## BEC2 observes asymmetric spin flipping transitions due to colored noise

### Qualitative explanation of the results

In order to gain a qualitative understanding of the results,
we present a simple semiclassical 1D model
of two thermal distributions of atoms trapped in Harmonic potentials,
V_{j}(x) = 1/2m_{j}Mω_{1}^{2}x^{2},
where M is the atomic mass, ω_{1}
is the trapping frequency of the m_{F} = 1 level
and m_{j} = m_{F} is either 1 or 2.
Each atomic distribution is represented in the figure on the right by a typical atom at an
average position, d_{i} =
√(k_{B}T/m_{j}Mω_{1}^{2}). |
Full size image |

The transition of a typical atom from m_{F} = 2
to m_{F} = 1 requires a photon energy
E_{2→1} = E_{12}^{0} + V_{2}(d_{2}) − V_{1}(d_{2}) =
E_{12}^{0} + 1/4k_{B}T and reduces
the energy of the atom relative to the trap bottom by 1/4k_{B}T.
A transition of a typical atom in m_{F} = 1 to m_{F} = 2 at
d_{1} = √2d_{2} requires a photon energy
E_{1→2} = E_{12}^{0} + V_{2}(d_{1}) − V_{1}(d_{1}) =
E_{12}^{0} + 1/2k_{B}T and increases
the energy of the atom by 1/2k_{B}T.

The relative transition rates depend on the number of
photons with the two energies E_{1→2} > E_{2→1} . If the
noise intensity increases with frequency in a wide enough
band (∼ k_{B}T) around f = E_{12}^{0}/*h*
(see blue arrows in the inset, Δf > 0),
the transition 1 → 2 is preferred and
the population of m_{F} = 1 is depleted, as in the plateau
on the right-hand side of the right figure below. If, by contrast, the intensity
decreases in this band (see red arrows in the inset above,
Δf < 0),
the transition 2 → 1 is preferred and the population
of m_{F} = 1 becomes dominant, as in the plateau on
the left-hand side.

Due to these effects, even when the noise intensity decreases by a few orders of magnitude (side peaks of the noise, inset below), the atoms are still sensitive to the details of the spectrum, as can be seen in the right figure below. This may prove to be a useful tool for characterizing noise features. Note also that the temperature determines the width of the spectral region the cloud samples, hence colder clouds are more sensitive to the fine details of the noise (e.g. 1 μK gives a resolution of 20 kHz). This effect is most noticeable at Δf ≈ 0.7MHz (right figure below) where the temperature band becomes much wider.

### The results

The left figure below presents the ratio R(t) = N_{1}/(N_{1} + N_{2}) as a
function of the time for which the noise is applied, for
different detuning Δf between the noise peak frequency
f_{0} and the Zeeman splitting E_{12}^{0}/*h* of the two trapped
levels at the magnetic field minimum. The inset presents
the spectrum of the noise we introduce. One may observe
a large difference in the evolution of R(t) between red- and
blue-detuned noise. When the noise is red-detuned
(Δf < 0), the m_{F} = 1 level is populated while when
the noise is blue-detuned (Δf > 0) the m_{F} = 1 level is
depleted. The solid curves are fits to the empirical form
R(t) = R_{∞} + (R_{0} − R_{∞})exp(−γt),
which represents an exponential convergence from the
initial value R_{0} ≡ R(t = 0) to an asymptotic value
R_{∞} ≡ R(t → ∞).

These results show that the relative transition rates between the levels strongly depend on the detuning of the noise and significantly differ from the expected rates in the case of white noise (dashed line) or a model where the external degrees of freedom are decoupled from the transition dynamics (0 ≤ R ≤ 1/2 band).

In the right figure we plot the measured value of R_{∞} for different
values of the center frequency of the applied noise.
The band and the three curves represent the theoretical
prediction for R_{∞}, as qualitatively described above.

Full size image |
Full size image |

More details on the experiment and the quantitative explanation are explained in our paper.