Decay of an isolated monopole into a Dirac monopole configuration

We study numerically the detailed structure and decay dynamics of isolated monopoles in conditions similar to those of their recent experimental discovery. We find that the core of a monopole in the polar phase of a spin-1 Bose-Einstein condensate contains a small half-quantum vortex ring. Well after the creation of the monopole, we observe a dynamical quantum phase transition that destroys the polar phase. Strikingly, the resulting ferromagnetic order parameter exhibits a Dirac monopole in its synthetic magnetic field. We observe quantitatively matching decay dynamics for both ferromagnetic and antiferromagnetic spin-spin interactions.

Figure 1

(a) Horizontally and (b) vertically integrated particle densities for the indicated waiting times after the creation ramp which produces the isolated monopole. Different colors correspond to particles in different $z$-quantized spin states with the color and intensity scales given in the bottommost panel. The peak column density is ${\tilde{n}}_{\mathrm{p}}=2.7\times {10}^{11}{\mathrm{cm}}^{-2}$ and the field of view is $15.5\times 15.5\mu {\mathrm{m}}^2$ in each panel. The white arrows indicate the location of a vortex line shown more clearly in Fig. 2.

Figure 2

(a) Spin density $|{\psi }^{†}\mathbf{F}\psi |$ and (b) particle density ${\psi }^{†}\psi$ of the condensate 200 ms after the creation ramp. The shown density range is $[n_{\min },n_{\max }]=[1.0,4.8]\times {10}^{-4}N/a_r^3$. The data in both panels are shown only for $\left|z\right|<1.5a_r$. The spin density is well depleted along the vortex, whereas the particle density is only partially depleted.

Figure 3

Projection of (a)–(c),(e),(f) the nematic vector $\widehat{\mathbf{d}}$ ($\uparrow$) and (d),(g) average spin $\widehat{\mathbf{S}}$ () of the condensate onto (a)–(d) the $xz$ and (e)–(g) $xy$ planes for different indicated waiting times after the creation ramp. The background color map shows (a)–(d) the $y$ component, (e),(f) the azimuthal component, and (g) the $z$ component of the spin. The core of the Alice ring is shown with black dots in (a), which shows a magnification of the center region of (b). The white arrow indicates the location of a vortex line shown more clearly in Fig. 2.

Figure 4

Temporal evolution of the relative population of the neutral state, $n_{\mathrm{ns}}\left(t\right)$ (solid line), and deviation of the order parameter from the created isolated monopole at $t=0, \epsilon \left(t\right)$ (dashed line). In addition to the standard parameters given in the text, neutral-state populations are also shown for (i) $q=0$ (), (ii) $\mathrm{\Gamma }=0$ (), and (iii) $c_2=-4\pi {\hslash }^2(a_2-a_0)/\left(3m\right)$ () in Eq. (1). Furthermore, we show $n_{\mathrm{ns}}\left(t\right)$ for cases in which the quadrupole field gradient is suddenly changed at $t=0$ from its standard value $b_{\mathrm{q}}^{\mathrm{s}}=3.7$ G/cm to $1.4b_{\mathrm{q}}^{\mathrm{s}}$ (), $0.5b_{\mathrm{q}}$ (), and $0.3b_{\mathrm{q}}$ (). All curves are interpolated using cubic splines for enhanced visual appearance.

Figure 5

(a) Horizontally and (b) vertically integrated particle densities produced in the strong magnetic field gradient $b_{\mathrm{q}}=11.1$ G/cm. All other parameters assume identical values to those specified in the main text. The peak column density is ${\tilde{n}}_{\mathrm{p}}=2.7\times {10}^{11}{\mathrm{cm}}^{-2}$ and the field of view is $22\times 22\mu {\mathrm{m}}^2$ in each panel.