A synthetic electric force acting on neutral atoms

Abstract

Electromagnetism is a simple example of a gauge theory where the underlying potentials (the vector and scalar potentials) are defined only up to a gauge choice. The vector potential generates magnetic fields through its spatial variation and electric fields through its time dependence¹. Here, we report experiments in which we have produced a synthetic gauge field. The gauge field emerges only at low energy in a rubidium Bose–Einstein condensate: the neutral atoms behave as charged particles do in the presence of a homogeneous effective vector potential². We have generated a synthetic electric field through the time dependence of an effective vector potential, a physical consequence that emerges even though the vector potential is spatially uniform.

Main

Gauge theories play a central role in modern quantum physics. In some cases, they can be viewed as emerging as the low-energy description of a more complete theory^3,4. Electromagnetism is the best known gauge theory and its gauge fields are the ordinary scalar and vector potentials. Magnetic fields arise only from spatial variations of the vector potential, whereas electric fields arise from both time variations of the vector potential and gradients of the scalar potential. These potentials are defined only to within a gauge choice, where for a charged particle the canonical momentum (the variable canonically conjugate to position) and the mechanical momentum (the mass times the velocity) are not equal. Our experiments² have realized a particular version⁵ of a class of proposals^{6,7,8,9,10,11} to generate effective vector potentials for neutral atoms through interactions with laser light, and have created synthetic magnetic fields¹² important for simulating charged condensed-matter systems with neutral atoms^13,14. Here we demonstrate the complementary phenomenon: a synthetic electric field generated from a time-dependent effective vector potential. Additionally, we make independent measurements of both the mechanical momentum and canonical momentum, where the latter is usually not possible.

The electromagnetic vector potential A for a charged particle appears in the Hamiltonian H=(p_can−q A)²/2m, where p_can is the canonical momentum, q is the charge and m is the mass. (p_can−q A=m v is the mechanical momentum for a particle moving with velocity v.) We recently demonstrated a technique to engineer Hamiltonians of this form for ultracold atoms, and prepared a Bose–Einstein condensate (BEC) at rest with an effective vector potential constant in time and space², corresponding to E=B=0, where E and B are the synthetic electric field and synthetic magnetic field for neutral atoms, respectively. In ref. 12, we made A depend on position, giving but E=0. Here we add time dependence to a spatially uniform vector potential , generating a synthetic electric field . The resulting force is distinct from that arising from gradients of scalar potentials ϕ(r), for example, from an external trapping potential. A revealing analogue is that of an infinite solenoid of radius r₀ as pictured in Fig. 1a: a magnetic field exists only inside the coil; however, a non-zero cylindrically symmetric vector potential extends outside the coil. Far from the coil A is nearly uniform, analogous to our uniform effective vector potential. When the current is changed in a time interval Δt, B changes with it and therefore A changes by ΔA. A charged particle on the axis feels an electric field during Δt, leading to , a change in the mechanical momentum even outside the solenoid.

Figure 1: Schematic diagram of the electric field generated by a time-varying vector potential.

a, Emulated system, showing the electric current flowing anticlockwise in the infinite solenoid (black coil) with radius r₀ and the real magnetic field B only inside the solenoid. The blue lines represent the vector potential A. A charged particle (red dot) located far from the coil experiences a nearly uniform A. b, Calculated time response of the synthetic vector potential and electric field for neutral atoms in our first measurement (see Fig. 3). The calculation includes the known inductive response time of the bias field B₀, which sets the detuning, and the calibration of detuning to vector potential shown in Fig. 2d.

We synthesize electromagnetic fields for neutral atoms by illuminating a ⁸⁷Rb BEC with two intersecting laser beams (Fig. 2a) that couple together three atomic spin states within the electronic ground state (Fig. 2b). The three new energy eigenstates, or ‘dressed states’, are superpositions of the uncoupled spin and linear-momentum states and have modified energy–momentum dispersion relations compared with those of uncoupled atoms. The dressed atoms act as particles with a single well-defined velocity v, which is the population-weighted average of all three spin components.

