## Making a monopole in an atomic gas

Some “grand unified theories” of physics predict the existence of the so-called magnetic monopole. No such particles have been seen, but analogs of monopoles may be observable in quantum fluids. Ray *et al.* created such an analog in a gas of ultracold ^{87}Rb atoms with three spin states at their disposal. The authors used a protocol involving external magnetic fields with particular spatial distributions to create and observe a monopole-like spin texture in the gas.

Science, this issue p. 544

## Abstract

Topological defects play important roles throughout nature, appearing in contexts as diverse as cosmology, particle physics, superfluidity, liquid crystals, and metallurgy. Point defects can arise naturally as magnetic monopoles resulting from symmetry breaking in grand unified theories. We devised an experiment to create and detect quantum mechanical analogs of such monopoles in a spin-1 Bose-Einstein condensate. The defects, which were stable on the time scale of our experiments, were identified from spin-resolved images of the condensate density profile that exhibit a characteristic dependence on the choice of quantization axis. Our observations lay the foundation for experimental studies of the dynamics and stability of topological point defects in quantum systems.

Two structures are topologically equivalent if they can be continuously transformed into one another (*1*, *2*), such as the letters O and P. Topological defects exist in a physical system if its state is not topologically equivalent to its ground state. Such defects can decay or disappear only as a result of globally nontrivial transformations, rendering them long-lived and ubiquitous in the universe.

Line defects are among the most common topological structures. In classical physics, for example, dislocations in a crystal lattice (*3*) can determine the strength and hardness of materials. In quantum physics, a line defect in a complex-valued order parameter is accompanied by a phase winding of an integer multiple of 2π. These quantized vortices are regarded as the hallmark of superfluidity (*4*, *5*) and constitute a versatile tool in the study of quantum physics. In contrast, the roles played by point defects in three-dimensional superfluids and superconductors remain less explored experimentally, although related objects such as skyrmion solitons and boojums at domain interfaces have been observed (*6*–*9*).

Homotopy theory (*2*, *10*) is a mathematical tool that classifies topological point defects according to the behavior of the order parameter on closed surfaces. Evaluation of the second homotopy group reveals whether point defects can occur. Nematic liquid crystals (*11*) and colloids (*12*) are examples of classical systems for which the second homotopy group is nontrivial and point defects have been observed [see also (*13*)]. Quantum systems described by multidimensional fields are also predicted to support point defects as stable elementary particles (*2*). The magnetic monopole (*14*, *15*) that emerges under broken symmetry in grand unified theories (*16*) is one such example.

The polar phase of a spin-1 Bose-Einstein condensate (BEC) permits the existence of topological point defects in the quantum mechanical order parameter (*17*, *18*). Although these defects are not elementary particles, they are analogous quantum objects often referred to as monopoles.

In our experiments, we create a topological point defect in the spin-1 order parameter of an ^{87}Rb BEC using a method originally suggested in (*19*) and used to create Dirac monopoles in a ferromagnetic BEC in (*20*) [see related work in (*21*)]. The key technical difference relative to (*20*) is that the condensate is initialized in its polar phase. This seemingly minor modification leads to a topological excitation with properties that are fundamentally different from those of the recently observed Dirac monopole. The Dirac monopole is not a pointlike topological defect in the order parameter, as the second homotopy group of the ferromagnetic phase contains only the identity element (*22*). Consequently, Dirac monopoles are attached to at least one terminating nodal line (*23*), which renders the energetics and dynamics of the excitation similar to those of vortices. No such nodal line is attached to the point defect structure we create here in the order parameter field, and hence we refer to it as an isolated monopole.

A spin-1 condensate can be described by the order parameter(1)where *n* is the particle density, φ is the scalar phase, and the spinor is represented by a normalized complex-valued vector ζ = (ζ_{+1} ζ_{0} ζ_{–1})* ^{T}*. Here, ζ

*= 〈*

_{m}*m*|ζ〉 is the

*m*th spinor component along the quantization axis

*z*. The most general polar order parameter, for which the local spin vanishes, is given by (2)where the Euler angles β(

**r**) and α(

**r**) refer to the spin rotation of a spinor (0 1 0)

*about the*

^{T}*y*and

*z*axes, respectively, and is a three-dimensional real-valued unit vector field known as the nematic vector. Equation 2 shows that the polar spin-1 condensate is simply described by the mean-field order parameter

_{}(3)with its topological properties determined by the factor (

*24*). Note that any unitary spin rotation imposed on the order parameter in Eq. 2 corresponds to an identical rotation of . Thus, the nematic vector follows adiabatic changes in the external magnetic field, much as the direction of the spin follows the field in the ferromagnetic case.

