# Observation of a superfluid Hall effect

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Edited by* Allan H. MacDonald, University of Texas, Austin, TX, and approved May 3, 2012 (received for review February 15, 2012)

## Abstract

Measurement techniques based upon the Hall effect are invaluable tools in condensed-matter physics. When an electric current flows perpendicular to a magnetic field, a Hall voltage develops in the direction transverse to both the current and the field. In semiconductors, this behavior is routinely used to measure the density and charge of the current carriers (electrons in conduction bands or holes in valence bands)—internal properties of the system that are not accessible from measurements of the conventional resistance. For strongly interacting electron systems, whose behavior can be very different from the free electron gas, the Hall effect’s sensitivity to internal properties makes it a powerful tool; indeed, the quantum Hall effects are named after the tool by which they are most distinctly measured instead of the physics from which the phenomena originate. Here we report the first observation of a Hall effect in an ultracold gas of neutral atoms, revealed by measuring a Bose–Einstein condensate’s transport properties perpendicular to a synthetic magnetic field. Our observations in this vortex-free superfluid are in good agreement with hydrodynamic predictions, demonstrating that the system’s global irrotationality influences this superfluid Hall signal.

Microscopically, the Hall effect (1) results from the Lorentz force **F** = *q***v** × **B** experienced by particles with charge *q* and velocity **v** moving in a uniform magnetic field **B**. In the plane perpendicular to **B** = *B***e**_{z}, this force acts on a current with density **J** = *J*_{x}**e**_{x} + *J*_{y}**e**_{y} to generate an electrochemical potential gradient normal to **J**, where the Hall part of the 2D resistivity tensor [1]is antisymmetric. In conventional metals and semiconductors, the Hall resistivity *ρ*_{xy} = *B*/*qn*(**r**) is related to the carriers’ charge *q* and density *n*(**r**), but not to the dissipative resistivity tensor , where is the 2 × 2 identity matrix. Typically, experiments measure a sample’s longitudinal and transverse voltages *V*_{xx} and *V*_{xy} (Fig. 1*A*) from which the resistivity tensor can be inferred (2). Here, we report an analogous transport measurement of the full resistivity tensor, including the antisymmetric contributions from the Hall effect, in a flattened elongated Bose-Einstein condensate (BEC) subjected to a synthetic magnetic field *B*^{∗}**e**_{z} (in which only the charge-field product *q*^{∗}**B**^{∗} is defined; ref. 3).

The transport characteristics of systems with many particles and sufficiently strong interactions (Coulomb repulsion in electron gases and plasmas, or *s*-wave contact interactions in BECs) resemble those of classical fluids and are described by hydrodynamics. These hydrodynamics can describe ultracold Bose (4) and Fermi (5) gases, or characterize the collective modes—plasmons (6) and magnetoplasmons (7)—of 2D electron gases (2DEGs). We show that a BEC in a synthetic magnetic field obeys hydrodynamic equations similar to those describing a 2DEG in a uniform magnetic field.

## Drude Model and Hydrodynamics

In a simple Drude model (8) description of a 2DEG in a magnetic field (Fig. 1*A*), scattering from impurities gives rise to resistance while a position-dependent potential *V*(**r**) controls the electron density. The collective electron dynamics can be expressed in terms of hydrodynamic continuity and Euler equations: [2][3]where **r** = (*x*,*y*); **p**_{m} = [*m*/*qn*(**r**)]**J** is the mechanical momentum of particles with mass *m*, charge *q*, and density *n*(**r**); Ω_{C} = *qB*/*m* is the cyclotron frequency for magnetic field strength *B*; *U*(**r**) = *qV*(**r**) is the potential energy; *g* = 2*πℏ*^{2}/*m* accounts for the effects of Fermi pressure in this noninteracting 2DEG (6); *τ* is the momentum relaxation time due to scattering from impurities; and *D*/*Dt* = ∂_{t} + *m*^{-1}[**p**_{m}(**r**)·∇] is the convective derivative. The matrix in Eq. **3** is proportional to the resistivity tensor [through the factor *m*/*q*^{2}*n*(**r**)] and has equal diagonal components from (conventional resistance) and antisymmetric off-diagonal components from (Hall resistance).

