Dispersive detection of radio-frequency-dressed states

We introduce a method to dispersively detect alkali-metal atoms in radio-frequency-dressed states. In particular, we use dressed detection to measure populations and population differences of atoms prepared in their clock states. Linear birefringence of the atomic medium enables atom number detection via polarization homodyning, a form of common path interferometry. In order to achieve low technical noise levels, we perform optical sideband detection after adiabatic transformation of bare states into dressed states. The balanced homodyne signal then oscillates independently of field fluctuations at twice the dressing frequency, thus allowing for robust, phase-locked detection that circumvents low-frequency noise. Using probe pulses of two optical frequencies, we can detect both clock states simultaneously and obtain population difference as well as the total atom number. The scheme also allows for difference measurements by direct subtraction of the homodyne signals at the balanced detector, which should technically enable quantum noise limited measurements with prospects for the preparation of spin squeezed states. The method extends to other Zeeman sublevels and can be employed in a range of atomic clock schemes, atom interferometers, and other experiments using dressed atoms.

Figure 1

Linear birefringence. An example level scheme for an $F=1\to F^{\prime }=1$ transition is shown on the left. Only $\sigma$ transitions are allowed for atoms in $|1,0\rangle$, and corresponding polarization components of near-resonant light will acquire a phase shift proportional to atom number. An initial ${45}^{\circ }$ polarization as shown on the right will become elliptical, measured by Stokes operator ${\widehat{S}}_z^{\prime }$ at the output.

Figure 2

Frequency dependence of spontaneous emission and atomic polarizability. (a) Experimental decay rates ${\gamma }_{\exp }$ (circles) for both clock states using light polarized at ${45}^{\circ }$ with respect to the quantization axis. The expected behavior (solid lines) for atoms in $F=1$ (blue group, right) and $F=2$ (red group, left) is based on measured light powers and beam sizes with 30% correction to one of the probe lasers, possibly due to slight misalignment. (b) The $k$-rank tensor contributions to the off-resonant D1 line polarizability (dashed, dash-dotted, and solid lines for $k=0,1,\text{and}2$, respectively). A single fit parameter was used to scale the measured linear birefringence (circles) to match the expected behavior of the $k=2$ terms. (c) Theoretical, off-resonant approximation, and experimental data for the figure of merit $\chi ={({\alpha }_F^{\left(2\right)}/{\alpha }_{J^{\prime }})}^2{I}_0/\gamma$, i.e., the ratio of squared polarizability to decay coefficient (per beam intensity $I_0=P/A$), which determines the maximally achievable, signal-to-noise-power ratio for fixed on-resonant optical density [see Eq. (B13)].

Figure 3

Principal behavior of zeroth (dashed red, $n=0$), first (dash-dotted green, $n=1$), and second (solid blue, $n=2$) harmonic signal components across rf resonance for the orthogonal case ($\beta =\pi /2$) plotted as $h_n^{\prime }={(\pm i)}^n\sqrt{2}{h}_n$ with $\phi =0,\alpha =\pi /4$.

Figure 4

Experimental setup. (a) A laser cooled rubidium sample is prepared in a superposition of two clock states by $\pi$-polarized optical pumping and microwave driving in a static field along $y$. After adiabatic dressing with a magnetic rf field along $x$ and optional rotation of the static field into the $z$ direction, linear birefringence of the sample is probed polarimetrically by two consecutive laser pulses propagating along $z$. (b) Main timing elements of the experimental procedure.

Figure 5

Typical experimental signals. (a) Single-sided, power spectral density of the amplified, high-pass filtered signal. Atomic signals arise at $\omega$ and $2\omega$. (b) Direct signal for $F=2$, recorded during a sweep of static field strength across the rf resonance in the orthogonal setting. Atoms are removed before a second probe pulse is used to determine the signal offset from imperfect detector balance (dashed line). The signal matches the theoretical response but shows probe induced decay. (c) Amplified signal for two-color measurement of both state populations in the parallel setting. Envelopes of the used temporal mode functions ${\tilde{u}}_{1,2}\left(t\right)$ are shown in red (first pulse, $F=2$) and blue (second pulse, $F=1$).

