\mathcal{P}, \mathcal{T}-odd Faraday rotation in intracavity absorption spectroscopy with molecular beam as a possible way to improve the sensitivity of the search for the time reflection noninvariant effects in nature
D.V. Chubukov, L.V. Skripnikov, A.N. Petrov, V.N. Kutuzov, L.N. Labzowsky
PP , T -odd Faraday rotation in intracavity absorption spectroscopy with molecularbeam as a possible way to improve the sensitivity of the search for the time reflectionnoninvariant effects in nature D. V. Chubukov , , L. V. Skripnikov , , A. N. Petrov , , V. N. Kutuzov , L. N. Labzowsky , Department of Physics, St. Petersburg State University,7/9 Universitetskaya Naberezhnaya, St. Petersburg 199034, Russia Petersburg Nuclear Physics Institute named by B.P. Konstantinov of NationalResearch Centre “Kurchatov Institut”, St. Petersburg, Gatchina 188300, Russia
The present constraint on the space parity ( P ) and time reflection invariance ( T ) violating electronelectric dipole moment ( e EDM) is based on the observation of the electron spin precession in anexternal electric field using the ThO molecule. We propose an alternative approach: observationof the P , T -odd Faraday effect in an external electric field using the cavity-enhanced polarimetricscheme in combination with a molecular beam crossing the cavity. Our theoretical simulation of theproposed experiment with the PbF and ThO molecular beams show that the present constraint onthe e EDM in principle can be improved by a few orders of magnitude.
I. INTRODUCTION
The existence of the electric dipole moment (EDM) for any particle or closed system of particles violates both thespace parity ( P ) and time-reversal ( T ) symmetries [1–3]. Up to date the most stringent experimental constraintsfor the particles’ EDMs are obtained for the electron ( e EDM) due to its strong enhancement in heavy atoms anddiatomic molecules. The most restrictive e EDM bounds were established in experiments with the ThO molecule( | d e | < . × − e cm [4]). Here e is the electron charge. Previously, accurate results were obtained on the Tlatom [5], YbF molecule [6] and HfF + cation [7]. For extraction of the e EDM values from the experimental data,accurate theoretical calculations are required. These calculations were performed for Tl [8–12], for YbF [13–16], forPbF [17–19], for ThO [20–23], and for HfF + [24–27]. In the same experiments it is possible to search for another P , T -odd effect: P , T -odd electron-nucleus interaction [28–30]. Effects originating from this interaction and from e EDM can be observed in an external electric field and cannot be distinguished in any particular atomic or molecularexperiment. However, they can be distinguished in a series of experiments with different species ( see, e.g. [25, 31]).Theoretical predictions of the d e value are rather uncertain. Within the Standard Model (SM) none of thempromises the e EDM value larger than 10 − e cm [32]. However, predictions of the SM extensions are many ordersof magnitude larger [33]. Different models for the P , T -odd interactions within the SM framework are discussed inRefs. [32, 34]. In modern experiments for the P , T -odd effects observation in atomic and molecular systems, eitherthe shift of the magnetic resonance [5] or the electron spin precession [4, 6, 7] in an external electric field is studied.Due to a large gap between the current experimental bound and the maximum SM theoretical prediction, alternativemethods for the observation of the P , T -odd effects are of interest. In Refs. [35, 36], it was mentioned the existence ofthe effect of the optical rotation of linearly polarized light propagating through a medium in an external electric field– the P , T -odd Faraday effect. The possibility of its observation was first studied theoretically and experimentallyin Ref. [37] (see the review on the subject [38]). Recently, a possible observation of the P , T -odd Faraday effectby the intracavity absorption spectroscopy (ICAS) methods [39–41] using atoms was considered [42]. In Ref. [39]an experiment on the observation of the P -odd optical rotation in the Xe, Hg, and I atoms was discussed. Thetechniques [39] are close