In 1973, Cohen-Tannoudji, Dupont-Roc and Fabre computed the higher-order radiative shift of a 'coherence resonance' — an optically detected level-crossing resonance, occurring near twice the radio-frequency (RF) angular frequency, of an atom 'dressed' by RF photons — and checked it by remeasuring the resonance centre of rubidium-87 with high precision as the RF intensity was varied. The centre follows a sixth-order perturbative expansion up to ω1/ω≃2\omega_1/\omega\simeq 2 (with ω1\omega_1 the Rabi angular frequency, proportional to the RF amplitude, and ω\omega the RF angular frequency) and then approaches a strong-coupling law ω0≃1.85 ω (2.405−ω1/ω)1/2\omega_0\simeq 1.85\,\omega\,(2.405-\omega_1/\omega)^{1/2}; but in exactly this region the resonance broadens, the measurement precision degrades, and the resonance disappears at ω1/ω=2.405\omega_1/\omega=2.405, the first zero of the Bessel function J0J_0 at which the dressed-atom g-factor vanishes. The question is whether the resonance-centre trajectory can be mapped experimentally with precision through this broadening and vanishing regime, so that the strong-coupling crossing law receives a quantitative test. No experiment is known that has mapped resonance-centre positions with precision in the strong-coupling regime.

Background

In 1973, Cohen-Tannoudji, Dupont-Roc and Fabre published two companion letters in which they computed quantum mechanically the higher-order terms of the radiative shift of the magnetic resonances of an atom strongly coupled to a radio-frequency (RF) field, and checked the result experimentally. The framework is the 'dressed atom' picture, in which the atom and the quantized RF field are treated as a single system: its energy levels, the 'dressed states' ∣k,n⟩|k,n\rangle (the Zeeman sublevel kk, a state of definite angular-momentum projection on the static field, accompanied by nn RF photons of angular frequency ω\omega), form a staircase of crossings and avoided crossings (pairs of levels that repel instead of crossing, because they are coupled) when plotted against the Larmor angular frequency ω0\omega_0 of the spin in the static field (Haroche 1971). Because arbitrarily many RF photons can be present with negligible spontaneous emission, the atom-field coupling can be made arbitrarily strong compared with the level spacings, a regime with no counterpart in optical spectroscopy.

A 'coherence resonance' is an optically detected level-crossing resonance that appears in transverse optical pumping, in which resonant light propagating perpendicular to the static field continuously creates an atomic alignment whose precession is monitored through the transmitted light. Coherence resonances are particularly valuable for testing higher-order radiative shifts because, over a large range of RF powers, they are shifted without being appreciably broadened. For an RF field linearly polarized perpendicular to the static field, the dressed levels originating from ∣+,n⟩|+,n\rangle (energy 12ω0+nω\frac{1}{2}\omega_0+n\omega) and ∣−,n+2⟩|-,n+2\rangle (energy −12ω0+(n+2)ω-\frac{1}{2}\omega_0+(n+2)\omega) cross at ω0≃2ω\omega_0\simeq 2\omega; the abscissa of this crossing gives the centre of the coherence resonance. The displacement of the centre from ω0=2ω\omega_0=2\omega is the analogue, for this two-quantum level-crossing resonance, of the Bloch-Siegert shift — the displacement of a resonance centre produced by the counter-rotating component of the drive, which the rotating-wave approximation (RWA, the standard simplification that retains only the near-resonant part of the coupling) discards. To lowest order the shift is ω0=2ω−ω12/(6ω)\omega_0=2\omega-\omega_1^2/(6\omega), where ω1\omega_1 is the Rabi angular frequency of the RF coupling, proportional to the RF amplitude.

