⚛️ quantum physics

Reconstructing the phonon distribution of trapped ions out of the Lamb-Dicke regime

This paper presents a novel method for reconstructing the phonon distribution of trapped ions beyond the Lamb-Dicke regime by recasting the problem as a filter-function inversion using composite pulses, enabling the characterization of states up to $n=250$ .

Original authors: Christophe H. Valahu (School of Physics, University of Sydney, NSW, Australia, Sydney Nano Institute, University of Sydney, NSW, Australia), Prachi Nagpal (School of Physics, University of Sydney, NSW

Published 2026-09-22

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CC BY 4.0

Original authors: Christophe H. Valahu (School of Physics, University of Sydney, NSW, Australia, Sydney Nano Institute, University of Sydney, NSW, Australia), Prachi Nagpal (School of Physics, University of Sydney, NSW, Australia), Teerawat Chalermpusitarak (School of Physics, University of Sydney, NSW, Australia, Sydney Nano Institute, University of Sydney, NSW, Australia), Maverick J. Millican (School of Physics, University of Sydney, NSW, Australia), Cameron McGarry (School of Physics, University of Sydney, NSW, Australia, Sydney Nano Institute, University of Sydney, NSW, Australia), Frank Scuccimarra (School of Physics, University of Sydney, NSW, Australia), Vassili G. Matsos (School of Physics, University of Sydney, NSW, Australia), Hon-Kwan Chan (School of Physics, University of Sydney, NSW, Australia), Ting Rei Tan (School of Physics, University of Sydney, NSW, Australia, Sydney Nano Institute, University of Sydney, NSW, Australia)

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 quantum physics, scientists often trap individual atoms, or ions, using invisible electric fields. These trapped ions do not sit still; they vibrate back and forth like tiny weights on a spring. In the language of quantum mechanics, these vibrations are not just continuous movements but come in discrete packets of energy called phonons. The number of these packets determines the state of the ion's motion. For decades, researchers have used these vibrating ions as a foundation for building quantum computers and simulating complex physical systems. To make these machines work, scientists must know exactly how many phonons are present in the vibration. However, a long-standing rule in this field, known as the Lamb-Dicke limit, has acted as a ceiling. This rule states that if the ion vibrates too vigorously—meaning it has too many phonons—the standard tools used to count them break down. The relationship between the vibration and the measurement becomes twisted and confusing, making it impossible to tell one high-energy state from another.

A team of researchers at the University of Sydney has now broken through this ceiling. They have developed a new method to count the vibrations of trapped ions even when they are moving far beyond the limits where previous techniques fail. By treating the measurement process as a problem of filtering signals, the team engineered a way to distinguish between different numbers of phonons, even when those numbers are very large. They successfully reconstructed the vibration patterns of ions containing up to 250 phonons, a range that was previously inaccessible. This achievement allows scientists to observe and control highly excited quantum states, opening the door to more powerful quantum simulations and more precise measurements of time and gravity.

The core of the problem lies in how scientists usually "listen" to these ions. To count the phonons, researchers typically shine a laser on the ion. This laser interacts with the ion's internal spin, a property similar to a tiny magnetic needle, and its motion. In the past, scientists relied on a simple relationship: the strength of the interaction would change in a predictable, straight-line way as the number of phonons increased. This worked well for small numbers of vibrations. But as the number of phonons grew, this relationship warped. The interaction strength would rise and fall in a complex, wavy pattern, causing different numbers of phonons to produce nearly identical signals. It became like trying to identify a person in a crowd by their height, only to find that many people of different heights were standing at the exact same spot due to a trick of perspective. The measurements became ambiguous, and the math used to reverse-engineer the vibration count from the data became unstable and prone to error.

To solve this, the researchers stopped trying to force the old, simple tools to work and instead redesigned the measurement process from the ground up. They adopted a framework known as filter-function design. Imagine trying to hear a specific note in a noisy room; you would use a filter to block out everything except that note. The team designed a series of such filters, but instead of blocking sound, they blocked specific numbers of phonons while letting others pass. They created these filters by using a technique called composite pulses. Instead of shining a single, steady laser pulse, they chopped the laser light into a rapid sequence of segments, each with a slightly different phase or timing. By carefully tuning the timing and the number of these segments, they could shape the laser's response so that it only triggered a strong signal for a very narrow range of phonon numbers.

The researchers did not rely on just one type of filter. They combined measurements from different types of laser interactions, known as sidebands, which respond to the motion in different ways. One type of interaction might be sensitive to low numbers of phonons, while another might be better at distinguishing high numbers. By mixing data from these different interactions, they built a complete picture that could distinguish every possible number of phonons in their target range. They tested this method on a single ion of ytterbium, trapped in a vacuum chamber at room temperature. They prepared the ion in specific states, forcing it to vibrate with exactly zero, twenty, or even two hundred and fifty phonons.

The results were striking. When they applied their new method, the reconstructed counts matched the prepared states with high precision, even at the highest energy levels. In contrast, when they tried to use the old, standard method without these new filters, the results became garbled and inaccurate as soon as the number of phonons exceeded about twenty. The new method remained reliable all the way up to 250 phonons. The team also demonstrated that this technique could track how these vibrations change over time. They prepared an ion with 100 phonons and watched as it slowly gained energy from its environment, a process known as heating. Their new filters allowed them to see the distribution of phonons broaden and shift in real-time, confirming that their method could capture the dynamic behavior of these quantum systems.

