Cavity-enhanced polarization rotation measurements for low-disturbance probing of atoms

We propose and demonstrate cavity-enhanced polarization-rotation measurement as a means to detect magnetic effects in transparent media with greater sensitivity at equal optical disturbance to the medium. Using the Jones calculus, we compute the effective polarization rotation effect in a Fabry-Perot cavity containing a magnetic medium, including losses due to enclosure windows or other sources. The results show that when measuring polarization rotation, collecting the transmitted light has advantages in simplicity and linearity relative to collecting the reflected light. We demonstrate the technique by measuring Faraday rotation in a $^{87}$Rb atomic ensemble in the single-pass and cavity-enhanced geometries, and observe enhancement in good agreement with the theoretical predictions. We also demonstrate shot-noise-limited operation of the enhanced rotation scheme in the small-angle regime.


Introduction
Dispersive measurement of atomic variables using near-resonant light is both a practical method of non-destructive probing [1,2] and a versatile technique for preparing non-classical states of matter [3][4][5]. When the atomic system is contained within an optical resonator, both the measurement uncertainty and the disturbance to the measured variable can be far below the intrinsic quantum noise, leading to strong squeezing and entanglement [6][7][8]. To date, most cavityenhanced probing techniques have been applied to non-magnetic transitions in atomic systems, of interest for atomic clocks [9,10]. There has been less work with cavity-enhanced measurement of atomic magnetization [11][12][13], of interest to magnetometers [14], gyroscopes [15], and instruments to search for physics beyond the standard model [16][17][18].
Polarization rotation (PR) measurements, in which a probe beam's linear polarization rotates upon propagating through a medium, are widely applied to detect magnetic effects in atomic media. These include Faraday rotation, in which the magnetic field produces circular birefringence via the Zeeman effect [19], and the so-called paramagnetic Faraday rotation, in which the spin polarization or magnetization of the medium is responsible for the rotation [20]. While it is to be expected that PR measurements can be enhanced by the Purcell effect created by cavity resonance, the methods used to date, including transmission [6,7] and Pound-Drever-Hall (PDH) [8,10] measurements of the line shift of a single cavity mode, are not directly applicable to PR, which involves two modes of different polarization. This intrinsically two-mode character of PR probing presents novel challenges in the design of cavity-enhancement (CE) methods.
In this work we propose and demonstrate a novel cavity-enhancement scheme to boost the sensitivity of PR measurements. We first describe the atom-light system using the Jones matrix formalism [21,22] to obtain analytic expressions for the cavity's output field in transmission Model for cavity-enhanced polarization-rotation measurements. A quasimonochromatic field in impinges upon an optical cavity formed by mirrors CM 1 and CM 2 , with intensity reflection (transmission) coefficients 1 ( 1 ) and 2 ( 2 ), respectively. An atomic medium in the cavity introduces a phase shift ∓ at for the ± polarizations on each pass (blue oval) between cavity mirrors. Optical windows with intensity transmission coefficients scatter light out of the cavity mode (red arrows), but do not produce back-reflection. Reflected or transmitted output field R or T , respectively, is analyzed by measurement of the Stokes parameter 2 . and in reflection. We note various inconveniences of a PDH-like PR measurement in reflection, and focus on the transmission geometry. We then report an experimental implementation, in which a resonant optical cavity is built around a 87 Rb vapor cell. We compare the polarization rotation angle with and without the cavity, and observe an enhancement factor in agreement with the theoretical model. We measure the noise arising in this implementation, and conclude that the probing system is shot-noise limited in a relevant regime for pulsed probing of cold atomic ensembles [23,24]. The new probing scheme will enable cavity-enhanced quantum-nondemolition measurement [25] of magnetic degrees of freedom in hot and cold atomic systems, creation of non-classical states of matter [24,26], and magnetic sensitivity beyond the standard quantum limit [27].

Model of cavity-enhanced polarization rotation
The components of the model are illustrated in Figure 1: an atomic medium is placed within a Fabry-Perot optical resonator that, apart from possible birefringent effects arising from the atomic medium itself, is polarization-independent. Horizontally polarized input light, with electric field in , impinges upon the cavity and output fields R (reflected) and T (transmitted), which are in general elliptically polarized, containing information about the magnetic conditions within the atomic medium. A balanced polarimeter (not shown in Figure 1) detects a Stokes component of R or T .

