Collective strong coupling of cold potassium atoms in a ring cavity

We present experiments on ensemble cavity quantum electrodynamics with cold potassium atoms in a high-finesse ring cavity. Potassium-39 atoms are cooled in a two-dimensional magneto-optical trap and transferred to a three-dimensional trap which intersects the cavity mode. The apparatus is described in detail and the first observations of strong coupling with potassium atoms are presented. Collective strong coupling of atoms and light is demonstrated via the splitting of the cavity transmission spectrum and the avoided crossing of the normal modes.


I. INTRODUCTION
The interactions between a single photon and atom in free space are typically very weak.
Jaynes and Cummings showed that the coupling matrix element, which we denote g, depends inversely on the square root of the volume occupied by the electromagnetic field [1].
Therefore it is advantageous for studies of cavity quantum electrodynamics (CQED) to confine the atom and light within an optical microcavity [2,3]. For initial conditions with the atom in its excited state and a photon number state |n of the cavity field, the Rabi oscillation frequency is equal to 2g(1 + n) 1/2 . For a small enough cavity, even the vacuum (n = 0) Rabi oscillation frequency can exceed the atomic and photonic decoherence rates (γ and κ, respectively in this work), and oscillatory excitation exchange between the atom and light can occur. The condition that g is large enough that vacuum Rabi oscillations persist over several cycles before damping is conventionally taken as the definition of the strong coupling regime of CQED.
The presence of vacuum Rabi oscillations can be detected through the spectral splitting of a weakly probed system [4]. The experimental observation of normal-mode splitting of cavity transmission spectra with a single or a few atoms was an important milestone in the historical development of CQED [5][6][7]. More recently a wide range of experiments have begun to study CQED with large atom number. The multi-atom extension of the Jaynes-Cummings Hamiltonian was provided by Tavis and Cummings [8], and later extended by Agarwal [9] to include damping. Ensemble CQED differs from single-atom CQED in some important ways. From a practical viewpoint, the vacuum Rabi frequency increases with atom number according to g → g N 1/2 , relaxing the technical constraints on the optical cavity design.
More fundamentally, a wealth of new physics can arise if the atomic density distribution extends in space across several optical wavelengths. This is associated with effective longrange interactions between atoms mediated by the quantum optical field [10,11]. Collective vacuum Rabi splitting in particular has been central to studies of optomechanical effects in ring cavities [12], atomic spin squeezing [13,14], cavity linewidth control [15], CQED with multiple atomic states [16] and cavity modes [17], and cavity Rydberg polaritons [18].
Here we present the first demonstration of collective strong coupling of cold potassium atoms, using a high-finesse ring cavity. Compared with more commonly used elements such as rubidium or caesium, potassium offers a choice of stable bosonic and fermionic isotopes with varying and tuneable atom-atom interactions [19]. The relatively small hyperfine splittings also make it potentially easier to reach a regime where multiple atomic states are mixed in the presence of strong light-matter coupling [16,20]. The outline of this paper is as follows. In Section II we describe our experimental apparatus, including the vacuum system (II A), the laser system (II B), and the ring cavity itself (II C). In Section III we demonstrate strong coupling on the D 1 lines of potassium-39, first through the observation of the vacuum Rabi splitting and its dependence on atom number, and second through the avoided crossing of the normal mode resonances across a range of cavity and probe laser detunings. A collective cooperativity of C = g 2 N/(κγ) > 100 is achieved, implying a large effective susceptibility for future studies of nonlinear optics with large dispersion.

