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arXiv:2405.17736 · Optics Express

Phonon number measurement using optimal composite pulses

Xie-Qian Li, Chun-Wang Wu, Ping‐Xing Chen

Published Nov 12, 2024• 1 citation

Abstract

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.

DOI: 10.1364/oe.537501

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Sideband cooling enables preparation of trapped ion motion near the ground state and is essential for many scientific and technological applications of trapped ion devices. Here, we study the efficiency of continuous and pulsed sideband cooling using both first- and second-order sidebands applied to an ion where the motion starts outside the Lamb-Dicke regime. We find that after optimizing these distinct cooling methods, pulsed and continuous cooling achieve similar results based on simulations and experiments with a

^{40}

Ca

^+

ion. We consider optimization of both average phonon number

\overline{n}

and population in the ground state. We also demonstrate the disparity between

\overline{n}

as measured by the sideband ratio method of trapped ion thermometry and the

\overline{n}

found by averaging over the ion's motional state distribution.

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

.

Sideband cooling enables preparation of trapped ion motion near the ground state and is essential for many scientific and technological applications of trapped ion devices. Here, we study the efficiency of continuous and pulsed sideband cooling using both first- and second-order sidebands applied to an ion where the motion starts outside the Lamb-Dicke regime. We find that after optimizing these distinct cooling methods, pulsed and continuous cooling achieve similar results based on simulations and experiments with a

^{40}\mathrm{Ca}^{+}

ion. We consider optimization of both average phonon number

\overline{n}

and population in the ground state. We also demonstrate the disparity between

\overline{n}

as measured by the sideband ratio method of trapped ion thermometry and the

\overline{n}

found by averaging over the ion's motional state distribution.

Low-entropy motional states of ion Coulomb crystals are an essential prerequisite for a plethora of applications and are typically prepared by laser cooling. As larger crystals are operated in the quantum regime, it remains unclear, and has recently become debated, whether increasing the ion number can be beneficial for cooling. Here, we investigate theoretically and experimentally many-ion sideband cooling and the role of collective effects in different spin-motion coupling regimes. For weak coupling, the many-body effects are insignificant. In the strong-coupling regime, however, the spin and motional subsystems undergo a coherent state swap, enabling cooling by a suitably timed laser pulse. Using planar Coulomb crystals with up to 91 ions, we demonstrate that the residual mean phonon occupation after one such pulse scales as

1/N^2

with the number of ions. By iterating the pulses, we measure mean phonon occupations

<2\cdot10^{-4}

. For large crystals in the coherent regime, we further show that the spin-motion dynamics becomes largely independent of the initial phonon statistics. Through spin measurements, the state-swap mechanism can be utilized to probe the phonon distribution in the mode.