Synthetic magnetic fields for ultracold neutral atoms
Neutral atomic Bose condensates and degenerate Fermi gases have been used to realize important many-body phenomena in their most simple and essential formsGreiner et al. (2003); Regal et al. (2004); Zwierlein et al. (2004), without many of the complexities usually associated with material systems. However, the charge neutrality of these systems presents an apparent limitation - a wide range of intriguing phenomena arise from the Lorentz force for charged particles in a magnetic field, such as the fractional quantum Hall states in two-dimensional electron systemsTsui et al. (1982); Laughlin (1983). The limitation can be circumvented by exploiting the equivalence of the Lorentz force and the Coriolis force to create synthetic magnetic fields in rotating neutral systems. This was demonstrated by the appearance of quantized vortices in pioneering experimentsZwierlein et al. (2005); Schweikhard et al. (2004); Madison et al. (2000); Abo-Shaeer et al. (2001) on rotating quantum gases, a hallmark of superfluids or superconductors in a magnetic field. However, because of technical issues limiting the maximum rotation velocity, the metastable nature of the rotating state and the difficulty of applying stable rotating optical lattices, rotational approaches are not able to reach the large fields required for quantum Hall physicsJuzeliūnas and Öhberg (2004); Jaksch and Zoller (2003); Sorensen et al. (2005). Here, we experimentally realize an optically synthesized magnetic field for ultracold neutral atoms, made evident from the appearance of vortices in our Bose-Einstein condensate. Our approach uses a spatially-dependent optical coupling between internal states of the atoms, yielding a Berry’s phaseBerry (1984) sufficient to create large synthetic magnetic fields, and is not subject to the limitations of rotating systems; with a suitable lattice configuration, it should be possible to reach the quantum Hall regime, potentially enabling studies of topological quantum computation.
In classical electromagnetism, the Lorentz force for a particle of charge moving with velocity in a magnetic field is . In the Hamiltonian formulation of quantum mechanics, where potentials play a more central role than fields, the single-particle Hamiltonian is , where is the vector potential giving rise to the field , is the canonical momentum and is the mass. In both formalisms, only the products and are important. To generate a synthetic magnetic field for neutral atoms, we engineer a Hamiltonian with a spatially dependent vector potential producing .
The quantum mechanical phase is the relevant and significant quantity for charged particles in magnetic fields. A particle of charge traveling along a closed loop acquires a phase due to the presence of magnetic field , where is the enclosed magnetic flux and is the flux quantum. A similar path-dependent phase, the Berry’s phaseBerry (1984), is the geometric phase acquired by a slowly moving particle adiabatically traversing a closed path in a Hamiltonian with position dependent parameters. The Berry’s phase depends only on the geometry of the parameters along the path, and is distinct from the dynamic contribution to the phase which depends upon the speed of the motion.
The close analogy with the Berry’s phase implies that properly designed position-dependent Hamiltonians for neutral particles can simulate the effect of magnetic fields on charged particles. We create such a spatially-varying Hamiltonian for ultracold atoms by dressing them in an optical field that couples different spin states. The appropriate spatial dependence can originate from the laser beams’ profileJuzeliūnas and Öhberg (2004); Juzeliūnas et al. (2006); Gunter et al. (2009) or, as here, a spatially-dependent laser-atom detuningSpielman (2009). An advantage of this optical approach compared to rotating gases is that the synthetic field exists at rest in the lab frame, allowing all trapping potentials to be time-independent.
The large synthetic magnetic fields accessible by this approach make possible the study of unexplored bosonic quantum-Hall states, labeled by the filling factor , the ratio of atom number to the number of flux quanta. The most outstanding open questions in quantum-Hall physics center on states whose elementary quasiparticle excitations are anyons: neither bosons nor fermions. In some cases these anyons may be non-abelian, meaning that moving them about each other can implement quantum gates, thus non-abelian anyons are of great interest for this “topological” quantum computationNayak et al. (2008). In electronic systems, the observed quantum-Hall state may be such a system, but its true nature is still uncertainRadu et al. (2008). In contrast, the bosonic quantum-Hall state with contact interactions has the same non-abelian anyonic excitations as the state in electronic systems is hoped toCooper (2008).
