2 6s 2 1 S$_{0}$ 11395 6s5d 1 D$_{2}$ 3 D$_{1,2,3}$ ∆= 93 18060 6s6p 1 P$_{1}$ 6s6p 3 P$_{0,1,2}$ { 35934 5d6d 3 D$_{1}$ 36447 3 S$_{1}$ Barium f + − 2 ≈ 1 Δ 2 | f$_{+}$ | 2 $^{f}$− 2 ω$_{eg}$ 2 Δ ω$_{eg}$ 2 ω$_{eg}$ 2 + Δ Ω$_{1}$ (Ω$_{1}$ + Ω$_{2}$ = ω$_{eg}$ ) FIG. 3. (color online). Partial energy-level diagram of bar- ium. Even (odd) parity states are in the left (right) column. Energies are in cm − $^{1}$. The close proximity of 6 s 6 p $^{1}$P$_{1}$ to the energy midpoint (Δ = 18060 − 35934 / 2 = 93 cm − $^{1}$) enhances the two-photon transition rate. FIG. 2. (color online). Plot of ∣ $_{∣}$f$_{+}$ − ∣ ∣ 2 on resonance (Ω$_{1}$ + Ω$_{2}$ = ω$_{eg}$ ) as a function of one photon’s frequency Ω$_{1}$. The two peaks correspond to the cases where the frequency of one of the photons matches the ground-to-intermediate transition. signal S ( δ ) on resonance is where and where D$_{ng}$ ( D$_{en}$ ) is the reduced dipole matrix ele- ment of the ground- to intermediate- (intermediate- to final-) state transition. 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 S ( δ ) = γ { W$_{+}$ ( δ ) + ν W$_{−}$ ( δ ) } , (8) f$_{+}$ = (Ω$_{1}$ − Ω$_{2}$) / 2 ( ω$_{ng}$ − Ω$_{1}$ + i Γ$_{n}$ / 2) ( ω$_{ng}$ − Ω$_{2}$ + i Γ$_{n}$ / 2) , (4) f$_{−}$ = ω$_{ng}$ − (Ω$_{1}$ + Ω$_{2}$) / 2 + i Γ$_{n}$ / 2 ( ω$_{ng}$ − Ω$_{1}$ + i Γ$_{n}$ / 2) ( ω$_{ng}$ − Ω$_{2}$ + i Γ$_{n}$ / 2) , (5) S cal = S ( δ$_{cal}$ ) = γW$_{−}$ (0) δ 2 cal Δ 2 + O ( δ 4 cal Δ 4 ) . (9) ν = S lim cal δ 2 cal Δ 2 . (10) 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). 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). 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 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 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 W$_{−}$ ( δ ) = 2 D 2 $_{en}$D 2 ng ¯ I$_{1}$ ¯ I$_{2}$ Γ ℏ $^{4}$Δ$^{2}$ϵ 2 0 c 2 ( 1 − δ 2 Δ 2 ) − 2 , (6) W$_{+}$ ( δ ) = δ 2 Δ 2 W$_{−}$ ( δ ) . (7)