Floquet realization and signatures of one-dimensional anyons in an optical lattice
We propose a simple scheme for mimicking the physics of one-dimensional anyons in an optical-lattice experiment. It relies on a bosonic representation of the anyonic Hubbard model to be realized via lattice-shaking-induced resonant tunneling against potential off-sets created by a lattice tilt and strong on-site interactions. No lasers additional to those used for the creation of the optical lattice are required. We also discuss experimental signatures of the continuous interpolation between bosons and fermions when the statistical angle is varied from 0 to . Whereas the real-space density of the bosonic atoms corresponds directly to that of the simulated anyonic model, this is not the case for the momentum distribution. Therefore, we propose to use Friedel oscillations in the density as a probe for continuous fermionization of the bosonic atoms.
Fundamental particles in nature are either bosons or fermions. Bosons obey Bose-Einstein statistics such that their joint wavefunction is symmetric with resepct to the exchange of two particles, whereas fermions obey Fermi-Dirac statistics and the wave function picks up a minus sign under particle exchange. In two dimensions, also anyons would be possible fundamental particles. They obey a fractional statistics that interpolates between bosonic and fermionic behavior Leinaas and Myrheim (1977); Goldin et al. (1981); Wilczek (1982); Tsui et al. (1982); Canright et al. (1989). If two anyons exchange their position, the wave function pics up a phase. Practically, anyons play a major role as quasiparticles of topologically ordered states of matter such as fractional-quantum-Hall states Laughlin (1983); Halperin (1984); Camino et al. (2005), with potential applications in robust topological quantum information processing Kitaev (2003); Das Sarma et al. (2005); Bonderson et al. (2006); Stern and Halperin (2006); Nayak et al. (2008); Alicea et al. (2011); Stern and Lindner (2013); Matthews et al. (2013). As shown by Haldane, for quasiparticles the concept of fractional statistics can be extended to arbitrary dimensions Haldane (1991). One-dimensional (1D) anyons have recently attracted an increased attention Ha (1994); Murthy and Shankar (1994); Wu and Yu (1995); Zhu and Wang (1996); Amico et al. (1998); Kundu (1999); Batchelor et al. (2006); Girardeau (2006); Calabrese and Mintchev (2007); del Campo (2008); Hao et al. (2008, 2009); Hao and Chen (2012); Wang et al. (2014a); Tang et al. (2015); Zhang et al. (2015), including two proposals for their implementation with bosonic atoms in an optical lattice Keilmann et al. (2011); Greschner and Santos (2015). These proposals are based on mapping the anyons via a generalized Jordan-Wigner transformation to bosons with a density-dependent tunneling parameter to be engineered by laser-dressing of internal atomic degrees of freedom. However, an experimental realization has not yet been achieved.
In the following, we propose a simple alternative scheme for the experimental realization of 1D anyons, based on time-periodic forcing. It is feasible in existing experimental setups and, in contrast to earlier proposals, does neither rely on the internal atomic structure nor require any lasers additional to those creating the optical lattice. Our scheme is based on engineering an occupation-dependent Peierls phase of the tunneling matrix elements by means of coherent lattice-shaking-assisted tunneling against potential offsets, which are created by a combination of a lattice tilt and strong on-site interactions.. The scheme, which is applicable in the low-density regime, also permits to effectively tune the interactions between the anyons. The fact that periodic forcing has recently been employed already experimentally for engineering both number-dependendent tunneling amplitudes Ma et al. (2011); Meinert et al. (2016) and non-number-dependent Peierls phases Aidelsburger et al. (2011); Struck et al. (2011, 2012, 2013); Aidelsburger et al. (2013); Miyake et al. (2013); Atala et al. (2014); Jotzu et al. (2014); Aidelsburger et al. (2015); Kennedy et al. (2015) (see also Ref. Eckardt (2015) for an overview of Floquet engineering in optical lattices) indicates that the proposed creation of number-dependent Peierls phases by such means is feasible.
