Gauge matters: observing the vortex-nucleation transition in a Bose condensate

While gauge invariance is central to our description of nature, specific physical situations lend themselves to a 'natural' choice of gauge, without violating gauge freedom. A Bose–Einstein condensate's (BEC's) order parameter includes a gauge-dependent phase. When a BEC is subjected to sufficiently rapid rotation (or a sufficiently strong artificial magnetic field⁶ , $\mathcal{B}$), the BEC exhibits vortices: points at which the density vanishes and the phase is singular. For a finite system, the structural phase transition from a state in which the phase varies smoothly to one with a single phase singularity occurs at a critical magnetic field ${\mathcal{B}}_{\mathrm{cr}}$, dependent on both particle–particle interactions and geometry [1]. In quantum gas experiments, images of time-of-flight (TOF) expanded clouds can reveal the vortex-nucleation transition [1–5] via the appearance of local minima in the imaged atomic density, each associated with a vortex core. Here, we detected this structural phase transition via an abrupt shape-change in TOF-expanded BECs accompanying the appearance of vortices. As we see below, this change depends upon the our experiment's geometry, which in turn implies a most natural gauge choice.

In the description of many-body systems subject to a (real or synthetic) uniform magnetic field, the vector potential is generally written in the gauge most convenient for the problem at hand. Although the symmetric and Landau gauges are common choices, there is no a priori 'natural' gauge. However, looking beyond the simple specification that the magnetic field is uniform, the physical mechanism that creates the magnetic field often suggests a specific gauge choice, for example: any magnetic field created by a single electric current has a natural gauge determined by the geometry in which the vector potential is proportional to the electrical current. In this sense, the symmetric gauge is a natural gauge for an infinite cylindrical solenoid (figure 1(a)), and the Landau gauge is a natural choice for two parallel counterflowing sheets of current (figure 1(b)). Analogous to an infinite solenoid, the synthetic magnetic field for rotating systems is most naturally expressed in the symmetric gauge $\mathcal{A}=(-\boldsymbol{\mathcal{B}}y{{\bf e}}_{x}+\mathcal{B}x{{\bf e}}_{y})/2$ because changes to the rotation rate lead to changes of $\boldsymbol{\mathcal{A}}$ having the form of the symmetric gauge. As in the case for two sheet currents, our experiment's [5] synthetic magnetic field $\boldsymbol{\mathcal{B}}$ is derived from an engineered vector potential [6] that is most naturally expressed by the Landau form $\boldsymbol{\mathcal{A}}=-\mathcal{B}y{{\bf e}}_{x}$, because the coupling between laser fields along ${{\bf e}}_{x}$ only affects the component of $\boldsymbol{\mathcal{A}}$ along ${{\bf e}}_{x}$ (figure 1(c)). The gauge freedom present in the initial formulation of the problem is inherited by the effective Hamiltonian describing laser dressed atoms. Two experiments starting with the same $\boldsymbol{\mathcal{B}}$ and ending with zero magnetic field can manifest different physical outcomes without violating gauge invariance, since the electric field $\boldsymbol{\mathcal{E}}=-{\rm d}\boldsymbol{\mathcal{A}}/{\rm d}t$ associated with the change [7] is determined by the physical mechanism that creates $\boldsymbol{\mathcal{B}}$. The natural gauge is the gauge for which the vector potential is zero when any created fields are 'off'. Any changes in the vector potential (which are the measureable quantities) are then directly proportional to the vector potential in our choice of gauge. We emphasize that any gauge will do, but in the sense described above, one can be most natural in many physical situations.

