Experimental Realization of a Dirac Monopole through the Decay of an Isolated Monopole

We experimentally observe the decay dynamics of deterministically created isolated monopoles in spin-1 Bose-Einstein condensates. As the condensate undergoes a change between magnetic phases, the isolated monopole gradually evolves into a spin configuration hosting a Dirac monopole in its synthetic magnetic field. We characterize in detail the Dirac monopole by measuring the particle densities of the spin states projected along different quantization axes. Importantly, we observe the spontaneous emergence of nodal lines in the condensate density that accompany the Dirac monopole. We also demonstrate that the monopole decay accelerates in weaker magnetic field gradients.

Popular Summary

All known magnets have two distinct ends: a north pole and a south pole. North and south poles by themselves—known as magnetic monopoles—have never been observed in nature, though their existence has been theoretically predicted for decades. Some of their properties have been deduced in recent experiments that simulate monopoles in a Bose-Einstein condensate, a dilute gas of atoms chilled to near absolute zero. One experiment investigated properties of a magnetic monopole first described by Dirac more than 80 years ago; another studied a different type of monopole in the quantum field that describes the gas. Several questions remain about the properties of these monopoles. Our new experiments in the condensate show that a quantum monopole decays into a Dirac monopole.

We create the quantum monopole by subjecting a Bose-Einstein condensate of rubidium atoms to a slowly varying magnetic field, which imprints magnetic order onto the condensate. We then record the resulting dynamics. The initially nonmagnetized phase of the condensate evolves into a magnetized, or ferromagnetic, phase. Consequently, the quantum monopole is destroyed and replaced by a Dirac monopole. A vortex line of vanishing particle density, akin to a tiny tornado, spontaneously appears in the new phase, closely agreeing with Dirac’s early theoretical work. We subsequently study the Dirac monopole in detail, providing, for the first time, experimental information on its full three-dimensional magnetic structure.

Our work provides the first experimental results on the stability of quantum monopoles and new evidence concerning the properties of Dirac monopoles in condensates. These findings establish fertile conditions for future studies of monopole dynamics in quantum matter.

Figure 1

(a–c) Control sequence of the external quadrupole magnetic field (blue thin arrows) during the creation process of an isolated monopole (Dirac monopole) in the polar (ferromagnetic) phase of the spin-1 BEC and (d) the theoretical spin configuration generating a Dirac monopole. The zero point of the quadrupole magnetic field, indicated by the black dot, is (a) well above, (b) approaching, and (c) in the middle of the condensate (shaded ellipse). The dashed arrow in (b–d) shows the path traced by the zero point as it is adiabatically brought into the condensate. In (d), the directions of the thick arrows indicate the direction of local spin, aligned with the quadrupole field, at selected points in space. (e) Mapping of a closed path on a spherical surface enclosing a solid angle $\mathrm{\Omega }$ from the real space to the spin space in the Dirac monopole spin configuration (see Sec. 2c). In the spin space, we illustrate how an operation $-R_z(\pi )$ can be used to implement the mapping.

Figure 2

Column particle densities in the three $-z$-quantized spin states for (a,b,e,f) the isolated monopole and (c,d,g,h) the dynamically formed Dirac monopole. The experimental data and the corresponding simulation results are shown in panels (a,c,e,g) and (b,d,f,h), respectively. The hold times are indicated in the bottom-right corner of each panel. The quadrupole field gradient is $b_{\mathrm{q}}=4.3\mathrm{G}/\mathrm{cm}$. The peak particle density is ${\tilde{n}}_{\mathrm{p}}=8.5\times {10}^8{\mathrm{cm}}^{-2}$ for the images along $y$ in (a–d) and ${\tilde{n}}_{\mathrm{p}}=1.0\times 10^9{\mathrm{cm}}^{-2}$ for the images along $z$ in (e–h). The field of view in each panel is $228\times 228\mu {\mathrm{m}}^2$.

Figure 3

Column particle densities imaged along $y$ in the three $-z$-quantized spin states during the decay of an isolated monopole for (a–d,i–k) experiments and (e–h,l–n) simulations. The hold times are indicated in the bottom-right corner of each panel. The quadrupole field gradient is $b_{\mathrm{q}}=4.3\mathrm{G}/\mathrm{cm}$. The peak particle density is ${\tilde{n}}_{\mathrm{p}}=8.5\times {10}^8{\mathrm{cm}}^{-2}$, and the field of view in each panel is $228\times 228\mu {\mathrm{m}}^2$.