to what is necessary for the P , T -odd Faraday effect observation. In Refs. [12, 43, 44] anaccurate evaluation of this effect oriented to the application of the techniques [39] was undertaken for the atomic caseand was extended to molecules in Ref. [19]. In the present paper, we consider PbF and ThO for the beam-based ICAS P , T -odd Faraday effect observation. According to our estimates, these molecules are promising candidate systemsfor such type of experiment (see below).As it was shown in earlier works [1], heavy atoms and molecules containing such atoms are promising systems tosearch for the P , T -odd effects. For the case of P , T -odd Faraday effect such systems should also satisfy the followingrequirements. The natural linewidth of the chosen transitions Γ nat (the collisional width is negligible for beam-basedexperiments) should be as small as possible, since it allows for the large saturating intensities at large detuningnecessary for the P , T -odd Faraday experiment (see sections II-III below). In other words, it allows reaching a bettersignal-to-noise ratio in such experiments. For this reason the most suitable are the transition from the ground to themetastable statem, X1 Π / → X2 Π / , in the PbF molecule and the transition from the ground to the metastablestate X Σ → H ∆ in the ThO molecule. The characteristics of these molecules are discussed in Section II. For themolecular case, the applied electric field E ext should be close to the saturating field E sat , which almost completelypolarizes a molecule. For diatomic molecules with total electronic angular momentum projection on the molecular a r X i v : . [ phy s i c s . a t o m - ph ] F e b axis, Ω, equal to 1 /
2, such as PbF, E sat is about 10 V/cm. Such a field can be created only within the space ofabout several centimeters. Diatomic molecules with Ω > / P - and P , T -odd effects was noted in Refs. [29, 36, 45].We can imagine an ICAS-beam experiment for the P , T -odd Faraday rotation observation as follows. A molecularbeam crosses the cavity in a transverse direction. Within the cavity it meets an intracavity laser beam. The crossingof these two beams is located in an electric field oriented along the laser beam. The detection of optical rotation(either using simple polarimetry, or phase-sensitive techniques) happens at the output/transmission of the cavity (thescheme of the proposed experimental setup is given in Fig. 1). FIG. 1: The principle scheme of the proposed experimental setup. A molecular beam crosses the cavity in a transversedirection. Within the cavity it meets an intracavity laser beam. The crossing of the two beams is located in an electric fieldoriented along the laser beam. The detection of the optical rotation happens at the output/transmission of the cavity.
Let us discuss the ultimate ICAS advances necessary for the proposed P , T -odd Faraday experiments. In Ref. [39]a possibility to have a total optical path length of about 100 km in a cavity of 1 m length was considered. Thisresults in 10 passes of the light inside the cavity and 10 reflections of the light from the mirrors. For a molecularbeam-based experiment with a beam of 1 cm in diameter, typical total optical interaction path-lengths are of about1 km, i.e. 10 times smaller. However, in another ICAS experiment [40] an optical path length of 7 × km fora cavity of the same size as in Ref. [39] was reported. This means that 700 times higher light-pass number insidea cavity may become realistic. Another important property of ICAS experiments is the sensitivity of the rotation-angle measurement. Using a cavity-enhanced scheme a shot-noise-limited birefringence-phase-shift sensitivity at the3 × − rad level was demonstrated [41]. We consider the above mentioned parameters used in ICAS experiments toassess the realizability of the proposed P , T -odd Faraday ICAS experiment for the search of the P , T -odd interactionsin molecular physics. II. P , T -ODD FARADAY EXPERIMENT ON MOLECULES The P , T -odd Faraday effect manifests itself as circular birefringence arising from the light propagating through amedium in an external electric field when the P , T -odd interactions