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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.

The Dirac monopole created in Ref. [15] is an analogue of the classical magnetic point charge considered by Dirac in the context of quantum mechanics [28]. It manifests itself as a point-like singularity in the so-called synthetic magnetic field [29], which is an effective gauge field for the scalar part of the order parameter arising naturally from its spin degrees of freedom. The Dirac monopole induces in the condensate order parameter a nodal vortex line with vanishing particle density extending from the location of the monopole to the boundary of the atom cloud. Thus the ferromagnetic order parameter supporting the Dirac monopole is energetically and dynamically reminiscent of a line defect. Critically, there is no topological point defect in the order parameter itself, as it is in a configuration topologically equivalent to the ground state. Indeed, the second homotopy group [2] for the ferromagnetic order parameter space is trivial and topological point defects are not permitted. Point defects may exist in the polar phase of a spin-1 condensate, however, as the second homotopy group for the polar order param- [30,31], is isomorphic to the additive group of integers, π 2 (G p ) ∼ = Z.
Topological point defects in the polar phase of a 87 Rb spin-1 BEC have been recently realized experimentally [27]. No nodal lines or other physical line-like objects are attached to this monopole, and we therefore refer to it as an isolated monopole. As Ref. [27] focused only on the first observation of the isolated monopole within a finite experimental resolution, there has been no detailed study of the fine structure of the defect. Furthermore, both the isolated monopole [5] and indeed the entire polar phase of the 87 Rb condensate [17] are expected to be unstable at low magnetic fields, prompting a study on the evolution of the isolated monopole after its creation.
In this Letter, we present computational results of the fine-grained structure and decay dynamics of the isolated monopole. We find that the created monopole exhibits a small half-quantum vortex ring [5] in its core. We show that beyond the experimentally accessed time scales [27] the polar order parameter evolves into a ferromagnetic order parameter, accompanied by the decay of the isolated monopole into a Dirac monopole in the resulting synthetic magnetic field. Importantly, this decay arises naturally from the thoroughly understood physics of the atom cloud without any phenomenological damping terms. Qualitatively similar dynamics are observed for both ferromagnetic and antiferromagnetic spin-spin interactions.
Theoretical background -The mean-field order parameter of a spin-1 condensate can be expressed as ψ(r) = n(r)ξ(r), where n(r) is the particle density and ξ(r) is a three-component complex spinor such that ξ(r) † ξ(r) = 1. The temporal evolution of the order parameter at low temperatures is accurately described by the differential equation where h(r) is the single-particle Hamiltonian, F = (F x , F y , F z ) is a vector composed of the dimensionless spin-1 matrices, Γ is the three-body recombination rate, and S(r) = ξ(r) † Fξ(r) is the local average spin. The coupling constants characterizing the atom-atom interactions are given by c 0 = 4π 2 (a 0 + 2a 2 )/(3m) and c 2 = 4π 2 (a 2 − a 0 )/(3m), where a f is the s-wave scattering length corresponding to the scattering channel with total two-atom hyperfine spin f . The single-particle Hamiltonian reads where m is the mass of the atoms, V (r) is an external optical trapping potential, µ is the chemical potential, g F is the Landé g-factor, µ B is the Bohr magneton, B is an externally applied magnetic field, and q is the strength of the quadratic Zeeman shift [32]. We assume that V (r) = m[ω 2 r (x 2 + y 2 ) + ω 2 z z 2 ]/2, where ω r and ω z are the radial and axial trapping frequencies, respectively.
In the pure polar phase with S(r) = 0, the order parameter can be expressed in the basis of the z-quantized spin states {|1 , |0 , |−1 } as [33,34] Thus in the Cartesian basis the polar order parameter reads ψ(r) = n(r)e iφ(r)d (r), whered = (d x , d y , d z ) T is a real unit vector known as the nematic vector. In the case of non-zero average spin, we investigate the nematic order through the magnetic quadrupole tensor [33] where {ξ i } are the components of the spinor in the Cartesian basis. For S(r) = 0, the vectord is defined as the eigenvector corresponding to the largest eigenvalue of Q.