Figure 2: Experimental setup for synthetic electric fields.

a, Physical implementation indicating the two Raman laser beams incident on the BEC (red arrows) and the physical bias magnetic field B₀ (black arrow). The blue arrow indicates the direction of the synthetic electric field E*. b, The three m_F levels of the F=1 ground-state manifold are shown as coupled by the Raman beams. c, Dressed-state eigenenergies as a function of canonical momentum for the realized coupling strength of ℏΩ_R=10.5E_L at a representative detuning ℏδ=−1E_L (coloured curves). The grey curves show the energies of the uncoupled states, and the red curve depicts the lowest-energy dressed state in which we load the BEC. The black arrow indicates the dressed BEC’s canonical momentum p_can=q*A*, where A* is the vector potential. d, Vector potentials as measured from the canonical momentum.

The dispersion relation of the lowest-energy dressed state changes near its minimum, from p²/2m to (p−p_min)²/2m* (Fig. 2c), where the minimum location p_min plays the role of q A. In addition, the mass m is modified to an effective mass m*>m, and both p_min and m* are under experimental control (not independently). We identify p_min=q*A*, the product of an effective charge q* and an effective vector potential A* for the dressed neutral atoms. As we change A*, we induce a synthetic electric field E*=−∂ A*/∂ t, and the dressed BEC responds as d(m*v)/dt=−∇ ϕ(r)+q*E*, where v is the velocity of the dressed atoms and m*v=p_can−q*A*. Here, Δ(m*v)=−q*(A_f*−A_i*) is the momentum imparted by q*E* .

We study the physical consequences of sudden temporal changes of the effective vector potential for the dressed BEC. These changes are always adiabatic such that the BEC remains in the same dressed state. We measure the resulting change of the BEC’s momentum, which is in complete quantitative agreement with our calculations and constitutes the first observation of synthetic electric fields for neutral atoms.

Our system (see Fig. 2a) consists of an F=1 ⁸⁷Rb BEC with about 1.4×10⁵ atoms initially at rest^15,16; a small physical magnetic field B₀ Zeeman-shifts each of the spin states m_F=0,±1 by E_0,±1. Here, B₀≈3.3×10⁻⁴ T and E₋₁≈−E₊₁≈g μ_BB₀≫|E₀| . The linear and quadratic Zeeman shifts are ℏω_Z=(E₋₁−E₊₁)/2≈h×2.32 MHz and −ℏε=E₀−(E₋₁+E₊₁)/2≈−h×784 Hz . A pair of laser beams with wavelength λ=801 nm, intersecting at 90° at the BEC, couples the m_F states with strength Ω_R. These Raman lasers differ in frequency by Δω_L≈ω_Z and we define the Raman detuning as δ=Δω_L−ω_Z. Here ℏΩ_R≈10E_L and |ℏδ|<60E_L, where E_L=ℏ²k_L²/2m=h×3.57 kHz and are natural units of energy and momentum.

When the atoms are rapidly moving or the Raman lasers are far from resonance (k_Lv or δ≫Ω_R), the lasers hardly affect the atoms. However, for slowly moving and nearly resonant atoms the three uncoupled states transform into three new dressed states. The spin and linear-momentum state |k_x,m_F=0〉 is coupled to states |k_x−2k_L,m_F=+1〉 and |k_x+2k_L,m_F=−1〉, where ℏk_x is the momentum of |m_F=0〉 along , and is the momentum difference between the two Raman beams. For each k_x, the three dressed states are the energy eigenstates in the presence of Raman coupling ℏΩ_R (see ref. 2), with energies E_j(k_x) shown in Fig. 2c (grey for uncoupled states, coloured for dressed states); we focus on atoms in the lowest-energy dressed state. Here the atoms’ energy (interaction and kinetic) is small compared with the ≈10E_L energy difference between the curves; therefore, the atoms remain within the lowest-energy dressed-state manifold⁵, without revealing their spin and momentum components.

