Phonon resonances in atomic currents through Bose-Fermi mixtures in optical lattices
We present an analysis of Bose-Fermi mixtures in optical lattices for the case where the lattice potential of the fermions is tilted and the bosons (in the superfluid phase) are described by Bogoliubov phonons. It is shown that the Bogoliubov phonons enable hopping transitions between fermionic Wannier-Stark states; these transitions are accompanied by energy dissipation into the superfluid and result in a net atomic current along the lattice. We derive a general expression for the drift velocity of the fermions and find that the dependence of the atomic current on the lattice tilt exhibits negative differential conductance and phonon resonances. Numerical simulations of the full dynamics of the system based on the time-evolving block decimation algorithm reveal that the phonon resonances should be observable under the conditions of a realistic measuring procedure.
pacs:67.85.Pq, 05.60.Gg, 72.10.-d
One of the most intriguing prospects opened up by recent advances in atomic physics is the possibility of studying many-body quantum systems. In particular, ultracold atoms confined to optical lattice potentials have been shown to be perfectly suitable for implementing physical models of fundamental interest not only to the field of atomic physics but also to condensed matter physics Lewenstein et al. (2007); Bloch et al. (2008). Specific examples of a highly versatile many-body system include Bose-Fermi mixtures in optical lattices, which have been used recently to analyze the effect of fermionic impurities on the superfluid to Mott-insulator transition Günter et al. (2006); Ospelkaus et al. (2006); Best et al. (2009). A further experimental setup closely related to condensed matter systems consists of ultracold atoms in tilted optical lattice potentials. Several fundamental quantum mechanical processes related to nonequilibrium transport of particles have been observed in this setup such as, e.g., Landau-Zener tunneling Zenesini et al. (2008), Bloch oscillations Ben Dahan et al. (1996); Roati et al. (2004), and processes analogous to photon-assisted tunneling Sias et al. (2008).
Of particular importance, collisionally induced transport of fermions confined to an optical lattice and coupled to an ultracold bosonic bath has been observed in an experimental setup of a similar type to the one considered in this article Ott et al. (2004). Furthermore, a closely related theoretical analysis of collision-induced atomic currents along a tilted optical lattice was provided by Ponomarev et al. Ponomarev et al. (2006) based on a random matrix approach. In essence it was pointed out in Ref. Ponomarev et al. (2006) that the atomic current exhibits Ohmic and negative differential conductance (NDC).
In the present article we study a natural extension of the mentioned experiments, namely lattice Bose-Fermi mixtures with the fermions confined to a tilted optical potential. Motivated by earlier considerations Bruderer et al. (2007, 2008) we show that this system is highly appropriate for exploring the effects of electron-phonon interactions on nonequilibrium electric transport through a conductor. Specifically, we demonstrate that electron-phonon resonances, predicted to exist in solids nearly 40 years ago Bryxin and Firsov (1972); Döhler et al. (1975); Bryksin and Kleinert (1997), yet for which there seems to be no conclusive experimental evidence Feng et al. (2003); Leo (2003), can be realized using ultracold atoms.
In our model the fermions take the role of electrons, whereas the bosons provide a nearly perfect counterpart to acoustic phonons in solids. The tilt imposed on the lattice potential of the fermions corresponds to an applied bias voltage to the system. The collisions between fermions and bosons result in fermion-phonon relaxation processes, which give rise to a net atomic current along the tilted lattice potential Ponomarev et al. (2006); Bruderer et al. (2008); Kolovsky (2008), as illustrated in Fig. 1. We show that in our case, where both the fermions and bosons are trapped in an optical lattice, phonon resonances in the atomic current occur. They arise as the momentum of the phonon emitted in the relaxation process approaches a so-called Van Hove singularity Van Hove (1953) at the upper edge of the phonon band. Moreover, we shall see that phonon resonances, at least on the level of approximations made in this article, are also expected to occur in Bose-Bose mixtures Catani et al. (2008).