are taken into account. Its origin is the same asfor the ordinary Faraday effect in an external magnetic field. In a magnetic field the Zeeman sublevels split in energy.Then the transitions between two states with emission (absorption) of the right (left) circularly polarized photonscorrespond to different frequencies since they occur between different Zeeman sublevels. This causes birefringence,i.e. different refractive indices n ± for the right and left photons. The same happens in an external electric field takinginto account the P , T -odd interactions. In this case, the level splitting is proportional to the linear Stark shift S ∆ .The rotation angle ψ ( ω ) of the light polarization plane for any type of birefringence looks like ψ ( ω ) = π lλ Re (cid:2) n + ( ω ) − n − ( ω ) (cid:3) , (1)where n ± are the refractive indices for the right and left circularly polarized light, l is the optical path length, ω isthe light frequency and λ is the corresponding wavelength. In the P , T -odd Faraday rotation case [12]Re (cid:2) n + ( ω ) − n − ( ω ) (cid:3) = ddω Re [ n ( ω )] S ∆ , (2)where n ( ω ) is the refractive index of linear polarized light. In the case of a completely polarised molecule the linearStark shift of molecular levels is determined by S ∆ = d e E eff , (3)where E eff is the internal molecular effective electric field acting on the electron EDM. If the molecule is not completelypolarized one introduces a corresponding polarization factor that depends on an external electric field E ext . To extract d e from the experimental data it is necessary to know the value of E eff which cannot be measured and should becalculated (see, e.g., Refs. [46, 47]).We evaluated the effective electric fields for the PbF molecule. The effective electric fields in the PbF moleculewere calculated within the relativistic coupled cluster with single, double and noniterative triple cluster amplitudesmethod using the Dirac-Coulomb Hamiltonian [19]. All electrons were included in the correlation treatment. For Pbthe augmented all-electron triple-zeta AAETZ [48] basis set was used. For F the all-electron triple-zeta AETZ [49–51]basis sets were used. The theoretical uncertainty of these calculations can be estimated as 5%. The value of E eff forthe ground electronic state is in good agreement with previous studies [17, 18].The rotation signal R ( ω ) in the experiment reads R ( ω ) = ψ ( ω ) N ev π , (4)where ψ is the rotation angle, N ev is the number of “events” in a statistical experiment. In the case under consideration N ev is the number of photons that had interacted with molecules, and then were detected. In principle, apart fromthe losses in the absorber inside the cavity, we have to take into account also the losses in the cavity itself, i.e. inthe mirrors. In this work we briefly discuss this part of the losses in section V, it changes as a function of intracavitylosses and strongly depends on cavity parameters.Expressed via the spectral characteristics of resonance absorption line the rotation signal reads R ( ω ) = π lλ ρe |(cid:104) i | r | f (cid:105)| h ( u, v ) (cid:126) Γ D × d e ( E i eff + E f eff )Γ D N ev , (5) ω = ω + ∆ ω. (6)Here ρ is the molecular number density, | i (cid:105) and | f (cid:105) are the initial and final states for the resonance transition, r is the electron radius-vector, Γ D is the Doppler width, E i eff and E f eff are the effective fields for the initial and finalstates, ω is the transition frequency, ∆ ω is frequency detuning; (cid:126) , c are the reduced Planck constant and the speedof light. Eq. (5) corresponds to the case of E1 resonant transition. For M1 transitions the factor e |(cid:104) i | r | f (cid:105)| shouldbe replaced by µ |(cid:104) i | l − g S s | f (cid:105)| where s , l are the spin and orbital electron angular momenta operators, respectively, g S = − . g factor and µ is the Bohr magneton. We employ the Voigt parametrization of thespectral line profile [1]: g ( u, v ) = Im F ( u, v ) , (7) f ( u, v ) = Re F ( u, v ) , (8) F ( u, v ) = √ πe − ( u + iv ) [1 − Erf( − i ( u + iv ))] , (9)where Erf( z ) is the error function, u = ∆ ω Γ D , (10)and v = Γ nat D . (11)Γ nat