An adjustable-length cavity and Bose–Einstein condensate apparatus for multimode cavity QED

A single-mode Fabry–Perót cavity possesses a geometrical configuration of mirrors such that families of transverse electromagnetic modes ${\rm TE}{{{\rm M}}_{l,m}}$ differing in the sum $l+m$ resonate at intervals in frequency much larger than κ. For example, the frequency spacing between the ${\rm TE}{{{\rm M}}_{0,0}}$ mode and the family of $l+m=1$ of modes, i.e., ${\rm TE}{{{\rm M}}_{1,0}}$ and ${\rm TE}{{{\rm M}}_{0,1}}$, must satisfy $|\Delta {{\omega }_{\{l+m=0\}-\{l+m=1\}}}|\equiv |{{\omega }_{\{l+m=0\}}}-{{\omega }_{\{l+m=1\}}}|\gg \kappa$, where ${{\omega }_{\{l+m\}}}$ is the frequency of all the modes in a family whose transverse field nodes sum to $l+m$. (In the absence of astigmatism and birefringence, ${{\omega }_{1,0}}={{\omega }_{0,1}}$.) The cavity is said to be in a single-mode configuration with respect to a pump field if the frequency detuning ${{\omega }_{P}}-{{\omega }_{C}}={{\Delta }_{\,C}}$ of the pump laser from the single cavity mode of interest is much smaller than the frequency spacing between all other modes, e.g., if $|{{\Delta }_{\,C}}|\ll \Delta {{\omega }_{\{l+m=0\}-\{l+m=1\}}}$. See figure 1 for sketch.

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Unlike single-mode cavities, multimode cavities support ${\rm TE}{{{\rm M}}_{l,m}}$ modes resonating at the same frequency, even if they possess differing numbers $l+m$ of transverse field nodes [19]. Interactions mediated by these modes are energy degenerate, since they possess the same frequency. However, the spatial range of the effective interaction is reduced due to the exchange of photonic excitation though superpositions of these disparate mode patterns: interferences among the mutually incommensurate Hermite–Gaussian—or equivalently, Laguerre–Gaussian—mode-functions ${{\Xi }_{l,m}}$ reduce the interaction range. Specifically, the infinite-range interactions in a single-mode cavity form a complete graph of pairwise interactions weighted by the product of a single-field mode amplitude ${{\Xi }_{l,m}}({{r}_{i}}){{\Xi }_{l^{\prime} ,m^{\prime} }}({{r}_{j}}){{\delta }_{l{{l}^{{\prime} }}}}{{\delta }_{m{{m}^{{\prime} }}}}.$ By contrast, the interactions in a multimode cavity are finite-ranged: considering the modes TEM_m,0, the interactions fall with distance as ${\rm sinc}(\zeta {{\rho }_{ij}})/{{w}_{0}}$, where w₀ is the TEM_0,0 mode waist, due to destructive interference among the Hermite–Gaussian cavity modes⁵ . This interaction approaches a δ-function in the limit $\zeta \to \infty$. Here, ζ is proportional to the number of degenerate cavity modes, r_i is the position of the atom i, and ${{\rho }_{ij}}$ is the separation between two atoms in the direction transverse to the cavity axis. While shorter-ranged in the transverse plane, the interaction range remains infinite and proportional to ${\rm sin} ({{k}_{0}}z)={\rm sin} (2\pi z/\lambda )$ along the cavity axis for the atomic cloud length and cavity geometry considered here. λ is the wavelength of the cavity resonance.

Multimode Fabry–Perót cavities possess special relationships between the cavity length L and the mirror radius of curvature R to ensure that the round-trip path-length of rays corresponding to distinct wavevectors ${\bf k}$ are equal. Common configurations are the concentric cavity $L=2R$ and the confocal cavity L = R. The mirror foci of concentric cavities meet at the cavity center; the mirrors of typical concentric cavities trace out but a small solid angle of a sphere of diameter $2R$. One can see that rays with ${\bf k}$ʼs of equal length but differing direction reflect onto themselves, and therefore modes with spot sizes smaller than the mirror diameter are simultaneously resonant. (In practice, this spot size is limited by aspheric aberration and poor mirror coating reflectivity or high absorption near the mirror edge.) The advantages of concentric cavities include small mode waists, providing larger C compared to other equal-length resonators configurations [20]. While concentric cavity modes are describable by simple Bessel functions, the condition $L=2R$ lies at the instability boundary of two-mirror resonators [19], forcing one to set the mirror length just shy of concentriccity, $L=2R-\epsilon$, to ensure stability, in practice.