In the low-energy limit, E<E_L, dressed atoms have a new effective Hamiltonian for motion along , H_x=(ℏk_x−q*A_x*)²/2m* (motion along and is unaffected); here we choose the gauge where the momentum of the m_F=0 component ℏk_x≡p_can is the canonical momentum of the dressed state. The red curve in Fig. 2c shows the eigenvalues of H_x for q*A_x*>0, indicating that at equilibrium p_can=p_min=q*A_x* (see ref. 2). Although this dressed BEC is at rest (v=∂ H_x/∂ℏk_x=0, zero group velocity), it is composed of three bare spin states each with a different momentum, among which the momentum of |m_F=0〉 is ℏk_x=p_can. None of its three bare spin components has zero momentum, whereas the BEC’s momentum—the weighted average of the three—is zero.

We transfer the BEC initially in |m_F=−1〉 into the lowest-energy dressed state with (see ref. 2 for a complete technical discussion of loading). At equilibrium, we measure q*A*=p_can, equal to the momentum of |m_F=0〉, by first removing the coupling fields and trapping potentials and then allowing the atoms to freely expand for a t=20.1 ms time of flight (TOF). Because the three components of the dressed state differ in momentum by ±ℏ2k_L, they quickly separate. Further, a Stern–Gerlach field gradient along separates the spin components. Figure 2d shows how the measured and predicted A* depend on the detuning δ. With this calibration, we use δ to control A*(t).

We realize a synthetic electric field E* by changing the effective vector potential from an initial value A_i* to a final value A_f*. We prepare our BEC at rest with , and make two types of measurement of E*. In the first, we remove the trapping potential and then change A* by sweeping the detuning δ in 0.8 ms, after which the Raman coupling is turned off in 0.2 ms. Thus, E* can accelerate the atoms unimpeded, and we measure the change of the BEC’s velocity from zero. Figure 3 shows the momentum Δp imparted to the atoms by E* as a function of the vector potential change q*(A_f*−A_i*), denoted by red and blue symbols for q*A_f*/ℏ=−2k_L,2k_L, respectively (see Methods for such a choice of A_f*). Owing to the large final detuning , the final atomic state is a nearly pure spin state, |m_F=+1〉 for q*A_f*/ℏ=2k_L or |m_F=−1〉 for q*A_f*/ℏ=−2k_L. For these undressed final states, m*=m and Δp=m*v=m v, equal to the change in mechanical momentum. We carried out a linear fit Δp=C q*(A_f*−A_i*) to the data and obtained C=−0.996(8), in good agreement with the expected C=−1.

Figure 3: Change in momentum from the synthetic electric-field kick.

Three distinct sets of data were obtained by applying a synthetic electric field by changing the vector potential from q*A_i* (between +2ℏk_L and −2ℏk_L) to q*A_f*. Circles indicate data where the external trap was removed right before the change in A*, where q*A_f*=±2ℏk_L (− for red, + for blue symbols). The black crosses, more visible in the inset, show the amplitude of canonical momentum oscillations when the trapping potential was left on after the field kick. The standard deviations are also visible in the inset. The grey line is a linear fit to the data (circles) yielding slope −0.996±0.008, where the expected slope is −1.

In the second measurement, we examined the time evolution of atoms that remain trapped and strongly dressed after being accelerated by E*. We changed A* in Δt≈0.3 ms but left the dressed BEC in the harmonic confining potential for a variable time before the TOF. As the BEC oscillated in the trap, we monitored the out-of-equilibrium canonical momentum p_can. It is our access to the internal degrees of freedom—here projectively measuring the composition of the Raman dressed state—that enables the determination of p_can. Figure 4a shows the time evolution of p_can for different A_i* all for A_f*≈0; as expected, p_can oscillates about q*A_f*. As Δt is small compared with the ≈25 ms trap period, the change of momentum is dominated by Δp=−q*(A_f*−A_i*), where the contribution from the trapping force is negligible. This translates into an oscillation amplitude Δp in both p_can and m*v=p_can−q*A_f* of dressed atoms; the solid crosses in Fig. 3 show the amplitude of the sinusoidal oscillations in p_can versus A_f*−A_i*≈−A_i*, proving that E* has imparted the expected momentum kick.