The initial atom number in the optically trapped ^{87}Rb BEC is *N* ≈ 2.1 × 10^{5} with calculated radial and axial Thomas-Fermi radii *R* = 7.2 μm and *Z* = 5.4 μm, respectively, and corresponding optical trapping frequencies ω* _{r}* ≈ 2π × 124 Hz and ω

*≈ 2π × 164 Hz, respectively. The creation process begins with aligned with a uniform magnetic field*

_{z}**B**

_{b}(

*t*) = (

*24*). We use

**B**

_{b}(

*t*) =

*B*(

_{z}*t*) here, but the experimental results are independent of the choice of direction. A quadrupole magnetic field

**B**

_{q}(

**r**) = of strength

*b*

_{q}= 3.7 G/cm is then introduced; the zero point (4)of the total magnetic field

**B**(

**r**,

*t*) =

**B**

_{q}(

**r**) +

**B**

_{b}(

*t*) is initially located well outside the condensate. We then change

**B**

_{b}until

**r**

_{0}lies near the center of the condensate (Fig. 1A). This “creation ramp” is carried out nearly adiabatically ( = –0.25 G/s)—that is, —thereby creating the isolated monopole structure in the order parameter field shown in Fig. 1, B and C. Nonadiabatic excitations and spin-exchange collisions are measured to be relatively small (~10%) for the experimental parameter values used here.

To select a quantization axis for imaging the monopole structure, we apply a “projection ramp” in which the magnetic bias field is rapidly increased to |**B**_{b}|/*b*_{q} >> {*R*, *Z*} along a direction of our choice, , leaving the nematic vector essentially unchanged. Subsequently, the spinor components quantized along this axis, 〈*m*_{p}|ζ〉, are spatially separated and imaged in both the vertical () and horizontal () directions (*24*). In Fig. 2, A and D, we show the corresponding experimentally obtained particle densities in the simple case . The theory [Eq. 2 with ] predicts hollow-core vortices of opposite unit circulations in the *m* = ±1 components along* z*, in agreement with the observed density “holes” in Fig. 2D. The unit phase winding and the opposite circulations of the two vortices are experimentally confirmed using interferometric techniques (*24*) (figs. S1 and S2). Furthermore, the data in Fig. 2, A and C, are in qualitative agreement with Eq. 2 because the particle density in the *m* = 0 component vanishes in the *z* = 0 plane, and the other two components accumulate in its vicinity. This agreement constitutes the primary evidence for the existence of the monopole.

We modeled the experimental creation and imaging process numerically by solving the full three-dimensional dynamics of the mean-field spinor order parameter from the spin-1 Gross-Pitaevskii equation (*19*). Figures 2 and 3 show one-to-one comparisons of the numerically obtained particle density distributions to the experimental results without any free parameters. The good quantitative agreement between the simulations and the experiments reinforces the congruence between the experiments and the results of the analytic theory, thereby providing complementary evidence for the realization of an isolated monopole structure in the order parameter. Discrepancies between the numerical and experimental results—for example, the density peak in the *m* = 0 component in Fig. 2F—may arise from the experimental noise and the choice of imaging technique that are not taken fully into account in the simulations (*24*).

Particle densities identical to those shown in Fig. 2 for our isolated monopole are expected for the topologically equivalent hedgehog monopole structure shown in Fig. 1D, as the only difference between the spinors of the two configurations is the sign of the *m* = ±1 components (see Eq. 2 and Fig. 1, C and D). In fact, after the projection ramp , the order parameter oscillates between the two configurations because of the 350-kHz Larmor precession of the nematic vector about . Because the other condensate dynamics occur on much longer time scales, the experiment also accurately produces the hedgehog monopole, as confirmed by the numerical simulations shown in fig. S3.