There are crucial advantages of our cold atom implementation over a solid-state system: First, neither impurities nor imperfections in the system suppress the resonances in the current Feng et al. (2003) and hence fermion-phonon scattering is the only relaxation process, which can be fully controlled via the bosonic system parameters Best et al. (2009); Ernst et al. (2009). Second, large lattice tilts can be achieved with an energy mismatch between neighboring fermion sites that exceeds the bandwidth of both the fermions and the phonons. This makes it possible to study the influence of the phonon density of states on the atomic current over the entire phonon band. In addition, it allows us to observe negative differential conductance Ponomarev et al. (2006); Bruderer et al. (2008); Kolovsky (2008), which is realized in a solid-state system with difficulty by resorting to semiconductor superlattices Esaki and Tsu (1970); Döhler et al. (1975); Leo (2003). Last, the parameters of our system can be chosen to ensure that Landau-Zener tunneling to higher fermion bands—often a significant effect in high field transport Leo (2003)—is negligible despite the large lattice tilts Bruderer et al. (2008) justifying the use of a tight-binding framework.
The key experimental techniques required for our scheme are twofold: first, the independent trapping of atoms of different species in species-specific optical lattice potentials Mandel et al. (2003); Lamporesi et al. (2010) and, second, the tunability of the boson-boson and boson-fermion interactions by Feshbach resonances Roati et al. (2007); Best et al. (2009). Also, we exploit the possibility of implementing low-dimensional systems by strongly increasing the depth of the optical potentials along specific directions. In this way effectively one-dimensional systems can be realized by tightly confining atoms to an array of tubes Bloch et al. (2008).
The structure of this article is as follows: In Sec. II we start from the Bose-Fermi Hubbard model and outline the effective description of the Bose-Fermi mixture in terms of Bogoliubov phonons and Wannier-Stark states. In Sec. III we derive a general expression for the drift velocity of the fermions and show that this expression encompasses the phenomena of negative differential conductance and phonon resonances. Section IV contains the results of a near-exact numerical simulation of a realistic experimental procedure to determine the dependence of the atomic current on the lattice tilt. We end with the conclusions in Sec. V.
Ii Effective model
The specific system we consider consists of a homogeneous, one-dimensional Bose-Fermi mixture of bosons and spin-polarized fermions, both trapped in separate optical lattice potentials. If the potentials are sufficiently deep so that only the lowest Bloch band is occupied then the Bose-Fermi mixture can be described by the Bose-Fermi Hubbard model Albus et al. (2003)
where denotes the sum over nearest neighbors. The operators () create (annihilate) a boson and, similarly, the operators () create (annihilate) a spinless fermion in a Wannier state localized at site . The bosonic and fermionic hopping parameters are and , respectively, and the on-site boson-boson and boson-fermion interactions are characterized by the energies and , both positive and independently tunable. In contrast to the bosons, the lattice potential of the fermions is assumed to be tilted with an energy splitting between adjacent sites.
At low temperatures and for sufficiently small boson-boson interactions most bosons are in the superfluid phase and accurately described by the phononic excitations of the superfluid. This description is obtained by transferring the bosonic part of the Hamiltonian into momentum space and adopting the Bogoliubov approach van Oosten et al. (2001), which results in the phonon Hamiltonian
Here, is the momentum running over the first Brillouin zone, i.e., with the lattice spacing, and the bosonic operators () create (annihilate) a Bogoliubov phonon. The excitation spectrum of the phonons is given by the Bogoliubov dispersion relation , where is the bosonic occupation number and is the dispersion relation of noninteracting bosons in the optical lattice. The dispersion of the Bogoliubov phonons, albeit not identical, is remarkably similar to the dispersion of acoustic phonons in a solid-state system with , shown in Fig. 2(a). The common features are a linear dispersion in the limit and the band structure with a gap at the boundary of the first Brillouin zone. It should be pointed out that the accuracy of the Bogoliubov description of phonons has been demonstrated experimentally Ernst et al. (2009), thus we have perfect knowledge of and control over the phonons in our system.