is the natural width. Finally, h ( u, v ) = ddu g ( u, v ) . (12)The comment can be made on the behavior of the spectral line shape for the considered P , T -odd Faraday effect.The behavior of the functions g ( u ) and f ( u ) with v (cid:28) h ( u ) = dgdu with v (cid:28) (a) (b)(c)Fig. 2: Behavior of the functions g ( u ), f ( u ) and h ( u ) with v (cid:28) P -odd), (b) behavior of the inverse of the absorption length L , (c) behavior of the rotation anglefor the Faraday effect (ordinary or P , T -odd). The function g ( u, v ) defines the behavior of the dispersion, the function h ( u, v ) determines the behavior of therotation angle with the detuning. The function f ( u, v ) defines the behavior of absorption and has its maximum at ω .We also introduce L ( ω ) = ( ρσ ( ω )) − – absorption length with some detuning from the resonance. The cross-section σ ( ω ) for the photon absorption by a molecule in case of E1 transition looks like σ ( ω ) = 4 π ω Γ D f ( u, v ) e |(cid:104) i | r | f (cid:105)| (cid:126) c . (13)Expressed via the absorption length at the arbitrary detuning, the rotation signal reads R ( ω ) = h ( u, v ) f ( u, v ) lL ( u, v ) d e ( E i eff + E f eff )2Γ D N ev . (14)The maximum of h ( u, v ) also coincides with ω . However, it has a second maximum [12], which allows to observe the P , T -odd Faraday effect off-resonance, in the region where absorption is small. In the following we choose ∆ ω = 5Γ D .At this detuning the absorption drops down essentially ( f ( u, v ) ∼ v/u ), but the rotation is still close to its secondmaximum ( h ( u, v ) ∼ /u ). Here we do not consider the hyperfine structure. If the hyperfine structure is resolved,it does not change the order-of-magnitude estimate for the rotation angle. However, the choice of certain hyperfinelevels depends on particular experiment. TABLE I: Parameters of transitions under investigation in molecular species. The adopted number density for different speciesis ρ ∼ cm − . Molecule Transition WavelengthLinewidth Effective field Absorption length λ , nm Γ nat , s − E eff , GV/cm L ( u = 5), cmPbF X1 Π / → X2 Π / . × × ThO X Σ → H ∆ × ×
1) One of promising candidates for the ICAS P , T -odd Faraday experiment with diatomic molecules is the PbFmolecule with the X1 Π / → X2 Π / transition ( λ = 1210 nm). The natural linewidth of the X2 state isΓ nat = 2 . × s − [52]. For PbF beam we adopt the transverse temperature of 1 K (e.g., in Ref. [53] the transversetemperature of the supersonic YbF beam was reported to be about 1 K) and the transverse Γ D = 4 . × s − . Ourcalculations give the following effective electric fields values: E eff ( Π +1 / ) = 38 GV/cm and E eff ( Π +3 / ) = 9 . ρ ∼ cm − . Then, according toEq. (13), the absorption length at dimensionless detuning u = 5, L ( u = 5) ∼ × cm. In Table I the parametersof the transition under investigation in PbF are listed.2) Consider the X Σ → H ∆ transition ( λ = 1810 nm) in ThO. This transition lies in the infrared region. Itis interesting to consider such a molecular system since the best constraint on the e EDM was obtained on ThO. Thenatural linewidth of the metastable H state is Γ nat = 5 × s − [4]. The effective electric field for the H state wascalculated in [20–23]. For the ThO beam ( T = 1 K) the transverse Γ D = 2 . × s − . In Ref. [54] the numberdensity of ThO molecular beam was reported to be about ρ ∼ (10 − ) cm − . We adopt the number densityof ThO molecules as ρ ∼ cm − . Then, according to Eq. (13), the absorption length at dimensionless detuning u = 5, L ( u = 5) ∼ × cm. In Table I the parameters of the transition under investigation in ThO are listed.In the following sections we will investigate theoretically in more detail the ICAS-beam P , T -odd Faraday experi-ment on the PbF and ThO molecules with the intensities near the saturation threshold. III. SHOT-NOISE LIMIT AND SATURATION LIMIT
