Control of dipolar relaxation in external fields

We study dipolar relaxation in both ultracold thermal and Bose-condensed Cr atom gases. We show three different ways to control dipolar relaxation, making use of either a static magnetic field, an oscillatory magnetic field, or an optical lattice to reduce the dimensionality of the gas from three-dimensional (3D) to two-dimensional (2D). Although dipolar relaxation generally increases as a function of a static magnetic-field intensity, we find a range of nonzero magnetic-field intensities where dipolar relaxation is strongly reduced. We use this resonant reduction to accurately determine the $S=6$ scattering length of Cr atoms: $a_6=103\pm 4a_0$. We compare this new measurement to another new determination of $a_6$, which we perform by analyzing the precise spectroscopy of a Feshbach resonance in $d$-wave collisions, yielding $a_6=102.5\pm 0.4a_0$. These two measurements provide, by far, the most precise determination of $a_6$ to date. We then show that, although dipolar interactions are long-range interactions, dipolar relaxation only involves the incoming partial wave $l=0$ for large enough magnetic-field intensities, which has interesting consequences on the stability of dipolar Fermi gases. We then study ultracold Cr gases in a one-dimensional (1D) optical lattice resulting in a collection of independent 2D gases. We show that dipolar relaxation is modified when the atoms collide in reduced dimensionality at low magnetic-field intensities, and that the corresponding dipolar relaxation rate parameter is reduced by a factor up to $7$ compared to the 3D case. Finally, we study dipolar relaxation in the presence of rf oscillating magnetic fields, and we show that both the output channel energy and the transition amplitude can be controlled by means of the rf frequency and Rabi frequency.

Figure 1

Comparison between the analytical calculation taking into account the effect of $a_6$ [Eqs. (17) and (18)] and calculations with no molecular potentials [Eqs. (9) and (10)].

Figure 2

Dipolar relaxation from the molecular physics point of view, illustrated for channel 2 (sketch); the molecular curves of the initial $l=0$ potential and the final $l=2$ potential cross at interparticle distance $R^{*}$.

Figure 3

Atom number and size of the cloud as a function of the delay time $t$ spent in $m_S=3$.

Figure 4

Dipolar relaxation rate parameter as a function of magnetic field for a $m_S=3$ BEC. Squares: experimental data points. Dashed line: calculation in the first-order Born approximation, with no molecular potential [Eqs. (9) and (10)]; the calculated rate parameter is divided by 2 to take into account the fact that the experiment is performed with a BEC.

Figure 5

Influence of initial and final molecular potentials on dipolar relaxation to state $|{\mathrm{out}}^{(1)}\rangle$. The figure shows the variation of the square of the radial integrals $I(\Delta E^{(1)})$ as a function of the energy gap $\Delta E^{(1)}$ (in G). The case where both potentials are ignored is the dotted-dashed line. The dashed curve corresponds to the case where the initial potential is taken into account: the strong diminution, as compared to the first curve, comes from the oscillations of the vibrational wave function. The final molecular potential induces an enhancement of the integral (solid line), which corresponds to an increase of the density probability close to the rotational barrier.

Figure 6

Contribution of the three different channels of dipolar relaxation, to states $|{\mathrm{out}}^{(1)}\rangle$, $|{\mathrm{out}}^{(2)}\rangle$, and $|{\mathrm{out}}^{(3)}\rangle$, and influence of the $C_6$ coefficient. The figure shows the variation of the relaxation rate parameter of the three final states (green dashed: $|{\mathrm{out}}^{(1)}\rangle$; blue long-dashed: to $|{\mathrm{out}}^{(2)}\rangle$; red dot-dashed: to $|{\mathrm{out}}^{(3)}\rangle$), together with their sum (thick solid line), as a function of the magnetic field. The central black solid line corresponds to the central $C_6$ value 733$a_0$, whereas the top dotted black line corresponds to 803$a_0$ (bottom dotted black line: 663$a_0$). The value of $a_4$ is set to 69a${}_0$.

Figure 7

Influence of the $a_6$ and $a_4$ values and comparison with experiment. The figure shows the variation of the dipolar relaxation rate parameter as a function of the magnetic field (in Gauss). Dots: experimental data points. (a) Straight lines: result of the calculation for various values of $a_6$: 94$a_0$, 98 to 107$a_0$ with step 1 ${\mathrm{a}}_0$, 110 and 114$a_0$. The $C_6$ coefficient is here equal to 733 ${\mathrm{a}}_0$ and $a_4=64a_0$. (b) Result of the calculation for various values of $a_4$ from 58$a_0$ to 70$a_0$ with step 1 ${\mathrm{a}}_0$. The $C_6$ coefficient is here equal to 733 ${\mathrm{a}}_0$ and $a_6=103a_0$.