The fermions confined to the tilted potential are represented by the eigenstates of the fermionic part of , i.e., the Wannier-Stark states Emin and Hart (1987); Glück et al. (2002). The unitary transformation that relates the Wannier operators to the Wannier-Stark operators is given by
where are Bessel functions of the first kind Watson (1995). It follows immediately from the properties of the Bessel functions that the Wannier-Stark states are centered at lattice site and have a width of the order of . By invoking the identities for the Bessel functions and one finds that the transformed fermionic part of takes the diagonal form
The spectrum of constitutes a so-called Wannier-Stark ladder Emin and Hart (1987); Glück et al. (2002) with a constant energy separation between adjacent sites, shown in Fig. 1. We emphasize that the Wannier-Stark states are stationary and hence a net current of fermions along the tilted lattice only develops in presence of additional relaxation processes. In particular, it results from the inverse of the transformation in Eq. (3) that a fermion initially localized at a single lattice site undergoes coherent Bloch oscillations with frequency Ben Dahan et al. (1996).
To rewrite the boson-fermion interaction in terms of Bogoliubov phonons and Wannier-Stark states we apply the same transformations as for the purely bosonic and fermionic parts of . Invoking the results established in Refs. van Oosten et al. (2001); Emin and Hart (1987) we find the fermion-phonon interaction Hamiltonian
with the corresponding matrix elements
where is the number of lattice sites. We note that coincides exactly with the description of the electron-phonon interactions in a solid-state system Emin and Hart (1987) and thus is perfectly adequate for investigating related phonon effects.
The fermion-phonon interaction describes the creation of a Bogoliubov phonon out of the superfluid phase and the reverse process, both caused by the hopping process of the fermions. These incoherent processes involving a single phonon enable fermions to make transitions between Wannier-Stark states separated by lattice sites. On the other hand, coherent processes resulting from , in which a virtual phonon is emitted and reabsorbed, lead to a phonon-mediated fermion-fermion interaction, a renormalized fermion hopping and a mean-field energy shift of the fermions Bruderer et al. (2007, 2008).
The effective fermion-fermion interaction, being short range, can be safely neglected if we assume a filling factor of the fermions much lower than 1. However, we have to take into account the renormalization of the fermion hopping when comparing our theoretical model to experimental or numerical results. As expected from theoretical considerations Bruderer et al. (2007, 2008) and confirmed by the numerical results in Sec. IV, the renormalization reduces the bare hopping. Last, the mean-field energy shift is explicitly given by , which is readily determined in the regime by taking the continuum limit 111Explicitly, the continuum limit of the sum over all momenta in the lowest band is .. For a single delocalized fermion with the shift is simply , whereas for a localized fermion with we find , with the speed of sound in the superfluid.
It is essential for our model that neither interactions with the fermions nor the trapping potential confining the fermions destroy the phononic excitations of the superfluid. These conditions are met by using species-specific optical lattice potentials Mandel et al. (2003); Lamporesi et al. (2010) and by limiting the number of fermions in the system Günter et al. (2006); Ospelkaus et al. (2006); Best et al. (2009). Moreover, we restrict our analysis to fermions moving slower than superfluid critical velocity, which is according to Landau’s criterion Landau (1941), to avoid excitations other than those caused by the hopping transitions. More precisely we consider the parameter regime , where is the maximal group velocity of the fermions in the lowest Bloch band.
Iii Atomic currents
Since the number of fermions is significantly smaller than number of bosons we effectively treat the superfluid as a phonon bath. Accordingly we describe the dynamics of the fermions by a master equation for the probabilities that a fermion occupies a Wannier-Stark state at site . The master equation reads
where are the rates for a phonon-assisted transition from site to site and the factors take the Pauli exclusion principle into account. The probabilities either describe the occupation of a single fermion or the distribution of an ensemble of fermions; in both cases we choose . The average position of the fermions at time is accordingly and the average drift velocity , which quantifies the atomic current along the lattice, is given by .
To obtain a useful expression for the drift velocity we exploit the fact that the system is homogeneous so that the transition rates only depend on the jump distance , where jumps with are defined to be downward the tilted lattice, as depicted in Fig. 1. By using Fermi’s golden rule based on the interaction Hamiltonian , i.e., to second order in the coupling , we find
where is the mean number of phonons with momentum in the superfluid at temperature and is Boltzmann’s constant. The rate () corresponds to the process, where a fermion jumps sites down (up) the lattice and thereby emits (absorbs) a single phonon of energy . The expressions for the jump rates and are valid as long as heating effects caused by the emitted phonons are negligible.