Confocal cavities with identical mirrors, by contrast, lie between stable resonator regimes, and so their length may be continuously tuned through the multimode configuration point L = R. Either Hermite–Gaussian or, equivalently, Laguerre–Gaussian transverse mode functions may be used to form a complete basis of supported modes [19]. In a ray-tracing picture, each degenerate mode traces an hour-glass, bow-tie pattern [21]. The frequency interval—free-spectral range—between sets of modes of equal parity is ${{\Delta }_{{\rm FSR}}}=c/2L$, with that between sets of opposite parity ${{\Delta }_{{\rm FSR}}}/2$. See figure 1. This unusual property will allow us to couple the atoms to purely even parity modes or purely odd parity modes. The degenerate modes of confocal cavities, while more difficult to work with in the framework of theories employing functional integration [7, 8], provide access to the same physics as concentric cavities. We have therefore chosen to build a cavity of the confocal configuration, though our system may easily be modified—without breaking vacuum in the ultracold atom production chamber—to increase the cavity length to the concentric configuration.

Our experimental apparatus, shown in figure 2, consists of two sections: an ultracold atom production section and a cavity science chamber. The science chamber, shown in detail in figure 3, houses the multimode cavity and the three stages of its vibration isolation stack. This section provides ample optical access for coupling light into and out of the cavity, for trapping the atoms within the cavity using various configurations of optical dipole traps (ODTs), and for imaging the intracavity atomic cloud.

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The vertically oriented mirrors of the high-finesse Fabry–Perót cavity are affixed to a mount constructed from non-magnetic stainless steel (SS) and Macor. This mount rests on a vibration-isolation stack within the UHV chamber for the purpose of damping environmentally coupled vibrations. Length stabilization is predominantly effected by a single-crystal PZT on which the lower mirror is affixed with UHV-safe epoxy⁶ . The upper mirror is mounted to a slip-stick piezo actuator⁷ with 3 mm of travel along the cavity axis. The displacements resulting from incremental actuation of the slip-stick piezo are hysteretic and non-repeatable, but the small variations may be easily compensated by applying small dc voltages to the stack piezo forming the very same actuator. Continuously tuning the length of the cavity by hundreds of microns is straightforward and travel is sufficiently linear so as to be able to observe spectra of well-coupled modes throughout this travel range, see figure 4.

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The rectangular cavity mount provides ample optical access, allowing beams to be directed at the atoms from any of the 24 viewports mounted on the spherical octagon UHV chamber. The mount hangs from the hooks of a heavy (3.5 kg) steel holder forming the second vibration isolation stage. This rests on the internal rim of a heavy (6 kg) steel cylinder forming the first vibration isolation stage. This cylinder then rests on a ledge around the inside of the UHV chamber. Small, several mm-cubed, Viton rubber pieces separate the metal pieces at each interface, thereby forming a three-stage low-pass vibration filter. The entire vibration-isolation stack can be modeled as a six-pole low-pass filter for vibrations transferred from the chamber to the cavity. Simulations indicate that all resonances lie below 200 Hz and that the cavity mount is well-protected from environmental vibrations above this cutoff. Figure 3 shows a schematic of the primary vibration isolation stack pieces. Not shown are the ion and non-evaporative getter pumps connected to the cavity science chamber, though these may be seen in figure 2. While there is a notable degree of initial outgassing from the Viton pieces and the epoxy used to construct the vibration stack and the cavity, the vacuum ion gauge nearest to the cavity reports a pressure of 1 × 10⁻¹⁰ Torr and the lifetime of our atomic cloud in the ODT within the cavity is 7 s. (The pressure in the production chamber is $2.5\times {{10}^{-11}}$ Torr, several times lower than in the cavity science chamber.) This amount of time is sufficient for evaporating the already ultracold gas to quantum degeneracy, as described below, and for studying multimode cavity QED physics. Viewports above and below the cavity mirrors provide optical access to the cavity mode. Locking light, probe light, and trapping light are coupled to the cavity through these viewports. We also detect the transmission of cavity fields through these viewports with either CCD cameras, single photon counters, or photodiodes. The upper viewport is 49 cm from the cavity mirror: to prevent the rapid divergence of the beam we installed an in-vacuum coupling lens to approximately collimate the vertical cavity output. The lower viewport is sufficiently close to the bottom mirror to render an in-vacuum lens unnecessary.

