Second-order response theory of radio-frequency spectroscopy for cold atoms

We present a theoretical description of the radio-frequency (rf) spectroscopy of fermionic atomic gases, based on the second-order response theory at finite temperature. This approach takes into account the energy resolution due to the envelope of the rf pulse. For a noninteracting final state, the momentum- and energy-resolved rf intensity depends on the fermion spectral function and pulse envelope. The contributions due to interactions in the final state can be classified by means of diagrams. Using this formalism, as well as the local density approximation in two and three dimensions, we study the interplay of inhomogeneities and Hartree energy in forming the line shape of the rf signal. We show that the effects of inhomogeneities can be minimized by taking advantage of interactions in the final state, and we discuss the most relevant final-state effects at low temperature and density, in particular the effect of a finite lifetime.

Figure 1

Rf spectroscopy of ultracold atoms. The hyperfine atomic levels $|1\rangle$, $|2\rangle$, and $|3\rangle$ are split by the Zeeman effect, such that a rf transition is allowed between states $|2\rangle$ and $|3\rangle$. The tunable many-body interaction between atoms of the cloud in states $|1\rangle$ and $|2\rangle$ shifts and broadens the level $|2\rangle$. The effects of this interaction are probed by comparing the frequency $\nu$ of the rf transition with the frequency ${\nu }_0$ of the noninteracting transition. Spin relaxation from $|2\rangle$ to $|1\rangle$ is forbidden by conservation of spin, meaning that the energy splitting between $|1\rangle$ and $|2\rangle$ is irrelevant. Levels $|1\rangle$ and $|3\rangle$ may be renormalized by interactions as well.

Figure 2

Diagrammatic representation of the double-imaginary time function (11). Diagrams (R) and (L) correspond to the first and second terms of Eq. (13), respectively, up to a minus sign (the correlation function equals minus the diagram, if standard diagrammatic rules are used). The two diagrams are identical, except for the direction of the arrows (R, right-handed, L, left-handed). The white dots denote the interaction of atoms with light, leading to transitions between internal states $|2\rangle$ and $|3\rangle$; the black dots denote the measured momentum distribution, and the hatched regions represent all interactions, including interactions with the state $|1\rangle$.

Figure 3

Leading term (I) and an example of vertex correction (II). In both cases, the two topologically inequivalent diagrams of kinds R and L must be considered. For a noninteracting final state, diagrams of type ${\mathrm{I}}^{\prime }$ are the only nonvanishing contributions. Double lines denote the Green's function, and single lines denote the noninteracting Green's function in the final state.

Figure 4

Resolution function for a rf pulse with a square (dashed line) and Gaussian (solid line) envelope in the case of a noninteracting final state. The width of the envelope in the time domain is fixed by $\mathrm{\Delta }{\nu }^{-1}$ in both cases (full width at half maximum in the Gaussian case). The curves are normalized to the peak maximum for easier comparison.

Figure 5

Energy-distribution curves for harmonically trapped ${}^{40}\mathrm{K}$ atoms in two dimensions: effect of the Hartree shift. The EDC (39) for a Gaussian pulse with $\mathrm{\Delta }\nu =5$ kHz are normalized and shown at different momenta (thick lines), for attractive (left) and repulsive (right) interaction. The thin lines show the intrinsic EDC corresponding to each curve $[\mathrm{\Delta }\nu =0$, Eq. (40)], divided by two for clarity. The other parameters are ${\omega }_r/2\pi =127$ Hz, $N_1=2000$, and $T=100$ nK.

Figure 6

Convergence of the measured EDC towards the intrinsic EDC (thin line) with improving the resolution by increasing the duration $\mathrm{\Delta }{\nu }^{-1}$ of the Gaussian pulse. The EDCs are shown for $g=-0.35$ and $k=6\mu {\text{m}}^{-1}$, with the same parameters as in Fig. 5.

Figure 7

(a) Radial density, (b) dispersion of the EDC maximum as a function of momentum, (c) Hartree “effective mass” as a function of $g$, and (d) width of the EDC. In all graphs, dotted lines correspond to $T=150$ nK, dashed lines to $T=100$ nK, thick solid lines to $T=50$ nK, and the thin solid lines give the analytical result for $T=0$ (see text). In (d), the width is shown as a function of $g$ for three momenta; the colors correspond to those used in Fig. 5. The model parameters are ${\omega }_r/2\pi =127$ Hz, $N_1=2000$, and $\mathrm{\Delta }\nu =5$ kHz. The minimum at $g=0$ in (d) corresponds to the resolution of 3.12 kHz.

Figure 8

(a) Evolution of the Hartree “effective mass” with ${\omega }_r$ for attractive (top curves) and repulsive (bottom curves) interaction and for three different temperatures. (b) Hartree “effective mass” as a function of $g$ for a $r^4$ trap and three temperatures. The thin solid line shows $1/(1+g/2)$ for easier comparison with Fig. 7. (Inset) Radial density for two values of $g$ and the same three temperatures.

Figure 9

(a) The function $\mathcal{I}(\alpha ,x)$ defined in Eq. (45) for various values of $\alpha$. Note that the cutoff is at $x\approx \alpha$. (b) Data of Ref. [24] (dots) and fits to Eq. (44) with $\mathrm{\Delta }\nu$ fixed to 7.1 kHz (lines). A constant background was added to Eq. (44) for fitting.

Figure 10

(a) Energy dependence of the resolution function (48) for a final-state scattering rate ${\mathrm{\Gamma }}_3=h\mathrm{\Delta }\nu$, corresponding to a lifetime ${\tau }_3=\mathrm{\Delta }{\nu }^{-1}/\left(4\pi \right)$, and for increasing measurement times (broader to narrower). The time $t$ is measured from the maximum of the Gaussian pulse envelope, such that $t=\mathrm{\Delta }{\nu }^{-1}$ corresponds to a measurement time one full width after the pulse maximum. (b) Full width at half maximum of the resolution function relative to the Fourier limited value and (c) maximum intensity as a function of the measurement time and scattering rate. The arrows pointing to the left in (c) indicate the time ${\tau }_3$, and those pointing to the right the time $\frac{1}{16ln2{\tau }_3\mathrm{\Delta }\nu }\mathrm{\Delta }{\nu }^{-1}$.

Figure 11

Dominant vertex corrections in the low-temperature low-density limit. The shaded boxes represent a particle-particle ladder series (pseudopotential). $1p$ and $1h$ stand for a particle or a hole in state $|1\rangle$, respectively, and similarly for the other states. The diagrams give a significant contribution only if the time ordering of the various vertices is such that all lines marked as particles go to the right (see Appendix pp3). The diagrams shown are right handed; there are two equivalent left-handed terms.

Figure 12

Contributions to the right-handed (R) and left-handed (L) diagrams of zeroth order in the interaction. The vertices are ordered horizontally by increasing imaginary time from left to right. $2p$ and $3p$ indicate particle lines in states $|2\rangle$ and $|3\rangle$, respectively, while $2h$ and $3h$ indicate hole lines.

Figure 13

First-order right-handed (R) and left-handed (L) contributions to the $\circ$-vertex corrections, which survive in the limit $k_{\text{B}}T\ll h{\nu }_0$. The imaginary times $\tau -{\tau }^{\prime }$, 0, ${\tau }^{\prime \prime }$, and $\tau$ are ordered horizontally as in Fig. 12. Any modification in the ordering of times produces at least one factor $e^{-\beta h{\nu }_0}$.