To determine in terms of transition rates we substitute the expression in Eq. (7) for and find after some rearrangement that
with the phonon emission rate at zero temperature. The first term in the sum in Eq. (9) vanishes if the fermions are degenerate at zero temperature with all lattice sites occupied, i.e., for and for with fixed by the number of fermions, whereas the second term represents finite temperature effects. For the system far from degeneracy and for small fermion occupation numbers we may make the approximation and use the condition , which yields for the drift velocity. We note that at this level of approximation the expression for the drift velocity and all subsequent analytical considerations apply, mutatis mutandis, also to lattice Bose-Bose mixtures Catani et al. (2008).
Taking the continuum limit of the sum over all phonon momenta (appearing in ) and integrating over the first Brillouin zone we finally obtain
with implicitly defined by . The latter relation also yields the maximum jump distance , which is compatible with energy conservation in the fermion-phonon relaxation process. We see from Eq. (10) that the total drift velocity is determined by the sum over all admissible jump distances weighted by the phonon density of states and the fermion-phonon coupling .
We first determine the mobility of the fermions in the Ohmic regime, which we define by the relation in the limit , where is approximately the ratio of the lattice tilt and the phonon bandwidth for the small values of we consider here. We find that for , with as previously defined. This sum is a Kapteyn series of the second kind, for which a closed form is known Watson (1995), yielding for the mobility
The mobility diverges in the limit , however, it is well defined in the regime . For the practically important case, where the potential depth for the fermions and hence the hopping parameter is varied, the mobility scales as in the limit .
The dependence of the drift velocity given in Eq. (10) on the full range of accessible lattice tilts is shown in Fig. 3. We see that the atomic current makes a sharp transition from Ohmic conductance to negative differential conductance. The main reason is that the width of the Wannier-Stark states, proportional to , and hence the overlap between states at different sites decreases as the tilt is increased. This reduced overlap results in a low drift velocity —in analogy to NDC in semiconductor superlattices Döhler et al. (1975); Leo (2003)—and is determined by the matrix elements . The drift velocity depends on the tilt as for each jump distance in the limit of small fermionic bandwidth , which can be readily achieved in cold atom systems. As can be seen in Fig. 3 the NDC is particularly pronounced due to the superposed contributions from different jump distances to the total current. These findings are in accordance with predictions for a free homogeneous superfluid Ponomarev et al. (2006); Bruderer et al. (2008); Kolovsky (2008), however, the result for in Eq. (10) describes additional features stemming from the influence of the phonon density of states.
Specifically, the drift velocity exhibits anomalies in the form of sharp peaks, which correspond to the anticipated electron-phonon resonances in a solid-state system Bryxin and Firsov (1972); Bryksin and Kleinert (1997). As shown in Fig. 2(b), the phonon density of states is approximately constant for small momenta, but exhibits a Van Hove singularity Van Hove (1953) at the edge of the first Brillouin zone. Therefore phonon resonances arise as the momentum of the emitted phonons approaches the edge of the phonon band. It can be found from Eq. (10) that the resonance condition is given by with and thus the current displays a peak at corresponding to each admissible jump distance . Since the fermion-phonon interaction provides the only relaxation process in the system these anomalies are directly reflected in the tilt dependence of the current. Further, the current vanishes as the lattice tilt exceeds the phonon bandwidth because there are no phonon states available in order to dissipate energy into the superfluid.
Let us briefly discuss the effect of an additional (shallow) harmonic trapping potential of the bosons on the phonon resonances. If the system is aligned along the -axis the potential takes the form with the trap frequency . In the experimentally relevant case, where the harmonic oscillator length satisfies the condition , the local density approximation (LDA) is applicable Albus et al. (2003); Krämer et al. (2003). The LDA consists of replacing the bosonic density by the position-dependent density , determined by , with the other parameters of the model unmodified. Consequently, the average drift velocity in Eq. (10) has to be replaced by , where denotes the normalized spatial distribution of the fermions. This averaging might cause some broadening of the phonon resonances. However, the position of the resonances and more generally (except for the prefactor ) depend on only through the expression . Thus the trap-induced broadening can be made arbitrarily small by reducing the boson-boson interaction energy , which sets the smallest energy scale close to resonance. This is partly explained by the fact that the resonances involve only phonons from the upper edge of the phonon band with wavelengths comparable to the lattice spacing .
