Single-site-resolved measurement of the current statistics in optical lattices
At present, great effort is spent on the experimental realization of gauge fields for quantum many-body systems in optical lattices. At the same time, the single-site-resolved detection of individual atoms has become a new powerful experimental tool. We discuss a protocol for the single-site resolved measurement of the current statistics of quantum many-body systems, which makes use of a bichromatic optical superlattice and single-site detection. We illustrate the protocol by a numerical study of the current statistics for interacting bosons in one and two dimensions and discuss the role of the on-site interactions for the current pattern and the ground-state symmetry for small two-dimensional lattices with artificial magnetic fields.
pacs:67.85.-d, 03.65.Vf, 05.60.Gg
Introduction.—Improved and new detection techniques have contributed significantly to the rapid progress in the study of strongly correlated states of ultracold atoms in optical lattices in recent years. The combination of a time-of-flight expansion followed by absorption imaging gives access to the momentum distribution and has revealed the superfluid to Mott insulator transition (Greiner et al., 2002). Going further, one can study the noise correlations in these absorption images, e.g., exhibiting the quantum statistics of bosons (Fölling et al., 2005) and fermions (Rom et al., 2006). In addition, the excitation spectrum of interacting bosons has been measured using momentum-resolved Bragg spectroscopy (Ernst et al., 2010). However, the most significant step in recent times has arguably been the introduction of single-site detection (using fluorescence microscopy) in optical lattices (Bakr et al., 2009; Sherson et al., 2010). This new method has been used to observe the shell structure of a Mott insulator (Bakr et al., 2010; Sherson et al., 2010), the propagation of single bosons (Weitenberg et al., 2011), a spin impurity (Fukuhara et al., 2013), and the structure of density correlations (Endres et al., 2011). In the future, it may provide new insights into the buildup of entanglement in many-body systems and the influence of measurements on quantum many-body dynamics (Daley et al., 2012; Keßler et al., 2013, 2012).
Here, we discuss a scheme for the spatially resolved measurement of the current between nearest-neighbor lattice sites. It relies on combining a bichromatic superlattice and single-site detection. Such a tool is especially timely in the context of the recent effort towards realizing gauge fields in optical lattices (for recent reviews, see (Dalibard et al., 2011; Lewenstein et al., 2012; Goldman et al., 2013)), with first implementations reported in (Aidelsburger et al., 2011; Struck et al., 2012; Jiménez-García et al., 2012; Aidelsburger et al., 2013; Miyake et al., 2013). These systems can be used to realize states that support equilibrium currents. Various methods have by now been proposed to detect interesting aspects of such systems, e.g., quantum Hall edge states and Chern numbers in topological insulators, using time-of-flight expansion (Scarola and Das Sarma, 2007; Alba et al., 2011; Zhao et al., 2011; Polak and Zaleski, 2013; Wang et al., 2013), light scattering (Douglas and Burnett, 2011; Goldman et al., 2012), or the time evolution of the real space density after a quench in the potential (Killi and Paramekanti, 2012; Killi et al., 2012; Goldman et al., 2013). The protocol discussed below has similarities with the latter approach. However, in addition to probing the current pattern (Killi and Paramekanti, 2012), it reveals the full spatially resolved current statistics of the quantum-many body state, from which correlation functions can be extracted.
Model and current operator.—We consider a tight-binding Hamiltonian for ultracold atoms in an optical lattice,
Here, is a bosonic or fermionic annihilation (creation) operator, is the possibly complex tunneling amplitude, and denotes pairs of nearest-neighbor lattice sites. The interaction part is a polynomial of local particle number operators, , and is the on-site energy at site . The (mass) current operator for this system can be defined using the local continuity equation for the particle density. For a lattice system it reads , here denotes the current from site to site and denotes the set of nearest-neighbor lattice sites of Using Heisenberg’s equation of motion gives , and accordingly the current operator between nearest-neighbor lattice sites and reads (for bosons and fermions)
Measurement protocol.—The protocol for measuring the eigenvalues of the current operator (2) is summarized in Fig. 1. The main idea is to use a bichromatic superlattice, which has already been realized experimentally (Sebby-Strabley et al., 2006; Trotzky et al., 2010), and to apply a beam-splitter operation to map the single-particle eigenstates of the current operator to the states localized at the left and right lattice site of each double well [Fig. 1(b)]. The measured value of the current operator is then essentially given by the difference of the particle numbers of both lattice sites. The particle number can be measured [Figs. 1(c) and 1(d)] in principle by the recently developed single-site imaging techniques (Bakr et al., 2009; Sherson et al., 2010), although at present these are still restricted to parity measurements (see the discussion on experimental details below).
