Origin of the excess phase noise on Ramsey recurrences in a Bose–Einstein condensate interferometer with state-dependent microwave potentials on an atom chip

The 2009 atom-chip experiment of Böhi et al. demonstrated a fully coherent trapped-atom Ramsey interferometer for a Bose–Einstein condensate in state-dependent microwave near-field potentials, with interference contrast reviving at each recurrence of the state-selective splitting. On the second and later recurrences, however, the Ramsey phase carried noise about three times larger than all measured sources (magnetic-field, microwave-power, and atom-number fluctuations plus quantum projection noise together account for only about 30% of the observed level); the authors hypothesized spin squeezing generated by state-dependent collisional dynamics. Follow-up work confirmed that this collisional nonlinearity is present and controllable — spin squeezing and multi-particle entanglement were generated on the same chip (Riedel et al., 2010) — and identified loss-induced collisional phase diffusion as the fundamental coherence limit of the unsplit two-component condensate (Li et al., 2020); but no published work quantitatively accounts for the excess noise on the split recurrences themselves. The open problem is to determine quantitatively the mechanism or mechanisms responsible for this excess phase noise — in particular whether state-dependent collisional dynamics (one-axis-twisting nonlinearity, spin squeezing, multimode dephasing) account for it.

Background

An atom chip is a microfabricated device whose current-carrying wires and microwave waveguides trap and manipulate ultracold atoms at distances of tens of micrometres from a surface, providing compact and robust control for clocks, interferometers, and quantum-information experiments. A Bose–Einstein condensate (BEC) is a dilute gas cooled so far that all atoms occupy a single quantum state, so that its collective dynamics are governed by wave mechanics and by collisions between the atoms. For entanglement-based technologies one needs a potential that depends on the internal (hyperfine) state of the atom, so that the two components of an atomic superposition can be moved independently while collisions between them remain controllable.

Böhi et al. (Nature Physics 5, 592 (2009)) created such state-dependent potentials with microwave near-fields radiated by an on-chip coplanar waveguide (CPW), a planar microwave transmission line. A microwave field slightly detuned from a hyperfine transition dresses the atomic states — that is, it mixes them into new energy eigenstates whose energies are shifted by an amount depending on the microwave field amplitude at the atom’s position and on the internal state — so the potential landscape differs from one internal state to the other. The qubit states were $|0\rangle = |F=1, m_F=-1\rangle$ and $|1\rangle = |F=2, m_F=+1\rangle$ of $^{87}$Rb, a pair with nearly identical magnetic moments, in a static magnetic microtrap ($N \approx 400$ atoms, trap frequencies 109 Hz axial, 500 Hz radial, bias field 3.23 G).

The experiment is a Ramsey interferometer: two resonant $\pi/2$ pulses separated by a delay $T_R$ convert the accumulated relative phase between $|0\rangle$ and $|1\rangle$ into a measurable population difference. Between the pulses, the CPW microwave was pulsed on, displacing the trap minimum for the dressed partner of $|0\rangle$ and setting its wavefunction oscillating (8.5 $\mu$m peak-to-peak at 116 Hz) while $|1\rangle$ stayed at rest. Whenever the two wavefunctions return to full overlap — at multiples of 8.6 ms — the Ramsey contrast revives; these revivals are called recurrences. The high contrast of the first recurrence proved that the joint evolution of internal and motional states is coherent; residual wavefunction distortion from the non-adiabatic switching of the potentials can in principle be suppressed by optimal control, in which the switching ramps are designed numerically (rather than chosen as simple ramps) so that the wavepacket stays close to the trap ground state.

For the second and later recurrences, however, the Ramsey fringe phase was noisy far beyond the known sources: measured magnetic-field fluctuations, microwave-power fluctuations, atom-number fluctuations, and quantum projection noise — the fundamental phase uncertainty $1/\sqrt{N}$ of $N$ uncorrelated atoms, about $0.02\pi$ here — together accounted for only about 30% of the observed noise, while without splitting the condensate comparable noise appeared only beyond several hundred milliseconds. The authors’ leading hypothesis was spin squeezing generated by the state-dependent collisional dynamics: when the two components have different, overlapping density distributions, elastic collisions create a nonlinear interaction of one-axis-twisting type (a term $\chi S_z^2$ in the collective-spin Hamiltonian, where $S_z$ measures the population imbalance), which correlates the atoms, squeezes one spin quadrature, and in a $\pi/2$-$\pi/2$ sequence manifests as increased phase noise. Mean-field shifts from the same collisions also cause multimode dephasing across the inhomogeneous condensate. Theory by Li, Treutlein, Reichel, and Sinatra (Eur. Phys. J. B 68, 365 (2009)) analyzed exactly this entangled spatial and spin dynamics in bimodal condensates, including particle losses. The same microwave-potential platform was the basis of a proposed two-atom quantum phase gate (Calarco et al., Phys. Rev. A 61, 022304 (2000); Treutlein et al., Phys. Rev. A 74, 022312 (2006)), in which two state-selective interferometers would collide their atoms to accumulate a conditional collisional phase shift.

Problem Statement

Identify, with a dedicated quantitative experimental test and theoretical modeling, the physical origin of the excess phase noise observed on the second and later Ramsey recurrences in the trapped-atom interferometer formed by a Bose–Einstein condensate in state-dependent microwave near-field potentials on an atom chip — the configuration of Böhi et al. (2009): $^{87}$Rb, $N \approx 400$ atoms, qubit states $|F=1, m_F=-1\rangle$ and $|F=2, m_F=+1\rangle$, a $\pi/2$–$\pi/2$ Ramsey sequence in which a pulsed on-chip microwave potential state-selectively splits the condensate and sets the dressed-state wavefunction oscillating (recurrences at multiples of 8.6 ms), where the measured technical fluctuations (magnetic field, microwave power, atom number) together with quantum projection noise account for only about 30% of the observed noise level. The answer must in particular decide whether the state-dependent collisional dynamics — the one-axis-twisting nonlinearity that generates spin squeezing, together with multimode mean-field dephasing and stochastic-loss-induced collisional phase diffusion — quantitatively account for the excess phase noise, or whether another mechanism (for example residual wavefunction distortion from the non-adiabatic splitting, or an uncharacterized technical noise source) dominates, attributing a quantitative fraction of the observed noise to each identified mechanism.

References

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