One characteristic feature of a quantum mechanical point defect is that arbitrary rotations of a properly chosen coordinate system, , can be compensated by rotations in the order parameter space,
, and vice versa. We study whether the created point defect has this property by imposing a spin rotation
on the spin state of the defect |ζ〉 such that we choose the direction of the projection ramp, , defined by the coordinate rotation (*x*_{p}, *y*_{p}, *z*_{p}) = _{p}^{–1}(*x*, *y*, *z*). The projection of the original spinor onto the new *z*_{p}-quantized basis is equal to the projection of the rotated spinor onto the *z*-quantized basis:(5)Thus, the rotational compensation property given above demands that there exists a rotation _{v} into a new coordinate system (*x*_{v}, *y*_{v}, *z*_{v}) = _{v}(*x*, *y*, *z*) such that Eq. 2, with (*x*, *y*, *z*) replaced by (*x*_{v}, *y*_{v}, *z*_{v}), yields the observed spinor components. Below, we analytically find the new coordinate system for both the hedgehog monopole and our isolated monopole in the case of an arbitrary projection axis, and show matching experimental observations.

The hedgehog monopole is characterized by the nematic vector , where the primed coordinates are defined as (*x*′, *y*′, *z*′) = (*x*, *y*, 2*z*). Because the radial vectors in any two rotated coordinate systems coincide, (*x*′_{p}, *y*′_{p}, *z*′_{p}) = (*x*′_{p}, *y*′_{p}, *z*′_{p}), we can choose (*x*′_{v}, *y*′_{v}, *z*′_{v}) = (*x*′_{p}, *y*′_{p}, *z*′_{p}) (i.e., _{v} = _{p}^{–1}). Together with Eq. 2, this shows that the vortices in the* m*_{p} = ±1 components of the hedgehog configuration always align with the projection axis . To find how the vortices will be oriented in the case of our isolated monopole, we make use of the property that the hedgehog monopole is obtained from the isolated monopole configuration by a continuous π-rotation about the *z* axis (Fig. 1, C and D); that is, and . By writing the observed spinor component as(6)we find that a proper choice of the new coordinate system is (*x*_{v}, *y*_{v}, *z*_{v}) = ℛ* _{z}*π(

*x*

_{p},

*y*

_{p},

*z*

_{p}). Thus, the vortices are aligned with .

The isolated monopole (Fig. 1, B and C) is topologically equivalent to the hedgehog structure (Fig. 1D) and has the same topological charge and stability properties. However, the fact that the projection axis and the vortex axis are not always aligned makes the isolated monopole an ideal object to demonstrate that the observed vortices are not technical artifacts of the projection ramp. The corresponding experimental results are shown in Fig. 4. In agreement with the result derived above, we observe that the two axes, and , are parallel when they lie in the *xy* plane (Fig. 4A) and rotate in opposite directions in the *xz* plane (Fig. 4B).

Both monopole structures are expected to exhibit an instability toward a formation of a vortex ring (*25*). Although this and other instabilities (*26*, *27*) occur slowly enough not to disturb the creation and imaging process (*24*), observation of the resulting decay dynamics and implementation of a system in their absence are interesting research directions. Furthermore, studies of the interaction between monopoles and other topological defects, such as domain walls and skyrmions (*7*), may yield additional insights into high-energy physics and cosmology (*28*). A related goal is to create a topological point defect that also generates the synthetic magnetic field of a monopole, thereby combining the scenarios of Dirac (*23*), ’t Hooft (*14*), and Polyakov (*15*). Finally, the observation of non-Abelian monopoles (*29*, *30*) remains an important goal.

## Supplementary Materials

www.sciencemag.org/content/348/6234/544/suppl/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S11

Reference (*31*)

## References and Notes

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**Acknowledgments:**Supported by NSF grant PHY-1205822, the Academy of Finland through its Centres of Excellence Program (grant 251748) and grants 135794 and 272806, the Finnish Doctoral Programme in Computational Sciences, and the Magnus Ehrnrooth Foundation. We thank CSC-IT Center for Science Ltd. (project no. ay2090) and Aalto Science-IT project for computational resources, M. Nakahara and M. Krusius for discussions concerning this work, and N. H. Thomas and S. J. Vickery for experimental assistance. All data used to support the conclusions of this work are presented in this manuscript and the supplementary materials. M.W.R. and D.S.H. developed and conducted the experiments and analyzed the data. E.R. and K.T. performed the numerical simulations under the guidance of M.M., who provided the initial ideas and suggestions for the experiment. All authors discussed both experimental and theoretical results and commented on the manuscript.