🔬 atomic physics

Multipassage Landau-Zener tunneling oscillations in transverse/longitudinal dual dressing of atomic qubits

This paper investigates multipassage Landau-Zener tunneling oscillations in rubidium and caesium atomic qubits subjected to a transverse/longitudinal dual-dressing configuration, revealing new coherence features and developing a specialized perturbation treatment to describe the system's adiabatic and non-adiabatic evolution outside the standard high-frequency Floquet paradigm.

Original authors: Alessandro Fregosi, Carmela Marinelli, Carlo Gabbanini, Valerio Biancalana, Maria Allegrini, Ennio Arimondo, Francesco Petiziol, Sandro Wimberger, Andrea Fioretti, Giuseppe Bevilacqua

Published 2026-09-09

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Alessandro Fregosi, Carmela Marinelli, Carlo Gabbanini, Valerio Biancalana, Maria Allegrini, Ennio Arimondo, Francesco Petiziol, Sandro Wimberger, Andrea Fioretti, Giuseppe Bevilacqua

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of atoms, scientists often treat particles like tiny magnets that can point in different directions, a property known as spin. These spins can be manipulated using magnetic fields, much like turning a dial to change a radio station. When physicists apply a steady magnetic field, the spins begin to wobble at a specific, predictable rate, similar to a spinning top precessing under gravity. This behavior is the foundation for technologies like atomic clocks and quantum sensors. However, when scientists add a second, oscillating magnetic field that changes rapidly, the rules become more complex. The interaction between the steady field and the shaking field can create a "dressed" state, where the atom's behavior is fundamentally altered by the surrounding electromagnetic environment. For decades, researchers have used a standard mathematical framework to predict how these atoms behave when the shaking happens very quickly. This framework works well when the shaking is fast compared to the atom's natural wobble, but it breaks down when the shaking is slow yet still very strong. Understanding this specific, difficult regime is crucial because it reveals new ways to control quantum systems, potentially leading to faster and more precise methods for processing information.

A team of researchers has now explored this tricky territory by studying rubidium and cesium atoms trapped in a laboratory setting. They set up an experiment where the atoms were subjected to two distinct magnetic fields: one that oscillated side-to-side and another that oscillated up-and-down, all while a steady magnetic field held them in place. The strength of these shaking fields was immense, with the atoms experiencing forces up to seven times stronger than their natural wobble. In this environment, the standard rules of physics that usually apply to fast shaking did not work. Instead of following the predictable patterns expected by conventional theory, the atoms exhibited a complex, rhythmic behavior that changed over time. The researchers measured the orientation of the atomic spins continuously, watching how they evolved over several milliseconds. They found that the spins did not just wobble at a single frequency; instead, they displayed a rich tapestry of oscillations, with the speed of the wobble speeding up and slowing down in a pattern that repeated itself.

The core of their discovery lies in how the atoms responded to the combination of these fields. When the shaking fields were weak, the atoms behaved as expected, maintaining a steady rhythm. But as the strength of the fields increased, the atoms entered a regime where the standard mathematical models failed to describe their motion. The researchers observed that the atoms underwent a series of rapid transitions, tunneling back and forth between different energy states multiple times. This process, known as multipassage tunneling, created interference patterns that showed up as distinct peaks and valleys in the spin's movement. Crucially, the team found that the atoms' behavior was not just a simple repetition of these tunnels. The speed at which the spins wobbled changed dynamically, speeding up and slowing down in sync with the oscillating fields. This variation was so significant that it created a new kind of rhythm, one that could not be explained by the old theories which assumed the shaking was either too fast to matter or too slow to cause such dramatic effects.

To make sense of these surprising results, the scientists developed a new theoretical approach. Since the standard methods for analyzing fast-shaking systems were useless here, they created a custom mathematical treatment that accounted for the strong, slow shaking. They combined this new theory with powerful computer simulations to model the atoms' behavior. The simulations confirmed that the complex patterns seen in the experiment were real and could be predicted by their new equations. They found that the atoms' motion was governed by a combination of three different frequencies: the frequency of the shaking fields themselves, a modified wobble rate caused by the fields, and a slower, rhythmic interference pattern that emerged from the repeated tunneling. This interference pattern, which appeared as a slow modulation of the faster wobbles, was a key signature of the new regime. The researchers were able to measure these frequencies directly, showing that the atoms' behavior matched their new theoretical predictions with high precision.

The experiment was conducted using two different types of atoms to ensure the results were robust. In one setup, they used a cloud of rubidium atoms cooled to temperatures just a fraction of a degree above absolute zero, holding them in a magnetic trap. In the other, they used cesium atoms in a gas-filled glass cell. Despite the different conditions, both groups of atoms showed the same complex behavior. The researchers monitored the atoms by shining a laser beam through them and measuring how the light's polarization rotated, which provided a direct window into the spin's orientation. They recorded the data over thousands of cycles, capturing the subtle shifts in the atoms' motion. The data revealed that the atoms were not just passively reacting to the fields; they were actively engaging in a complex dance of energy exchange, with the strength of the fields dictating the tempo of the interaction.