Formally, the protocol relies on the time evolution of noninteracting atoms in a symmetric double-well potential, The difference in the atom number at the two wells oscillates in time and can be expressed as
where Thus, the current can be obtained as the density difference, , for suitable chosen evolution times , .
The previous expression for shows that the current operator has discrete eigenvalues, just like the particle number operator. This situation, surprising at first sight, can also be understood as follows: The eigenvalues of the current operator are given by the difference in the density of atoms going to the right and left times the velocity. This is seen by diagonalizing the current operator (2), which yields with and . The operators and have a simple meaning: They correspond to right- and left-going atoms (Foo, ). Since the total particle number in the double well, , commutes with , we can assume a situation of fixed . Then the spectrum of the current operator is , with for bosons and for fermions due to the Pauli principle.
Bosons in 1D.—We first consider the current statistics of the homogeneous one-dimensional (1D) Bose-Hubbard model,
with real tunneling amplitude and on-site interaction strength . To keep the numerics manageable, we focus on the 1D case, even though we believe that the qualitative features of the local properties we are going to discuss should not be dependent on dimensionality in a significant way. The two phases of the Bose-Hubbard model exhibit a characteristic atom number statistics at a single site: a Poisson distribution for the superfluid state () and a fixed atom number in the Mott insulating regime () for integer filling ; see, e.g., Refs. (Jaksch et al., 1998; Bloch et al., 2008).
The corresponding current statistics is presented in Figs. 2(a)-2(d) at filling . For , the atoms are in a product state of coherent states at the individual sites. Thus, and are also Poisson distributed, with mean . The current (2), being the difference between two Poisson distributed variables, is then given by the so-called Skellam distribution, , where denotes the modified Bessel function. When increasing , the distribution becomes more and more concentrated at the eigenvalues This is a consequence of the Mott insulating state being a superposition of the eigenstates corresponding to , . Note that, in general, the current eigenvalues are even multiples of for the Mott insulator at arbitrary integer filling.
Figure 2(e) shows the interaction dependence of current correlation functions, which might be detected using the scheme proposed here. The variance of the current increases monotonically with the on-site interaction strength, from for to at integer filling deep in the Mott insulating regime. For the current correlation between neighboring pairs of lattice sites, , becomes negative for intermediate with a minimum close to the superfluid-to-Mott-insulator transition, while it vanishes for and . In contrast, for a half-filled lattice, decreases asymptotically as one goes into the hardcore boson limit ().
We note that the correlations of the current into and through a lattice site might be accessed by an extended version of the measurement scheme using a triple well-superlattice structure; see Supplemental Material (Sup, ).
Bosons in a synthetic magnetic field.—We consider interacting bosons subject to a uniform synthetic magnetic field perpendicular to the 2D lattice. The corresponding Hamiltonian reads, in Landau gauge,
Here, are the integer and coordinates of the lattice sites, and the phase , which a boson picks up when circulating in an anticlockwise direction around a unit cell, encodes the effect of the magnetic field. For a charged particle, would equal the number of flux quanta per unit cell. The single-particle spectrum of (5) is given by the famous fractal “Hofstadter butterfly” (Hofstadter, 1976).
Below we discuss relatively small 2D lattices, which might be realized first in experiments (with a suitable superlattice structure dividing the entire lattice into such small plaquettes as implemented for lattices in (Aidelsburger et al., 2011)). A similar system has been realized with Bose-Einstein condensates in rotating lattices (Tung et al., 2006; Williams et al., 2010), where the “Lorentz force” is replaced by the Coriolis force (Cooper, 2008). The creation and observation of topological states in this setup have been theoretically studied in (Barberán et al., 2006; Hafezi et al., 2007; Palmer et al., 2008; Umucalilar et al., 2008; Umucalilar and Mueller, 2010; Graß et al., 2012). Moreover, a transition between ground states of different rotational symmetry at discrete rotation frequencies was found (Bhat et al., 2006, 2006), which leads to a discontinuity in the edge current. Here, we study the effects of finite on-site interactions on such transitions (these previous works (Bhat et al., 2006, 2006) discussed only the limit of hardcore bosons).
The results obtained from exact diagonalization for a lattice are summarized in Fig. 3. Note that we restrict ourselves to the interval , as the Hamiltonian is invariant under , and only changes the magnetic field direction. Figure 3(a) shows the current and density profile of the ground state for different interaction strengths and ’s. The current patterns found for are similar to those of (not shown), whereas additional current configurations appear for large . Note that the central current reverses sign beyond some critical flux number , and this value decreases for larger .