Jones Matrix description
We treat the probe light as quasi-monochromatic, and assume it is spatially matched to the cavity's fundamental TEM 00 mode. At any given point in space and in a given direction (forward or backward propagating), the field is completely specified by the amplitudes of circular left (L) and circular right (R) polarizations. Because the cavity per se is polarization-insensitive and the atoms are not, it is more convenient to describe the field in terms of its amplitudes for + and − polarization, i.e. the circular polarizations with respect to the atomic quantization axis, chosen to be the forward propagation direction.  1 indicates the 2 × 2 identity matrix. 0 is the phase shift acquired on a round trip of the cavity, including polarization-independent phases acquired in propagation through the atomic medium, cell windows, and upon reflection from mirrors. and are the intensity reflection and transmission coefficients of the th mirror.
is the intensity transmission coefficient of the windows, assumed equal. Reflection from a window is assumed to fall outside the TEM 00 mode, and thus does not propagate further within the model. at is the relative phase between + and − polarizations upon passing through the atomic medium, equal to half the geometrical polarization rotation angle. Because the PR is of magnetic origin, the same matrix applies for either forward or backward propagation through the medium. and (±) indicates the electric field amplitude for the ± polarization at . For example, the input field, immediately before reaching the input mirror and propagating in the forward direction, is , where 0 is the amplitude of the input field. Each optical transformation, e.g. transmission through the medium, propagation by a distance, or reflection from a mirror, is represented by a matrix that acts on the Jones vector. These matrices are given in Table 1.
It is convenient to use Stokes components [22,28] to describe the polarization at the detection stage. Considering a pulse of duration and effective beam cross-section eff , the Stokes parameter , in photon-number units, is whereˆis the corresponding matrix, given in Table 1. Considering that the input light is fully polarized in the horizontal direction, i.e., in 1 = in 0 , the angle of polarization rotation is = arcsin( out 2 / out 0 ). The superscript "in" refers to the forward-propagating field that enters the system, and "out" refers to the output field, which could be in the forward or backward direction, depending on where the detection is placed.

Single pass and cavity schemes
We consider three cases: single pass (i.e. without cavity), cavity in reflection and in transmission. We will be interested in two figures of merit, the rotation gain ≡ / at and the system efficiency sys ≡ out 0 / in 0 .

Single pass PR measurement
The output field is from which SP = at and thus SP = 1. The system efficiency is SP sys = 2 .

Cavity enhanced PR measurement
We repeat the calculation of the output field as in previous section, but include now the cavity mirrors. The input field is considered as in the previous section, and the output field T ( R ) from cavity in transmission (in reflection) is: The geometric series can be evaluated analytically, to find For simplicity and since it is the highest-sensitivity scenario, in what follows we consider only the on-resonance case, that is 0 = 0. To quantify the performance of the cavity in boosting the rotation signal, we define where R, T is the cavity enhancement factor, i.e. rotation gain factor. The system efficiency is in analogy with the single-pass case.