Iv Numerical simulation
The theoretical results derived in the previous section are, strictly speaking, valid for stationary atomic currents in a homogeneous system. We now show based on numerical simulations that the predicted negative differential conductance and the phonon resonances are observable in a system of finite size under the conditions of a realistic measuring procedure. In order to measure the atomic current in an experimental setup we envisage a procedure consisting of the following three steps:
Initially, both the bosons and fermions are prepared in a horizontal optical lattice and the total system is in equilibrium 222 Our numerical simulations show that the total equilibration of the system is not an essential requirement for the procedure to work. We obtain similar results if the fermions are immersed into the superfluid in a nonadiabatic way.. The fermions are each localized in separate sites sparsely distributed through the fermionic lattice of sufficient depth so that holds. In this configuration the fermions are automatically cooled by the surrounding superfluid Daley et al. (2004), which is only slightly distorted Bruderer et al. (2008).
Subsequently, the lattice of the fermions is tilted for a fixed evolution time of the order of to let them evolve. To obtain a detectable displacement of the fermions the fermionic hopping may have to be increased by reducing the depth of the lattice.
Finally, the spatial distribution of the fermions is detected, e.g., by in situ single-atom resolved imaging Bakr et al. (2009); Sherson et al. (2010). From the difference between the initial and final distributions it should be possible to extract a reliable estimate for the atomic current. Alternatively, the momentum distribution and hence the drift velocity of the fermions may be determined directly by a time-of-flight measurement.
To demonstrate the feasibility of this procedure we have simulated the fully coherent dynamics at zero temperature of both the bosons and fermions based on the complete Bose-Fermi Hubbard model given in Eq. (1). To this end we have used the time-evolving block decimation (TEBD) algorithm Vidal (2003), which is essentially an extension to the well-established density matrix renormalization group (DMRG) method White (1992); Schollwöck (2005). The TEBD algorithm permits the near-exact dynamical simulation of quantum many-body systems far from equilibrium, which is crucial for our purposes. The experimental procedure was simulated for a system consisting of lattice sites with a bosonic filling factor and a single fermion initially located at the center of the lattice. We used box boundary conditions and the evolution time was limited to to minimize finite-size effects, primarily reflections of phonons from the boundaries.
Figure 4 shows the displacement of the fermion as a function of the lattice tilt for a set of realistic experimental parameters, in particular, for three different values of the fermion hopping . The main features predicted by our theoretical analysis can be clearly recognized, namely the dominant NCD-peak, the phonon resonance at corresponding to the jump distance and the suppression of the current for . In addition, for sufficiently large values of the phonon resonance at corresponding to the jump distance is visible. For lower values of the phonon resonance for may still lead to a characteristic drop in the current once the lattice tilt exceeds half the phonon bandwidth. The resonances for higher values of the jump distance seem to be masked mainly by broadening effects for the parameter regimes tested.
The ratio between the height of the phonon resonance and the NDC peak, i.e., their relative visibility , depends nontrivially on the fermion hopping . The hopping enters the expression for in Eq. (10) through the matrix elements in the form of , where the parameter depends on the lattice tilt. Close to resonance we have , whereas for small tilts we obtain and hence in the superfluid regime. Accordingly, the height of the phonon resonance () varies only slowly with as opposed to the height of the NDC peak (), which is characterized by the oscillatory nature of the Bessel functions. Thus a careful choice of allows us to optimize the visibility , i.e., to minimize the height of the NDC peak, by tuning close to a zero of the relevant Bessel functions. For the increasing values of considered in our simulation the first zero of the Bessel function is approached, which explains the improved visibility for higher , shown in Fig. 4.
The general broadening of the phonon resonances may be caused by finite-size effects and multiphonon processes. The finite-size effects include reflections of phonons from the boundary of the system that in turn affect the motion of the fermion. Multi-phonon processes, which have been neglected in our theoretical analysis, also cause some broadening of the single-phonon resonances. In particular the continuous drop in the current to zero can be explained by the emission of several low-energy phonons during a jump process, which is allowed even if the lattice tilt exceeds the phonon bandwidth.
