show a slight increase in these frequencies as compared to the twocomponent case, with the spin oscillations remaining dominant.
An interesting perspective when dealing with multicomponent strongly interacting gases is a case where interactions are slightly imbalanced and a particular symmetry is broken. Here, we analyze the threecomponent case with broken SU(3) symmetry. It is useful, in this context, to rewrite the SU(3) permutation operator in terms of raising and lowering operators. These are defined as T ^{±} = (λ^{1} ± iλ^{2})/2, V^{±} = (λ^{4} ± iλ^{5})/2 and U^{±} = (λ^{6} ± iλ^{7})/2, where once again λ^{i} are the GellMann matrices. We still focus the particular case of N = 4, with N_{↑} = 2, N_{→} = 1 and N_{↓} = 1.
Below we rewrite the permutation operator with these modifications, including an additional symmetrybreaking parameter 1/η.
The inclusion of η above means we are explicitly breaking the symmetry of the system by changing the energy contribution of turning ↑〉 into →〉 and viceversa. In Fig. 5(e–h) we show the result of breaking the SU(3) symmetry (by making η = 0.5) on the dynamics. While the effects in the ↑〉 and →〉 components are subtle  an increase in the spin excitation peak as seen in panels (f) and (g)  in ↓〉 it is more drastic, with the spin contributions being distributed over different low frequencies. The remaining components still preserve isolated peaks for spin oscillations. These results point to the possibility of measuring the independence of spin and density dynamics even in a context where internal symmetries are not perfectly preserved. Naturally, different outcomes for the spin excitations can be expected by choosing a different value of η, or by breaking the symmetry in a different interaction channel.
SU(4)
We now examine the effect of applying our formalism to the case where the number of particles N matches the number of internal components. To that end, we consider the SU(4) fermionic gas with N = 4 and internal states labeled as ↑〉, ↗〉, ↘〉 and ↓〉. The number of particles in each state is thus given by \({N}_{\uparrow }={N}_{\nearrow }={N}_{\searrow }={N}_{\downarrow }=1\) (the socalled 1 + 1 + 1 + 1 infinitely repulsive system with different masses is known to have interesting properties, which were described in^{66}). We rewrite the permutation operator for the SU(4) system as
where now λ represents the vector spanning the 15 SU(4) generators^{67}. In the following, we focus on describing the results only for the ↑〉 and ↗〉 components.
It becomes clear that the behavior 〈x^{2}(t)〉 as a function of time is the same for both components shown in Fig. 6 (this also holds for the other two components not shown). This is expected since the number of internal components matches the total number of particles in the system. Moreover, the frequency spectrum shows that the only contributions in the oscillations stem from the density excitations, as opposed to the previous cases. This allows us to interpret the dynamics of the SU(N) system with strong interactions as the one expected for a gas of impenetrable bosons, as long as the number of particles matches the number of internal components. This conclusion is in agreement with the observation that breathing mode frequencies of the SU(N) Fermi gas with strong repulsion approaches that of a TonksGirardeau gas^{46}.
In the models considered here, a vanishing spin signal in the spectrum can additionally be obtained by taking a balanced system with a lower number of internal components (e.g. a 2 + 2 SU(2) system). This can be explained as a result of the symmetric perturbation to the potential that determines the initial state of the system. Turning off the barrier in this particular case has no effect on the ground state of the spin chain, which remains unchanged even as the density oscillations take place. However, for a matching number of particles and internal components, the results described in this section are the only possible outcomes for a system initialized in the ground state.
Discussion
We have presented an analysis of the dynamics of density and spin fluctuations in trapped fewbody systems with SU(N) symmetry. In the limit of strong interactions, the Hamiltonian of this system can be mapped into a spin chain. This mapping allows for a straightforward generalization to systems with more internal components. In addition, it takes into account the geometry of the trapping potential into the set of exchange coefficients of the spin chain.
The dynamics of the system is obtained after a quench in the trapping potential, where a Gaussian barrier in the center of the harmonic trap is suddenly switched off. This simple protocol is particularly interesting from an experimental point of view, since it requires only minor modifications to the potential, without resorting to spinselective traps. The sudden change induces the motion of the spatial degrees of freedom, which in turn are reflected in a timedependence of the exchange coefficients of the spin chain. It is important to point out that, since the system is initialized in the ground state of the spin chain, the motion observed in this sector is only possible due to the quench in the potential. By monitoring the time evolution of the breathing modes given by the oscillations in 〈x^{2}(t)〉, we describe the excitation spectrum of SU(2), SU(3) and SU(4) systems. Moreover, in the particular case where the number of internal components exactly matches the number of particles, we see that the spin excitations are completely washed out, and the only contributions are due to density oscillations that agree with those of a spinless Fermi gas. While the theoretical scheme presented here may in principle be hard to generalize to a large number of particles (specially due to the difficulties in calculating the dynamical exchange coefficients over long periods of time), a numerical manybody approach employing the same dynamical protocol may be able to do so. Aside details regarding the true strength of the interactions between nonidentical particles, we expect a similar behavior to arise in this context, provided that the same essential ingredients are maintained.