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In this work, we studied ⁸⁷Rb BECs subject to uniform laser-induced synthetic magnetic fields. The constituent bosons experienced a Lorentz force, just as would charged particles in a magnetic field. This synthetic magnetic field was continuously tunable from 0 to above ${\mathcal{B}}_{\mathrm{cr}}$. BECs are characterized by the complex-valued order parameter $\psi ({\bf r})=\sqrt{{\rho }_{c}({\bf r})}\mathrm{exp}[{\rm i}\phi ({\bf r})]$ with condensate density ${\rho }_{c}({\bf r})$ and phase $\phi ({\bf r})$. The Gross–Pitaevskii equation (GPE) gives the time dependence of $\psi ({\bf r})$. The single-valuedness of the phase lends the superfluid system one of its defining properties: irrotationality. The current density ${\bf J}({\bf r})=\mathbb{R}{\rm e}[{\psi }^{*}({\bf r})\hat{{\bf v}}\psi ({\bf r})]$,where $\hat{{\bf v}}=[-{\rm i}{\hslash }\boldsymbol{\nabla }-\boldsymbol{\mathcal{A}}({\bf r})]/m$, describes the flow of particles and links the local velocity to its phase $\phi ({\bf r})$ via ${\bf v}({\bf r})=[{\hslash }\boldsymbol{\nabla }\phi ({\bf r})-\boldsymbol{\mathcal{A}}({\bf r})]/m$ for particles of mass m. Notice that irrotationality applies to the local per-particle canonical momentum ${\bf p}({\bf r})={\hslash }\boldsymbol{\nabla }\phi ({\bf r})$, i.e., $\nabla \times {\bf p}({\bf r})=0$, but neither to velocity nor mechanical momentum ${{\bf p}}_{{\rm m}}({\bf r})=m{\bf v}({\bf r})={\bf p}({\bf r})-\boldsymbol{\mathcal{A}}({\bf r})$.

We prepared trapped, equilibrated⁷ BECs in one of three configurations: (i) $\mathcal{B}=0$, a reference case described by standard superfluid hydrodynamics (figures 2(a), (d)); (ii) $0\lt \mathcal{B}\lt {\mathcal{B}}_{\mathrm{cr}}$, described by modified superfluid hydrodynamics (figures 2(b), (e)); and (iii) $\mathcal{B}\gt {\mathcal{B}}_{\mathrm{cr}}$, with vortices (figures 2(c), (f)). For all cases, we initiated TOF by abruptly removing the confining potential $V({\bf r})$ and rapidly making the vector potential $\boldsymbol{\mathcal{A}}=0$. Effectively⁸ , this rapid turn-off left $\phi ({\bf r})$ unaltered and mapped the gauge-dependent canonical momentum ${\bf p}$ just before TOF (at ${t}_{{0}^{-}}$) onto the gauge-independent mechanical momentum just after TOF began (at ${t}_{{0}^{+}}$): ${\bf p}({t}_{{0}^{-}})={{\bf p}}_{{\rm m}}({t}_{{0}^{+}})$. Measuring the shearing motion in TOF, which has contributions from the initial velocity field ${\bf v}({\bf r})$ and the position-dependent electric force, allows us to distinguish between cases (i), (ii), and (iii).

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Case (i): the $\mathcal{B}=0$ expansion of a repulsively interacting BEC released from a harmonic trap is well-studied: zero-point energy is typically negligible and interactions between the atoms dominate. In the ground state, the constant (in situ) phase gives ${{\bf p}}_{{\rm m}}({t}_{{0}^{+}})=0$. The familiar inverted-parabola Thomas–Fermi profile of harmonically trapped BECs [8] is preserved by the interaction-driven expansion during TOF [9].

Case (ii): modest magnetic fields ($0\lt \mathcal{B}\lt {\mathcal{B}}_{\mathrm{cr}}$) alter the interaction-dominated TOF expansion, as was demonstrated in vortex-free rotating BEC systems [10]. The in situ phase $\phi ({\bf r})$ is everywhere well-defined and the canonical momentum ${\bf p}({\bf r})$ differs from the mechanical momentum ${{\bf p}}_{{\rm m}}({\bf r})$ due to the presence of the vector potential that gives $\boldsymbol{\mathcal{B}}=\boldsymbol{\nabla }\times \boldsymbol{\mathcal{A}}({\bf r})$. The $t={0}^{-}$ canonical momentum in the Landau gauge

$$\begin{eqnarray}&&{{\bf p}}^{{ii}}=-\mathcal{B}\left({\displaystyle \frac{\tilde{\epsilon }+1}{2}}\right)\left(y{{\bf e}}_{x}+x{{\bf e}}_{y}\right),\end{eqnarray} \tag{ 1 }$$

where $\boldsymbol{\nabla }({xy})=(y{{\bf e}}_{x}+x{{\bf e}}_{y})$, also defines ${{\bf p}}_{{\rm m}}({t}_{{0}^{+}})$, where $-1\lt \tilde{\epsilon }\lt 1$ is a trap- and cyclotron-frequency-dependent anisotropy parameter (see equation (5) and [11]). Here the anisotropy is large, so that $\tilde{\epsilon }$ is nearly −1; compared to previous measurements [6] in a cylindrically symmeteric trap ($\tilde{\epsilon }=0$), the canonical momentum components are small. (In the corresponding symmetric gauge expression $\tilde{\epsilon }+1$ is replaced by $\tilde{\epsilon }$.)

