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where 

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and where D$_{ng}$ ( D$_{en}$ ) is the reduced dipole matrix ele- ment of the ground- to intermediate- (intermediate- to final-) state transition. 

Equations (4) and (5) show that when the two pho- tons have the same frequency (i.e., when Ω$_{1}$ = Ω$_{2}$), then f$_{+}$ = 0, but f$_{−}$ ̸ = 0. Figure 2 shows this dis- tinction at Ω$_{2}$ = Ω$_{1}$ = ω$_{eg}$/ 2: for exchange-symmetric photon states, the transition rate is identically zero; for -antisymmetric states, the rate is ∝ 1 / Δ $^{2}$, where Δ ≡ ω$_{ng}$ − ω$_{eg}$/ 2 is the separation between the energy of the intermediate level ω$_{ng}$ , and the halfway point ω$_{eg}$/ 2. This fact motivated the choice of atomic species in the experiment: in barium, Δ is relatively small (93 cm − $^{1}$), providing resonant enhancement to W$_{−}$ ( Fig. 3). 

Measurements of the transition rate made at laser fre- quencies near the halfway point probe the exchange sym- metry of photons. It is convenient to express the rates (3) in the case where the two laser frequencies are symmetri- cally detuned by a small amount δ (where δ ≪ Δ) about the energy midpoint, δ ≡ | ω$_{eg}$/ 2 − Ω$_{1}$ | = | ω$_{eg}$/ 2 − Ω$_{2}$ | . Assuming | Ω$_{1}$ + Ω$_{2}$ − ω$_{eg}$ | ≪ Γ, the transition rates W + − expressed as functions of δ are 

Because hamiltonians which treat multiple photons iden- tically cannot mix exchange-symmetric and exchange- antisymmetric subspaces of the photon state space [11], there is no interference between the two amplitudes in the total excitation rate. Thus the measured fluorescence 

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signal S ( δ ) on resonance is 

where γ is the product of the number of atoms and the total detection efficiency, and ν is a small dimensionless parameter that characterizes the degree of violation of be statistics. 

Measuring ν is achieved by first calibrating the appa- ratus at δ = δ$_{cal}$ . Expanding S ( δ ) about δ = 0, for ν ≪ δ 2 $_{cal}$/ Δ 2 ≪ 1, the calibration signal is 

Measuring at the halfway point ( δ = 0), the signal is S lim = S (0) = γν W$_{−}$ (0), so that 

The previous version of the experiment [12] measured ν < 10 − 7 at the 90% confidence level, using barium in a vapor cell and pulsed lasers to drive the 6 s $^{2 1}$S$_{0}$ → 5 d 6 d $^{3}$S$_{1}$ two-photon transition. As described below, the ex- periment’s sensitivity was limited by the relatively large bandwidth of the lasers ( ≈ 3 GHz). This letter describes a new experiment that avoids these limitations by us- ing continuous-wave lasers, an in-vacuum power build-up cavity ( pbc ), and an atomic beam ( Fig. 4). 

A run begins with the lasers locked to pbc modes that are separated by 4 × c/ 2 L = 2 GHz ( Fig. 5, left). The cavity and lasers are then swept over L1, acquiring a cali- bration spectrum with δ$_{cal}$ = 2 π × 1 GHz. The “Charyb- dis” laser is moved to the same longitudinal pbc mode as the “Scylla” laser. Then, both lasers are tuned back- and-forth over the limit point L2 many times, acquiring data with δ$_{lim}$ ≤ 2 π × 2 MHz. During the ≈ 1 hour that a run can take, the lasers’ intensities in the pbc may droop a few percent due to mechanical drift and dye consump- tion. The pbc transmission signals of both lasers are 

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