We, moreover, discuss experimental signatures of the anyonic model in its ground state using exact diagonalization. Considering small chains, as they can be realized in quantum-gas microscopes Islam et al. (2015); Fukuhara et al. (2015), we monitor experimentally measurable observables that directly reflect anyonic properties and are invariant under the Jordan- Wigner transformation from anyons to bosons. This excludes the momentum distribution, which is altered by the transformation so that the measurable bosonic momentum distribution does not correspond to that of the anyons. It includes, however, on-site densities and their correlations, as well as the second Rényi entropy of the subsystem given by the first sites. We show that in small systems Friedel oscillations can serve as a signature for the continuous fermionization occurring when the statistical angle is varied from to .
The Hubbard model of one-dimensional lattice anyons with on-site interactions Keilmann et al. (2011) takes the form
Here the annihilation and creation operators, and , for anyons at site obey the commutation relations and , which are parametrized by the statistical angle . Here for , respectively, so that on-site the particles behave like bosons. Thus even for , these lattice anyons are just pseudofermions, since many of them are allowed to occupy the same site. Following references Kundu (1999); Keilmann et al. (2011), the anyonic model can be mapped to the bosonic model
via the generalized Jordan-Wigner transformation . Here the anyonic exchange phase has been translated to a density-dependent Peierls phase: when tunneling one site to the right (left), a boson pics up a phase given by () times the number of particles occupying the site it jumps to (from). Thus, if two particles pass each other via two subsequent tunneling processes to the right (left), the many-body wave function pics up a phase of (). These tunneling processes are illustrated in Fig. 1(a).
For the realization of the number-dependent tunneling phase, we consider bosons in a tilted periodically forced lattice described by the Hamiltonian
Here and denote the bare tunneling and interaction parameters, characterizes a strong potential tilt, captures possible weak additional on-site potentials, and incorporates a homogeneous time-periodic force of angular frequency with vanishing cycle average . It can be implemented as an inertial force , with lattice constant , by shaking the lattice position back and forth. We require the resonance conditions
as well as the high-frequency conditions
where we have introduced the small interaction detuning . The largest share of the on-site energy is then given by , so that tunneling is energetically suppressed. Namely, when a particle tunnels from to this energy changes by with . However, coherent tunneling processes can be induced by time-periodic forcing as -“photon” processes, where the drive provides or absorbs energy quanta . They are described by an effective tunneling matrix element Eckardt and Holthaus (2007); Sias et al. (2008), which, through , will depend on the occupation numbers. Such number-dependent resonant tunneling has recently been investigated both experimentally Ma et al. (2011) and theoretically Bermudez and Porras (2015).111An alternative approach for Floquet engineering number-dependent tunneling matrix elements relies on a modulation of the interaction strength Gong et al. (2009); Rapp et al. (2012); Greschner et al. (2014); Liberto et al. (2014); Wang et al. (2014b); Meinert et al. (2016). As we will show now, it can be used to achieve the number-dependent tunneling phases appearing in Eq. (2).
Using the time-periodic unitary operator
where so that , we can perform a number-dependent gauge transformation. It integrates out the strong on-site terms as well as the periodic force, but leads to number-dependent tunneling terms in the new Hamiltonian . Averaging over the rapidly oscillating phase factor by integrating over one driving period (corresponding to the leading order of a high-frequency approximation Goldman and Dalibard (2014); Bukov et al. (2015); Eckardt and Anisimovas (2015)), we obtain the effective time-independent Hamiltonian
It contains the number-dependent tunneling parameter
and the tunable interaction parameter , which can take both negative and positive values.
The effective tunneling matrix elements should reproduce the number-dependent tunneling parameters of Eq. (2). We restrict ourselves to the low-density regime, where the dominant processes involve one or two bosons. These processes are those depicted in Fig. 1(a) as well as the hermitian conjugated processes for tunneling leftwards. As illustrated in Fig. 1(b), these processes are associated with different potential energy changes . Tunneling rightwards from a singly or doubly occupied site onto an empty site corresponds to or , respectively, and should be described by the parameter,
with real tunneling amplitude and arbitrary Peierls phase reflecting the freedom of gauge. Tunneling rightwards from an empty site onto an occupied site is associated with and the corresponding tunneling parameter should carry an additional phase ,
for the (integrated) driving force (other choices are possible). This ansatz already ensures that . The additional constraint defines lines in the - plane, as can be seen in panel (c) of Fig. 1. The thickness and the color of the plotted lines represents the tunneling amplitude and the statistical angle , respectively. The variation of and along the lines is also plotted in panels (d) and (e). Whereas lines 2 and 3 cover the full range , line 1 roughly allows to realize .