Alignment of the mirrors for optimal multimode operation is performed before insertion of the cavity mount into the UHV chamber. The inset in figure 3 shows the cavity outside the UHV chamber. We first epoxy the lower mirror to the 5 mm-long cylindrical single-crystal fast piezo, which itself has been epoxied to the SS mount connecting to the Macor base of the cavity mount. We next align the upper mirror to this fixed lower mirror. To optimize the angle and transverse position of the upper mirror, the edge of its flat side is first epoxied to the end of a 3.9 cm-long SS cylinder possessing an inner diameter sufficiently large so as not to obscure cavity transmission. This SS cylinder is then slipped through a 1.5 mm-larger-diameter mounting ring. This ring is screwed onto the face of the vertically mounted slip-stick piezo actuator. The opposite end of the SS cylinder is temporarily held by a positioning stage, and the gap between it and the mounting ring allows us to adjust both the three-axis position and angle of the mirror while the epoxy filling the gap between the cylinder and ring sets. For maintaining proper alignment, we monitor cavity transmission resonances at 780 nm while the epoxy sets, adjusting the positioning stage as needed. We ensure optimal alignment of the mirror with respect to the lower mirror with this procedure, maximizing mode density at the critical length for multimode operation while ensuring good coupling as the slip-stick actuator tunes the cavity toward and away from the L = R condition.

In-vacuum length stabilization of the cavity is achieved by deriving an error signal using the Pound–Drever–Hall method [22, 23] with 1560 nm light resonant with the ${\rm TE}{{{\rm M}}_{0,0}}$ mode of the cavity. Fast feedback is applied to the small single-crystal piezo on the lower mirror. Slow, but large-dynamic-range feedback is applied to the piezo stack in the slip-stick translation stage that actuates the upper mirror. Measurements of the vibration characteristics of the cavity were obtained by loosely locking an optical interferometer to the flat backfaces of each mirror. The amplitude and phase excursions of the interference fringes, when driven by a swept sine-wave, provided an estimate of the transfer functions of the two actuators. Two prominent features were found: a 120 Hz mechanical resonance originating from the slip-stick translation stage and a 3.5 kHz dispersive resonance arising from the coupling of the single-crystal PZT and its mount. An electronic feedback controller was designed based on these observations, incorporating an integrator at dc and open-loop dc gain only limited by the constituent electronic components. Two additional poles in the transfer function between 10 and 100 Hz serve to lower the gain profile rapidly over the next two decades. Two zeros in the transfer function near 1 kHz recover phase margin and result in a stable unity gain point near 10 kHz. The electronic controller also incorporates active resonant filters to provide additional gain to combat the noise induced by the resonance at 120 Hz and to provide the appropriate phase and gain to cancel the 3.5 kHz dispersive resonance. The combination of these elements allows us to achieve length stabilization with rms deviation of the cavity length of 1.8(2) pm (71(5) kHz at 780 nm) as measured in via the in-loop error signal. This results in a cavity stability nearly 4× smaller than the intrinsic cavity linewidth $2\kappa$ at 780 nm, whose measurement we provide in the next section.

Light at the wings of the 1064 nm 9.8 W ODT transport beam 'tweezer' heats the cavity when this beam is focused 40 cm away in the ultracold atom production chamber. The heating ceases after the focus is translated closer to the cavity, but in the meantime, thermal expansion causes the cavity length to decrease. The length can decrease further than the dynamic range of the fast piezo, at which point cavity lock is lost. To counter this drift, we complement the fast piezo feedback with an extremely slow feedback—unity gain point of 1 Hz—to the piezo stack internal to the slip-stick positioning stage. This prevents the dc component of the fast piezo feedback voltage from railing. These two feedback loops provide robust locking of the cavity length throughout the 16 s BEC experimental cycle over the course of an afternoon.

Our cavity enjoys a single-atom cooperativity greater than unity, as we will show below. To the best of our knowledge, only two other in situ adjustable-length high-finesse cavities have been built for cavity QED research, though for single-mode operation [24, 25]. Strongly coupled cavity QED with a family of near-degenerate higher-order modes and a single atom was explored in reference [26]. Near-confocal cavities with thermal atoms were explored in references [27–29, 38, 39].

