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arXiv:2609.24147

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

Christophe H. Valahu, Prachi Nagpal, Teerawat Chalermpusitarak, Maverick J. Millican, Cameron McGarry, Frank Scuccimarra, Vassili G. Matsos, Hon-Kwan Chan, Ting Rei Tan

quant-phSubmitted Sep 21, 2026• Announced Sep 22, 2026• 0 citations• Revised Oct 15, 2026 (v2)
v1 · Sep 21, 2026v2 · Oct 15, 2026
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Abstract

Characterizing the phonon distribution of trapped-ion motional states is essential for many applications in quantum information processing, but existing methods become ill-conditioned beyond the Lamb-Dicke regime. We overcome this limitation by recasting phonon-state reconstruction as a filter-function inversion problem. Using composite pulses on the carrier and multiple sidebands, we engineer well-conditioned filters in phonon space to characterize pure and mixed states, study heating dynamics, and reconstruct Fock states up to $n=250$.

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Characterizing the phonon distribution of trapped-ion motional states is essential for many applications in quantum information processing, but existing methods become ill-conditioned beyond the Lamb-Dicke regime. We overcome this limitation by recasting phonon-state reconstruction as a filter-function inversion problem. Using composite pulses on the carrier and multiple sidebands, we engineer well-conditioned filters in phonon space to characterize pure and mixed states, study heating dynamics, and reconstruct Fock states up to n=250n=250n=250.
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The measurement of phonon numbers in laser-cooled ions is pivotal for determining whether an ion is in its ground state. Commonly employed experimental techniques include the analysis of red-to-blue sideband ratios and the adiabatic evolution on the red-sideband transition. We introduce a theoretical approach utilizing composite pulses that enables the direct measurement of the population in a specific Fock state, eliminating the need for state evolution fitting. This method offers a more straightforward quantification of higher Fock state populations compared to the red-sideband adiabatic evolution technique. By applying quantum optimal control strategies, we enhance the fidelity of the unitary operations executed by the composite pulses. By utilizing numerical calculation methods of quantum optimal control, we have overcome the limitations of analytical calculations that require weak coupling and first-order Lamb-Dicke expansion, allowing our method to achieve sufficient accuracy in strong coupling regime. Furthermore, we also present a data processing method that allows the estimated value of the phonon number to be closer to the true value. An illustrative example of its efficacy in high Fock state measurement is provided.
A method for gaining information about the phonon-number moments and the generalized nonlinear and linear quadratures in the motion of trapped ions (in particular, position and momentum) is proposed, valid inside and outside the Lamb-Dicke regime. It is based on the measurement of first time derivatives of electronic populations, evaluated at the motion-probe interaction time t=0. In contrast to other state-reconstruction proposals, based on measuring Rabi oscillations or dispersive interactions, the present scheme can be performed resonantly at infinitesimal short motion-probe interaction times, remaining thus insensitive to decoherence processes.
A method for the experimental reconstruction of the quantum state of motion for a single trapped ion is proposed. It is based on the measurement of the ground-state population of the trap after a sudden change of the trapping potential. In particular, we show how the $Q(\ensuremath{\alpha})$ function and the quadrature distribution $P(x, \ensuremath{\theta})$ can be measured directly. In an example we demonstrate the principle and analyze the sensitivity of the reconstruction process to experimental uncertainties as well as to finite grid limitations. Our method is not restricted to the Lamb-Dicke Limit and works in one or more dimensions.
A method for gaining information about the phonon-number moments and the generalized nonlinear and linear quadratures in the motion of trapped ions (in particular, position and momentum) is proposed, valid inside and outside the Lamb-Dicke regime.It is based on the measurement of first time derivatives of electronic populations, evaluated at the motion-probe interaction time τ = 0.In contrast to other state-reconstruction proposals, based on measuring Rabi oscillations or dispersive interactions, the present scheme can be performed resonantly at infinitesimal short motion-probe interaction times, remaining thus insensitive to decoherence processes.
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