Lifetime of a Polarized-Fermi-Gas in the Excited-State

In all measurements effectively the reduction of the lifetime of the sample relative to the

“natural” lifetime of the atoms in the optical lattice is measured. Thus it is important to understand the fundamentally limiting timescale of the lifetime of excited state atoms. In figure 6.5 the lifetime of spin-polarized ground state atoms and spin-polarized excited state atoms is shown. A rather long lifetime ofτ_g ≈18.5 s for the ground state atoms is measured.

A simple exponential decay is used to determine the value forγ_g = 1/τ_g = 1/18.5 s. The lifetime for the excited state isτ_e= 2.3 s. Additionally, a transfer of atoms from the excited state into the ground state is observed. This ground state atom growth is approximated by a simple exponential growth:

N_g^RAP(t) =N0

1−exp−γ_g^RAPt (6.13)

The following rates are extracted from the data:

γ_e^RAP = 0.428/s±0.123/s γ_g^RAP = 0.228/s±0.020/s γ_g = 0.054/s±0.034/s

(6.14)

Here the subscript “RAP” indicates whether a RAP has been performed before the atoms were counted. γ_g is the ground-state loss rate obtained from the measurement without a transfer of atoms into the excited state.

Time (s) Number of Atoms (103 )

0 2 4 6 8

0 2 4 6 8 10 12 14 16 18 20

Excited−State after RAP Ground−State after RAP Ground−State w/o RAP

τ=4.4s τ=2.3s

τ=18.5s

Figure 6.5. | Lifetime of a spin-polarized gas in deep 3D-lattice. The lifetime of ground-and excited-state atoms for a spin-polarized gas with ground-and without a transfer to the excited state by RAP. Additionally exponential decay regressions are performed and their corresponding 1/e-lifetimes are given. See main text for details and discussion.

A significantly lower lifetime of the excited state compared to the ground state atoms is observed. This lifetime is also smaller than the lifetime of the excited state for a free atom which is on the order of tens of seconds. The reduction of the lifetime is attributed to the influence of the electromagnetic field of the lattice laser beams. The atoms are trapped by the dipole force which is based on the mixing of energy states. While small this mixing effectively lowers the lifetime of the atoms.

The excited state atoms are not lost from the trap but decay into the ground state. This is seen by the growth of ground state population. The rate of ground state population growth is smaller than the decay of the excited state into the ground state. This is explained in terms of density losses resulting from the inelastic collision of excited state atoms and atoms decayed from the excited state into the ground state: after a decay into the ground state, the atoms are no longer in the same quantum state. Tunneling between lattice sites is therefore allowed. A ground state atom and an excited state atom can scatter inelastically (see section 2.3.2) and both are lost from the trapping potential.

To conclude: the dynamics of the atom number is governed by three different processes:

• Excited state atoms decay into the ground state.

• Inelastic scattering of ground and excited state atoms.

• Ground and excited state atoms loss due to single particle processes (e.g. background gas collisions).

These processes are simulated by a rate-equation model which incorporates onsite density dependent loss between excited-state and ground-state atoms. The model consists of two coupled differential equations.

The first equation describes the time evolution of the number of excited state atomsN_e. It assumes a single-particle loss due to decay into the ground-state with a rate γe. Further, it is assumed that the excited state atoms are lost from the trap due to the same processes which remove ground-state atoms from the trap. This single particle decay is incorporated by a term with the rate γg. γg is taken from the fit as shown in equation 6.14. The last term describes inelastic scattering between ground- and excited-state atoms (i.e. a density term):

dNe

dt =−γ_gNe−γeNe−γegNeNg (6.15) The second rate equation describes the number of ground-state atoms. Atoms from the ground state are lost by single-particle decay with a rate γg and two-body losses between ground and excited state atoms as before. The gain of atoms from the decay of the excited state is modeled with the rateγ_e:

Time (s) Number of Atoms (103 )

0 5 10 15

0 2 4 6 8 10 12 14 16 18 20

Observed Excited−State Fitted Lifetime−Model Observed Ground−State Fitted Lifetime−Model

Figure 6.6. | Simulation of the Lifetime of a Spin-Polarized Gas in Deep 3D-Lattice.

Shown is the fit of a lifetime model incorporating density dependent losses to experimental lifetime data. See main text for details.

dN_g

dt =−γ_gN_g+γ_eN_e−γ_egN_eN_g (6.16) Hereγ_g= 0.054/s is the single-particle loss-rate of ground-state atoms as shown in equation 6.14. The loss-rateγ_eg describes onsite inelastic scattering.

This model neglects excited-state decay into higher bands (Lamb-Dicke regime) and it assumes the sameγ_g for the ground- and excited-state.

The lifetime model is fitted to the data. The regression yields the following values (for completeness alsoγ_g is given but is held constant during the fit):

γ_g = 0.0540/s γ_e= 0.1558/s γeg = 0.0045/s

(6.17)

The experimental data together with the fitted lifetime model is shown in figure 6.6. The data and the lifetime model seem to agree well and the behavior for longer time scales is

reasonable. The lifetime model presented here shows the best fit results compared to simpler models which do not include a density-dependent term. See appendix A for details.

To conclude this section a lifetime measurements in a deep optical lattice of ¹⁷³Yb in the metastable excited state was presented. The behavior of the atom-number was simulated using rate equations. A density-dependent loss-term has to be incorporated in order to describe the observed particle number dynamics. This means the loss process between excited and ground state atoms plays a significant role on the time scales considered here.