A second strong-coupling effect is the renormalization of the dressed atom's g-factor (the constant relating magnetic moment to angular momentum, and hence the Larmor frequency to the static field): to lowest order in the static field the effective g-factor is multiplied by J0(ω1/ω)J_0(\omega_1/\omega), where J0J_0 is the zeroth-order Bessel function. As the RF amplitude grows, the coherence-resonance centre therefore moves toward ω0=0\omega_0=0, and the resonance disappears exactly at the first zero of J0J_0, ω1/ω=2.405\omega_1/\omega=2.405, where the dressed levels become tangent at ω0=0\omega_0=0; the ordinary one-quantum resonance vanishes at the same point. The 1973 letters compared the measured centre with three theoretical curves: curve A, the lowest-order shift ω0=2ω−ω12/(6ω)\omega_0=2\omega-\omega_1^2/(6\omega); curve B, a sixth-order perturbative expansion that represents the data up to ω1/ω≃2\omega_1/\omega\simeq 2; and curve C, a strong-coupling formula obtained by treating the Zeeman interaction as a perturbation of the strongly coupled atom-RF Hamiltonian (Landré et al. 1970), ω0=ω [J0(ω1/ω)/S(ω1/ω)]1/2≃1.85 ω (2.405−ω1/ω)1/2\omega_0=\omega\,[J_0(\omega_1/\omega)/S(\omega_1/\omega)]^{1/2}\simeq 1.85\,\omega\,(2.405-\omega_1/\omega)^{1/2}, where SS is a double sum of products of Bessel functions.

The underlying theory of periodically driven quantum systems is Floquet theory, the expansion of the driven state in Fourier harmonics of the drive (Shirley 1965); semiclassical analytical expressions for the Bloch-Siegert shift were developed in parallel (Hannaford, Pegg and Series 1973; Series 1978; Arimondo and Moruzzi 1982). RF spin dressing later became a working tool across physics. The J0J_0 g-factor renormalization and its collapse at the first zero have been verified with dressed neutrons (Muskat, Dubbers and Schärpf 1987) and with a radio-frequency-dressed Bose-Einstein condensate of chromium atoms (a macroscopically occupied atomic quantum gas), in which the renormalization was followed up to the degeneracy point at Ω/ω=2.4\Omega/\omega=2.4 in that paper's notation (Beaufils et al. 2008). Dressed-spin precession has been measured in polarized helium-3 up to dressing amplitudes of order 1.4-1.6 (Esler et al. 2007; Chu et al. 2011). Bloch-Siegert shifts have been observed in strongly driven Rydberg atoms, highly excited atoms with large dipole matrix elements (Stoneman, Thomson and Gallagher 1988), and analyzed theoretically for the strongly driven two-level atom (Wei, Windsor and Manson 1997). 'Magic' RF dressing, which shifts clock transitions to field-insensitive operating points, has been proposed for optical clocks (Zanon-Willette, de Clercq and Arimondo 2012, 2013), and spin-dressed relaxation and frequency shifts are analyzed for neutron electric-dipole-moment searches (Swank et al. 2018). Modified Zeeman hyperfine spectra demonstrating the g-factor modification were observed already by Haroche, Cohen-Tannoudji, Audoin and Schermann (1970). Modern reviews are due to Silveri et al. (2017) and Allegrini and Arimondo (2024); a proposal for optical detection of the Bloch-Siegert effect in atomic vapours has been put forward by Sudyka, Pustelny and Gawlik (2016).

Problem Statement

In the 1973 experiment the coherence resonance of the RF-dressed rubidium-87 atom could be followed practically over all its domain of existence, but with decreasing precision precisely in the strong-coupling region where the deviation from perturbation theory is largest; for the ordinary one-quantum resonance the quality factor collapses so rapidly that its centre cannot be measured at all for ω1/ω>1\omega_1/\omega>1. The open question is: can the centre ω0\omega_0 of the radiatively shifted coherence resonance of a strongly RF-driven atom be determined experimentally with precision throughout the strong-coupling regime — from ω1/ω≃2\omega_1/\omega\simeq 2, where the sixth-order perturbative expansion ceases to represent the data, up to the vanishing point ω1/ω=2.405\omega_1/\omega=2.405 — so that the strong-coupling crossing law, in its full curve-C form ω0=ω [J0(ω1/ω)/S(ω1/ω)]1/2\omega_0=\omega\,[J_0(\omega_1/\omega)/S(\omega_1/\omega)]^{1/2} (with SS the double sum of products of Bessel functions of Landré et al. 1970), whose asymptotic behaviour near the vanishing point is ω0≃1.85 ω (2.405−ω1/ω)1/2\omega_0\simeq 1.85\,\omega\,(2.405-\omega_1/\omega)^{1/2}, receives a quantitative experimental test, with the behaviour of the centre tracked while the resonance line simultaneously broadens and distorts? 'With precision' is understood in the standard experimental sense: measured centres with stated uncertainties tight enough to discriminate the strong-coupling law (curve C) from the lower-order perturbative predictions (curves A and B); the asymptotic square-root law is not a separate alternative target but the near-vanishing-point form of the same curve-C law.

References

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