A clear signature of the continuous fermionization of the system with increasing is the formation of a Fermi sea in the momentum distribution of the anyons (see e.g. Ref. Tang et al. (2015)). However, in an experiment one cannot measure the anyonic momentum distribution, but only the bosonic one, which, due to both the Jordan-Wigner transformation and the gauge transformation (6) differs from that of the anyons. Therefore, in the following we will consider only such observables that are invariant under these transformations. These include the densities , the two-particle correlations , and also the second Rényi entropy characterizing the purity of the reduced density matrix of the subsystem given by the first sites , .222 The Fock states for 1D anyons () and bosons () obey the Jordan-Wigner transformation . It corresponds to independent site-local transformations , so that tracing out a site is identical for bosons and 1D anyons, . In the following, we will focus on ground-state properties, so that is an entanglement entropy (as it has been measured recently in a bosonic chain Islam et al. (2015)).
We compute , , and using exact diagonalization both for the ideal model (2) and the effective Hamiltonian (7). We consider bosons on sites, corresponding to a density of . The effective tunneling matrix elements (8) were obtained for the driving function corresponding either to path 1 of Fig. 1(c) or to path 2, in case the desired statistical angle is not available in path 1. Despite the fact that they reproduce the ideal tunneling matrix elements only for processes involving one and two particles, we find very good agreement between the ideal and the effective model: In Figs. 2(a) and (b), we plot and for the effective model with and various anyonic angles . The results match very well with those obtained for the ideal model shown in Figs. 2(c) and (d). For non-zero on-site interactions, the agreement is equally good, so that we only plot the results for the ideal model in Fig. 2(e) and (f).
In Fig. 2(c), we can observe that the density distribution flattens in the center, when the statistical angle is switched on. This effect can be understood, by noting that the scattering properties resulting from the density dependent tunneling resemble those of repulsive on-site interactions Greschner and Santos (2015), which favor a flat density. For large , the density becomes modulated, with one maximum for each particle in the system. These oscillations correspond to Friedel oscillations, which are a hallmark of fermionic behavior Friedel (1958). They are a finite-size effect induced by the hard-wall boundary conditions (as they can be realized in quantum-gas microscopes). Their wavelength is roughly given by , with Fermi wave number , corresponding to the average particle distance, which in our system is given by lattice constants. Generally, Friedel oscillations occur in the vicinity of localized defects. Their build-up allows us to monitor the continuous fermionization of the 1D anyons in a system of bosons with number-dependent tunneling. The oscillations are also clearly visible in the entanglement entropy [Fig. 2(d)]. An intuitive explanation is that the maximum corresponds to the position of a particle, whose delocalization within the maximum contributes to the entanglement between the left and right subsystems.
In Figs. 2(e,f), showing data for a system with significant on-site interactions , the fermionic signatures occur already for smaller . This observation is consistent with the well-known fact that increasing the on-site interactions is another way of approaching fermionic behavior Paredes et al. (2004). This is illustrated also in Fig. 2(g,h) showing data for plain bosons () and different . In the hard-core limit (), the bosons can be mapped to fermions. Despite the fact that 1D anyons only become pseudofermions for , the data for in Figs. 2(c,d) agrees very well with that for in Figs. 2(g,h). This suggests that for low densities pseudofermions behave like true fermions. This is confirmed also by the correlations shown in Fig. 2(i). Their diagonal elements , which are a measure for double occupation, vanish for , even though pseudofermions locally behave like bosons. Simulations show that pseudofermions behave like fermions up to a filling of about one third sup (). This is also the filling, where the low-density description, stating that driven bosons behave like anyons, is found to break down sup ().
