Mach-Zehnder interferometer for in-situ characterization of atom traps

1. Problem Statement

The precise control and characterization of trapping potentials are critical for the performance of quantum simulators, quantum sensors, and atom interferometers. While techniques exist to measure trap frequencies (e.g., ballistic excitation, parametric heating), they often require:

• Displacing the trap or loading atoms off-center.
• Transferring momentum to the atoms.
• Adiabatic deformation or switching off the trap.
• Long observation times.

Furthermore, accurately characterizing weak anharmonicities (cubic and quartic deviations from a perfect harmonic potential) in optical dipole traps remains challenging. Existing methods struggle to provide high-precision, in-situ measurements of both the fundamental trap frequency and the magnitude of these anharmonic terms without perturbing the system significantly.

2. Methodology

The authors propose a novel in-trap Mach-Zehnder Interferometer (MZI) scheme that utilizes two internal atomic states with different trapping potentials.

• Core Concept: The scheme employs an atom with two internal states, $|1\rangle$ and $|2\rangle$, which experience different trapping potentials $V_1(z)$ and $V_2(z)$. These potentials differ either in their trap frequencies ($\omega_1 \neq \omega_2$) or in the constant forces acting on them ($F_1 \neq F_2$), causing their potential minima to be spatially separated.
• Interferometer Sequence:
  1. Preparation: An atomic cloud is prepared at rest in state $|2\rangle$ at the minimum of $V_2$.
  2. Splitting ($t=0$): A short light pulse (approximated as instantaneous) transfers a portion of the population to state $|1\rangle$. This creates a second wavepacket that begins to oscillate in $V_1$, while the original wavepacket remains in $V_2$.
  3. Evolution ($0 < t < T$): The two wavepackets evolve independently. Due to the different potentials, they undergo different center-of-mass (CM) motions and "breathing" (width oscillations).
  4. Recombination ($t=T$): A $\pi$-pulse swaps the populations. The wavepacket in $|1\rangle$ moves to $V_2$, and the one in $|2\rangle$ moves to $V_1$.
  5. Final Evolution ($T < t < 2T$): The wavepackets continue evolving for another time $T$.
  6. Readout ($t=2T$): A final $\pi/2$-pulse mixes the states, and the population of state $|2\rangle$ is measured.
• Signal Analysis: The population $P_2(T)$ follows an interference pattern:
  $$P_2(T) = \frac{1}{2} [1 + C(T) \cos(\phi(T) + \Delta\Phi_P)]$$
  where $C(T)$ is the contrast and $\phi(T)$ is the phase. The signal exhibits sharp peaks (or zero-crossings with high slope) when the wavepackets perfectly overlap in position and width at $t=2T$. This occurs when the evolution time $T$ matches integer multiples of the trap periods ($T = n \cdot 2\pi/\omega$).

3. Key Contributions

• Non-Destructive Characterization: The method requires no momentum transfer, trap switching, or displacement of the trap center. It operates entirely in-situ within the trap.
• High-Precision Frequency Extraction: By analyzing the slope of the signal near the resonance points (where wavepackets overlap), the trap frequency can be determined with extremely high precision.
• Anharmonicity Bounds: The technique allows for the estimation of upper bounds on cubic ($\alpha$) and quartic ($\beta$) anharmonic terms by observing how the measured frequencies shift as the trap separation (controlled by an external force) changes.
• 3D Generalization: The authors extend the 1D model to 3D, addressing the issue of transversal breathing modes that can degrade contrast. They propose a robust method to isolate the longitudinal frequency by varying the applied force and identifying signal crossings.

4. Results

• Simulation Parameters: The study uses $^{87}\text{Rb}$ atoms in a crossed optical dipole trap (1064 nm, 1 W power, 100 $\mu$m waist).
• Frequency Precision:
  • For a trap frequency of $\omega \approx 2\pi \times 100$ Hz and an interrogation time of $2T = 100$ ms, the method achieves a relative uncertainty of $\Delta\omega/\omega \approx 4.3 \times 10^{-6}$.
  • This is an order of magnitude better than previous ballistic techniques (which achieved $\sim 7.8 \times 10^{-5}$).
• Anharmonicity Bounds:
  • By extrapolating the frequency dependence on the trap separation distance $|z_{0,1} - z_{0,2}|$ to zero separation, the authors determined upper bounds for the anharmonic coefficients in the optical dipole potential:
    • Cubic term: $|\alpha| < 4.63 \times 10^{-16} \text{ J}\cdot\text{m}^{-3}$
    • Quartic term: $|\beta| < 5.63 \times 10^{-12} \text{ J}\cdot\text{m}^{-4}$
  • These values are consistent with theoretical Taylor expansions of the dipole potential.
• Robustness: The method remains accurate even in the presence of position-dependent detuning caused by large forces, provided the pulse parameters are optimized or the specific resonance peaks (quasi-Gaussian) are identified.

5. Significance

• Enhanced Sensor Performance: High-precision characterization of trap potentials directly improves the sensitivity of atom interferometers used for measuring gravity, rotation, and fundamental constants.
• Quantum Simulation: Accurate knowledge of the potential landscape is essential for realizing complex quantum simulators and matter-wave circuits (atomtronics).
• Thermodynamics: The technique enables the determination of thermodynamic properties of trapped Bose-Einstein Condensates (BECs) by precisely mapping the trap geometry.
• Scalability: The method is applicable to various trapping geometries (magnetic, optical dipole, atom chips) and does not require complex hardware modifications, making it a versatile tool for next-generation quantum devices.

In conclusion, this paper presents a powerful, non-invasive interferometric technique that significantly advances the metrology of atom traps, offering a path to ultra-precise quantum sensing and simulation.