Measuring the polarization of electromagnetic fields using Rabi-rate measurements with spatial resolution: Experiment and theory

When internal states of atoms are manipulated using coherent optical or radio-frequency (rf) radiation, it is essential to know the polarization of the radiation with respect to the quantization axis of the atom. We first present a measurement of the two-dimensional spatial distribution of the electric-field amplitude of a linearly polarized pulsed rf electric field at $\sim 25.6\mathrm{GHz}$ and its angle with respect to a static electric field. The measurements exploit coherent population transfer between the $35s$ and $35p$ Rydberg states of helium atoms in a pulsed supersonic beam. Based on this experimental result, we develop a general framework in the form of a set of equations relating the five independent polarization parameters of a coherently oscillating field in a fixed laboratory frame to Rabi rates of transitions between a ground and three excited states of an atom with arbitrary quantization axis. We then explain how these equations can be used to fully characterize the polarization in a minimum of five Rabi-rate measurements by rotation of an external bias field, or, knowing the polarization of the driving field, to determine the orientation of the static field using two measurements. The presented technique is not limited to Rydberg atoms and rf fields but can also be applied to characterize optical fields. The technique has the potential for sensing the spatiotemporal properties of electromagnetic fields, e.g., in metrology devices or in hybrid experiments involving atoms close to surfaces.

Figure 1

Finite-element simulation of the electric field ${\vec{F}}_{\text{bias}}$ (gray arrows) created by applying a potential difference $V_{23}$ between cylindrical electrodes $E2$ and $E3$ (green boxes). The dc and rf electric-field vectors ${\vec{F}}_{\text{bias}}$ and $\vec{F}$ at the position ($y=-1\mathrm{mm}, z=5\mathrm{mm}$) are indicated by the full and dotted black arrows, respectively. The parallel ($F_{\parallel }$) and perpendicular ($F_{\perp }$) components of $\vec{F}$ with respect to ${\vec{F}}_{\text{bias}}$ are indicated by the blue and orange arrows.

Figure 2

(a) Measured detunings ${\mathrm{\Delta }}_0$ of the center transition frequencies of the parallel (solid, orange circles) and perpendicular transitions (open, blue circles) between $35s$ and $35p$ as a function of the applied bias potential difference. The vertical bars indicate the fitted FWHM of the transitions. Black solid and dashed lines are fits to Eq. (2). (b) rf spectrum of the parallel (upper orange) and perpendicular (lower blue) transitions for $V_{23}=-0.4$ V, see red arrow in (a). (c) Calculated Stark effect of the $(35,s,0)$ (red), $(35,p,0)$ (orange), and $(35,p,\pm 1)$ (dashed-blue) states. The energies of the $n=34,m=0$ and $\left|m\right|=1$ manifolds are indicated by black and dashed-black lines, respectively. The parallel ($\pi$) and perpendicular ($\sigma$) transitions are indicated by orange and blue arrows, respectively.

Figure 3

Measured field strengths of the rf-field component (a) parallel and (b) orthogonal to the dc electric field for the maximal rf amplitude applied. The separation between adjacent contour lines corresponds to 1 mV/cm. (c) Population in $|g_0\rangle$ for the measurement at the frequency of the parallel transition as a function of the amplitude of the applied pulse at the position indicated by $\times$ in (a). Data (black dots) and fit of the analytic function [Eq. (A2)] (red curve) used to extract ${\mathrm{\Omega }}_{\parallel }$. (d),(e) Population in $|g_0\rangle$ for the measurement at the frequency of the perpendicular transition (black dots) as a function of the amplitude of the applied pulse at the positions indicated by $▵,◯$ in (b). Red curve: fit of Eq. (A2) to the data; dashed blue curve: result of an exact numerical simulation (see Appendix) which indicates a splitting of the perpendicular transition occurring in the white shaded region in (b). The splitting is $3.7\text{MHz}$ at the position marked $▵$ in (b). (f) Spatial distribution of the angle $\mathrm{\Theta }$ between ${\vec{F}}_{\text{bias}}$ and $\vec{F}$, extracted from measurements presented in (a) and (b). (g) Distribution of total electric-field strength $|\vec{F}|$, inferred from measurements presented in (a) and (b) [color scale as in panels (a) and (b)].

Figure 4

(a) Polarization ellipse (red) of an excitation field $\vec{F}\left(t\right)$ at ${\vec{r}}_0$. The angles $\alpha$ and $\beta$ indicate the angles that rotate the laboratory frame $\mathcal{L}$ (black) into the bias-field frame $\mathcal{B}$ (black, dashed). (b) Level diagram of a four-level system with common state $|g_0\rangle$ coupled to three states $|e_{-}\rangle ,|e_{+}\rangle$, and $|e_{\pi }\rangle$. The transitions are indicated by blue and orange arrows for ${\sigma }_{+/-}$ and $\pi$ polarized light, respectively. For each transition $[\gamma =(+,-,\pi \left)\right]$, the detuning (${\mathrm{\Delta }}_{\gamma }$) from the drive frequency $\omega$ and the Rabi frequency (${\mathrm{\Omega }}_{\gamma }$) are introduced.