Improving observability of the Einstein–de Haas effect in a rubidium condensate

The main obstacle in the experimental realization of the Einstein–de Haas effect in a Bose-Einstein condensate is the need for very precise control of the extremely small (of the order of tens of $\mu \mathrm{G}$) external magnetic field. In this paper, we numerically study the response of a rubidium condensate to a magnetic field that is linearly dependent on time. We find a significant transfer of atoms from the initial maximally polarized state to the next Zeeman component at magnetic fields of the order of tens of milligauss. We propose an experiment in which such a time-dependent magnetic-field-based scheme could enable the observation of the Einstein–de Haas effect in a rubidium atom condensate.

Figure 1

Maximal transfer, within 50 ms, to the $m_F=0$ Zeeman component as a function of the magnetic field. Resonant lines correspond to the initial number of atoms equal to $N_{+1}=100,95,90,85,80$ (from right to left). The frequencies of an axially symmetric cigar-shaped trap are ${\omega }_{x,y}=2\pi \times 6400$ Hz and ${\omega }_z=2\pi \times 1600$ Hz. The maximal density in each case is about ${10}^{15} {\mathrm{cm}}^{-3}$.

Figure 2

Resonant magnetic field (in milligauss) as a function of the initial number of atoms in the $m_F=+1$ Zeeman component. Lines correspond to resonances that differ by an amount of excitation energy deposited in the axial direction. The upper, middle, and lower lines (which are fits to numerical data marked by bullets) correspond to resonances responsible for the transfer of atoms to the first, third, and fifth axially excited state, respectively. Red bullets (leftmost) are at the values of resonant magnetic fields assuming no contact interactions are present. Note that the solid red lines approach the red bullets (leftmost).

Figure 3

Typical density of the $m_F=0$ component, integrated along the radial direction, corresponding to the upper (left frame) and middle (right frame) sets of points in Fig. 2.

Figure 4

Relative population of the $m_F=0$ Zeeman component (solid red line) and the magnetic field (dashed blue line) as functions of time. We want to stress that (i) both for axial and spherical traps there is a significant resonant transfer of atoms to other Zeeman components (compare upper and middle frames), (ii) a relatively large transfer of atoms to higher excited states is also possible for a sufficiently chosen magnetic-field ramp—see the lower panel and (5). Upper frame: the initial number of atoms in the $m_F=+1$ state is $N_{+1}=100$. The value of the magnetic field changes in time as $B\left(t\right)=B_{\mathrm{ini}}-\alpha t-\beta t^2$, where $B_{\mathrm{ini}}=-7.9$ mG, $\alpha =6$ mG/s, and $\beta =3 \mathrm{mG}/s^2$. The trap parameters are the same as those in Fig. 1. Middle frame: the initial number of atoms in the $m_F=+1$ state is $N_{+1}=50$. The frequency of a spherically symmetric trap is ${\omega }_{x,y,z}=2\pi \times 5000$ Hz, which results in the maximal density of about ${10}^{15} {\mathrm{cm}}^{-3}$. The value of the magnetic field changes in time as $B\left(t\right)=B_{\mathrm{ini}}-\alpha t$, where $B_{\mathrm{ini}}=-10.65$ mG and $\alpha =6$ mG/s. Lower frame: same as the middle frame, but $B_{\mathrm{ini}}=-25.4$ mG and $\alpha =1.2$ mG/s.

Figure 5

Typical density in the $xz$ plane integrated along the $y$ direction (left frame) and the phase in the $xy$ plane (right frame) of the $m_F=0$ component, corresponding to the resonance, as in Fig. 4 (lower frame), characterized by the quantum numbers $n_r=m_{\mathrm{rot}}=n_z=1$.

Figure 6

Test of the sensitivity of the atomic transfer to a small variation of the magnetic-field sequence. The solid black line shows the result for $\alpha =6$ mG/s (the same as in Fig. 4, upper frame). When additional fluctuations of the magnetic field at the level of $1\%$ are added, the transfer remains almost unchanged (red dotted line). The other two curves show how sensitive the process is to the small change of deterministic parameters. Even about $10\%$ deviation in the $\alpha$ parameter does not significantly modify the transfer—see the dotted-dashed blue line ($\alpha =6.5$ mG/s). The other curve (dashed green line) is for $\alpha =4.0$ mG/s.