The critical value of can be identified via the change in the ground-state rotational symmetry or via the first discontinuity of the edge current. We define the edge current as the sum of all currents along the boundary, counted in an anticlockwise direction; see Fig. 3(c). The rotational symmetry is best discussed in the symmetric gauge; see (Sup, ). For a square lattice, the Hamiltonian (5) commutes with the rotation by , . Thus, a nondegenerate ground state is an eigenstate of , with eigenvalue , . Additional transitions in the ground-state symmetry (discontinuities of the edge current) show up for intermediate . The critical flux numbers for these transition points hardly change for . The discussed current patterns and the edge current can be measured using the proposed protocol, which therefore provides a means of studying the flux and interaction dependence of such transitions.
We now turn to spatial current correlations. An illustrative example is displayed in Fig. 4, for a half-filled lattice. Such bosonic flux ladders exhibit a transition from a Meißner phase [Fig. 4(a)] to a vortex phase [Fig. 4(b)] when is increased (Orignac and Giamarchi, 2001; Cha and Shin, 2011). For the situation shown in Fig. 4(a), we address the question of whether the currents at A, B, and C are correlated. This is done by constructing the joint probability distribution of the eigenvalues from an ensemble of snapshots, since the currents and can be measured simultaneously. Figures 4(d) and 4(e) display two examples of these joint probability distributions. The measured values of and at the two far-removed links A and C are (to a very good approximation) independent of each other, i.e., the joint probability distribution is just a product of the eigenvalue distribution of and [cf. Fig. 4(e)]. In contrast, the current operators and for nearby sites are correlated as the joint probabilities and are clearly different even though and are almost equal; see Fig. 4(d). The dependence of the current correlation on the flux number is shown in Fig. 4(f). We find a positive correlation for [parameter regime with the current pattern shown in Fig. 4(a)], which becomes stronger with increasing on-site interaction strength. In contrast, the average current [inset in Fig. 4(f)] hardly changes with the interaction strength for . For larger flux values, the correlation falls off to values around zero.
Discussion of experimental details.—Let us consider the combination of the proposed measurement protocol with gauge fields created by laser-assisted tunneling (Jaksch and Zoller, 2003; Gerbier and Dalibard, 2010; Kolovsky, 2011) implemented in (Aidelsburger et al., 2011, 2013; Miyake et al., 2013). The required 2D lattice consists of alternating columns with different on-site energies (and may trap different internal states (Jaksch and Zoller, 2003; Gerbier and Dalibard, 2010)). Tunneling between different columns is only nonzero when it is driven by additional light fields (which also imprint the phase on the tunneling amplitude), while bare tunneling exists within each column. Thus, the bichromatic superlattice for the current measurement could be applied in the direction of the columns, while the tunneling between the columns is inhibited by switching off the driving laser fields.
All of the present experiments on single-site-resolved detection only resolve the parity of the atom number at any lattice site (Bakr et al., 2009; Sherson et al., 2010). For a single or two coupled 1D chains, considered in Figs. 2 and 4, one might let the atoms expand into another direction before the detection process (similar to (Weitenberg et al., 2011)) to avoid double or higher occupancies. However, for true 2D configurations, one is currently restricted to small filling factors, until the parity problem can be circumvented via alternative approaches. We have further investigated these limitations of the measurement protocol for the configuration shown in Fig. 4, by numerically simulating situations with parity detection only. We also included residual interactions, , and timing errors during the evolution in the double-well potential. We find that the current pattern and the current correlation can still be observed in the presence of parity detection, even though the absolute value of the observed current decreases by up to 25%. The influence of the residual interaction is of the order of a few percent for . A timing error leads to a change in the current and current correlation of less than and , respectively, for .
Conclusions.—We have analyzed a protocol for the site-resolved measurement of the current operator in optical lattices. Using already available experimental techniques, it can be employed for interacting bosons at small filling factors. It can, in principle, be extended to fermions and possibly also to situations with different species. Measuring the statistics and spatial structure of currents seems a promising tool to study the physics of interacting ultracold atoms subject to gauge fields.
Acknowledgments.—We thank the DFG for support in the Emmy-Noether programme and the SFB/TR 12.
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Supplemental Material for “Single-site-resolved measurement of the current statistics in optical lattices”
Stefan Keßler and Florian Marquardt
Institute for Theoretical Physics, Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany
Max Planck Institute for the Science of Light, Günther-Scharowsky-Straße 1/Bau 24, 91058 Erlangen, Germany
Appendix A Extended Measurement scheme
In the main text we describe a projective measurement of the current operator between two nearest-neighbor lattice sites. From the measurement outcomes one can calculate the expectation values of any sum of current operators of the form (2), as, for instance, the edge current. Here, we discuss an extended setup with the double-well potential replaced by a triple-well potential [cf. Fig. 1(b)]. This modification allows to measure the statistics of the (one-dimensional) current through a lattice site, . The obtained statistics can be used (together with the one of the “usual” setup) to evaluate, e.g., the variance of the (one-dimensional) current into a lattice site, . This variance cannot be conceived from the statistics of alone as it involves the expectation value .