The cavity finesse was measured by means of ring-down spectroscopy of the cavity transmission [30]. The decay times obtained from these measurements, in conjunction with the cavity lengths obtained from measurements of ${{\Delta }_{{\rm FSR}}}$, reveal a finesse of the ${\rm TE}{{{\rm M}}_{0,0}}$ mode at 780 nm to be 5.72(2) $\times {{10}^{4}}$. Similar measurements reveal the finesse at 1560 nm to be 1.22(1) $\times {{10}^{4}}$ for the same ${\rm TE}{{{\rm M}}_{0,0}}$ resonance. The resulting κʼs are $2\pi \;\times$ 132(1) kHz and $2\pi \;\times \;617(3)$ kHz, respectively. The mirrors exhibit a total round trip loss of 54.9(2) ppm at 780 nm.

We probe the cavity by locking it to a weak 1560 nm laser coupled to a ${\rm TE}{{{\rm M}}_{0,0}}$ resonance while addressing the cavity with 780 nm light produced by doubling this same laser. Using the slip-stick translation stage, we measured transmission spectra in the three regimes of intermode spacing of interest to our future cavity QED experiments: confocal degeneracy of one family of modes, mode separation larger than $2\kappa$ and on the order of an atomic linewidth, and modes far apart from one another and satisfying the single-mode criterion $|{{\Delta }_{C}}|\ll \Delta {{\omega }_{\{l+m=0\}-\{l+m=1\}}}$ with ${{\Delta }_{C}}\leqslant 10$ MHz. The results of these measurements are shown in figure 4, and the higher-order mode peak heights vary due to imperfect mode-matching of the input mode, which nominally has a Gaussian waist slightly bigger than the spot size of the ${\rm TE}{{{\rm M}}_{0,0}}$ mode at the mirror face. The use of the locking laser as the source for the probe laser ensures that the cavity remains stable with respect to the probe light despite the potential drift of the locking laser. The frequency drift of this narrow, temperature-stabilized fiber laser is slow and fractionally insignificant in the large atomic detuning limit.

Under the paraxial approximation, the transverse mode frequencies of a Fabry–Pérot cavity made from parabolic mirrors are

$$\begin{eqnarray}&&{{f}_{nlm}}=\frac{c}{2L}\left( n+\frac{m+1/2}{\pi }{{\theta }_{m}}+\frac{l+1/2}{\pi }{{\theta }_{l}} \right),\end{eqnarray} \tag{ 1 }$$

where ${{\theta }_{i}}={\rm arccos} \sqrt{(1-\frac{L}{{{R}_{i1}}})(1-\frac{L}{{{R}_{i2}}})}$ and R_ij is the radius of curvature of the jth mirror in the i direction [19]. The ideal confocal cavity corresponds to the condition ${{\theta }_{l}}={{\theta }_{m}}=\pi /2$ such that the resonance frequency is unchanged as long as $n+(m+l)/2={\rm constant}$.

In practice, however, highly polished mirror substrates are spherical rather than parabolic, and this spherical aberration reduces the number of supportable higher-order modes [20]. Moreover, mirror substrates may not be cylindrically symmetric due to mounting stress or the opposing mirrors may be mounted off-axis and at a relative angle. Consequently, they exhibit astigmatism, meaning that the cavity has different radii of curvature along different transverse axes. In this case ${{\theta }_{l}}\ne {{\theta }_{\,m}}$, with, say, ${{\theta }_{\,m}}$ corresponding to the direction with maximum R and ${{\theta }_{l}}$ corresponding to the direction with minimum R, or vice versa. In such cases, it is no longer possible to ensure that f_nlm only depends on $n+(m+l)/2$: the two transverse degrees offreedom are no longer equivalent. The astigmatic cavity then exhibits two confocal degeneracy points: one with degenerate l modes and ${{\theta }_{l}}=\pi /2$, and one with degenerate m modes and ${{\theta }_{m}}=\pi /2$. Equation (1) may now be rewritten in terms of a frequency shift proportional to $m+l$ combined with a correction along one of the transverse axes:

$$\begin{eqnarray}&&{{f}_{nlm}}=\frac{c}{2L}\left( n+\frac{m+l}{\pi }{{\theta }_{l}}+\frac{l}{\pi }\xi +\frac{2{{\theta }_{l}}+\xi }{\pi } \right),\end{eqnarray} \tag{ 2 }$$

where ${{\theta }_{m}}={{\theta }_{l}}+\xi$.