Methods-The nematic vector behaves identically to the average spin under rotations in spin space, and hence it also follows adiabatic changes in the external magnetic field. Consequently, the method originally developed in Ref. [8] for the adiabatic creation of Dirac monopoles in the ferromagnetic phase can be used to create isolated monopoles in the polar phase, as realized in Ref. [27]. In brief, the condensate is subjected to an external magnetic field B(r, is a quadrupolar magnetic field with gradient b q and B b (t) = B z (t)ẑ is a spatially homogeneous bias field. In the beginning of the simulation, the condensate is in the spin state |0 , yielding a nematic vectord(r) =ẑ. At the initial bias field B z = 1 G, the field gradient is linearly ramped from zero to b q = 3.7 G/cm in 10 ms and the bias field is subsequently decreased to B z = 10 mG in 10 ms. The monopole is created by decreasing the value of the bias field linearly to zero at a rateḂ z (t) = −0. 25   FIG. 1. (a) Horizontally (b) and vertically integrated particle densities for the indicated waiting times after the creation ramp producing the 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 isñp = 2.7 × 10 11 cm −2 and the field of view is 15.5 × 15.5 µm 2 in each panel. The red arrows indicate the location of a vortex line shown more clearly in Fig. 2 G/s. This part of the control protocol is referred to as the creation ramp and it ideally results ind =B q (r). After the creation ramp the temporal evolution continues with the quadrupole field and optical trap intact. The data in both panels is shown only for |z| < 1.5ar. The spin density is well depleted along the vortex, whereas the particle density is only partially depleted.
The initial particle number is 2.1 × 10 5 and the optical trapping frequencies are ω r = 2π × 124 Hz and ω z = 2π × 164 Hz. We take the other parameters according to 87 Rb such as the literature values for the atom loss arising from the three-body recombination Γ = × 2.9 × 10 −30 cm 6 /s [35,36], the quadratic Zeeman shift q = 2π × 2.78 MHz/T [17], and the scattering lengths a 0 = 5.387 nm and a 2 = 5.313 nm. The computational volume considered is 24×24×24a 3 r , where a r = /(mω r ) = 1 µm and the corresponding size of the computational grid is 200×200×200 points. In order to enhance the numerical emulation of the experimental conditions [27], we add spatially uncorrelated complex noise to the spinor components at each grid point prior to the creation ramp. The amplitude of the noise is uniformly distributed to introduce 0-1% fluctuations in the local particle density for each spinor component.
Results-We numerically integrate Eq. (1) and apply the control protocol described above with the initial conditiond =ẑ. Figure 1 shows the resulting spin-contrast images of the condensate particle densities for different waiting times after the creation ramp. As in Ref. [27], the condensate particle densities just after the creation ramp are in good agreement with Eq. (3) andd =B q (r). Thus the particles almost entirely reside in the so-called neutral state which corresponds to the zero eigenvalue of the local Zeeman Hamiltonian. However, 50 ms after the creation ramp, the polar phase has noticeably decayed towards the local ferromagnetic strong-field seeking state (SFSS), i.e., the spin state that minimizes the local Zeeman energy. This decay begins at the top and bottom edges of the condensate where the magnetic field is strongest and continues gradually until the condensate resides almost entirely in the SFSS.

noise.
