<p>We study a full model of interaction of a pair of two-level atoms initially in atomic spin superposition states. The exact general dynamics is a superposition of energy-conserving red-sideband (co-rotating) and blue-sideband (counter-rotating) qubit state transition processes. Three important dynamical properties arise: First, the full model of atom–atom interaction has an internal non-vanishing <i>residual detuning</i> parameter which characterizes the coupling regimes and determines the nature of the dynamics even at resonance; second, the collective spin population inversion describing the exchange of spin excitations in the blue-sideband transitions undergoes normal periodic Rabi oscillations with peaks at <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4933_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$\pm 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>±</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> in the strong coupling regime, but develops fast oscillations of progressively diminishing amplitudes in the weak coupling regime, eventually vanishing at extremely weak coupling; third, at resonance, the individual atom spin population inversions undergo a beat phenomenon of periodic amplitude-modulated oscillations with maximum peaks <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11128_2025_4933_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">$\pm 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>±</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and time period determined by the difference of blue- and red-sideband Rabi frequencies. In off-resonance interactions at intermediate coupling, the beat phenomenon persists over the entire range of the atom–atom frequency of detuning parameter. The periodicity of the beat phenomenon may be interpreted as quantum collapses and revivals of the envelope of amplitude-modulated oscillations. Our analysis establishes significant dynamical differences between two alternative full Hamiltonian models: One model generates entangled Bell states, while the other generates entangled Mølmer–Sørensen states, thereby realizing distinct Bell and Mølmer–Sørensen quantum gates.</p>

Superposition of red- and blue-sideband processes in interacting qubits: effects of residual detuning

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2025年9月23日

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Joseph Akeyo Omolo,
Onyango Stephen Okeyo,
Christopher Mayero

摘要

We study a full model of interaction of a pair of two-level atoms initially in atomic spin superposition states. The exact general dynamics is a superposition of energy-conserving red-sideband (co-rotating) and blue-sideband (counter-rotating) qubit state transition processes. Three important dynamical properties arise: First, the full model of atom–atom interaction has an internal non-vanishing residual detuning parameter which characterizes the coupling regimes and determines the nature of the dynamics even at resonance; second, the collective spin population inversion describing the exchange of spin excitations in the blue-sideband transitions undergoes normal periodic Rabi oscillations with peaks at $\pm 1$ $\pm 1$ in the strong coupling regime, but develops fast oscillations of progressively diminishing amplitudes in the weak coupling regime, eventually vanishing at extremely weak coupling; third, at resonance, the individual atom spin population inversions undergo a beat phenomenon of periodic amplitude-modulated oscillations with maximum peaks $\pm 1$ $\pm 1$ and time period determined by the difference of blue- and red-sideband Rabi frequencies. In off-resonance interactions at intermediate coupling, the beat phenomenon persists over the entire range of the atom–atom frequency of detuning parameter. The periodicity of the beat phenomenon may be interpreted as quantum collapses and revivals of the envelope of amplitude-modulated oscillations. Our analysis establishes significant dynamical differences between two alternative full Hamiltonian models: One model generates entangled Bell states, while the other generates entangled Mølmer–Sørensen states, thereby realizing distinct Bell and Mølmer–Sørensen quantum gates.