Our cavity is found to display a small amount of astigmatism—perhaps due to mounting stress or imperfections in the manufacturing—with major and minor axes located at ∼45° to the science chamber principle axes; see figure 3. The two radii of curvature, measured by the length of the cavity at the two degeneracy conditions, are ${{R}_{1}}=9.959(1)$ mm and ${{R}_{2}}={{R}_{1}}+8.6(1.1)$ μm. The respective spectra for these lengths are shown in figures 4(b) and (c). They are characterized by a single peak of nearly degenerate modes next to a broad shoulder of modes extending out to either lower or higher frequencies. The modes around the sharp peak are associated with the family of modes satisfying the degeneracy condition of one of the astigmatic axes, while the broad pedestal of nearly degenerate modes to the side is associated with the unmet degeneracy condition of the other astigmatic axis. Changing the cavity length from that shown in figures 4(b)–(c) reverses the family of modes—l versus m—that satisfies the degeneracy condition. The mode spectra of each family exhibit small deviations from the description of an ideal, fully linear, astigmatic cavity. This is likely due to spherical aberrations and mirror imperfections [21].

The cavity is effectively single mode at a large distance from confocallity, and the difference between the major and minor radii of curvature becomes negligible compared to $L-R$. The astigmatism then appears as a dispersion of spectrally isolated families of modes with constant $l+m$, as in figure 4(e). We note that much larger splittings between families of modes are possible via further translation of the positioning stage.

The maximum number of degenerate modes in the cavity is limited by the mirror imperfections which cause the finesse to decrease with increasing transverse mode index and by spherical aberrations which introduce non-linearities in f_nlm. We will assess in future work where the limit in the number of supportable modes lies for this cavity. However, we can obtain a rough estimate by noting the largest index mode we image on a CCD camera at a length just shy of confocallity. Unlike for a concentric cavity, the waists of the modes of a confocal cavity do not diverge at the mirror upon reaching the L = R degeneracy point, and so we can assume that any mode supported at $L=R\pm \epsilon$ will likely remain so at L = R. We find that l and m of up to order 50 each may be observed, providing a rough estimate of $\operatorname{lm}/4\approx {{10}^{3}}$ as the number of same-parity modes supported within a ∼50 MHz bandwidth in figures 4(b) and (c).

The cavity exhibits no measurable birefringence; it is therefore capable of supporting both linearly and circularly polarized modes since there is no frequency splitting between modes of orthogonal polarization.

The high reflectivity of the cavity mirrors is obtained via a multilayer dielectric stack which acts as an interference filter. The characteristics of this filter, unlike a simple metal mirror, are strongly wavelength dependent [30]. The penetration depth of light into the dielectric stack is different at 780 versus 1560 nm. As a result, the effective length of the cavity is different at these two wavelengths. To first order in small changes $\delta L$ of the cavity length, and ignoring the Gouy phase, the line shift of the cavity obeys the following relation: $\delta \nu /\nu =\delta L/L$ with $\delta {{\nu }_{780}}=2\times \delta {{\nu }_{2\times 780=1560}}$. This is true in the absence of any dispersion in the effective cavity length. However, real cavity coatings exhibit dispersion and ${{L}_{1560}}={{L}_{780}}+\Delta L$, where $\Delta L$ is a required correction to obtain the frequency relationship between cavity resonances at the two wavelengths: $\delta {{\nu }_{780}}=(2+{{\epsilon }_{D}})\delta {{\nu }_{1560}}$. In addition to modifying the behavior of the cavity with respect to length changes, the dispersion of the cavity length also results in an absolute frequency offset ${{\Delta }_{D}}$ between the frequencies at which the 1560 nm laser and the doubled light are resonant with the cavity. In order to have resonant probe light at 780 nm and locking light at 1560 nm, sidebands must be introduced onto the 1560 nm laser at either ${{\Delta }_{D}}$ or ${{\Delta }_{D}}-{{\Delta }_{FSR}}/2$.

