Dressed-State Spectroscopy of Proton Spins in Water Beyond the Rotating-Wave Approximation

Abstract

The quantum Rabi model provides the framework for describing a two-level system interacting with a strong oscillating field beyond the rotating-wave approximation. We report the first experimental observation of the resulting dressed states of proton spins in water, realized using a Rabi-type setup with a strong off-resonant magnetic dressing field. The measured resonance spectrum exhibits multiple spin-state transitions involving several dressing-field quanta, including higher-order resonances predicted by the quantum Rabi model. The dressed-state energies show excellent agreement with theoretical expectations, extending dressed-state spectroscopy to proton spins and opening new possibilities for precision spin manipulation in nuclear magnetic resonance and related precision measurements.

Figures (4)

• Figure 1: Schematic of the experimental setup (not to scale). The protons in flowing water enter the setup from the left. After polarization (orange), they continue into the interaction region, which is shielded by a four-layer mu-metal shield (gray). The innermost layer has a diameter of 180mm and a length of 360mm. The constant main magnetic field $B_0$ (blue) is in the $z$-direction. The oscillating fields for the spin dressing $B_d$ (red) and the spin flip $B_1$ (green) are oriented in the water flow direction along the $x$-axis and can be applied via their corresponding coils. A pulse-NMR system (purple) is used to measure the spin polarization.
• Figure 2: (a) Energy level diagram, calculated from Eq. (\ref{['eq:QRM_matrix']}). It shows the normalized dressed state energy as a function of the dressing parameter $x$ for $y=0.8$. The labels indicate the corresponding Fock state at $x=0$. The vertical dashed line indicates the position of a level crossing at $x\approx2.2$. (b) Calculated transition frequencies for the energy levels shown. Since negative frequencies (dotted levels) cannot be distinguished in our measurements, they are mapped to positive values.
• Figure 3: Measurements for the configuration $\omega_0 = 2 \pi \times 1000~\mathrm{Hz}$ and $\omega_d = 2 \pi \times 350~\mathrm{Hz}$, corresponding to $y\approx2.86$. (a) and (b) Two measured spectra (black dots) at $x=3.3$ ($B_d = 26.9µT$) and $x=5.5$ ($B_d = 44.8µT$), fitted with the multi-Gaussian function given in Eq. (\ref{['eq:multiGaussian']}) (blue lines). The residuals are shown below each spectrum and confirm the quality of the fit used to extract the resonance frequencies. The scaled transition probabilities are shown at the corresponding calculated transition frequencies (orange vertical bars). (c) Energy level diagram showing the transition frequencies as a function of the dressing parameter $x$. The calculated energies are shown together with the fitted resonant transition frequencies (black dots). Error bars are smaller than the marker size. The purple and green lines distinguish the two families of transition branches and highlight (avoided) level crossings. The vertical dashed lines indicate the positions of the spectra shown in (a) and (b).
• Figure 4: Density map of the dressed spin states. It shows the spin polarization (color scale) as a function of the probe frequency and the dressing parameter $x$. This representation corresponds to the raw spectra as shown in the two examples in Fig. \ref{['fig:spectra']} with only a baseline correction applied. The calculated energy levels are overlaid as semi-transparent solid lines to guide the eye.