To engineer a vector potential , we illuminate a Rb BEC with a pair of Raman laser beams with momentum difference along (Fig. 1a). These couple together the three spin states, and , of the electronic ground state (Fig. 1b), producing three dressed states whose energy-momentum dispersion relations are experimentally tunable. Example dispersions are illustrated in Fig. 1c. The lowest of these, with minimum at , corresponds to a term in the Hamiltonian associated with the motion along , , where is an engineered vector potential that depends on an externally controlled Zeeman shift for the atom with a synthetic charge , and is the effective mass along . To produce the desired spatially-dependent (Fig. 1d), generating , we apply a Zeeman shift that varies linearly along . The resulting is approximately uniform near , at which point . (Here, the microscopic origin of the synthetic Lorentz forceCheneau et al. (2008) is optical along , depending upon the velocity along ; the force along is magnetic, depending upon the velocity.) By these means, we engineer a Hamiltonian for ultracold atoms, that explicitly contains a synthetic magnetic field, with vortices in the ground state of a BEC. This is distinctly different from all existing experiments, where vortices are generated by phase imprintingLeanhardt et al. (2002); Andersen et al. (2006), rotationSchweikhard et al. (2004); Madison et al. (2000); Abo-Shaeer et al. (2001), or a combination thereofMatthews et al. (1999). Each of these earlier works presents a different means to impart angular momentum to the system yielding rotation. Fig. 1e shows an experimental image of the atoms with . Fig. 1f, with , shows vortices. This demonstrates an observation of an optically induced synthetic magnetic field.
We create a Rb BEC in a 1064 nm crossed dipole trap, loaded into the lowest energy dressed stateLin et al. (2009a) with atom number up to , and a Zeeman shift MHz, produced by a real magnetic bias field . The nm Raman beams propagate along and differ in frequency by a constant , where a small Raman detuning largely determines the vector potential . The scalar light shift from the Raman beams, combined with the dipole trap gives an approximately symmetric three-dimensional potential, with frequencies Hz. Here, and are the appropriate units for the momentum and energy.
The spin and momentum states coupled by the Raman beams can be grouped into families of states labeled by the momentum . Each family ,,, is composed of states that differ in linear momentum along by , and are Raman-coupled with strength . For each , the three dressed states are the eigenstates in the presence of the Raman coupling, with energies Lin et al. (2009a). The resulting vector potential is tunable within the range . In addition, includes a scalar potential Spielman (2009). , , and are functions of Raman coupling and detuning , and for our typical parameters , reducing from Hz to Hz. The BEC’s chemical potential kHz is much smaller than the kHz energy separation between dressed states, therefore the BEC only occupies the lowest energy dressed state. Further, it justifies the harmonic expansion around , valid at low energy. Hence, the complete single-atom Hamiltonian is , where is the external potential including .
The dressed BEC starts in a uniform bias field , at Raman resonance (), corresponding to Lin et al. (2009a). To create a synthetic field , we apply a field gradient such that , ramping in 0.3 s from to a variable value up to 0.055 Tm, and then hold for to allow the system to equilibrate. The detuning gradient generates a spatial gradient in . For the detuning range in our experiment, is approximately constant, leading to an approximately uniform synthetic field given by (Fig. 1d). To probe the dressed state, we switch off the dipole trap and the Raman beams in less than s, projecting each atom into spin and momentum components. We absorption-image the atoms after a time-of-flight (TOF) ranging from 10.1 ms to 30.1 ms (Fig. 1ef).
For a dilute BEC in low synthetic fields, we expect to observe vortices. In this regime, the BEC is described by a macroscopic wave function , which obeys the Gross-Pitaevskii equation (GPE). The phase winds by around each vortex, with amplitude at the vortex center. The magnetic flux results in vortices and for an infinite, zero temperature system, the vortices are arrayed in a latticeYarmchuk et al. (1979) with density . For finite systems vortices are energetically less favorable, and their areal density is below this asymptotic value, decreasing to zero at a critical field . For a cylindrically symmetric BEC, is given by ln where is the Thomas-Fermi radius and is the healing lengthLundh et al. (1997). is larger for smaller systems. For our non-cylindrically symmetric system, we numerically solve the GPE to determine for our experimental parameters (see Methods).