We find the factor ${{\epsilon }_{D}}$ by varying the frequency of the locking sideband required to address the cavity resonance at 1560 nm and then measuring the resultant translation of the 780 nm cavity resonance when the cavity is locked at this new frequency: ${{\epsilon }_{D}}=5.83(5)\times {{10}^{-4}}$. This information, together with the laser frequency and ${{\Delta }_{D}}$, provides sufficient information to infer that the effective length difference $\Delta L=2.78(3)\;\mu {\rm m}$. This length difference is beneficial for locking because the cavity is not simultaneously degenerate for both wavelengths. Instead, the 1560 nm resonances are separated by ∼4.5κ when the cavity is degenerate at 780 nm, at the $L={{R}_{2}}$ condition.

Once the atomic cloud is positioned between the mirrors in the science chamber, a second 1064 nm laser beam crosses it to confine the atoms. This beam is circular with a Gaussian waist (1/e radius) of 48 μm. Forced evaporation in this crossed ODT produces a nearly pure BEC of $4.5(4)\times {{10}^{4}}$ atoms at density $4.7(5)\times {{10}^{14}}$ cm⁻³, as shown in figure 5. The trap frequencies are 145(5), 115(1), and 167(2) Hz in $\hat{x}$, $\hat{y}$, and $\hat{z}$, respectively. We expect to achieve pure condensates with 10⁵ atoms after performing additional evaporation optimization. The total time to achieve a BEC within the cavity, starting from MOT loading, is 16 s.

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For ease of coupling to the cavity mode at 780 nm, we can also load a thermal gas into an intracavity FORT formed with the 1560 nm locking light in the TEM_0,0 mode. To do so, we increase the power of the locking laser until the steady state circulating power provides a trap depth of 5.2(1.3) μK. The transport ODT brings atoms into the science chamber and holds the cloud within the cavity mode. Addition of a second ODT beam crossing the first increases the mode matching between the 1064 nm trap and the 1560 nm cavity mode. Optical evaporation with the crossed 1064 nm beams loads atoms in the intracavity FORT, resulting in $2.5(2)\times {{10}^{5}}$ atoms at 1.8(5) μK confined in the 1560 nm standing-wave TEM_0,0 cavity mode. The temperature and atom number uncertainties, as well as the relatively high final temperature, are dominated by the lack of intensity stabilization and dynamical intensity adjustment of the FORT power, all of which will be added in the near future.

The atoms in the FORT are thermal and the in-trap density distribution obeys Boltzmann statistics with $\rho ({\bf x})={\rm exp} (-V({\bf x})/T)$, where $V({\bf x})$ is the standing-wave optical potential of the FORT. The resulting atomic density distribution is Gaussian, assuming a harmonic approximation to the potential. Such an approximation is valid as long as the atoms are sufficiently cold so as to localize close to the potential minima. We choose to operate at larger trap depth, though lower atom number, in the experiments discussed below. Since the atoms are loaded into the FORT from an ODT with waists larger than $\lambda /2$, the atoms will occupy multiple wells of the FORT. The waist of the 1560 nm ${\rm TE}{{{\rm M}}_{0,0}}$ cavity mode is $\sqrt{2}$ larger than the 780 nm ${\rm TE}{{{\rm M}}_{0,0}}$ mode waist: the FORT waist is $35\sqrt{2}=50$ μm. For the trap parameters used below—${{V}_{{\rm FORT}}}=22(6)$ μK and $T=5(1)$ μK—the atomic density distribution has a transverse waist of ∼17 μm, which is smaller than the 35 μm waist of the 780 nm mode.

The first measurement is of the transmission of the cavity as atoms are transported into the cavity by the ODT tweezer. As shown in figure 6 for both the single mode and the mulitmode configurations, at t = 0, light is injected into the cavity on resonance and the transmission is recorded on a single photon counter. As the atoms enter the cavity, the atomic ensemble-cavity interaction dispersively shifts the cavity line out of resonance with the probe laser and the transmission drops smoothly [34–36]. After a hold time, the atoms are rapidly removed from the cavity by a resonant laser oriented transverse to the cavity axis. After this the transmission sharply returns to the original value. For the single-mode configuration, 10% of photons leaking from the cavity mode are detected by the single-photon detector. This detection efficiency includes loss due to transmission, mode-matching, and the imperfect quantum efficiency of the detector. In the multimode configuration, the detection efficiency is ∼10% for the lowest order modes, but decreases for higher order modes due to poor mode-matching into the detector.