To compare trapped interacting BECs with the 2DEGs described above, the superfluid hydrodynamic (SFHD) equations may be extended (9, 10) to include a time-independent artificial magnetic field **B**^{∗} = *B*^{∗}**e**_{z}. The resulting 2D continuity and Euler equations are identical to Eqs. **2** and **3**, complete with the off-diagonal antisymmetric components of the resistivity tensor—the Hall term—proportional to the cyclotron frequency Ω_{C} = *q*^{∗}*B*^{∗}/*m* (*SI Text*). The BEC’s trap provides the external potential energy with trap frequencies *ω*_{x,y} along **e**_{x,y}. The interaction term *gn*(**r**) accounts for the mean-field energy due to contact interactions in the BEC. In principle, the resistivity tensor’s diagonal components are zero due to the BEC’s superfluidity; however, atom loss can imitate damping and we retain a phenomenological dissipative term proportional to *τ*^{-1}. Though the systems with which we work are fully three-dimensional, the dynamics of interest are in the plane perpendicular to *q*^{∗}**B**^{∗} and are described by these 2D equations. For quantitative comparisons to data, calculations were performed using the 3D version of Eqs. **2** and **3** (*SI Text*).

## Superfluidity in an Artificial Magnetic Field

By expressing the SFHD equations in terms of the gauge-invariant mechanical momentum **p**_{m}, which describes the actual motion of particles, we depart from the conventional expressions in terms of the canonical momentum **p** = **p**_{m} + *q*^{∗}**A**^{∗} (where **A**^{∗} is the artificial vector potential such that **B**^{∗} = ∇ × **A**^{∗}), which is directly related to the gradient of the superfluid order parameter’s phase. The distinction between **p** and **p**_{m} has tangible consequences when considering the condition of irrotationality ∇ × **p** = 0. When a synthetic magnetic field is present, irrotationality requires equilibrium fluid flow, with ∇ × **p**_{m} = -*q*^{∗}**B**^{∗}. This condition is relaxed at places like vortex cores, where superfluid density is zero; however, in our trapped BEC, vortices are energetically allowed only when Ω_{C} is greater than a critical value (11, 12). Unless noted, we remained in the fully irrotational regime where Ω_{C} is below this critical value.

In this experiment, we studied mass transport in a BEC subject to a synthetic magnetic field **B**^{∗} created using Raman dressing (3, 11). Previous experiments used rotating traps to exploit the equivalence between the Coriolis and Lorentz forces (13), methods that use atom-light coupling (14) instead of rotation allow for greater geometric freedom and potentially increased field strengths (15). The Raman technique used here enabled our experiment’s highly anisotropic BECs, whose 4∶1 aspect ratio (Fig. 1*B*) mimicked a typical bar Hall geometry and facilitated identification of the Hall response.

## Results and Discussion

Our experiments began with a nearly pure ^{87}Rb BEC of about 2 × 10^{5} atoms in the internal state (16) confined in a crossed optical dipole trap with frequencies (*ω*_{x},*ω*_{y},*ω*_{z})/2*π* ≈ (11.3,42.5,90) Hz (Fig. 1*B*). Laser beams counterpropagating along **e**_{x} Raman-coupled spin states *m*_{F=1} = 0, ± 1 of differing momentum to create new dressed eigenstates—spin-momentum superpositions (11). The dressed atoms experienced a synthetic vector potential (11), which, in the presence of an appropriate real magnetic field gradient, gave *q*^{∗}**A**^{∗} = (-*q*^{∗}*B*^{∗}*y*,0,0) and the associated synthetic charge-field product *q*^{∗}**B**^{∗} = *q*^{∗}*B*^{∗}**e**_{z}.

### Probing the Hall Response.

To measure transport, we generated a mass current (a velocity field) in the BEC by perturbatively modulating the trap along **e**_{x} (Fig. 1*C*). We modified the potential energy *U*(*x*,*y*,*z*) = (*κ*_{x}*x*^{2} + *κ*_{y}*y*^{2} + *κ*_{z}*z*^{2})/2, where is the spring constant along **e**_{i}, by adding a time-dependent drive *δU*_{x} = (*δκ*_{x}*x*^{2}/2) sin(*ωt*) with amplitude *δκ*_{x} and drive frequency *ω*. To ensure linear response, we chose *δκ*_{x} ≤ 0.13*κ*_{x}; to avoid transients, we smoothly increased *δκ*_{x} from zero in 500 ms. After driving in steady state for a time *t*_{mod}, we measured the atomic density, either in situ or after time-of-flight (TOF). We characterized the system’s temporal response to the drive by repeating this procedure for 64 equally spaced values of *t*_{mod}, the longest of which encompassed eight cycles of modulation.