One of the most striking findings was the presence of these "Rabi-like" oscillations, which acted as a slow beat underlying the faster wobbles. In the standard view of these systems, such slow beats are often ignored or considered a minor detail. However, in this study, they were a dominant feature, shaping the entire evolution of the quantum state. The researchers showed that these oscillations were not random noise but a structured interference effect, resulting from the atoms passing through the same energy barriers multiple times. This repeated passage created a phase relationship between the different paths the atoms could take, leading to constructive and destructive interference that modulated the spin's amplitude. The team's ability to track this phase directly, rather than just measuring the final state, provided a much clearer picture of the quantum dynamics at play.

The implications of this work extend beyond just understanding these specific atoms. The study demonstrates that the standard tools used to engineer quantum systems, which rely on the assumption of fast shaking, have limits. When the shaking is strong but slow, entirely new behaviors emerge that require different theoretical tools to understand. The researchers' new method for analyzing these systems opens the door to controlling quantum states in ways that were previously thought impossible. By manipulating the amplitude and phase of the oscillating fields, it may be possible to steer the atoms into specific states with high precision, offering new possibilities for quantum computing and sensing. The study also highlights the importance of continuous monitoring in quantum experiments. By watching the atoms evolve in real-time, the researchers could see details that would have been missed if they had only looked at the beginning and end of the process.

In the end, the paper presents a detailed map of a previously uncharted region of quantum physics. It shows that when atoms are subjected to strong, slow magnetic shaking, they do not simply follow the old rules. Instead, they enter a dynamic regime where multiple frequencies interact to create complex, time-varying patterns. The researchers successfully measured these patterns, developed a new theory to explain them, and confirmed their findings with computer simulations. This work does not just add a small detail to existing knowledge; it challenges the way scientists think about controlling quantum systems in strong fields. It suggests that there is a rich landscape of behaviors waiting to be explored, where the interplay between different frequencies can be harnessed to create new forms of quantum control. The ability to observe and predict these behaviors brings scientists one step closer to mastering the quantum world, turning complex interactions into useful tools for the future.

Technical Summary: Multipassage Landau-Zener Tunneling Oscillations in Transverse/Longitudinal Dual Dressing of Atomic Qubits

Problem and Context
This work investigates the time evolution of a non-resonant dressed-atom qubit subjected to a dual-dressing configuration involving two electromagnetic fields: one oscillating parallel (longitudinal) and the other orthogonal (transverse) to a static quantization magnetic field. While Floquet engineering is a established technique for analyzing quantum systems under periodic driving—typically relying on high-frequency expansions to derive effective Hamiltonians—this study addresses a regime where the driving frequency is significantly lower than the qubit's Larmor precession frequency ( $\omega \ll \omega_0$ ). In this low-frequency, strong-drive regime, standard Floquet high-frequency expansion (HFE) approaches are inapplicable. Furthermore, while previous studies on dual-dressing have explored adiabatic behaviors, this configuration allows for the exploration of non-adiabatic features and strong nonlinear responses that deviate from traditional micromotion patterns. The specific challenge lies in characterizing the multipassage Landau-Zener-Stückelberg-Majorana (LZSM) tunneling process when controlled by an additional dressing field that modulates both the amplitude and phase of the tunneling, particularly when the Rabi frequencies of the driving fields exceed the Larmor frequency by a factor of up to seven.

Methodology
The research combines experimental measurements with theoretical modeling and numerical simulations.

Experimental Setup: Experiments were conducted using two distinct atomic systems:
1. Rubidium-85 ( $^{85}\text{Rb}$ ): Cold atoms trapped in a magneto-optical trap (MOT), prepared in the $F_g=3$ hyperfine state, and monitored during a 10 ms free-fall.
2. Cesium-133 ( $^{133}\text{Cs}$ ): Atoms in a vapor cell with buffer gas, prepared in the $F_g=4$ hyperfine state.
  In both setups, atoms were spin-polarized along the x-axis using a circularly polarized pump laser. Subsequently, two radiofrequency fields (1–10 kHz range, 0–50 $\mu$ T amplitude) were applied along the x and z axes (XZ configuration) or y and z axes. The system was subjected to static magnetic fields in the $\mu$ T range.
- Detection: The time evolution of the atomic spin coherence was monitored via the Faraday effect. A probe beam propagating along the y-axis (for Rb) or x-axis (for Cs) measured the polarization rotation, corresponding to the expectation values $\langle \sigma_y(t) \rangle$ or $\langle \sigma_x(t) \rangle$ , respectively. A balanced polarimeter provided high-resolution detection of amplitude and phase.
Theoretical Framework:
- Hamiltonian: The system is modeled by a time-dependent Hamiltonian $H(t) = \frac{1}{2}\vec{h}(t) \cdot \vec{\sigma}$ , where the effective field $\vec{h}(t)$ includes the static Larmor term and the transverse/longitudinal oscillating components.
- Approaches: Due to the low-frequency, strong-drive nature ( $\omega \ll \omega_0, \Omega_x, \Omega_z$ $ω ≪ ω_{0}, Ω_{x}, Ω_{z}$ ), the authors employed:
  1. Numerical Simulations: Direct integration of the Schrödinger equation to capture exact dynamics.
  2. Adiabatic Perturbation Theory: Valid for weak dressing parameters, analyzing the time-periodic eigenvalues (dressed Larmor frequency $\Omega_{Ld}$ ).
  3. Modified Floquet Treatment: An ad-hoc perturbative expansion in the limit of large $\Omega_z/\omega$ , utilizing a time-dependent rotating frame where Fourier components involve Bessel functions $J_n(\Omega_z/\omega)$ . This yields an effective Floquet Hamiltonian and a kick operator.