It is well known that the SFSS in the quadrupole field B q (r) corresponds to the synthetic magnetic field of a Dirac monopole [8]. However, the final order parameter in our case does not have the terminating doublequantum vortex observed in Ref. [15]. Instead, there is one single-quantum vortex that reverses its circulation at the monopole, a scenario that has previously been shown to minimize the mean-field energy in the case of a Dirac monopole [37]. We confirm the presence of this single-quantum vortex in Fig. 2 where it is visible as a line of suppressed spin density. We have verified that the phase winding along this vortex line reverses its sign near the origin where the magnetic field vanishes (data not shown). The orientation of the vortex depends on the particular realization of the applied noise. Figure 3 shows the nematic vector and selected components of the spin vector during the decay of the isolated monopole. A ferromagnetic ring with a well-defined polarization is clearly visible just after the creation ramp [see Fig. 3(c) and 3(d)] although it is so small that it was not distinguished within the finite experimental resolution of Ref. [27]. This ring resides at the monopole core and retains it size during the temporal evolution. Since the nematic vector is observed to rotate by π about the ferromagnetic core [see Fig. 3(a)], the ring is identified as a half-quantum vortex ring, or Alice ring, discussed in Ref. [5]. We determine the radius of the Alice ring from the behavior of the nematic vector to be roughly 0.2 µm which exceeds neither the spin healing length / 2m|c 2 |n(0) ≈ 4 µm nor the density healing length / 2m|c 2 + c 0 |n(0) ≈ 0.3 µm. Thus the structure essentially manifests itself a point defect. The subsequent decay of the polar phase destroys the Alice ring and eventually the characterization of the condensate using the nematic vector becomes obscure. We therefore do not show the nematic vector but rather the local spin for long evolution times. Ultimately, the local spin aligns with the external magnetic field as shown in Fig. 3(g) and 3(h). Figure 4 shows the fraction of particles in the neutral state and the deviation of the order parameter from the initial isolated monopole state during the decay. The relative population of the neutral state is given by n ns (t) = 1 N (t) dr|Π ns (r, t)ψ(r, t)| 2 , where N (t) is the total number of atoms and Π ns (r, t) is a projection to the neutral state [13]. The deviation of the order parameter from the initial isolated monopole state at t = 0 is characterized by (t) = 1 − | dr ψ * (r, t)ψ(r, 0)|/ N (0)N (t). We observe that just after the creation, ramp roughly 90% of the atoms reside in the neutral state, in agreement with the experimentally obtained value in Ref. [27]. The decay of the isolated monopole into the ferromagnetic phase is characterized by an approximately linear temporal decrease in the population of the neutral state. Figure 4 also shows the results obtained for the three additional cases: (i) eliminating the quadratic Zeeman potential, (ii) eliminating three-body recombination, and (iii) reversing the sign of the spin-spin interaction strength. None of these changes leads to a significant effect on the decay dynamics, indicating that the decay is not originating from these terms. We also studied the creation and decay of the isolated monopole with parameters corresponding to 23 Na atoms and obtained qualitative agreement with 87 Rb atoms (results not shown). (1) are shown with red circles, blue crosses, and green diamonds, respectively. All curves are interpolated using cubic splines for enhanced visual appearance.
Conclusions-Our numerical studies suggest that the isolated monopole structure observed in Ref. [27] contains a small Alice ring. This vortex ring is destroyed by a subsequent dynamical phase transition into a ferromagnetic order parameter supporting a Dirac monopole. Although the quadrupole field has been observed to stabilize the polar phase of a 87 Rb condensate if the field zero is well outside the condensate [27], our simulations reveal that after the field zero is brought into the condensate, the polar phase decomposes on a time scale of 100 ms. We attribute this behavior to the spatially varying magnetic field and the linear Zeeman interaction. Neither the spin-spin interactions, quadratic Zeeman effect, nor three-body recombination have a significant effect on the decay dynamics. These studies set the stage for the dynamics of topological point defects in quantum fields. Finding ways to extend the lifetime of the defect and thereafter to study the dynamics of multiple interacting point defects remain future challenges. This research has been supported by the Academy of Finland through its Centres of Excellence Program (Project No. 251748) and Grants No. 135794 and No. 272806, the U.S. National Science Foundation (NSF PHY-1208522), Finnish Doctoral Programme in Computational Sciences, the Magnus Ehrnrooth Foundation, and the Emil Aaltonen Foundation. CSC -IT Center for Science Ltd. (Project No. ay2090) and Aalto Science-IT project are acknowledged for computational resources.