Sensitivity and sensitivity enhancement at equal disturbance
The shot-noise limited sensitivity to rotation angle can be quantified using the propagation of error formula. Writingˆa t for the estimate of at , the mean squared error (MSE) ofˆa t is where the approximation assumes 1 and that shot noise dominates over electronic and technical noise.
Meanwhile, for linear-optical disturbance effects such as optical pumping, the disturbance to the atoms is proportional to the spatially-averaged probe intensity. Defining as 2 int 0 the photon number seen by the atoms, int 0 can be expressed in terms of in 0 , which through sys is related also to out 0 . In the SP case ( int 0 ) SP = ( in 0 ) SP . In the CE case, the forward-propagating intracavity field int is Assuming at 1 and 0 = 0, i.e., cavity resonance, evaluating again analytically the geometric series, using Equation 1 and multiplying by two to account for the fact that the intra-cavity light passes the atoms both in the forward and backward directions, we obtain For a comparison at equal disturbance, we take ( int 0 ) CE = ( int 0 ) SP , which implies the relation between the input powers. Using Equation 8, the CE versus SP MSE ratio at equal disturbance is then 2.6. Results and comparison between cavity schemes Figure 2 shows the rotation gain , the system efficiency sys and sensitivity enhancement R, T for the CE PR measurement, as a function of mirror reflectivities and for different values of at .
The at values studied are between 10 −3 and 10 −2 radians. As one might expect, and as predicted by Equation 7, the cavity produces a strong enhancement when the + and − polarizations are simultaneously resonant with the cavity. This is possible when F at 1. We note that in applications to atomic media, at can usually be made small by choice of detuning from resonance, with no reduction in sensitivity to the rotation-producing effect, e.g., spin polarization or magnetic field. This is because increased detuning reduces both at and the disturbance due to inelastic scattering, in such a way that the constant-disturbance sensitivity [29] is maintained [30]. We consider one-sided cavities ( 1 = 0.99 for cavity in transmission or 2 = 1 in reflection) to measure the PR from one cavity output. The greater complexity of the reflected signal is evident.
Moreover, the reflected one shows a non-linear relationship of to at , which manifests as a at -dependent gain R , whereas T is nearly at -independent. We can see that the two configurations have quite different behaviors. In reflection, shows a large enhancement around a sharp, multi-lobed resonance, which moreover changes both shape and "position," i.e., the 1 value at which the resonance occurs, with at . In other words, has a notably nonlinear relationship to at and this is reflected also in the sensitivity enhancement R . The resonance is due to destructive interference between the immediate reflection at the in-coupling mirror, which does not carry any information about the atoms, and the field exiting the cavity, which depends on at .
In the transmitted signal, the rotation gain grows monotonically with 2 and is only weakly dependent on at . This means is a nearly linear function of at . The system efficiency T sys shows a broad and nearly at -independent resonance, and coincides with fairly large values of rotation gain T and large sensitivity gain T . The maximum rotation gain in reflection is larger than that obtained in transmission. This apparent advantage is perhaps illusory: the strongest rotation gains occur together with the smallest efficiencies. In this regime, the rotation angle = arcsin( out 2 / out 0 ) is large because 0 is small, due to the destructive interference upon reflection from a cavity near critical coupling. Such "dark fringe" interferometric techniques can be advantageous, especially in high-power interferometry, when the detectors' optimal power levels are below the input power [31]. In probing atomic systems, however, the input power is limited by saturation of the atomic medium [29,32,33], leading to power levels that are easily detected with typical photodetectors. In this scenario, there is no advantage to reducing 0 .

Experimental system
The experimental system is shown in Figure 3. Isotopically-enriched 87 Rb vapor with no buffer gas is housed in a glass vapor cell with a single-pass length (interior dimension of the cell) of 15 cm and with wedged windows, which are anti-reflection coated for 780 nm. The cell is housed in a solenoid to produce an adjustable magnetic field along the cell axis (not shown in the diagram). Cavity mirrors CM 1 (input) and CM 2 (output), each have a 0.35 m radius of curvature, and are spaced to create a near-concentric cavity with an optical length of 0.661(3) m, measured from free-spectral range (FSR) measurements. CM 2 is mounted on a piezo-electric transducer (PZT) for precise control of the cavity length. The nominal TEM 00 mode waists at the center of the resonator are 141(3) µm for 780 nm and 200(4) µm for 1560 nm. Due to the quasi-concentric geometry, higher transverse modes are separated in frequency with respect to the TEM 00 mode [34]. In what follows, we use superscripts and to indicate probe and locking light, at wavelengths = 780 nm and = 1560 nm, respectively. The mirrors have ( ) instead F ( ) with a photodiode placed in reflection (not shown in the schematic). A removable "flip mirror" inside the cavity is used to bypass CM 2 and thereby switch from cavity-based to single-pass measurement. The cavity length is stabilized using an auxiliary Pound-Drever-Hall (PDH) lock. 1560 nm light from a single-frequency erbium-doped fiber laser (EDFL) is phase modulated at 50 MHz with a waveguide electro-optic modulator (EOM), circularly polarized, mode-matched to the cavity and injected into CM 1 . The reflected power is collected on a fast photodiode and demodulated to obtain the PDH error signal. A proportional-integral-derivative (PID) controller feeds back to the PZT to maintain cavity resonance. The EDFL is frequency-doubled and its second harmonic is stabilized by modulation-transfer spectroscopy [35] to the 85 Rb D 2 line ( = 3 → = 4). The probe light, from a external-cavity diode laser (ECDL), is stabilized with respect to the EDFL second harmonic by an offset lock, tuneable over ≈ 2 GHz around the 87 Rb D 2 line ( = 1 → = 2). An acousto-optic modulator (AOM) is used to chop the probe light into pulses. The polarization is purified with a pair of polarizing beamsplitter cubes, and half of the probe power is split off and detected on a reference photodiode (RPD) to infer in 0 . The remaining beam is spatially matched to the TEM 00 cavity mode. The locking and probe beams are shaped by independent telescopes and combined at a dichroic mirror before the cavity (not shown in the diagram).