Figure 4

Experimental column particle densities of $-z$-quantized spin states showing nodal lines associated with the Dirac monopole. In (a–o), the first, second, and third columns correspond to the top images of ${\zeta }_{-1}$, ${\zeta }_0$, and ${\zeta }_{+1}$ spinor components, respectively; the fourth column shows the related composite side images; and the fifth column shows the corresponding numerically obtained spin densities $\tilde{s}=|{\mathrm{\Psi }}^{†}\mathbf{F}\mathrm{\Psi }|$ before the projection ramp. In (a–e), the monopole is located in the upper half of the condensate, in (f–j) in the middle, and in (k–o) in the lower half. The solid red arrows in (c,g,k) indicate the locations of the nodal lines, and the dashed red arrows in (e,j,o) indicate the direction of the vorticity of the nodal line. In (p–w), each set of two adjacent images is extracted from an individual experiment in which the monopole is located in the middle of the condensate, such that (p,r,t,v) show the column particle density of the ${\zeta }_0$ spinor component and (q,s,u,w) show the corresponding composite top image, respectively. The data for (p,q) are the same as in Fig. 2. The hold times are indicated in the bottom-right corner of each panel. The fields of view are as in Fig. 2, and the peak particle densities are (a–c,f–h,k–m,p,r,t,v) ${\tilde{n}}_{\mathrm{p}}=5.0\times 10^8{\mathrm{cm}}^{-2}$, (d,i,n) ${\tilde{n}}_{\mathrm{p}}=8.5\times 10^8{\mathrm{cm}}^{-2}$, and (q,s,u,w) ${\tilde{n}}_{\mathrm{p}}=1.0\times 10^9{\mathrm{cm}}^{-2}$. The simulation data in (e,j,o) is shown for a region of $15\times 15\mu {\mathrm{m}}^2$ with a thickness of $5.8\mu \mathrm{m}$ along $y$. The shown minimum and maximum spin densities are ${\tilde{s}}_{\min }=2.1\times 10^8{\mathrm{cm}}^{-3}$ and ${\tilde{s}}_{\max }=10.1\times 10^8{\mathrm{cm}}^{-3}$.

Figure 5

Experimental column particle densities of (a–h) $x$- and (i–p) $-y$-quantized spin states. The first, second, and third columns correspond to spinor components ${\zeta }_{-1}$, ${\zeta }_0$, and ${\zeta }_{+1}$, respectively, and the fourth column shows the corresponding composite image. Here, the hold time is $t_{\mathrm{hold}}=200\mathrm{ms}$. The red arrows indicate the locations of the nodal lines. The fields of view are as in Fig. 2, and the peak particle densities are (a–c,e–g,i–k,m–o) ${\tilde{n}}_{\mathrm{p}}=5.0\times 10^8{\mathrm{cm}}^{-2}$, (d,l) ${\tilde{n}}_{\mathrm{p}}=8.5\times 10^8{\mathrm{cm}}^{-2}$, and (h,p) ${\tilde{n}}_{\mathrm{p}}=1.0\times 10^9{\mathrm{cm}}^{-2}$.

Figure 6

Magnetization parameter $M_z$ as a function of hold time $t_{\mathrm{hold}}$. Blue crosses and red circles represent the mean value of ten individual experiments with magnetic field gradients $b_{\mathrm{q}}=4.3\mathrm{G}/\mathrm{cm}$ and $b_{\mathrm{q}}=2.2\mathrm{G}/\mathrm{cm}$, respectively. The error bars indicate the uncertainty of 2 standard deviations of the means. The solid and dashed lines are piecewise linear fitting functions with constant ends.

Figure 7

(a–d) Experimental column particle densities of $-z$-quantized spin states and (e–h) numerical column particle densities with a circularly rotating field zero in the $xy$ plane corresponding to the magnetic field peak-to-peak amplitude of 1 mG and 60-Hz oscillation frequency. Here, the hold time is $t_{\mathrm{hold}}=100\mathrm{ms}$. The first, second, and third columns correspond to the spinor components ${\zeta }_{-1}$, ${\zeta }_0$, and ${\zeta }_{+1}$, respectively, and the fourth column shows the corresponding composite images. In each panel, the peak particle density is ${\tilde{n}}_{\mathrm{p}}=1.0\times 10^9{\mathrm{cm}}^{-2}$, and the field of view is $228\times 228\mu {\mathrm{m}}^2$.