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Observation of isolated monopoles in a quantum field

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Science  01 May 2015:
Vol. 348, Issue 6234, pp. 544-547
DOI: 10.1126/science.1258289

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 87Rb 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 (69).

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 87Rb 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 parameterEmbedded Image(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 ζ0 ζ–1)T. Here, ζm = 〈m|ζ〉 is the mth spinor component along the quantization axis z. The most general polar order parameter, for which the local spin vanishes, is given byEmbedded Image (2)where the Euler angles β(r) and α(r) refer to the spin rotation of a spinor (0 1 0)T about the y and z axes, respectively, and Embedded Image 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 parameterEmbedded Image(3)with its topological properties determined by the factor Embedded Image (24). Note that any unitary spin rotation imposed on the order parameter in Eq. 2 corresponds to an identical rotation of Embedded Image. Thus, the nematic vector Embedded Image 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 87Rb BEC is N ≈ 2.1 × 105 with calculated radial and axial Thomas-Fermi radii R = 7.2 μm and Z = 5.4 μm, respectively, and corresponding optical trapping frequencies ωr ≈ 2π × 124 Hz and ωz ≈ 2π × 164 Hz, respectively. The creation process begins with Embedded Image aligned with a uniform magnetic field Bb(t) = Embedded Image (24). We use Bb(t) = Bz(t) Embedded Image here, but the experimental results are independent of the choice of direction. A quadrupole magnetic field Bq(r) = Embedded Image of strength bq = 3.7 G/cm is then introduced; the zero pointEmbedded Image (4)of the total magnetic field B(r,t) = Bq(r) + Bb(t) is initially located well outside the condensate. We then change Bb until r0 lies near the center of the condensate (Fig. 1A). This “creation ramp” is carried out nearly adiabatically (Embedded Image = –0.25 G/s)—that is, Embedded Image—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.

Fig. 1 Schematic representation of the experiment.

(A) Magnetic field lines as Bz is decreased. The zero point of the magnetic field is shown as a black dot. (B to D) Cross sections through the condensate in the xy′ plane (B) and in the xz′ plane (C) showing the nematic vector field (thick arrows) defining our isolated monopole structure, which is related to the hedgehog monopole structure (D) by a rotation of π about the z′ axis, ℛz(π). The primed coordinates are defined as x′ = x, y′ = y, and z′ = 2z; the gray arrows depict magnetic field lines.

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 |Bb|/bq >> {R, Z} along a direction of our choice, Embedded Image, leaving the nematic vector essentially unchanged. Subsequently, the spinor components quantized along this axis, 〈mp|ζ〉, are spatially separated and imaged in both the vertical (Embedded Image) and horizontal (Embedded Image) directions (24). In Fig. 2, A and D, we show the corresponding experimentally obtained particle densities in the simple case Embedded Image. The theory [Eq. 2 with Embedded Image] 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 Embedded Image vanishes in the z = 0 plane, and the other two components Embedded Image accumulate in its vicinity. This agreement constitutes the primary evidence for the existence of the monopole.

Fig. 2 Experiment compared to numerical simulations following a projection ramp along –z.

(A and D) Experimentally obtained images of the condensate taken along the horizontal (y) axis (A) and the vertical (z) axis (D). (B and E) Results of the corresponding numerical simulations. In each panel, the top image gives a false-color composite, in which the color intensity represents the particle density of each spinor component integrated along the respective imaging axis. The lower three sets of images show the densities for the individual components. (C and F) Quantitative comparison of experimental (solid lines) and simulated (dashed lines) column density, Embedded Image, for cross sections. The field of view is 288 μm × 288 μm for images along the horizontal axis and 219 μm × 219 μm for those along the vertical axis. The peak column density in all images is Embedded Image = 12.9 × 108 cm–2. Color and intensity scales are shown at bottom of figure.

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).

Fig. 3 Experiment compared to numerical simulations following a projection ramp along –y.

(A) Experimentally obtained images of the condensate taken along the horizontal (y) axis. (B) Results of the corresponding numerical simulations. See Fig. 2 for further description. (C and D) As above, but for images taken along the vertical (z) axis. The field of view is 288 μm × 288 μm in (A) and (B), 219 μm × 219 μm in (C) and (D). The peak column density is Embedded Image = 12.9 × 108 cm–2.

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 Embedded Image, the order parameter oscillates between the two configurations because of the 350-kHz Larmor precession of the nematic vector about Embedded Image. 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, Embedded Image, can be compensated by rotations in the order parameter space, Embedded Image , and vice versa. We study whether the created point defect has this property by imposing a spin rotation Embedded Image on the spin state of the defect |ζ〉 such that we choose the direction of the projection ramp, Embedded Image, defined by the coordinate rotation (xp, yp, zp) = Embedded Imagep–1(x, y, z). The projection of the original spinor onto the new zp-quantized basis is equal to the projection of the rotated spinor onto the z-quantized basis:Embedded Image(5)Thus, the rotational compensation property given above demands that there exists a rotation Embedded Imagev into a new coordinate system (xv, yv, zv) = Embedded Imagev(x, y, z) such that Eq. 2, with (x, y, z) replaced by (xv, yv, zv), 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 Embedded Image, where the primed coordinates are defined as (x′, y′, z′) = (x, y, 2z). Because the radial vectors in any two rotated coordinate systems coincide, Embedded Image(xp, yp, zp) = Embedded Image(xp, yp, zp), we can choose (xv, yv, zv) = (xp, yp, zp) (i.e., Embedded Imagev = Embedded Imagep–1). Together with Eq. 2, this shows that the vortices in the mp = ±1 components of the hedgehog configuration always align with the projection axis Embedded Image. 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, Embedded Image and Embedded Image. By writing the observed spinor component asEmbedded Image(6)we find that a proper choice of the new coordinate system is (xv, yv, zv) = ℛzπ(xp, yp, zp). Thus, the vortices are aligned with Embedded Image.

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 Embedded Image derived above, we observe that the two axes, Embedded Image and Embedded Image, are parallel when they lie in the xy plane (Fig. 4A) and rotate in opposite directions in the xz plane (Fig. 4B).

Fig. 4 Experimental results for different choices of the projection axis.

(A) The angle of the vortices in the |m = ±1〉 states, φv, resulting from projections in the xy plane with azimuthal angle φp. Condensates are imaged along the z axis and φv is extracted from the alignment of the density profile in the |m = 0〉 state, as shown in the insets (see also figs. S4 to S7). Typical uncertainties are indicated by the error bars shown. The dashed line shows the theoretical result. The black arrows in the insets show the projection axes, zp, and the chevrons show the expected orientation of the vortex axes, zv. (B) Same as (A) but for angles θv resulting from projections in the xz plane with polar angle θp and imaging axis y.

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

  1. See supplementary materials on Science Online.
  2. 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.
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