Figure 6

Scan across rf resonance. (a) Experimental mode amplitudes (small circles) at $2\omega$ arising from an equal superposition of the two clock states together with fitted $h_2$ functions (solid lines, left axis) and their ratio (dashed line, right axis) are shown as a function of static field strength in the parallel setting. The stronger signal (upper red curve) arises from atoms in $F=2$ due to larger ${\xi }_F\left(0\right)$. (b) Measured variance $\langle {\left(\mathrm{\Delta }{\sigma }_z\right)}^2\rangle$ as a function of static field amplitude. The noise is reduced (red circles) by synchronizing with an ac mains signal compared to asynchronous measurements (blue crosses). Deliberately introduced static field noise (black diamonds) leads to peaked behavior following the derivative of signal ratio.

Figure 7

Experimental detection of Rabi cycles. (a) The populations of both clock states $|1,0\rangle$ (blue crosses) and $|2,0\rangle$ (red circles) are measured as a function of microwave pulse duration. (b) Normalized population difference (circles) together with model function (solid line). (c) The residuals indicate an oscillating noise amplitude. Each data point corresponds to a single experimental cycle.

Figure 8

Scaling of detection noise with probe power. The linear dependence confirms shot-noise limited performance for both probe lasers. Electronic noise is negligible in the typical operating range of a few hundred microwatts probe power. The shot-noise scaling can be used to calibrate the electronic gain $g_{\mathrm{el}}=U/S_z=2\hslash {\omega }_{L}U/\mathrm{\Delta }P$, relating output voltage $U$ to photon flux or light power difference $\mathrm{\Delta }P$ incident on the two detectors. For pure shot noise from the input light field of power $P$ and known quantum efficiency of the detector $\eta <1$ (electrons per photon), the electronic gain can be measured as $g_{\mathrm{el}}=2\sqrt{\eta \hslash {\omega }_L\zeta \langle S_{UU}\rangle /P}\approx 1.3\times {10}^{-13}\mathrm{V}/\mathrm{Hz}$, assuming quantum efficiency $\eta =0.86$ and an estimated noise power correction factor $\zeta \approx 0.5$ due to aliasing.

Figure 9

Analysis of technical noise for (anti)symmetric superpositions of the two clock states. (a) State preparation noise is identified by driving the clock transition with odd multiples of $\pi /2$ pulse areas and quantifying the quadratic scaling of variance for 100 measurements. For a $\pi /2$ pulse, we estimate an uncertainty contribution of $\mathrm{\Delta }{\sigma }_z=\sqrt{2.4\times {10}^{-7}}\pi /2\approx 0.08\%$. (b) Experimental data for variance of atom number difference show quadratic scaling with total atom number $n$ for two experimental conditions. Signals were measured at constant magnetic field, fulfilling the rf resonance condition only for atoms with $F=2$ (black squares). A small magnetic-field shift introduced between the two probe pulses allows for resonant measurements on both states and reduces technical noise introduced by field fluctuations (blue circles). The model fits (solid lines) separate photon shot-noise equivalent (dotted) and technical noise. Dashed lines indicate the estimated level of state preparation noise above photon shot noise.

Figure 10

Dependence of $2\omega$-signal phases $2{\phi }_{1,2}$ on half-wave plate angle in the parallel setting (corresponding to $\phi /2$). (a) Experimental data are shown together with linear fits with slopes $\pm (3.99\pm 0.01)\mathrm{deg}/\mathrm{deg}$, matching the expected value of $\pm 4$. (b) The strong anticorrelation of residuals shows that the dominant uncertainty stems from the wave-plate setting.

Figure 11

Demonstration of state-to-quadrature mapping. Two probe pulses of different duration are sent through the atomic cloud with temporal overlap. (a) The IQ-modulated rf signal from the atomic response (oscillating black curve) is shown together with total light power $P_1+P_2$ (upper red curve). The light pulse edges have been shaped to suppress excitation of Larmor precession in the effective field, which can occur at higher laser powers. (b) The rf response is demodulated with $10\text{−}\mathrm{kHz}$ bandwidth centered at $2\omega =360\mathrm{kHz}$ and separated into two orthogonal quadratures $I_U\left(t\right)$ and $Q_U\left(t\right)$. The light polarization was adjusted to obtain out-of-phase responses from atoms in $F=1$ (thick blue line) and $F=2$ (thin red line), i.e., by choosing ${\phi }_2-{\phi }_1\approx \pi /2$.