Beyond the Spin Model Approximation for Ramsey Spectroscopy

Ramsey spectroscopy has become a powerful technique for probing nonequilibrium dynamics of internal (pseudospin) degrees of freedom of interacting systems. In many theoretical treatments, the key to understanding the dynamics has been to assume the external (motional) degrees of freedom are decoupled from the pseudospin degrees of freedom. Determining the validity of this approximation—known as the spin model approximation—has not been addressed in detail. Here we shed light in this direction by calculating Ramsey dynamics exactly for two interacting spin-1/2 particles in a harmonic trap. We focus on $s$-wave-interacting fermions in quasi one- and two-dimensional geometries. We find that in one dimension the spin model assumption works well over a wide range of experimentally relevant conditions, but can fail at time scales longer than those set by the mean interaction energy. Surprisingly, in two dimensions a modified version of the spin model is exact to first order in the interaction strength. This analysis is important for a correct interpretation of Ramsey spectroscopy and has broad applications ranging from precision measurements to quantum information and to fundamental probes of many-body systems.

Figure 1

(a) Ramsey spectroscopy of two interacting spin-1/2 particles. (b) In a harmonic trap the spectrum degeneracy allows near-resonant mode-changing collisions coupled to the spin dynamics.

Figure 2

Ramsey dynamics [see Eq. (4)] with $\delta =0$: (a) 1D spin model, exact solution, and projection of population onto initial mode (here $n_1=10$ and $n_2=0$), with $u_{\uparrow \downarrow }^{1,0}\approx 0.2{\omega }_z$. Dephasing of the exact dynamics results from mode changes. (b) Thermal averages in two dimensions: spin model vs effective spin model, at different temperatures, with $u_{\uparrow \downarrow }^{1,0}\approx 0.04{\omega }_{\perp }$. For both figures: ${\theta }_1={\theta }_2=\pi /3$, with thermally averaged inhomogeneity $⟨\mathrm{\Delta }\mathrm{\Omega }⟩/⟨\mathrm{\Omega }⟩=0.3$. ${\theta }_i=\mathrm{\Omega }{t}_i$ are bare pulse areas.

Figure 3

Thermally averaged frequency shifts: (a) 1D spin model and exact solution vs population excitation fraction (number of atoms in $\uparrow$ divided by the total number of atoms) after the first pulse, at intermediate and long times. Here, $u_{\uparrow \downarrow }\equiv u_{\uparrow \downarrow }^{1,0}\approx 0.2{\omega }_z$. (b) Frequency shifts for 2D spin model vs effective spin model, with $u_{\uparrow \downarrow }\equiv u_{\uparrow \downarrow }^{1,0}\approx 0.04{\omega }_{\perp }$. For both figures: $T=208\hslash {\omega }_{\perp }/k_B$, $\theta ={\theta }_1={\theta }_2$, ${\omega }_z={\omega }_{\perp }=700\times 2\pi \mathrm{\text{Hz}}$, and thermally averaged inhomogeneity $⟨\mathrm{\Delta }\mathrm{\Omega }⟩/⟨\mathrm{\Omega }⟩=0.3$. ${\theta }_i=\mathrm{\Omega }{t}_i$ are bare pulse areas.