Rotation gain measurements
To measure polarization rotation, horizontally-polarized probe pulses of 20 µs duration, tuned to resonance with the cavity and detuned 1.7 GHz to the red of the = 1 → = 0 transition in 87 Rb, are sent through the system. out 2 is detected with the differential photodiode (DPD) [36] and in 0 is detected with the reference photodiode. The PR angle is then estimated aŝ = arcsin( 2 / 0 ). Probe pulses of 1.4 × 10 7 photons were used for the single-pass probing, and 1.1 × 10 7 photons for the cavity-enhanced probing. With these numbers, the spatially-averaged probe intensity seen by the atoms is the same in the two scenarios. We note that, in the low-power, offresonance probing regime employed here, the PR angles are negligibly affected by probe-induced nonlinearities such as saturation of the optical transition or optical pumping, so the photon number does not affect the rotation gain. Keeping the mean intensity equal is important, however, for a fair comparison of signal to noise ratios under conditions of equal illumination and thus equal perturbation to the atomic medium.
To induce polarization rotation via the Faraday effect, a magnetic field B is applied to the atoms in a range of 0 to 10 Gauss. No optical pumping is applied to the vapor. Figure 4 shows a comparison of the measured rotation angles for single-pass and cavity-enhanced probing. The observed rotation gain is linear in the applied field and in agreement with predictions from the model of subsection 2.2.
With the setup coefficients given in subsection 3.1, the parameters SP sys and T sys can be estimated. We used the definition given in subsection 2.3 and subsection 2.4, where we consider 0 = 0 (resonant cavity condition) and SP 1, in agreement to the experimental data (inset Figure 4). We obtain SP sys = 0.945 (12) and T sys = 0.08 (3). The parameter T can be also calculated, with the values obtain for SP sys and T sys and with the values of exp T or th T written in the caption of Figure 4. Using Equation 11, we have th T = −11.8(10) dB and exp T = −12.5 (14) dB, in good agreement with our prediction.

Noise measurements
The resonant cavity, in addition to enhancing the polarization rotation signal, can be expected to introduce additional technical noise, for example fluctuation of the cavity transmission due to fluctuations in the cavity length. To assess these noise effects, we measure the detection noise in the CE and SP cases as a function of input photon number. We follow a similar procedure to the one detailed in [36]: we send light pulses of 100 µs duration with different beam powers. There is no magnetic field applied on the atoms, so the expected FR angle is zero. We partition each pulse in segments of different lengths from 1 µs to 100 µs, and extract the variance of the detected out 2 and the mean of the detected power out 0 (inferred from the RPD). Then we fit with the model var( out 2 ) = + out 0 where , , and describe the strength of the electronic noise, shot noise, and technical noise contributions, respectively. Figure 5 shows the results in both SP and CE scenarios. We can appreciate a clear increase of the technical noise in the CE scheme, which narrows the region of shot-noise limited detection. The CE scheme is nonetheless shot-noise limited in a range of photon numbers suitable for probing of cold atomic ensembles [37].

Conclusions and outlook
In this work we have analyzed and demonstrated cavity-enhanced polarization rotation measurements. We found general expressions, computed using the Jones calculus, for the rotation enhancement for both the forward and backward collection scenarios. Using forward collection, we demonstrate experimentally an enhancement of exp T = 16(2), in good agreement with theory predictions. Noise characterization measurements show the system to be shot noise limited in a relevant range of pulse energies. The technique is readily applicable to a variety of material systems, including hot, cold and ultra-cold atomic and molecular ensembles, and will enable CE quantum non-demolition measurements of magnetic degrees of freedom, with potential applications in atomic magnetometers [14], gyroscopes [15], and instruments to search for physics beyond the standard model [16][17][18].