Case (iii): vortices significantly affect the TOF expansion. At the location of each vortex, the phase is singular (figure 2(c)), with a $2\pi$-winding around it. As an example, a vortex centered at the origin of a cylindrically symmetric system contributes a phase $\phi ({\bf r})={\mathrm{tan}}^{-1}(y/x)$, and the $t={0}^{-}$ canonical momentum in the Landau gauge is approximately

$$\begin{eqnarray}&&{{\bf p}}^{{iii}}={\displaystyle \frac{{\hslash }\left(-y{{\bf e}}_{x}+x{{\bf e}}_{y}\right)}{{x}^{2}+{y}^{2}}}-\mathcal{B}\left({\displaystyle \frac{\tilde{\epsilon }+1}{2}}\right)\left(y{{\bf e}}_{x}+x{{\bf e}}_{y}\right).\end{eqnarray} \tag{ 2 }$$

As before, this in-trap canonical momentum becomes the mechanical momentum ${{\bf p}}_{{\rm m}}({t}_{{0}^{+}})$ as TOF begins.

We estimate the relative shear of these configurations from their momenta at ${t}_{{0}^{+}}$. Case (i) is shear free. For case (ii), the momentum along ${{\bf e}}_{x}$ has typical scale $| {{\bf p}}_{{\rm m}}\cdot {{\bf e}}_{x}| \approx (\tilde{\epsilon }+1)\mathcal{B}y/2$. In the presence of many vortices (case (iii), in the diffused vorticity limit [12]) this increases by ${\hslash }{N}_{{\rm v}}y{/R}^{2}$, where R is the characteristic system size and ${N}_{{\rm v}}$ is the total number of vortices. The difference is proportional to ${\hslash }{n}_{{\rm v}}y$, where ${n}_{{\rm v}}$ is the areal vortex density, leading to an abrupt increase in the shearing momentum when vortices enter (figures 2(g)–(i)). Microscopically, this increased velocity originates from the spatial variations in phase associated with the vortices (figure 2(c)).

The transition between configurations (ii) and (iii) at ${\mathcal{B}}_{\mathrm{cr}}$ occurs when the system can lower its energy by admitting a vortex; this critical field depends on the system's trap and interaction parameters [1]. The critical cyclotron frequency ${\Omega }_{{\rm C}}^{\mathrm{cr}}={\mathcal{B}}_{\mathrm{cr}}/m$, above which vortices are energetically stable [1, 13, 14] may be estimated as ${\Omega }_{{\rm C}}^{\mathrm{crit}}=(5{\hslash }{/{mR}}_{\perp }^{2})\mathrm{ln}({R}_{\perp }/\xi )$ (note that in rotating system experiments, the cyclotron frequency ${\Omega }_{{\rm C}}$ differs by a factor of two from the rotation frequency, commonly referred to as Ω.) ${R}_{\perp }={\left[4\mu /m({\omega }_{x}^{2}+{\omega }_{y}^{2})\right]}^{1/2}$ is the mean transverse Thomas–Fermi radius in a harmonic potential with frequencies ${\omega }_{x,y,z}$. The healing length $\xi ={({{\hslash }}^{2}/2m\mu )}^{1/2}$ sets the characteristic vortex core size, where $\mu ={n}_{0}(4\pi {{\hslash }}^{2}{a}_{s}/m)$ is the central mean-field energy, ${a}_{{\rm s}}$ is the s-wave scattering length [15], and n₀ is the central density. For the parameters in this experiment, ${\Omega }_{{\rm C}}^{\mathrm{cr}}/2\pi =12.0(1.1)$ Hz.