We assume a laser configuration, which allows to create a triple-well superlattice structure in one spatial direction, such that the dynamics within each triple well is described by . We restrict to the case of a real tunneling amplitude assuming the superlattice to be applied in the direction of the columns, where no phase is imprinted on the tunneling amplitude (see discussion of experimental details in the main text). Experimentally, the triple-well superlattice might be realized by the use of a bichromatic lattice with an additional laser beam with half the wavelength of the short lattice, as discussed in (Schlagheck et al., 2010). The diagonalization of the three-site Hamiltonian yields with and . Making use of the expression for the time-dependent annihilation operator, , the current through lattice site 2, , at time zero (ramp up of the triple-well potential) expressed in terms of the one-particle density matrix after an evolution time equals
At time points , , only the diagonal terms of the one-particle density matrix contribute and the current is given by
The density difference on the right-hand side can be measured by a single-site-resolved measurement of the atom number. The meaning of Eq. (S.3) is rather simple: For the time span the single-particle eigenstates corresponding to the current eigenvalues 0, and are mapped on the states localized at the lattice sites 1,2, and 3, respectively. Note that a symmetric triple-well potential cannot be used for the direct detection of the current into lattice site 2, , as this current operator and three-site Hamiltonian have the common eigenstate Therefore, there is never a time for which this difference of current operators can be expressed as a combination of single-site densities.
Figure S1 presents the variance of the current through and into a lattice site for the 1D Bose-Hubbard model, which is also considered in the main text. The mean value of both currents vanishes for the ground state since ; see also Figs. 2(a-d). Figure S1 shows a drastic interaction dependence for a lattice with integer filling For the superfluid state, the fluctuation of the current into a lattice site, and thus , is much stronger than the fluctuation of the current through a lattice site. In contrast, they are the same in the Mott insulating regime in the limit .
Appendix B Rotational symmetry
Here, we discuss the ground-state rotational symmetry for a many-body system described by Hamiltonian (5); see also (Bhat et al., 2006). We consider a square lattice of lattice sites [for a rectangular lattice the following reasoning is essentially the same replacing by , which results in two eigenvalues ]. An anticlockwise rotation of the coordinate system (passive rotation) by an angle maps the lattice sites onto themselves, , with the and components Applying four times is equivalent to the identity transformation and thus the possible eigenvalues of are given by the fourth roots of one, with
Let us now consider the Hamiltonian (5) for this square lattice. It reads in the symmetric gauge [in Eq. (5) we used the Landau gauge], which identifies the center-of-mass of the lattice as a special point:
The transformation of the annihilation (creation) operator under the rotation, , makes it directly apparent that this Hamiltonian commutes with , i.e., . Thus, for an eigenstate of , is also an eigenstate of with the same eigenvalue . If is nondegenerate, then is an eigenstate of , too.
In the main text, we discuss the rotational symmetry and the edge current of the ground state of Hamiltonian (5) as function of the parameters and . A change in the rotational symmetry of a (nondegenerate) ground state happens by an exact level crossing of the two lowest eigenenergies (as these states correspond to two different irreducible representations of the rotation; see (Landau and Lifshitz, 1977)). This implies that the ground-state energy is nonanalytic at the crossing point and we also observe a discontinuity in the edge current.
Appendix C Timing error
Let us discuss on more general grounds the effect of an imprecisely chosen evolution time in the double-well potential [Fig. 1(b)] on the current measurement. We consider an evolution time , where is the (dimensionless) timing error. The actually measured density difference between the left and the right well is given by Eq. (3) in the main text. For a small timing error it yields up to second order in :
The comparison with the ideal case, , shows that there are two contributions that lead to an error in the current measurement: one proportional to the true value of the current and another proportional to the initial density difference between the two lattice sites. The first term is just a relative change of the current by , which should be very small in an experimental realization of the measurement protocol. The second term seems to be more severe since it is linear in the timing error and does not depend on the value of the current. However, for the evaluation of the average current this error is suppressed for a system with an approximately homogeneous density distribution or in case that the distribution of timing errors over different measurement runs is roughly symmetric with respect to Indeed, we have found only a small effect of the timing error on the current and the current correlations for the results shown in Fig. 4 (as reported in the main text), where the density distribution is roughly homogeneous.