Ultracold atom interferometry with pulses of variable duration

We offer interferometry models for thermal ensembles with one-body losses and the phenomenological inclusion of perturbations covering most of the thermal atom experiments. A possible extension to the many-body case is briefly discussed. The Ramsey pulses are assumed to have variable durations and the detuning during the pulses is distinguished from the detuning during evolution. Consequently, the pulses are not restricted to resonant operation and give more flexibility to optimize the interferometer to particular experimental conditions. On this basis another model is devised in which the contrast loss due to the unequal one-body population decays is canceled by the application of a nonstandard splitting pulse. For the importance of its practical implications, an analogous spin-echo model is also provided. The developed models are suitable for the analysis of atomic clocks and a broad range of sensing applications; they are particularly useful for trapped-atom interferometers.

Figure 1

Ramsey interferometry with variable-duration pulses. (a) Pulses are generally non-$\pi /2$ with durations $T_{p1}$ and $T_{p2}$. (b) Spin-echo sequence with variable-duration pulses.

Figure 2

Ramsey interferometry in the Bloch vector representation. The fans of vectors illustrate phase diffusion. The black circular arrows show the rotation by the effective torque acted on the vector. A fringe in $P_z$ is produced by varying $T$. In the phase-Ramsey method the fringe is produced at a fixed $T$ by varying the phase of the second $\pi /2$ pulse. (a) Initially the atoms are in state $\left|1\right.〉$, state $\left|2\right.〉$ is unpopulated. (b) After the first $\pi /2$ pulse the effective torque brings the vector to the phase plane ($P_z=0$). (c) Without interrogation the system relaxes, undergoing population loss and phase destruction, resulting in phase diffusion. (d) The second $\pi /2$ pulse applies an effective torque, projecting the vectors onto the axis of population difference.

Figure 3

Equalizing the state $\left|1\right.〉$ and $\left|2\right.〉$ instantaneous population decays by a variable splitter pulse to enhance $P_z$. The model is evaluated for the following parameters: $\mathrm{\Delta }=0$ rad ${\mathrm{s}}^{-1}, T_{p2}=\pi /\left(2{\mathrm{\Omega }}_R\right), {\gamma }_1=1.5{\mathrm{s}}^{-1}, {\gamma }_2=0.5{\mathrm{s}}^{-1}, {\gamma }_d=0{\mathrm{s}}^{-1}, \phi =2\pi \times 2$ rad, and $\mathrm{\Omega }=2\pi \times 1$ rad ${\mathrm{s}}^{-1}$. ${\rho }_{11}$ (blue) and ${\rho }_{22}$ (red) are from Eqs. (13), $P_z$ (dashed) from Eq. (10). (a) $T_{p1}=\pi /2$: equally split populations ${\rho }_{nn}$ at the end of free evolution, at $T$; (b) $T_{p1}=\pi /3$: unequal splitting to produce a crossing of the population decays before the arrival of the second pulse; (c) $T_{p1}=\pi /2$: after the complete sequence, $P_z$ has a monotonic $sech\left[({\gamma }_1-{\gamma }_2)T/2\right]$ envelope at ${\gamma }_1\ne {\gamma }_2$ according to Eq. (7c); (d) $T_{p1}=\pi /3$: after the complete sequence the peak of visibility is observed at $T=T_{\text{optimal}}\approx 1.1$ s as expected from Eq. (14).

Figure 4

Ramsey spectra. The following values are used: $\mathrm{\Omega }=2\pi \times \left(510\text{Hz}\right), T=5$ ms. The measurable $P_z(\mathrm{\Delta },T)$ with the Ramsey condition $\phi =\mathrm{\Delta }$ [the one-body model of Eqs. (5)] is plotted in panel (a). Losses are neglected. The Rabi pedestal in panel (a) has a narrow line shape at $T=0$ and has a single-peak coming from Eq. (10), where the pulses are assumed to be $\pi /2$ for all $\mathrm{\Delta }$. Whereas in panel (c), corresponding to the model of Eqs. (5), the pulses are $\pi /2$ only at resonance; at $\mathrm{\Delta }\ne 0$ they become non-$\pi /2$ pulses distorting the spectrum. Panels (b) and (d) are differences $P_z-{\mathcal{I}}_z$ of panels (a) and (c), correspondingly.

Figure 5

Spin-echo interferometry in the Bloch vector representation. $\varphi$ is the phase of the second $\pi /2$ pulse. As in Ramsey interferometry (Fig. 2), $\varphi$ modulation can be used at a fixed $T$ to record a $P_z$ fringe. Insets (a), (b), and (c) show dynamics identical to that of Figs. 2, 2, and 2. In panel (d) the spin-echo pulse implements phase reversal, and the ensemble starts rephasing. (e) After the second $\pi /2$ pulse the Bloch vectors are refocused. (f) The ensemble spin spread is nullified.