We prepared BECs with $N\approx 1.4(3)\times {10}^{5}$ in the f = 1 ground state hyperfine manifold at the intersection of two $\lambda =1.064\mu {\rm m}$ laser beams (figure 1(c)). The resulting potential was approximately harmonic and had measured frequencies $\{{\omega }_{x},{\omega }_{y},{\omega }_{z}\}/2\pi =\{10.1(1),47.3(3),90(1)\}\;\mathrm{Hz}$ (see supplementary data). We implemented an artificial magnetic field [5, 16] using the combination of two counterpropagating ${\lambda }_{{\rm R}}=790.1$ nm Raman lasers (figure 1(d)) traveling along $\pm {{\bf e}}_{x}$ in conjunction with a (real) magnetic field ${\bf B}\cong ({B}_{0}+{B}^{\prime }y){{\bf e}}_{y}$, giving a gradient in detuning from Raman resonance ${g}_{F}{\mu }_{{\rm B}}{B}^{\prime }/h$ along ${{\bf e}}_{y}$ ranging from 0 to $640\;\mathrm{Hz}\;\mu {{\rm m}}^{-1}$. This gave effective cyclotron frequencies ranging from ${\Omega }_{{\rm C}}/2\pi =0$ to 20 Hz.

To study the evolution of the density distribution after mean-field-driven expansion, we released the atoms from the trap, and adiabatically transformed the Raman-dressed superposition into a single Zeeman level for imaging [17] in the first 2 ms of TOF (see supplementary data). The cloud expanded for a total of 36.2 ms TOF before being imaged along ${{\bf e}}_{z}$ (figures 1(j)–(l)).

Figures 2(a)–(f) shows computed in situ phase and density distributions for a range of cyclotron frequencies. Vortices nucleate only above ${\Omega }_{\mathrm{cr}}$, here only in panels (c) and (f). For this geometry, the system's ground state consists of a linear chain of vortices; larger cyclotron frequencies or less anisotropy would result in a regular vortex lattice. (In the experiment, the vortices have not equilibrated to their ground state configuration.) Figures 2(g)–(l) depicts calculated and measured TOF densities; the cloud's shear increases monotonically with increasing cyclotron frequency ${\Omega }_{{\rm C}}=\mathcal{B}/m$ (e.g., there is a small shear in figures 2(h), (k) and a large shear for figures 2(i), (l)). For strong artificial fields the observed density distributions were irregularly fragmented (figure 2(l)), while the computed clouds were ordered. These observations are both consistent with the presence of vortices — disordered in the case of experiment — whose characteristic phase gradients cause density modulations after TOF. The generic ${{\bf e}}_{y}$-aligned stripes present in TOF (figures 2(i), (l)) result from the predominantly ${{\bf e}}_{y}$-expansion from the anisotropic trap.

We obtained TOF images at various artificial field strengths, and fit the resulting 2D column densities to a sheared Thomas–Fermi profile

$$\begin{eqnarray}&&{n}_{2{\rm D}}^{\mathrm{TOF}}={n}_{0}^{\mathrm{TOF}}{\left[1-{\left({\displaystyle \frac{x}{{R}_{x}}}\right)}^{2}-{\left({\displaystyle \frac{y}{{R}_{y}}}\right)}^{2}+{a}_{{xy}}\left({\displaystyle \frac{{xy}}{{R}_{x}{R}_{y}}}\right)\right]}^{3/2},\end{eqnarray} \tag{ 3 }$$

where ${n}_{0}^{\mathrm{TOF}}$ is the central density; ${R}_{x,y}$ are Thomas–Fermi radii; and a_xy is a dimensionless shear parameter. We obtain a measure of the spatial irregularity from the fit residuals $\mathcal{F}={{\displaystyle \sum }}_{x,y}{[{n}_{2{\rm D}}^{\mathrm{meas}}(x,y)-{n}_{2{\rm D}}^{\mathrm{fit}}(x,y)]}^{2}$. Figure 3(a) shows that $\mathcal{F}$ sharply increases (see supplementary data) around ${\Omega }_{{\rm C}}/2\pi =15$ Hz, an indication of the transition to the BEC's vortex phase.

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Additional evidence for the entrance of vortices into this anisotropically trapped BEC is the behavior of the shear parameter a_xy: figure 3(b) shows a_xy sharply increasing above ${\Omega }_{{\rm C}}/2\pi =15$ Hz, in reasonable agreement with the predicted critical cyclotron frequency ${\Omega }_{{\rm C}}^{\mathrm{crit}}/2\pi =12.0(1.1)\mathrm{Hz}$. We compare this result to two calculations.