The build up of Friedel oscillations requires the system to be quantum degenerate, i.e. temperatures well below the Fermi energy , so that the thermal wavelength is large compared to the mean particle distance. Computing canonical expectation values for anyons on sites for the finite temperature (corresponding to an von Neumann entropy per particle of , which is accessible in a system of spinless bosons), we find well pronounced Friedel oscillations sup (). In an experiment, the low-entropy state of the effective Hamiltonian (7) has to be prepared starting from a low-entropy state of the undriven system. Let us assume that initially and the system is prepared in a Mott-insulator state , with the set containing lattice sites and denoting the vacuum state. This is the asymptotic ground state in the presence of an external potential in the limit . For finite , this is also the ground state of the effective model (7), with . Thus, we can adiabatically melt the Mott insulator into the ground state of by smoothly ramping up the forcing, i.e. the , and then continuously switching off the external potential . In order to minimize the mass transport during this adiabatic process, it is useful that contains equally spaced lattice sites, so that describes a superlattice. We have simulated this protocol integrating the time evolution of the full time-dependent Hamiltonian (3) and find excellent agreement between the final state and the ground state of [Fig. 3]. This confirms both a description in term of the effective Hamiltonian and the proposed preparation scheme.
In summary, we proposed a simple scheme for the realization of 1D anyons, which is feasible in existing experimental setups. It is based on Floquet engineering a system of bosonic atoms with number-dependent tunneling phases. We, moreover, showed that Friedel oscillations can serve as a directly measurable signature for the continuous fermionization of the anyons.
Acknowledgements.We thank Sebastian Greschner, Axel Pelster, and Luis Santos for useful discussions. CS is grateful for support by the Studienstiftung des deutschen Volkes. This work was supported by the DFG via the Research Unit (Forschergruppe) FOR 2414.
Behavior with Respect to Density
Let us first explore how the system behaves when the filling is increased. In Fig. 4, we plot the ground-state density profile for a small chain of sites with open boundary conditions for various particle numbers between and . We compare results for 1D anyons (dashed blue lines) with , which are pseudofermions that behave on site like bosons, with results obtained for both free fermions (solid black lines) and bosons described by the effective Hamiltonian (7) given in the main text (dotted red lines).
The effective bosonic Hamiltonian is constructed to reproduce the anyonic behavior for single- and two-particle processes. Accordingly, its ground state is expected to mimic that of the anyonic model for low densities. The simulations presented in Fig. 4, which are based on exact diagonalization, confirm this behavior. We find almost perfect agreement between bosons and anyons, until the low-density approximation breaks down abruptly at a filling of .
In Figure 2(i) of the main text, we can observe that the probability for finding two pseudofermions on the same site practically vanishes like for actual fermions. The results presented in Fig. 4 show that pseudofermions mimic almost perfectly the behavior of free fermions for low densities, up to a filling of . Here the fermionic density is given by
are the wave functions of the single-particle eigenstates with wave numbers and energies .
An intuitive explanation for the excellent agreement between pseudofermions and fermions at low densities can be given as follows. The number-dependent tunneling terms appearing in the bosonic representation (2) of the anyonic model (1) give rise to two-particle scattering, which is described by a scattering length that diverges for (see Ref.  of the main text). Thus, in the low-density regime, which is governed by two-particle processes, we recover the physics of hard-core bosons, which in turn can be mapped to (actual) fermions.
Let us finally investigate the effect of a finite temperature. For a system of anyons on lattice sites with , we compute the canonical expectation value for the density profile by using exact diagonalization for computing not only the ground state, but also the excited states of the interacting system. We compare results obtained for four different temperatures . The corresponding (von Neumann) entropies per particle are calculated to be given by , , , and (in units of the Boltzmann constant), respectively. Figure 5 shows how the density profile of the anyonic model changes when the temperature is increased. One can clearly see that the Friedel oscillations are preserved as long as .
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- (65) Supplemental material, contains an investigation of the role of density and temperature.