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In a second series of experiments, we measure the dispersive shift of the cavity resonance with atoms confined within the intracavity FORT and with a pump field injecting 1.7(3) photons when on resonance with the cavity. In the single-mode cavity case of figure 7, we observe that coupling of the atomic ensemble to the cavity shifts the cavity resonance by $\delta {{\omega }_{c}}=7.66(2)\;{\rm MHz}\gg \kappa$, as expected from considering the normal-mode spectrum of the Tavis–Cummings model of single-mode, multi-atom cavity QED [34, 36]. Using numerical simulations of the optical Bloch equations [36], we determine that the observed polariton shift is only ∼1% smaller than it would be in the weak-driving limit. This limit is defined as that of a pump field injecting much less than one intracavity photon when the pump is on resonance with the cavity. Nonlinear saturation effects are absent in this limit.

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We use the single-mode cavity dispersive shift to obtain a measurement of the single atom coupling constant g₀ via $\delta \omega _{c}^{{\rm SM}}={{N}_{{\rm eff}}}g_{0}^{2}/{{\Delta }_{A}}$, where ${{N}_{{\rm eff}}}={{\alpha }_{0,0}}N$ is the effective number of atoms coupled to the cavity ${\rm TE}{{{\rm M}}_{0,0}}$ mode and ${{\Delta }_{A}}={{\omega }_{c}}-{{\omega }_{a}}$. The overlap ${{\alpha }_{0,0}}$ of the atomic cloud confined within the intracavity 1560 nm standing-wave FORT with the electric field of the 780 nm cavity standing-wave is given by [37]:

$$\begin{eqnarray}&&{{\alpha }_{0,0}}=\frac{1}{2}\frac{1+{{{\rm e}}^{-4/\eta }}}{1+2/\eta }=0.5(1),\end{eqnarray} \tag{ 3 }$$

with $\eta ={{V}_{{\rm FORT}}}/{{k}_{B}}T$. The trap depth ${{V}_{{\rm FORT}}}$ of our FORT is 22(6) μK and the atoms are at $T=5(1)$ μK. The value of g₀ is inferred from the measurements $N=2.0(2)\times {{10}^{5}}$ and ${{\Delta }_{A}}=8.7(1){\rm GHz}$ and the use of the appropriate transition matrix elements for the D2-line of ⁸⁷Rb at this detuning, the polarization of the pump beam, and the direction of the quantization axis set by an applied magnetic field. We determine ${{g}_{0}}=2\pi \times 1.0(1)$ MHz on the $|5{{S}_{1/2}},F=2,{{m}_{F}}=-2\rangle \to |5{{P}_{3/2}},F=3,{{m}_{F}}=-3\rangle$ cycling transition or ${{g}_{0}}=2\pi \times 7.2(8)\times {{10}^{5}}$ Hz on the $|5{{S}_{1/2}},F=1,{{m}_{F}}=-1\rangle \to |5{{P}_{3/2}},F=2,{{m}_{F}}=-2\rangle$ transition. The expected theoretical value of the coupling is $g_{0}^{{\prime} }=\sqrt{{{\mu }^{2}}{{\omega }_{c}}/\hbar {{\epsilon }_{0}}V}$, where μ is the transition matrix element, $V=\pi w_{0}^{2}\;L/4$ is the mode volume and ${{w}_{0}}=\sqrt{R\lambda /2\pi }=35$ μ m is the mode waist ($1/e$ mode radius). Based on the measurements of the cavity R and L mentioned above, $g_{0}^{{\prime} }=2\pi \times 1.47$ and $g_{0}^{{\prime} }=2\pi \times 1.04$ MHz respectively for the two transitions. We note that the single-atom cooperativity when driving on the $|5{{S}_{1/2}},F=2,{{m}_{F}}=-2\rangle \to |5{{P}_{3/2}},F=3,{{m}_{F}}=-3\rangle$ transition is $C\approx 2.5$.

The disparity between the inferred g₀ʼs and the corresponding theoretical $g_{0}^{{\prime} }$ʼs is likely due to an overestimate of ${{\alpha }_{0,0}}$. The harmonic approximation to V(x) is not quite accurate for these small ratios of trap depth-to-temperature. A breakdown of this approximation would tend to lower ${{\alpha }_{0,0}}$, which in turn would increase the inferred g₀ʼs to be closer to the expected $g_{0}^{{\prime} }$ʼs. A more accurate calculation of ${{\alpha }_{0,0}}$, together with improved stability of the FORT power and better measurements of the intratrap T and N of the atoms, will allow us to obtain a more accurate value of g₀.