Figure 8

Filled squares and circles: loss parameter as a function of temperature for two slightly different magnetic fields close to the minimum of dipolar relaxation (filled circles: 3.8 G; filled squares: 3.9 G). The lowest temperature corresponds to a pure BEC, whereas for other data points the gas is fully thermal. The figure shows the factor of 2 reduction of dipolar relaxation when BEC is reached as well as the fact that dipolar relaxation does not depend on temperature for thermal gases.

Figure 9

Calculation of dipolar relaxation to state $|{\mathrm{out}}^{(1)}\rangle$ starting from either $l=0$ (red solid line) or from higher partial waves (blue dotted line: from $l=2$; green dotted line: from $l=4$. Dipolar relaxation through even larger partial waves lead to even smaller contributions).

Figure 10

(a) Absorption picture taken 5 ms after the optical lattice has been released adiabatically (band-mapping procedure). (b) (black) Cut of picture (a) along the axis of the lattice, as compared to (grey) a cross section of Bose-condensed atoms 5 ms after having been submitted to a brief pulse of the optical lattice, thus producing diffraction peaks at $\pm 2\hslash k_L$: The band-mapping procedure shows that all atoms lie inside the first Brillouin zone, showing no population in higher lattice bands. (c) Cut of picture (a) perpendicular to the optical lattice axis, from which we deduce the average temperature in the lattice.

Figure 11

Heating of the atoms in the lattice due to dipolar relaxation, without losses. (a) Typical data for the number of atoms (black diamonds) and temperature (red dots) of the cloud as a function of the time spent in $m_S=3$ in the lattice; the lattice depth is 35 $E_R$, the Larmor frequency is ${\omega }_0/2\pi =100$ kHz. The straight line is a linear fit to the data used to infer the heating rate. The dotted line is a straight line to guide the eye. (b) Temperature of the cloud for $t=0.5$ ms (red dots) and $t=20$ ms (black diamonds), as a function of ${\omega }_0/2\pi$. Dashed lines are guides for the eye.

Figure 12

(a) Dipolar relaxation loss parameter as a function of magnetic field, in the lattice (filled triangles) and without the lattice (open triangles). The dashed vertical lines represent (left) the magnetic field above which excitation by dipolar relaxation of the first vibrational band becomes energetically allowed (first threshold); (right) the second threshold. The red solid line is the result of the 2D theory for the specific orientation of the magnetic field in the experiment, and the black dotted line the result of the 3D theory. (b) Results of the 2D theory for a magnetic field parallel to the 2D planes (black line) and perpendicular to them (grey line).

Figure 13

Dipolar relaxation in the presence of rf from the molecular physics point of view, illustrated for channel 2 (sketch); the difference of energy between initial and final channels is controlled by a combination of the Larmor frequency and of the rf frequency and $R^{*}$ is modified accordingly.

Figure 14

Rf-assisted dipolar relaxation loss parameter as a function of the Rabi frequency for a thermal gas in $\log {}_{10}$ scale. Inset (in linear scale): rf-assisted dipolar relaxation loss parameter as a function of the Rabi frequency for a thermal gas (black filled circles) and for a BEC (red squares). The measured loss rate parameter is roughly a factor of 2 smaller for a BEC than for a thermal gas. Solid line: theoretical results for a thermal gas.

Figure 15

Rf-assisted dipolar relaxation loss parameter for three different rf frequencies (symbols: experimental data; lines: theory).

Figure 16

Rf-assisted dipolar relaxation loss parameter as a function of Rabi frequency for two different values of the bias field (circles: ${\omega }_0/2\pi =$100 kHz; diamonds: ${\omega }_0/2\pi =$1.21 MHz). Losses are much suppressed when the rf frequency is much smaller than ${\omega }_0$, indicating that rf-assisted losses with the absorption of a small ($<$5 in the present case) number of rf photons becomes energetically impossible. Solid lines: corresponding theoretical results.

Figure 17

Rf-assisted dipolar relaxation loss parameter for two different orientations of a low static magnetic field (black circles: static field parallel to rf field; red dots: static field perpendicular to rf field); lines are guides for the eye.