Nonadiabatic losses from radio-frequency-dressed cold-atom traps: Beyond the Landau-Zener model

Nonadiabatic decay rates for a radio-frequency-dressed magnetic trap are calculated using Fermi's golden rule: that is, we examine the probability for a single atom to make transitions out of the dressed trap and into a continuum in the adiabatic limit, where perturbation theory can be applied. This approach can be compared to the semiclassical Landau-Zener theory of a resonant dressed atom trap, and it is found that, when carefully implemented, the Landau-Zener theory overestimates the rate of nonadiabatic spin-flip transitions in the adiabatic limit. This indicates that care is needed when determining requirements on trap Rabi frequency and magnetic-field gradient in practical atom traps.

Figure 1

Bare and adiabatic potentials $V$ as a function of atomic position $z$ for an atom with total angular momentum $F=1$ in an rf-dressed cold-atom trap. The gravitational potential is included and results in the slope of the coincident potentials labeled $m_F=0$ and $m_F^{\prime }=0$. Blue dashed lines show the potentials given in Eq. (7), with the crossing point necessary for Landau-Zener theory, and labeled $m_F=-1,0,1$. The solid black lines show the adiabatic potentials in the dressed state basis given by Eq. (11) and labeled $m_F^{\prime }=-1,0,1$. To give a concrete example, we use values taken from Ref. [16], i.e., for a ${}^{87}\mathrm{Rb}$ atom and a magnetic-field gradient of $B^{\prime }=1.1$ T/m resulting in a detuning which varies linearly with position. The Rabi frequency is set to ${\mathrm{\Omega }}_0/2\pi =8$ kHz for this graph.

Figure 2

Scaled decay rate ${\mathrm{\Gamma }}_n/{\omega }_z$ is shown as a function of the harmonic-oscillator label $n$ for different models. The solid line with circles shows the result for the horizontal trapping model ($h$) as given by Eq. (32). The dashed line with asterisks indicates the scaled vertical trapping decay rates ($v$) as given by Eq. (47). These vertical trapping decay rates are not a smooth function of $n$. We can also include nonadiabatic potentials based on including part of ${\widehat{V}}_B$ and resulting in modified parameters given in the Appendix. For the vertical trapping model, the effect is a shift of the location of the “dips” (dotted line). However, in the case of the horizontal trapping model, the inclusion of these corrections produces no visible change to the solid ($h$) line and is not shown. The chained lines (${\text{LZ}}_h$ and ${\text{LZ}}_v$) indicate the result of a Landau-Zener calculation given by Eq. (51) (with $\epsilon$ set to zero in the “$h$” case). For the parameters used (i.e., from Fig. 1) the trap frequency was ${\omega }_z/2\pi =0.93$ kHz for the horizontal trapping model [Eq. (23)] and corresponding Landau-Zener model. For the case of the vertical trapping model and its corresponding Landau-Zener curve, the trap frequency is found from Eq. (39) to be ${\omega }_z/2\pi =0.87$ kHz. The points marked with small squares are given by the analytic approximation Eq. (34). The calculations are done for the $F=1$ hyperfine ground state of ${}^{87}\mathrm{Rb}$ with the parameters given in Fig. 1. This corresponds to $\eta ={\mathrm{\Omega }}_0/\left(\alpha a_z\right)\sim 2.9$ for the horizontal trapping model and, for the vertical trapping model, $\eta \sim 2.8$ [Eq. (49)] and $\epsilon =Mg/\left(\hslash \alpha \right)=0.28$.

Figure 3

Schematic diagram showing the key wave functions and potentials $V_{i,f}\left(z\right)$ in the vertical trapping model. The wave functions are scaled to equal maximum height for comparison. The ground-state wave function of the harmonic oscillator is shown centered at $z_0$, Eq. (37), in its potential $V_i\left(z\right)$, Eq. (36). The energy of the ground state is $E_0$, as given by $V_0+\hslash {\omega }_z/2$, Eqs. (38),(39). An energy resonant Airy function is shown, which is an eigenstate of the linear potential $V_f\left(z\right)$, Eq. (41). The potential “wall” on the left is located at $z\to -\infty$ in the calculations. The eigenstate is associated with a turning point at $z_{\kappa }$. The parameters for this figure are as in Fig. 1.

Figure 4

Scaled decay rates ${\mathrm{\Gamma }}_n/{\omega }_z$ as a function of the parameter $\eta$ for the ground state (${\mathrm{\Gamma }}_0/{\omega }_z$, top), the first excited state (${\mathrm{\Gamma }}_1/{\omega }_z$, middle), and the fifth excited state (${\mathrm{\Gamma }}_5/{\omega }_z$, bottom) of the initial harmonic trap. The prediction of the vertical trapping model, Eq. (47), solid line, is shown with the predictions of the horizontal trapping model, Eq. (32), dashed line, and the Landau-Zener model, Eq. (51), chained line. The dotted lines indicate the effect of nonadiabatic potentials which produce corrections to ${\omega }_z$ from Eq. (A9) and to ${\mathrm{\Gamma }}_n$ as described in Appendix pp1-s2 and in Fig. 2. The calculations are done for the $F=1$ hyperfine ground state of ${}^{87}\mathrm{Rb}$ with $\epsilon \sim 0.20$ in the vertical trapping model case.

Figure 5

Examination of the sensitivity of the wave functions in the vertical trapping model. The trapped harmonic-oscillator ground-state wave function ${\mathrm{\Phi }}_n\left(z\right)$, associated with $m_F^{\prime }=1$, is displayed with dashed lines for two different Rabi frequencies: ${\mathrm{\Omega }}_0/\left(2\pi \right)=6.0$ kHz and 6.8 kHz. Other parameters are fixed and are the same as in Fig. 1. Also shown are the two Airy functions associated with $m_F^{\prime }=0$ eigenstates matching the energies of the corresponding harmonic-oscillator ground states. We see that there is a substantial shift in the peaks of the Airy functions between the two Rabi frequencies shown, whilst the change in the location of the oscillator ground state, $z_0$, is very small. This shows how the overlap integral Eq. (46) can be sensitive to parameters. The wave functions have been scaled so that they reach a value of unity at the maximum height.

Figure 6

Ground-state decay rate contours for different models as a function of Rabi frequency and field gradient. The contours show where the scaled decay rate ${\mathrm{\Gamma }}_0/{\omega }_z={10}^{-3}$, i.e., where decay takes place after about 160 oscillations in the trap. The decay rates generally increase towards the top left of the figure. The solid contour line ($v$) shows the result from the vertical trapping model, as given by Eq. (47). The dotted contour line shows the effect of including the nonadiabatic potentials from ${\widehat{V}}_B$ in the vertical trapping model case (see Appendix pp1-s2). The dashed contour line corresponds to the ground-state decay rate from the horizontal trapping model ($h$), as given by Eq. (32). It is a suitable approximation to the boundary displayed by the vertical trapping model. The effect of including the nonadiabatic potentials is not shown for the horizontal trapping model as there is no visible difference from the dashed contour line. The chained contour line corresponds to the Landau-Zener decay rate, Eq. (51), with a trap frequency calculated using Eq. (39). The results presented in this figure are for ${}^{87}\mathrm{Rb}$ and $F=1$.