In order to compare the numerical and the theoretical results, at least qualitatively, we adapt our expression for the drift velocity to a system of finite size. In addition, we introduce the observed broadening effects characterized by the energy into the theory. Explicitly, we calculate the average drift velocity by using the expression with the rates . The rates reduce to the previous expression in absence of broadening, i.e., . The average displacement of the fermion (in units of lattice sites) after the evolution time is then approximately given by .
We fit this theoretical model to the numerical results with the broadening , the maximal jump distance , and the effective fermionic hopping as free parameters. The first two parameters allow us to extract a quantitative value for the broadening and to determine the dominant hopping process in the experiment. The fermionic hopping needs adjustment because the bare hopping is renormalized by coherent phonon processes, which lead to a reduced hopping as discussed in Refs. Bruderer et al. (2007, 2008). This is in direct analogy to the increased effective mass of polarons due to the drag of the phonon cloud Mahan (2000).
As can be seen in Fig. 4 the fit describes the numerical results very accurately with moderate broadening and a minor reduction of the fermionic hopping . Further, we find that and thus the dominant transport processes are nearest- and next-nearest-neighbor hopping; this is consistent with the absence of higher-order phonon resonances noted earlier. The high level of agreement between our theoretical model and the numerical results is partly explained by two observations: First, the numerical results show that the fermion reaches a constant drift velocity on a time scale much shorter than the evolution time , thus transient effects due to the sudden tilting of the lattice are negligible. Second, the drift velocity remains approximately constant over the entire evolution time, as illustrated in Fig. 5, resulting in a stationary atomic current.
Since our numerics simulates the full dynamics of the system we can monitor the average position of the fermion at different evolution times and extend the range of parameters considered so far. In particular, we are interested in the atomic current in presence of strong boson-fermion and boson-boson interactions, where our theory, assuming a superfluid phase, is no longer applicable. Figure 5 shows the displacement of the fermion as a function of the lattice tilt for the interaction strengths and for different evolution times . We see that the NDC peak and the phonon resonance are not noticeably affected by the presence of strong interactions, thus they both seem to be robust features of the atomic current. We note that the additional peaks in the displacement-tilt dependence (located between the NDC peak and the phonon resonance) are due to finite-time effects, which is revealed by a more detailed analysis of the time dependence of the numerical results. Furthermore, we observe an approximately linear increase of the displacement of the fermion with the evolution time—a finding also confirmed for weaker interactions.
In our analytical and numerical investigation of phonon-assisted atomic currents along a tilted potential we have shown that Bose-Fermi mixtures in optical lattices lend themselves naturally to investigate nonequilibrium transport phenomena present in solid-state systems. In more detail, we have formulated an effective model for the Bose-Fermi mixture describing the bosons and fermions in terms of Bogoliubov phonons and Wannier-Stark states, respectively, with a generic fermion-phonon interaction of an identical type to the one encoutered in solids.
We have studied the dependence of the atomic current on the lattice tilt from first principles and found that our model accommodates negative differential conductance and phonon resonances. To demonstrate that these features are observable by using ultracold atoms in the context of a finite-size system and a realistic measuring procedure we have calculated the atomic current numerically by using the TEBD algorithm including the full dynamics of both the bosons and the fermions. Our numerical results show that the phonon resonance at the boundary of the phonon band is a robust phenomenon that occurs over a wide range of system parameters despite broadening, which might be increased by finite temperature effects Bruderer et al. (2008); Kolovsky (2008) not directly taken into account in our model. Finally, we note that in an obvious extension of this work we will investigate the effect of the transition of the bosons from the superfluid to the Mott insulator regime on the atomic current through the system.
Acknowledgements.M.B. thanks the Swiss National Science Foundation for the support through the project PBSKP2/130366. M.B., W.B. and A.P. acknowledge financial support from the German Research Foundation (DFG) through SFB 767. S.R.C. and D.J. thank the National Research Foundation and the Ministry of Education of Singapore for support. D.J. acknowledges support from the ESF program EuroQUAM (EPSRC Grant No. EP/E041612/1).
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