Key Contributions

Extension of LZSM Interferometry: The study extends standard LZSM multipassage interferometry by introducing a longitudinal dressing field ( $\Omega_z$ ) that controls the tunneling process. This creates a "dual-dressing" scenario where the energy gap is periodically modulated in a complex manner, leading to multiple avoided crossings within a single driving period.
Direct Coherence Monitoring: Unlike traditional LZSM studies that focus on transition probabilities between eigenstates, this work directly measures the full wavefunction evolution, including the phase, by monitoring spin coherences. This reveals dynamics that are distinct from standard probability readouts.
Theoretical Adaptation for Low-Frequency Regimes: The authors developed a specific perturbative dressing treatment for the non-adiabatic, low-frequency regime where standard Floquet engineering fails. This includes an analytical solution for the spin's time evolution based on a modified Floquet approach.
Characterization of Multi-Frequency Oscillations: The work identifies and characterizes oscillations at multiple frequencies arising from different quantum contributions: the dressing frequency, the time-dependent dressed Larmor frequency, and "Rabi-like" oscillations associated with multi-passage interference.

Results

Complex Spin Dynamics: Experimental data revealed quasi-periodic spin precession with three distinct patterns: the fundamental dressing period, shorter-period structures reflecting large-amplitude coherence oscillations, and chirped frequency patterns.
High Rabi Ratios: Experiments achieved Rabi frequencies up to 7.4 times the Larmor frequency (and up to 10 in some cases), significantly exceeding previous continuous-wave strong-drive experiments (which typically reached ~4x).
Dressed Larmor Frequency ( $\Omega_{Ld}$ ): The time evolution of the spin coherence was found to be dominated by the instantaneous dressed Larmor frequency $\Omega_{Ld}(t) = \sqrt{\Omega_x^2 \cos^2(\omega t) + [\omega_{0z} + \Omega_z \cos(\omega t + \Phi_{0z})]^2}$ . The experimental extraction of the dressed frequency from zero-crossings of the polarimeter signal showed remarkable agreement with theoretical predictions without adjustable parameters.
Stückelberg and Rabi-like Oscillations:
- Stückelberg Oscillations: These appear as high-frequency oscillations in the coherence signal that match the frequency of the dressed Larmor precession. The paper notes these nearly periodic oscillations do not appear in previous Stückelberg results (e.g., from atom optics interferometry) because the Landau-Zener tunneling rate is small compared to the oscillation frequency.
- Rabi-like Oscillations: These are identified as low-frequency oscillations (e.g., 0.25 kHz) that modulate the high-frequency Larmor signal. In the occupation probability ( $P_+$ ), these Rabi-like oscillations appear clearly as the primary feature, while in the coherence ( $\langle \sigma_x \rangle$ ), they manifest as an amplitude modulation of the dominant Larmor oscillations. These oscillations arise from the interference of quantum amplitudes in multi-passage Landau-Zener tunneling and produce periodic, large modifications of the coherence time dependence, particularly when the longitudinal dressing amplitude is large.
Non-Adiabatic Features: The data confirmed that the system operates in a non-adiabatic regime where the tunneling probability and interference patterns are significantly modified by the longitudinal field, deviating from the standard adiabatic LZSM behavior.

Significance
The paper claims that this work opens a new exploration direction within the broad area of LZSM interferometry. By demonstrating continuous monitoring of qubit coherences in a dual-dressing regime, the study provides direct access to the phase of the wavefunction, offering new perspectives for quantum control. The ability to manipulate the dynamics of atomic qubits through transverse and longitudinal dressing expands the toolbox for quantum state manipulation. Specifically, the rich dynamics associated with the XZ configuration suggest potential avenues for quantum control accelerations and the development of alternative fast quantum logic gates based on non-adiabatic LZSM transitions. The work emphasizes that the transition probability (monitoring the z-component of the Bloch vector) and coherence (monitoring the xy-plane evolution) provide complementary but distinct views of the system's evolution, with the latter revealing the dominance of dressed Larmor oscillations and the former acting as a strong probe for Rabi-like oscillations.

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