A. Vacuum system
An overview of the vacuum system is shown in Fig. 1(a). The system is split into two main sections: a relatively high-pressure collection chamber housing the potassium vapour source, and a low-pressure science chamber containing the high-finesse ring cavity. A narrow graphite transfer tube (Goodfellow, 494-159-79 [21]) supports the differential pressure required to keep the science chamber clean. This tube is 100 mm long, with an inner diameter of 3 mm, and is mounted in a stainless steel tube which is welded into a blank ConFlat flange. The transfer tube maintains a calculated pressure ratio of 340:1 between the collection and science chambers. In a first generation apparatus, in which a single chamber housed both the potassium source and the cavity [22,23], we achieved collective strong coupling but found that the cavity finesse degraded over the time scale of a few weeks. We have been operating the two-chamber apparatus for around 1 1 /2 years with no detectable decrease in finesse.
Potassium atoms are released into the collection chamber from alkali metal dispensers (SAES, K/NF/4.5/25/FT10) mounted on an electrical feedthrough, and aimed at the walls of the surrounding stainless steel cross. The cross is kept heated, along with the rest of the source side of the apparatus, in order to maintain a high enough potassium vapour pressure. Potassium-39 atoms from the thermal background are cooled in a two-dimensional

B. Laser system
The trapping and cooling laser subsystem employs three home-built external cavity diode lasers of the kind described in [24]. One laser serves as a master, locked to the potassium-39 hyperfine ground state crossover resonance using sub-Doppler magnetically-induced dichroism [25]. A slave laser is offset-locked to the master using a side-of-filter technique [26].
The master-slave beat note is mixed with a voltage-controlled oscillator (VCO) whose output frequency is tuned with an analogue output from a computer control card (National Instruments, PCI-6733). The slave laser is stabilized near the D 2 F = 2 ↔ F ′ = 3 cooling transition (here F is the total electronic plus nuclear angular momentum, and primes denote excited states), and a fraction of the light is shifted by 2 × 227 MHz with a double-passed acousto-optic modulator (AOM) for repumping on the F = 1 ↔ F ′ = 2 transition [27].
The cooling and repumping beams are then re-combined and injected into a home-built AOMs are in double-pass configuration. AOM1 tunes the probe beam, whose transmitted power is detected at the APD, AOM2 tunes the ring cavity, and AOM3/AOM4 are reserved for future experiments.
In order to stabilize both the science cavity and probe laser to arbitrary detunings, we have built a Fabry-Perot transfer cavity based on the design in [28]. The design exploits the degeneracy of transverse modes to sub-divide the free spectral range (FSR) into an integer number r of resonances, which are equally spaced by FSR/r. In our case the cavity is The use of a modulation frequency equal to half the mode spacing results in a distinctive square-wave shape of the error signals, with locking points of alternating slopes separated by the modulation frequency [23]. This separation sets the coarse resolution of the laser system. Fine tuning is provided by AOMs which can span neighbouring lock points. Some of the 770 nm light is used to stabilize the transfer cavity itself using sub-Doppler frequency modulation spectroscopy [24] to control a piezo ring actuator behind one mirror. A small fraction of the 770 nm light is shifted to the D 1 F = 1 ↔ F ′ = 2 transition for probing the cavity, with the rest of the light shifted to F = 2 ↔ F ′ = 2 for future experiments. We use 250 µW of 852 nm light to stabilize the ring cavity using Pound-Drever-Hall locking, but several mW are available if we wish to produce an intracavity optical dipole trap in the future.