Figure 4: Oscillating atoms in the trapping potential after application of a synthetic electric-field pulse.

a,b, Left panels: the vector potential is changed from q*A_i*=0.75ℏk_L (red circles), 0.25ℏk_L (black circles) and ≈0 (green circles), all to q*A_f*≈0, and from q*A_i*=0.75ℏk_L to q*A_f*=0.35ℏk_L (blue symbols). The measured momentum for all circles is the canonical momentum p_can, and that for the squares is the mechanical momentum m v. In b, p_can oscillates about whereas m v oscillates about zero. Right panels: Energy–momentum dispersion curves for uncoupled states (grey) and dressed states (coloured). The arrows indicate oscillations of p_can about q*A_f* for atoms in the lowest-energy dressed state.

We repeated the experiment with a non-zero q*A_f*/ℏ≈0.35k_L, and observed, as expected, that the oscillations in p_can were offset from zero (Fig. 4b). This illustrates that the observed quantity is not the mechanical momentum m v, which should oscillate about zero. We also measured m v, where v is the population-weighted average velocity of all spin components (see Methods); although m v does indeed oscillate about zero, the oscillation amplitude is smaller than that of p_can. Given the increased effective mass, m*/m≈2.5, the trap frequency ν_x along should be reduced by from that for undressed atoms, and the oscillation amplitude of m v should be reduced by m/m*=0.39(1) from that of p_can. Our results show that ν_x² is reduced by a factor of 0.38(4), as expected, but the momentum oscillation amplitude is reduced by 0.30(2), slightly less than predicted (see Methods).

Here we have demonstrated the effects of spatially homogeneous synthetic electric fields; however, this technique is generally applicable to create spatially varying forces. Indeed, as the effective vector potential A* is parameterized by the Raman detuning δ and coupling Ω, it can be locally patterned through suitable spatially inhomogeneous magnetic bias fields or vector light shifts. Our capability of measuring both the canonical momentum and the mechanical momentum m v is essential. The former characterizes the effective vector potential for dressed spin states, and the latter demonstrates that the dressed atom behaves as a usual particle with an effective mass m* and a well-defined velocity v. For atoms initially at rest, as the vector potential is changed with the canonical momentum remaining fixed, the electric field results in a mechanical momentum. For azimuthal vector potentials, such electric-field-induced mechanical momenta can be used to identify the superfluid fraction of cold-atom systems¹⁷. In addition, time-varying, alternating vector potentials provide a unique way to drive the trapped BEC. The BEC’s response is a measurement analogous to the a.c. transport coefficients of condensed-matter systems.

Methods

Example TOF images of the dressed state. We measured the momentum of each bare spin component of the dressed state after TOF, during part of which a Stern–Gerlach gradient was applied. Figure 5a–c shows example images of the data in Figs 2d, 3 and 4a, respectively. For the calibration of the vector potential A* from the detuning δ, we use dressed states at equilibrium where p_can=q*A* and measure p_can, which is defined as the momentum of |m_F=0〉. The mechanical momentum m v is the population-weighted average momentum over all three m_F states, which is nearly zero at equilibrium. This is shown at ℏδ=−1.7E_L and q*A*=0.56ℏk_L in Fig. 5a. As A* is changed from the initial value q*A_i*=0.56ℏk_L to the final value q*A_f*=−2ℏk_L, immediately after the trap turnoff, the atoms are accelerated unimpeded by the resulting synthetic electric field E*; the final atomic state becomes a nearly pure spin state, |m_F=−1〉, with the momentum Δp=m v=−q*(A_f*−A_i*)≈2.56ℏk_L, as illustrated in Fig. 5b. When the induced E* is applied to trapped atoms, the dressed state is driven out of equilibrium where both p_can and m v oscillate with time (see Fig. 5c).