When the geometry of the trap deformation is well-overlapped with one of the system’s low-energy collective excitations (17, 18)—eigenmodes of the SFHD equations—the response is enhanced as *ω* approaches that mode’s resonance frequency. Such collective modes have been measured in both Bose (19⇓–21) and Fermi (22, 23) gases. In this work, we predominantly drove the quadrupole-like “*X*_{2}” eigenmode (whose frequency is interaction-dependent) corresponding to compression and expansion primarily along **e**_{x} (an oscillatory response), with smaller out-of-phase contributions, due to interactions, along **e**_{y} and **e**_{z} ( and responses). When the synthetic magnetic field is nonzero, correlated transport along **e**_{x} and **e**_{y} appears (an response) resulting from mixing with the scissors mode (21). This *B*^{∗}-dependent correlated transport is the Hall effect. Measurements of these second-order moments probe the response of the local chemical potential to particle currents, averaged over the system. The quantitative values of these responses depend upon the equation of state, represented by the interaction term (*gn*(**r**)) in Eq. **3**. Had we excited the “*X*_{1}” dipole (sloshing) mode, we would have observed the Hall effect, but would not have probed internal properties related to interactions, as described by Kohn’s theorem (24).

Using absorption imaging, we measured the time-dependent column density distributions *n*(*x*,*y*) either in situ (to directly determine the density response) or after TOF (to determine the momentum response). For in situ measurements, we imaged the atoms immediately following release from the trap (*t*_{off} < 1 μs). For TOF measurements, we allowed the atoms to expand freely for 36.2 ms before imaging. During the first 2 ms of TOF, we deloaded (25) the Raman-dressed atoms into the bare state before removing the Raman dressing fields.

These density distributions offer snapshots of the dynamically evolving BEC at *t*_{mod}. For each *t*_{mod}, we extract three independent second-order moments of the density distribution, , , and : [4]where *x*_{i}∈{*x*,*y*} was measured from the center of the density distribution .

### Temporal Response.

Fig. 2 shows *n*(*x*,*y*) at three times during the modulation cycle without (Figs. 2 *A*–*C*) and with (Figs. 2 *D*–*F*) the synthetic magnetic field. Although the compression of the cloud along the drive direction **e**_{x} was always the most evident signal, a nonzero synthetic magnetic field (Ω_{C} ≠ 0) tilted the clouds as the velocity field responded to the drive current, macroscopically manifesting the BEC’s superfluid Hall resistivity. In Fig. 2 *D*–*F*, the Hall signal oscillates about zero and reaches its minimum, zero, and maximum, respectively. The temporal responses of the second-order moments are plotted in Fig. 2 *G* (Ω_{C} = 0) and *H* (Ω_{C} ≠ 0) for *ω*/2*π* = 12 Hz. This modulation frequency is above resonance (Fig. 3): close enough for a strong signal, but far enough to spectrally resolve the natural mode from the driven mode. Although and were not significantly affected by the introduction of the synthetic field, a weak Ω_{C} dependence of the parallel and perpendicular moments arises from variations in the effective trapping frequencies as field strength increases (3, 10).

### Spectral Response.

Fig. 3 shows the spectrum of the response across the low-frequency *X*_{2} resonance. For each modulation frequency *ω*, we measured the periodically varying second-order moments , , and for eight complete cycles. Using a lock-in technique referenced to *ω*, we separately extracted the in-phase and out-of-phase components of the response (Fig. 3, *Upper*), from which we determine the second-order moments’ amplitude responses , , and (Fig. 3, *Lower*). We used the TOF method to ensure a strong signal even far from resonance. At a moderate synthetic magnetic field strength [Ω_{C}/2*π* = 10.5(5) Hz], no vortices entered the BEC and Eqs. **2** and **3** apply. We simultaneously fit the six *ω* dependent in- and out-of-phase responses to the solutions of the linearized SFHD equations (propagated through TOF without linearization, see *SI Text*) and extracted three free parameters: the phenomenological damping term *τ*^{-1} = 4.4(4) s^{-1}, which is the diagonal part of the resistivity matrix in Eq. **3**; the effective trap frequency, *ω*_{x}/2*π* = 6.07(2) Hz; and the drive strength *δκ*_{x}/*κ*_{x} = 0.074(2). The latter two parameters are in reasonable agreement with the directly measured quantities *ω*_{x}/2*π* = 6.1(2) Hz and *δκ*_{x}/*κ*_{x} = 0.10(1). The small value of the damping term modifies these fits only very near resonance and does not allow us to distinguish between possible dissipation mechanisms, such as atom loss, heating, or trap inhomogeneity.