This work is significant because it removes a major barrier to using trapped ions for advanced quantum tasks. Many promising applications, such as simulating complex materials or improving the accuracy of atomic clocks, require ions to be in highly energetic states. Until now, scientists could not reliably verify that these states were what they claimed to be. By proving that they can reconstruct the phonon distribution for arbitrary states over a vast range, the researchers have provided a robust tool for the next generation of quantum experiments. The method is not limited to just counting; it can also be used to characterize mixed states, where the ion is in a combination of different vibration levels, and to study the fundamental limits of how quantum systems interact with their environment. The ability to see clearly into these high-energy regimes means that the potential of trapped ions as a resource for quantum technology is no longer capped by the limitations of measurement.

Technical Summary: Reconstructing the Phonon Distribution of Trapped Ions Out of the Lamb-Dicke Regime

Problem Statement
Characterizing the phonon distribution of trapped-ion motional states is critical for quantum information processing, simulation, and metrology. However, existing reconstruction methods rely on the Lamb-Dicke approximation, where spin-motion coupling scales linearly or monotonically with phonon number. Beyond this regime (at high phonon numbers, $n$ ), the coupling becomes highly nonlinear and can develop near-degeneracies. This leads to three primary challenges:

Indistinguishability: Different phonon numbers produce nearly identical measurement responses.
Informational Incompleteness: Standard protocols using single sidebands often fail to generate a full-rank filter matrix, making unique state determination impossible.
Ill-Conditioning: Even when distinguishable, the reconstruction becomes highly sensitive to measurement noise, leading to large errors in the estimated distribution.

Methodology
The authors propose a phonon-number reconstruction protocol based on a filter-function framework. The goal is to estimate the phonon population vector $\mathbf{P}$ from a set of measurements $\mathbf{M}$ via the linear relation $\mathbf{F}\mathbf{P} = \mathbf{M}$ , where $\mathbf{F}$ is a filter matrix. To ensure reliable reconstruction beyond the Lamb-Dicke regime, the authors engineer $\mathbf{F}$ to be both full-rank (ensuring distinguishability) and well-conditioned (minimizing noise amplification).

Key methodological components include:

Composite-Pulse (CP) Sequences: Instead of primitive pulses, the authors use CP sequences with $J=7$ phase segments. This creates narrowband filters in Rabi-frequency space, which map to specific regions in Fock space.
Multi-Sideband Combination: The protocol combines measurements from multiple spin-motion interactions (carrier $k=0$ , first blue-sideband $k=1$ , and second blue-sideband $k=2$ ). Since the coupling strengths $\Omega_k(n)$ have distinct nonlinear dependencies for different $k$ , combining them resolves degeneracies that exist within a single sideband.
Optimization: A greedy A-optimal design algorithm selects the specific pulse durations and sideband orders to minimize the trace of the estimator covariance matrix for a target phonon range $[n_{min}, n_{max})$ .
Experimental Implementation: The protocol is demonstrated on a $^{171}\text{Yb}^+$ ion in a Paul trap. The system utilizes stimulated Raman transitions to drive the spin-motion interaction. Critical to the success is the precise calibration of the Lamb-Dicke parameter ( $\eta \approx 0.085$ ) and a closed-loop servo system to stabilize Rabi frequencies against laser intensity noise.
Reconstruction Algorithm: The phonon distribution is recovered via constrained convex optimization, minimizing $\|\mathbf{F}\mathbf{P} - \mathbf{M}\|_2^2$ subject to physical constraints ( $0 \le P(n) \le 1$ and $\sum P(n) \le 1$ ).

Key Results
The authors experimentally demonstrated the reconstruction of phonon distributions for various states:

Fock States: They successfully reconstructed pure Fock states up to $n = 250$ , well beyond the Lamb-Dicke regime. The protocol correctly identified target states across the range $0 \le n < 100$ .
Mixed States: The method was applied to thermal and coherent states with mean phonon numbers up to $E[n] = 36$ . The optimal protocol yielded mean and variance values consistent with theory, whereas a "primitive" protocol (using only the first sideband without CP modulation) failed significantly, producing rank-deficient matrices and large deviations.
Heating Dynamics: The authors measured the heating dynamics of Fock states prepared at $n=10$ and $n=100$ . They observed the broadening of phonon distributions over time, with rates consistent with a model involving bosonic loss and gain channels.
Performance Comparison: The optimized filter family (combining multiple sidebands and CP sequences) provided full-rank, well-conditioned matrices. In contrast, primitive protocols were shown to be rank-deficient (e.g., rank 54 for a 100-state range) and ill-conditioned, leading to informational incompleteness and high variance.

Significance and Claims
The paper claims to introduce and experimentally demonstrate a protocol that overcomes the limitations of the Lamb-Dicke approximation for phonon-state reconstruction. The primary significance lies in:

Scalability: Enabling the characterization of motional states with high phonon numbers ( $n=250$ ), which are essential for advanced quantum resources but previously inaccessible to standard tomography.
Robustness: Providing a method that remains reliable for both pure and mixed states over arbitrary phonon ranges by explicitly engineering well-conditioned filter functions.
Direct Observation: Allowing for the direct observation of highly excited motional dynamics, such as heating rates in high- $n$ Fock states.
Thermometry: Serving as a thermometry technique that does not require a priori assumptions about the phonon distribution, applicable to precision spectroscopy and atomic clocks.

The authors modestly note that while they have demonstrated reconstruction of the diagonal elements (phonon distribution), the framework could potentially be extended to reconstruct full density matrices. They also emphasize that the current implementation is limited to sideband orders $k \in \{0, 1, 2\}$ due to the weakening of higher-order interactions relative to decoherence rates.

Technical Summary: Reconstructing the Phonon Distribution of Trapped Ions Out of the Lamb-Dicke Regime

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