Our results indicate that the decoupling of spin and density dynamics, rather than being exclusively a bulk effect in manybody ensembles, can occur in fewbody systems under fairly simple conditions. We point out that multicomponent fermionic systems are the object of current experimental investigation, and the scheme presented in this work could serve as a starting point for experiments with manybody ensembles. The formalism we explored can also be used to predict the behavior of quantum gases with different atomic species (e.g. a bosonic mixture), or generalized to more involved quench protocols, simply by mapping the time evolution of the spatial orbitals into the exchange coefficients of the spin chain under consideration.
Methods
Details for strong interactions
The exchange coefficients in Eq. (2) are determined by the geometry of the trapping potential, and can be calculated as^{33}
where the spinless fermion wave function Φ(x_{1}, ..., x_{i}, ..., x_{N}) is built as the Slater determinant of the N lowest occupied orbitals of the potential V(x). The corresponding energy of this state is given by E_{0} in Eq. (2), which is calculated as the sum of the energies of each occupied level. This wave function can be simply viewed as the antisymmetrized version of the TonksGirardeau wave function for impenetrable bosons^{68}. Methods for obtaining the exchange coefficients in different trapping geometries are available and can efficiently calculate α_{i} for systems with up to N ~ 30^{69,70}.
The spin densities given by Eq. (3) require calculating the spatial distribution of each individual particle in the trap, which is given by
The expressions above involve multidimensional integrals of the spinless fermion wave function Φ(x_{1}, ..., x_{N}), which for large systems can become increasingly hard to calculate. Moreover, if we consider timedependent orbitals ϕ_{j}(x, t), these expressions must be evaluated for all times after the sudden change in the trapping potential. Therefore, we explore the determinant form of the spinless fermion wave function Φ(x_{1}, ..., x_{N}) to rewrite these equations in a shape which is suited to calculations in a dynamical context. Equivalent forms of these expressions have been presented in^{53,69,70}. The individual onebody densities can be written as
while the exchange coefficients may be obtained from
where the matrix B(x) is composed by the singleparticle states superpositions \({b}_{mn}(x)={\int }_{\infty }^{x}\,dy\,{\varphi }_{m}^{\ast }(y){\varphi }_{n}(y)\), and the subscript jk on the right side of Eq. (13) indicates that the jth row and kth column are removed. Although we omit the time in these expressions, we consider the singleparticle orbitals ϕ_{i}(x) to be timedependent. In Eq. (3), we calculate the spin densities by combining the singleparticle spatial distributions with the probability of finding a given spin component at each site. This last quantity can be calculated, for a given spin state \(\psi (t)\rangle ,{\rm{as}}\,{m}_{\uparrow ,\downarrow }^{i}(t)=\langle \psi (t\mathrm{)(1}\pm {\sigma }_{z}^{i}\mathrm{)/2}\psi (t)\rangle \).
Details of the quench protocol
The trapping potential at t = 0 is given by
where V_{0} determines the height of the Gaussian peak and s sets its width. The system is therefore separated in an effective doublewell by taking ω_{0} = 1, V_{0} = 25 and s = 0.1. The initial spinless fermion wave function Φ(x_{1}, ..., x_{N}, t = 0) is constructed with the single particle orbitals obtained by numerical diagonalization, using the N_{s} = 35 lowest energy states of the harmonic oscillator. We note that, since the Gaussian peak is large compared to the individual densities of the orbitals, the ground state is quasidegenerate (the two lowest energy states have nearly the same distribution, with opposite parity).
For t > 0, we make V_{0} = 0 and the spinless fermion wave function is constructed by considering the timeevolved single particles orbitals ϕ_{j}(x, 0) according to
where \({c}_{n}=\int \,{\psi }^{\ast }(x)\varphi (x,\mathrm{0)}\,dx\) and ψ_{n}(x) are the eigenstates of the harmonic oscillator, with ε_{n} the corresponding eigenvalues.
Floquet quasienergy spectrum for a time dependent hamiltonian
To obtain the frequency of spin oscillations for the timedependent spin chain, we focus on finding the Floquet modes for the periodic Hamiltonian and calculating the gaps in the Floquet quasienergy spectrum^{71}. Generally speaking, we are interested in finding the time evolution operator at any time t by solving the following differential equation for U(t, t_{0}):
By obtaining and diagonalizing U(T, 0) (where T is the period of the Hamiltonian) we obtain a set of quasienergies ε_{n} and Floquet modes at t = 0, which we write as u_{n}(0)〉. The time evolution of the system is then given by
where u_{n}(t)〉 denotes the timeevolved Floquet modes and c_{n} = 〈u_{n}(0)ψ(0)〉. Once we have calculated ε_{n}, we can easily obtain the energy gaps that contribute to the time evolution of the system by considering the dominant contributions given by c_{n}. In a small system, this is a fairly straightforward process, since the initial state projects only onto a few Floquet modes.
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Acknowledgements
Parts of this work have been published, with modifications, in ref.^{50}. The authors thank Xiaoling Cui, XiWen Guan and David Petrosyan for their useful comments on the manuscript. Leonardo Fallani, Jacopo Catani and Artem Volosniev are also thanked for interesting discussions on spincharge separation. The following agencies  Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES), the Danish Council for Independent Research DFF Natural Sciences and the DFF Sapere Aude program  are gratefully acknowledged for financial support.
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Barfknecht, R.E., Foerster, A. & Zinner, N.T. Dynamics of spin and density fluctuations in strongly interacting fewbody systems. Sci Rep 9, 15994 (2019). https://doi.org/10.1038/s41598019523922
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DOI: https://doi.org/10.1038/s41598019523922
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