### Dependence on Artificial Field Strength.

Although the above data are well characterized by and are in good agreement with a SFHD description and its TOF propagation, in situ measurements provide a clearer demonstration of Hall physics. For different values of Ω_{C}, we measured the BECs’ time-dependent density profiles and calculated normalized second-order moments , , and from fits to skewed Thomas–Fermi distributions with radii *R*_{x} and *R*_{y} (*SI Text*). As in Fig. 2, the modulation drive frequency was *ω*/2*π* = 12 Hz.

Fig. 4 shows the amplitude response of the normalized second-order moments along with simultaneous fits to the linear-response predictions of SFHD using a single fit parameter, *δκ*_{x} (with the previously measured *ω*_{x} and *τ*^{-1} as fixed parameters). For small Ω_{C}, we find good agreement between our measurements and SFHD. The nonlinearity of at high fields is caused by the shift of the *X*_{2} resonance away from the fixed drive frequency *ω* as Ω_{C} increases (3, 10). For small Ω_{C}, where the resonance is nearly constant, the slope of vs Ω_{C} measures an analogue of the Hall resistivity *ρ*_{xy} = *m*Ω_{C}/*q*^{2}*n* (Eq. **1**) averaged over the cloud’s changing density (assuming the BEC’s resistivity has units for which the artificial charge *q*^{∗} is equivalent to the electron charge *q* = -e). The good agreement between our measurements and the irrotational SFHD model demonstrates the utility of this Hall technique as a simple, quantitative way to probe microscopic properties, like the superfluid Hall resistivity, of a quantum degenerate gas. Beyond Ω_{C}/2*π* ≈ 15 Hz, indications of vortices were evident in TOF; therefore, we expect that the predictions of vortex-free SFHD no longer apply. In the large-*B*^{∗} limit of “diffused vorticity,” (26) in which many vortices have entered the BEC and “ordinary” fluid behavior with ∇ × **p**(**r**) = *q*^{∗}**B**^{∗} is valid, calculations show that the superfluid Hall signal approaches zero, but does not become negative as in some superconducting materials (27, 28). The intermediate regime, in which several vortices are present but the limit of diffused vorticity is not valid, is an interesting system for future study.

### Conclusions.

Inspired by condensed-matter techniques, we have demonstrated Hall physics in a BEC and showed that it is sensitive to the properties of an interacting, irrotational BEC. This macroscopic Hall measurement technique can be extended to probe the microscopic properties of more complicated ultracold configurations, such as 2D systems with vortices (arising from thermal fluctuations, ref. 29, or synthetic magnetic fields, ref. 11), systems with spin-orbit coupling (30), and quantum Hall systems (31).

## Acknowledgments

We appreciate conversations with J. V. Porto. This work was partially supported by Office of Naval Research, Army Research Office with funds from both Defense Advanced Research Planning Agency’s Optical Lattice Emulator program and the Atomtronics Multidisciplinary University Research Initiative; and the National Science Foundation through the Physics Frontier Center at JQI. L.J.L. acknowledges support from Natural Sciences and Engineering Research Council; K.J.-G. acknowledges Consejo Nacional de Ciencia y Tecnologia; and M.C.B. acknowledges National Institute of Standards and Technology-American Recovery and Reinvestment Act.

## Footnotes

- ↵
^{1}To whom correspondence should be addressed. E-mail: ian.spielman{at}nist.gov.

Author contributions: L.J.L. led the data taking effort in which K.J.-G., R.A.W., M.C.B., A.R.P., W.D.P., and I.B.S. participated; L.J.L. and I.B.S. performed numerical and analytic calculations; I.B.S. suggested the initial measurements; and L.J.L., K.J.-G., R.A.W., M.C.B., A.R.P., W.D.P., and I.B.S. wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1202579109/-/DCSupplemental.

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