The first calculation, the usual hydrodynamic description [11, 12], valid only in the absence of vortices, is modified to include the artificial magnetic field from a Landau-gauge vector potential [16], predicting the in situ density

$$\begin{eqnarray}&&n({\bf r})={n}_{0}\left[1-{\left({\displaystyle \frac{x}{{\tilde{R}}_{x}}}\right)}^{2}-{\left({\displaystyle \frac{y}{{\tilde{R}}_{y}}}\right)}^{2}-{\left({\displaystyle \frac{z}{{\tilde{R}}_{z}}}\right)}^{2}\right]\end{eqnarray} \tag{ 4 }$$

and the canonical momentum ${{\bf p}}^{{ii}}$ (equation (1)). Here, ${\tilde{R}}_{i}={[2\mu /m{\tilde{\omega }}_{i}^{2}]}^{1/2}$ are modified Thomas–Fermi radii with effective trapping frequencies ${\tilde{\omega }}_{i}$, where ${\tilde{\omega }}_{x}^{2}={\omega }_{x}^{2}+{\Omega }_{{\rm C}}^{2}{[(1+\tilde{\epsilon })/2]}^{2}$, ${\tilde{\omega }}_{y}^{2}={\omega }_{y}^{2}+{\Omega }_{{\rm C}}^{2}{[(1-\tilde{\epsilon })/2]}^{2}$, and ${\tilde{\omega }}_{z}={\omega }_{z}$. The anisotropy parameter

$$\begin{eqnarray}&&\tilde{\epsilon }={\displaystyle \frac{{\tilde{\omega }}_{x}^{2}-{\tilde{\omega }}_{y}^{2}}{{\tilde{\omega }}_{x}^{2}+{\tilde{\omega }}_{y}^{2}}}\end{eqnarray} \tag{ 5 }$$

can be obtained self-consistently from these equations [12, 16]. These distributions are propagated in TOF using the hydrodynamic equations for comparison to the measured distributions. From these calculated TOF distributions, we extracted the shear parameter and compared it with the measurement, as shown in figure 3(b).

The second calculation uses the GPE, which is valid for all our cases. We numerically found the GPE ground state at each $\boldsymbol{\mathcal{B}}$ and then used the time-dependent GPE to calculate TOF evolution (see supplementary data available at stacks.iop.org/njp/17/065016/mmedia). We find excellent agreement both with the amount of shear in the cloud for all configurations and in the location of the critical artificial field strength for vortex nucleation.

To understand the impact of the experimentally dictated natural gauge, we investigated the effects of different gauge choices. We performed GPE calculations as described above to obtain a_xy for two alternate gauge choices with the same $\boldsymbol{\mathcal{B}}$: the symmetric and Landau gauges. We compared these to the Raman system described above (which does not have a simple analytical form, but is well-approximated for small deviations in momentum and position by a Landau gauge; figure 4(a) shows the Raman gauge deviates only slightly from the ideal Landau gauge choice). Though the Landau and symmetric gauge atomic systems are identical before release, their TOF responses differ because of their different electric field impulses at turn-off, which reflect each system's natural gauge (figure 4(b)). For $\mathcal{B}\lt {\mathcal{B}}_{\mathrm{cr}}$, this electric field impulse produces a significantly larger shearing when the symmetric gauge is natural. This behavior reflects the fact that the gauge-dependent in situ canonical momentum, which becomes ${{\bf p}}_{{\rm m}}({t}_{{0}^{+}})$, is different in the two cases. Each system reacted to the entrance of vortices with a marked change in a_xy, here positive, as set by the vortices' direction of circulation, but with a significantly different character.

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In conclusion, we showed that the shape of the cloud observed after TOF is strongly affected by the presence of vortices and signals the vortex nucleation transition. By exploiting the connection between phase and velocity, we were able to extract topological features of our system's order parameter from the TOF density distribution. Our experiment's anisotropic geometry and Landau-like natural gauge led to a signal that made this transition particularly clear. By comparing to what would have happened in the symmetric gauge, we showed that the gauge choice can make a significant difference in the experimental outcome. In the future, techniques that match a system's geometry to the artificial field's natural gauge might be used to selectively excite edge modes in cold-atom quantum-Hall-like systems [18–20].

Acknowledgments