Figure 5: Example TOF images of the dressed state.

a, A dressed state in equilibrium, where p_can is equal to the vector potential q*A*. Here ℏδ=−1.7E_L and correspondingly q*A*=0.56ℏk_L. Images of this type provide the calibration of A* versus detuning δ shown in Fig. 2d. b, A synthetic electric field E* is applied to the atoms in a, by changing A* from the initial q*A_i*=0.56ℏk_L to the final q*A_f*=−2ℏk_L; the atoms then acquire a momentum Δp≈2.56ℏk_L. c, Out-of-equilibrium dressed state where the atoms oscillate in the trap after application of E* by changing A* from q*A_i*=0.75ℏk_L to q*A_f*≈0. Owing to a larger density of the sample than those in a, scattering halos between |m_F=0〉 and |m_F=1〉 are visible, indicating interaction during the TOF.

Dynamic change of effective vector potentials. In our first measurement of synthetic electric fields, we observed the momentum imparted by the field kick to the atoms, resulting from a change in the effective vector potential from an arbitrarily chosen q*A_i* to q*A_f*=±2ℏk_L (see Fig. 3). In principle we could use any A_f* and observe a momentum kick q*(A_i*−A_f*); however, in general the effective vector potential also depends on the strength of the Raman coupling Ω_R. As a result, extra synthetic electric fields typically appear when Ω_R is adiabatically turned off. There are three specific cases for which A* does not depend on Ω_R : when the detuning δ=0 and . For the former case, only when |q*A_i*|<ℏk_L is there no extra electric force during the removal of Ω_R, where the final atomic state is |m_F=0〉. For |q*A_i*|>ℏk_L, the final atomic state is |m_F=±1〉 with an extra momentum of imparted. Thus in our experiment we changed the vector potential from q*A_i* to q*A_f*=±2ℏk_L by changing the detuning from δ_i to a large ; the subsequent turnoff of Ω_R then exerted no extra forces.

Control of Raman detuning. In all of our experiments, we set the Raman detuning δ away from resonance through small changes of the bias magnetic field, and hold the 2.32 MHz frequency difference between the Raman beams constant. Because all temporal changes in δ lead to synthetic electric fields, bias magnetic field noise and relative laser frequency noise can lead to motion in the trap or heating. We phase locked the two Raman beams and observed no change in the heating, showing that relative laser frequency noise is not important in our experiment. However, it is very sensitive to ambient magnetic-field noise, here tied to the 60 Hz line. This noise gives rise to intractable dynamics of the canonical momentum of the dressed state, where δ is held constant after the loading. We measured the field noise from the state decomposition of a radiofrequency-dressed state (no Raman fields) nominally on resonance and then feed-forward cancelled the field noise. This reduced the ∼0.2 μT root-mean-square magnetic-field noise at 60 Hz by about a factor of 20, and the remaining root-mean-square field noise is ∼0.03 μT (including all frequency components up to ≈5 kHz). All of our measurements were made by locking to the 60 Hz line before loading into the dressed state.

Momentum measurements of the dressed state. The Raman dressed state is a superposition of spin and momentum components; its canonical momentum p_can is the momentum of |m_F=0〉. Experimentally, we fit the m_F=0 density distribution after the TOF to a Thomas–Fermi profile¹⁸ and identify p_can as the centre of the distribution. The mechanical momentum of the dressed state m v was measured by a population-weighted average over all three spin states including every pixel with discernible atoms in the image. This takes into account the modification of the TOF density distribution for all m_F states due to interactions during the TOF. Although interactions can exchange momentum between spin states, the total momentum is conserved. Our imaging sensitivity to the m_F=±1 atoms is the same to within 5%, which is insufficient to explain the 0.30(2) reduction factor in the oscillation amplitude of the mechanical momentum, smaller than the predicted value, m/m*=0.39(1).