Geometric Ramsey Interferometry

• Geometric Ramsey Interferometry is a quantum metrological technique that leverages adiabatic evolution and dark-state manipulation in multilevel systems to acquire nontrivial geometric phases.
• It generalizes conventional Ramsey protocols by integrating geometric phase contributions, enhancing error resilience and offering versatile control in both ultracold atomic and qudit implementations.
• The approach facilitates holonomic quantum gate operations and improved phase readout, evidenced by high fringe contrast and robust performance across varied experimental settings.

Geometric Ramsey interferometry is a quantum metrological technique in which the accumulated phase is of geometric origin, rather than exclusively dynamical, and where quantum state manipulation is realized through adiabatic evolution in multilevel systems. This approach generalizes conventional Ramsey interferometry, traditionally performed with resonant electromagnetic pulses connecting two quantum states, by introducing nontrivial geometric phase contributions in dark-state or higher-dimensional subspaces. Recent implementation in tripod schemes with ultracold strontium atoms, as well as in qudit systems using single nuclear spins, has established geometric Ramsey interferometry as robust, versatile, and sensitive to purely geometrical features of quantum evolution (Madasu et al., 2023, Godfrin et al., 2018).

1. Bare-State Hamiltonians and Dark-State Manifolds

The geometric Ramsey protocol in tripod schemes begins with three long-lived ground states $|1\rangle$, $|2\rangle$, $|3\rangle$ resonantly coupled to a common excited state $|e\rangle$ via three coherent laser fields, each characterized by a complex Rabi frequency $\Omega_a(\mathbf{r},t)$ ($a=1,2,3$). The bare-state Hamiltonian in the interaction picture is

$$H_{\text{bare}} = \frac{1}{2} \left[ \Omega_1 |e\rangle\langle 1| + \Omega_2 |e\rangle\langle 2| + \Omega_3 |e\rangle\langle 3| + \text{h.c.} \right].$$

Diagonalization yields two bright states (with energy gaps $\pm\Omega/2$; $$\Omega = \sqrt{|\Omega_1|^2 + |\Omega_2|^2 + |\Omega_3|^2}$$) and two degenerate dark states at zero eigenvalue. In adiabatic conditions, population remains confined to the dark-state manifold, thereby protecting against radiative decay.

For generalized Ramsey protocols with qudits (e.g., a single $I=3/2$ nuclear spin), the elementary basis states $|m\rangle$ ($m=-3/2, -1/2, +1/2, +3/2$) are manipulated with transition-selective pulses, and level splitting is engineered via the system's hyperfine and quadrupolar interactions (Godfrin et al., 2018). Multichromatic or monochromatic pulse sequences effect rotations in chosen subspaces but typically preserve nontrivial geometric connectivity in the projective Hilbert space.

2. Effective Hamiltonians and Geometric Phases

By projecting the full Hamiltonian onto the dark-state subspace, the effective evolution is governed by a $2\times2$ gauge Hamiltonian: $$H_{\mathrm{eff}} = \frac{(\hat{p}-\hat{A})^2}{2m} + \hat{Q} + \hat{w},$$ where:

• $$\hat{A}_{\mu\nu} (\mathbf{r}, t) = i \hbar \langle D_\mu | \nabla D_\nu \rangle$$ is the non-Abelian vector (Berry) connection,
• $$\hat{Q}_{\mu\nu} (\mathbf{r}, t) = (\hbar^2/2m) \langle \nabla D_\mu | \nabla D_\nu \rangle$$ is the geometric scalar potential,
• $$\hat{w}_{\mu\nu} (\mathbf{r}, t) = -i\hbar \langle D_\mu | \partial_t D_\nu \rangle$$ is the time-dependent gauge potential.

In the tripod scheme, during the free-evolution interval (lasers off), $\hat{w} \to 0$ and $\mathbf{\Omega}_a \to 0$, but the adiabatic connection pins the system at $(\theta, \phi) \to (\pi/2, \pi/2)$. The geometric scalar term remains nonzero, producing a relative energy shift $\Delta E_{\text{geo}} = 2p_r^2/m$, where $p_r = \hbar k$ is the photon recoil, and resulting in a geometric phase

$$\phi_{\mathrm{geo}} = \frac{(Q_{22} - Q_{11}) T}{\hbar} = + \frac{2 p_r^2 T}{\hbar m}.$$

This phase is expressible as an integral of the Berry connection around the parameter-space loop traced by the sequence: $$\phi_{\text{geo}} = \oint_C \mathrm{Tr}[A \cdot dR] = \iint_S F_{\theta\phi} d\theta\, d\phi, \quad F = dA + A \wedge A.$$

In multilevel systems, the geometric phase for a cyclic evolution is $\gamma_G = i \oint_C \langle \psi | d\psi \rangle$, which for a two-level Bloch-sphere path reduces to half the solid angle subtended ($\gamma_G = (\Omega/2)$). For higher-spin subspaces, the accrued geometric phase is proportional to the spin difference.