C. Ring cavity
In CQED experiments with single atoms in the strong coupling regime, the Fabry-Perot geometry is preferred for geometrical reasons -it is relatively straightforward to produce a small open mode volume, and therefore large coupling strength g, in the gap between a pair of parallel mirrors. In contrast, ensemble CQED relaxes the constraints on mode volume, making ring geometries viable alternatives. The demonstration of collective atomic recoil lasing with cold atoms [29] relied intrinsically on the presence of distinct counter-propagating travelling wave modes in a triangular ring cavity. The cavity-enhanced quantum memory of Ref. [30] also exploited such modes for phase-matched four-wave mixing. Bow-tie cavities have been used for making quantum non-demolition measurements [31] and for creating cavity Rydberg polaritons [18].
Our ring cavity is shown in detail in Fig. 1(b)  To characterize the cavity further we exploit the inherent astigmatism of the ring geometry. Because of the 45 • angle of incidence on the curved mirror, the effective radius of curvature is R = ROC/ √ 2 along the tangential plane and R ⊥ = ROC √ 2 in the sagittal plane. This in turn leads to different Gouy phases, splitting the degeneracy of higher-order (transverse) Hermite-Gaussian cavity modes [32]. In Figure 3 we show a transmission spectrum where the incident probe beam has been misaligned deliberately in order to excite numerous transverse modes. For our geometry the resonance frequencies are given by, ω q m,n = FSR q + (m + 1/2) cos −1 (1 − L/R ) 2π Here (q, m, n) are the longitudinal, tangential, and sagittal mode numbers, respectively, L is the total round-trip length of the cavity, and FSR = 2π c/L is the free spectral range. The last term in Eq. (1) describes a π phase shift for antisymmetric tangential modes in a cavity with an odd number of mirrors [32]. For simplicity we have omitted the unknown net phase shift due to the dielectric mirror coatings, which leads to an offset of ∼ 1500 MHz between s-and p-polarizations in our cavity.
In principle one can keep fixed either the probe laser frequency or the cavity length, and scan the other to determine the free spectral range (and therefore the cavity length).
However the piezo scans of our laser and cavity are not linear enough over the required few-GHz range to accurately do this. Instead we match a total of 15 transverse modes, with splittings ranging from 9 -1200 MHz. The Pound-Drever-Hall sidebands provide a local frequency calibration. An ABCD matrix calculation is then performed using L as a free parameter to match the observed splittings. The fitting is most tightly constrained by the resonance pairs with smallest splittings, but all of the splittings are consistent. We obtain L = 9.51(5) cm and FSR = 2π × 3151(16) MHz. We have included the effect of a 0.5% uncertainty stemming from the uncertainty on ROC as specified by the manufacturer. This value of L is a percent or two smaller than the design length, but we do observe that the cavity mode is not perfectly centred on the mirrors. Given this value of FSR, we calculate a finesse of F = 1710(60). Knowing L we can also infer the cavity mode spot size, and thus the Rabi frequency 2g between a single atom and photon. In everything that follows, we restrict ourselves to the TEM 00 spatial mode of the cavity. The calculated 1/e 2 intensity radii are w = 90.2(5) µm and w ⊥ = 128.0(3) µm. The electric dipole moment for the D 1 transitions (wavelength λ = 770.1 nm [33] and natural atomic linewidth γ = 2π × 2.978(6) MHz [34]) is d = [3ǫ 0 γλ 3 /(4π 2 )] 1/2 = 2.905 ea 0 (here e is the electron charge and a 0 is the Bohr radius).
the cavity mode volume for a peak-normalized field mode function E(x), and ω c is the cavity resonance frequency.