For synthetic fields greater than the critical value, we observe vortices that enter the condensate and reach an equilibrium vortex number after s. Due to a shear force along when the Raman beams are turned off, the nearly-symmetric insitu atom cloud tilts during TOF. While the vortices’ positions may rearrange, any initial order is not lost. During the time of our experiment, the vortices do not form a lattice and the positions of the vortices are irreproducible between different experimental realizations, consistent with our GPE simulations. We measure as a function of detuning gradient at two couplings, and (Fig. 2). For each , vortices appear above a minimum gradient when the corresponding field exceeds the critical field . (For our coupling is only approximately uniform over the system and is the field averaged over the area of the BEC.) The inset shows for both values of plotted versus , the vortex number for a system of area with the asymptotic vortex density, where () is the Thomas-Fermi radius along (or . Since the system size, and thus , are approximately independent of , we expect this plot to be nearly independent of Raman coupling. Indeed, the data for and only deviate for , likely due to the intricate dynamics of vortex nucleationMurray et al. (2009).
Figure 3 illustrates a progression of images showing vortices nucleate at the system’s edge, fully enter to an equilibrium density and then decay along with the atom number. The time scale for vortex nucleation depends weakly on , and is more rapid for larger with more vortices: It is s for vortex number , and increases to s for . For ( near ), the single vortex always remains near the edge of the BEC. In the dressed state, spontaneous emission from the Raman beams removes atoms from the trap, causing the population to decay with a 1.4(2) s lifetime, and the equilibrium vortex number decreases along with the BEC’s area.
To verify the dressed state has reached equilibrium, we prepare nominally identical systems in two different ways. First, we vary the initial atom number and measure as a function of atom number at a fixed hold time s. Second, starting with a large atom number, we measure both and , as they decrease with (Fig. 3). Figure 4 compares versus measured with both methods, each at two detuning gradients corresponding to fields . The data show as a function of is the same for these preparation methods, providing evidence that for s, has reached equilibrium. As the atom number falls, the last vortex departs the system when the critical field – increasing with decreasing – surpasses the actual field.
In conclusion, we have demonstrated optically synthesized magnetic fields for neutral atoms resulting from the Berry’s phase, a fundamental concept in physics. This novel approach differs from experiments with rotating gases, where it is difficult to add optical lattices and rotation is limited by heating, metastability, and the difficulties to add large angular momentum, preventing access to the quantum-Hall regime. A standout feature in our approach is the ease to add optical lattices. For example, the addition of a 2D lattice makes it immediately feasible to study the fractal energy levels of the Hofstadter butterflyHofstadter (1976). Further, a 1D lattice can divide the BEC into an array of 2D systems normal to the field. A suitable lattice configuration allows access to the quantum-Hall regime, with an ensemble of 2D systems each with atoms, and with a realistic nK interaction energy.
We thank W.D. Phillips for discussions. This work was partially supported by ONR, ARO with funds from the DARPA OLE program, and the NSF through the JQI Physics Frontier Center. R.L.C. acknowledges the NIST/NRC postdoctoral program and K.J.G. thanks CONACYT.
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.1 Dressed state preparation
We create a Rb BEC in a crossed dipole trapLin et al. (2009b), with atoms in . The quadratic Zeeman shift is for MHz, where is the Land -factor. To maintain at the BEC’s center as we ramp the field gradient , we change by as much as . Simultaneously, we decrease the dipole beam power by , producing our Hz trap frequency along . Additionally, the detuning gradient makes the scalar potential anti-trapping along , reducing from 70 Hz to 50 Hz for our largest . Spontaneous emission from the Raman beams decreases the atom number to for , with a condensate fraction of .
.2 Numerical method
We compare our data to a finite temperature 2D stochastic GPE (SGPEBlakie et al. (2008)) simulation including the dressed state dispersion that depends on through the detuning gradient . We evolve the time-dependent projected GPE
projects onto a set of significantly occupied modes, and parameterizes the 2D interaction strength. The SGPE models interactions between the highly-occupied modes described by and sparsely occupied thermal modes with dissipation and an associated noise term. We approximately account for the finite extent along by making depend on the local 2D density . For low temperatures this 2D model correctly recovers the 3D Thomas-Fermi radii, and gives the expected 2D density profile. These quantitative details are required to correctly compute the critical field or number for the first vortex to enter the system, which are directly tied to the 2D condensate area.