3. Ramsey Protocols and Phase Readout

In geometric Ramsey interferometry with tripod schemes, the basic protocol follows a $\pi/2$ pulse (beam splitter), followed by a free-evolution time $T$ (no laser fields), and a $-\pi/2$ pulse for recombination. The output port’s population exhibits Ramsey fringes: $$P = \frac{1}{2} [ 1 + \cos(\phi_{\mathrm{dyn}} + \phi_{\mathrm{geo}}) ],$$ where $\phi_{\mathrm{dyn}}$ (dynamical phase) accrues during pulses, and $\phi_{\mathrm{geo}}$ (geometric phase) accrues during free evolution. Symmetric pulse design and beam-off conditions are used to isolate the geometric phase experimentally.

In qudit-based protocols, a variety of Ramsey-type experiments are generalized:

• Single-transition Ramsey: Measures geometric phases via phase sweeps over two-level subspaces; the fringe period reveals the phase accumulated.
• Double-transition Ramsey: Used to measure two-qubit gate phases (e.g., iSWAP); involves shelving and unshelving population to enable multi-arm interference and to extract gate phase offsets.
• Double-Hadamard Ramsey: Assesses multistate coherence in an equal-amplitude superposition among three levels, with visibility decay revealing coherence times.

4. Experimental Realizations

The tripod scheme has been demonstrated in a degenerate Fermi gas of $^{87}$Sr atoms ($N \simeq 4.5 \times 10^4$, $T \simeq 50$ nK), using the $689$ nm intercombination line to couple ground states differentiated by $m_F$ quantum number to a common excited state. Three laser beams impart controlled momentum kicks, establishing phase connections. Pulse shapes are Gaussian ($\sigma_t = 2.5\, \mu$s), with free-evolution times up to $10\, \mu$s. Detection employs time-of-flight fluorescence imaging, mapping populations in $|3\rangle, |1\rangle, |2\rangle$ to distinct momentum peaks. Observed Ramsey-fringe contrast reaches $\approx 0.8$, with decay time $\tau \approx 23\, \mu$s (Madasu et al., 2023).

For qudit experiments, a single TbPc$_2$ molecular magnet is placed in a microtransistor, operated at $40$ mK. Nuclear spin transitions are driven by a microwave antenna and detected via conductance jumps in a spin-coupled readout dot. Pulse calibration involves Rabi oscillation measurements and field sweeps. Multiple protocols probe geometric phases (fringe slopes $= 1$ for spin-$\frac{1}{2}$, $=2$ for spin-$1$ paths), gate-phase offsets ($\phi_\text{gate} = \pi/2$, swap fidelity $\approx 93\%$), and coherence times ($T_2^3 \sim 90\, \mu$s).

5. Robustness, Multi-Port Operation, and Outlook

A defining feature of geometric Ramsey interferometry in tripod systems is robustness against pulse-timing errors: the output is governed by geometrical properties of the parameter-space trajectory, not temporal pulse details. Figure 1 of (Madasu et al., 2023) establishes that for a pulse width $3\, \mu$s $< \sigma_t < 15\, \mu$s, the dark-state splitting angle $\theta$ remains near $\pi/2$, demonstrating geometric insensitivity.

Multiple input and output channels are accessible: final readout in the tripod scheme can be selected via phase jumps or free-evolution time, allowing for bilateral atomtronic switches. More generally, extension to Ramsey–Bordé or Mach–Zehnder–type sequences in the same subspace enables large-area, recoil-sensitive interferometry, harnessing geometric phase for improved robustness and signal discrimination.

In qudit schemes, Ramsey interferometry naturally extends to arbitrary Hilbert-space dimensions, permitting direct access to geometric phases, gate benchmarking, and coherence in multiplet superpositions. Measured geometric phases in higher spin subspaces enable holonomic quantum gate operation and provide new pathways for error-protected encodings and quantum sensing in molecular systems (Godfrin et al., 2018).

6. Mathematical Summary: Core Equations

Concept	Equation(s)	Regime/Interpretation
Tripod Hamiltonian	$$H = \frac{\hbar}{2} \sum_a \Omega_a |e\rangle\langle a| + \text{h.c.}$$	Light-matter interaction (bare basis)
Dark state projector	$$\hat{P}_D = \sum_{\mu=1,2} |D_\mu\rangle\langle D_\mu|$$	Adiabatic subspace restriction
Berry connection	$A_{ij} = i \langle D_i | \nabla D_j \rangle$	Non-Abelian geometric phase accumulation
Geometric phase (Ramsey)	$$\phi_{\text{geo}} = \oint_C \mathrm{Tr}[A \cdot dR]$$	Closed path in parameter space ($\theta$, $\phi$)
Qudit geometric phase	$\gamma_G = i \oint_C \langle \psi | d\psi \rangle$	General cyclic evolution in Hilbert space

7. Significance and Prospects

Geometric Ramsey interferometry realizes interferometers whose phase sensitivity is inherently geometric, not reliant on poorly controlled dynamical contributions. This endows the protocol with robustness to certain operational errors, multi-port versatility, and suitability for quantum metrology and information processing where geometric phases can implement holonomic gates or encode error protection. The demonstrated applications in ultracold atomic gases and single-molecule magnets illustrate both the fundamental principles and the feasibility of geometric interferometry in high-coherence, multilevel quantum systems (Madasu et al., 2023, Godfrin et al., 2018). A plausible implication is that further development of geometrical control protocols could enhance decoherence resilience and support scalable architectures for quantum technologies beyond the qubit paradigm.