III. COLLECTIVE STRONG COUPLING
As discussed above, collective strong coupling between the cavity field and the atomic ensemble is evidenced by the normal mode or vacuum Rabi splitting of the cavity transmission spectrum. Given a number density of atoms ̺(x), the effective number of atoms in the cavity mode is N = dx ̺(x)|E(x)| 2 [35] and the vacuum Rabi frequency becomes G = g (ξN) 1/2 [9]. The factor ξ = 5/18 is the relative oscillator strength averaged over all of the F = 1 ↔ F ′ = 2 transitions. Our cloud has an approximately spherical Gaussian density distribution, with a root-mean-squared size, σ ∼ 0.8 mm, which is large compared to w and w ⊥ and small compared to the corresponding Rayleigh ranges. Then N ≈ (π 3 /2) 1/2 ̺(0) σ w w ⊥ , and a typical peak density of 10 9 cm −3 gives N ≈ 4 × 10 4 and G = 2π × 9 MHz, which is well into the regime of collective strong coupling.
We begin each experimental run by collecting several million atoms in the 3D-MOT, and then blocking the pushing beam with the shutter. The repumping light is extinguished 100 µs before the cooling light, in order to optically pump atoms into the F = 1 ground states.
The weak probe light (typically on the order of 1 nW before the cavity) and the magnetic field gradient are left on during the entire experimental cycle. The probe frequency is swept for 100 µs and then the atoms are recaptured. Separate time-of-flight measurements yield a temperature of ∼ 700 µK, and show that the cloud expansion is negligible over the duration of the probe scan. The transmitted probe signal at the APD is recorded and averaged on a digital oscilloscope. Example transmission spectra are shown in Fig. 4, for the case where the cavity is on resonance with the free-space atomic transition. The probe power was < 1 nW before the cavity. The intra-cavity atom number N was varied by translating the centre of the MOT through the cavity mode using an added uniform magnetic field. The transmission spectra are well described by the CQED prediction [9], Here ∆ c is the detuning between the probe laser and the uncoupled cavity, and ∆ a is the probe-atom detuning, which are equal for the data in Fig. 4. Equation (2) assumes that the atomic excited-state population is negligible. With atoms in the cavity, the normalmode splitting is apparent; as G is increased, the resonance frequencies approach ±G, the widths approach (κ + γ)/2, and the amplitudes approach (1 + γ/κ) −2 . Fits to Eq. When the cavity is detuned from resonance with the uncoupled atomic transition, the atoms induce a dispersive shift to the cavity resonance in addition to the splitting just described [3]. By taking two-dimensional scans over ∆ c and ∆ a , it is possible to map out the avoided crossing of the normal modes induced by the coupling [36]. This is shown in Fig. 5 for larger MOT number. In (a) we show the cavity transmission spectra, with the cavity-atom detuning (∆ a − ∆ c ) increasing vertically. Note that at large probe detunings, the incident probe power is reduced due to the finite bandwidth of the AOM. This will be compensated in future experiments with an active feedback system. Here we are not concerned with the peak heights, and simply normalize all of the traces to the maximum incident power. When we track the peak positions, we clearly see the avoided crossing, as shown in Fig. 5(b). The data are again well described by the theory in [9], which gives the normal mode resonance frequencies, For these data G = 2π × 18.1(7) MHz, implying N = 1.01(8) × 10 5 . The observed splitting corresponds to a collective cooperativity of C = G 2 /(κγ) = 119 (9). The cooperativity is the central parameter describing the dominance of the atomic coupling with the cavity mode over the continuum of free-space modes, as well as the onset of optical nonlinearities [2,3,36].

IV. OUTLOOK
We have described an apparatus for studying ensemble cavity QED in the regime of collective strong coupling. Potassium-39 atoms are cooled in a 2D-MOT and transferred to a 3D-MOT overlapping the mode of a high-finesse ring cavity. We have characterized the properties of the cavity which are relevant to understanding the atom-light coupling.
We have demonstrated collective strong coupling through observations of the vacuum Rabi splitting of the cavity transmission spectrum for varying numbers of atoms. Finally, we have observed the avoided crossing of the normal modes of the coupled system.
We next aim to control the group index and optical gain of the atomic medium. It is well known that electromagnetically induced transparency (EIT) can lead to large refractive group indices [37]. We will apply the laser system described in [38] to our potassium MOT.
We can estimate the group index of the intracavity EIT medium as n g ∼ (2G/Γ) 2 , where Γ is the EIT linewidth. For our current conditions, Doppler broadening of the two-photon transition limits Γ ∼ 2π × 0.6 MHz, but standard methods could reduce the MOT temperature to ∼ 30 µK [39][40][41], for which Γ ∼ 2π × 0.1 MHz. At that level the magnetic field variations due to the MOT gradient over the size of the atom-light overlap region will dominate, giving Γ ∼ 2π × 0.3 MHz. This implies a group index of several 10 4 , allowing wide-ranging control over the light scattering dynamics in the cavity [42], with minimal absorption losses, and in the strong coupling regime. We can also study lasing with the cold potassium atoms as the gain medium [43][44][45][46][47]. For superradiant (slow-light) lasing, the group index is approximately equal to κ/GBW where GBW is the gain bandwidth [46,48]. In this case lasing on the lower-finesse p-polarized mode of the ring cavity would be advantageous. Finally we note that our ring geometry makes our system attractive for studying the dynamics of anomalous dispersion [49] as applied towards superluminal enhancement of rotation sensing in a ring laser gyro [50].