The signal (R) to noise (F) ratio in this section we will write via the number of “events” N ev : RF = ψN ev π √ N ev = ψ √ N ev π . (15)Here ψ is the rotation angle of light polarization plane. The number N ev in the P , T -odd Faraday experiment shouldbe defined as a number of photons which have interacted with molecules and then detected. The total number ofphotons N phot involved in the experiment may be larger than the number of involved molecules N mol , may be smallerthan N mol , may be equal to it. We will be interested in the case when N phot (cid:29) N mol .For shot-noise limited measurement, the condition RF > RF > n times the statistically improved signal-to-noise ratio RF = ψ √ nN ev π , (16)in principle, can be made arbitrary large. This means that the shot-noise limited measurement without observation ofthe angle ψ in any particular measurement, in principle, is also possible. In this case one should collect the statisticsfrom many measurements. The same way of collecting statistics is used in the ACME experiments [4].For the shot-noise limited measurement we have to make the number N ev (i.e. the number of photons) as largeas possible. However, this number is limited by the saturation effects. In section IV we show that for suggestedexperiments we do not reach this limit.For laser beams of high intensity the laws of nonlinear optics should be applied. The refractive index n ( ω ) dependson the intensity of the light I ( ω ) in the following way [55]: n ( ω ) = n ( ω )1 + I ( ω ) /I sat ( ω ) , (17)where n ( ω ) is the refractive index for weak light and I sat ( ω ) is the saturation intensity. When the light intensityexceeds the saturation one, I ( ω ) > I sat ( ω ), both absorption and dispersion decrease. Equation (17) is derived withinthe two-level model of an atom (a molecule) which is valid for the resonant processes of our interest.It is instructive to look at Eq. (17) from the point of view of Einstein relations between the spontaneous andstimulated emission [56]: W st if = W fi = π c (cid:126) ω J ( ω ) W sp if , (18)where W sp if is the spontaneous probability (transition rate) for transition between the initial ( i ) and final ( f ) states(which can be approximated as natural linewidth for the transition Γ nat ), W st if stands for stimulated emission and W fi corresponds to the absorption probability. Equation (18) is written for the polarized anisotropic (laser beam)radiation with frequency ω , J ( ω ) dω = I ( ω ). The dimensionless coefficient at W sp if defines the “number of photons inthe field” N . When a certain transition i → f is considered, dω ∼ Γ nat . Then the number N defines actually therelative importance of the spontaneous and stimulated emission. If N <
1, the spontaneous emission dominates, for
N > I = I sat in Eq. (17) according to Ref. [55] corresponds to N ≈ W st if ≈ W sp if .Taking into account the detuning from the resonance and the Doppler width, one can also represent the stimulatedemission and absorption probabilities in terms of the absorption cross-section as it was done, for instance, in Ref. [57]: W st if = W fi = σ ( ω ) I ( ω ) (cid:126) ω . (19)Then, the saturation intensity which reduces the refractive index down to one-half can be expressed as follows: I sat = (cid:126) ωστ s , (20)where τ s is the saturation time constant (or the effective lifetime or the recovery time). It is the time for the moleculesto become excited and to decay again. This time can be approximated as τ s ≈ (Γ nat ) − . As it was noted in Ref. [57],from Eqs. (19)-(20) clear physical meaning of the saturation intensity follows. It means one photon incident on eachatom or molecule, within its cross-section σ , per the recovery time τ s .Substituting the absorption cross-section from Eq. (13) to Eq. (20) one obtains the expression for the saturationintensity: I sat ( ω, u ) = (cid:126) ω Γ D πc f ( u, v ) . (21)The most important feature is that for any intensity I (cid:62) I s the effect of saturation does not arise instantaneouslyand takes the saturation time t sat ∼ (cid:16) W st if (cid:17) − for its formation. For the off-resonance measurement t sat can be largeenough.It is interesting to compare the resonance and large-detuned cases in terms of signal-to-noise ratio. Then, thefigure-of-merit is as follows. One should consider the next ratio: ψ ( u = 5) (cid:112) I sat ( u = 5) ψ ( u = 0) (cid:112) I sat ( u = 0) ∼ v lL ( u = 5) (cid:114) u v . (22)For the case of ThO, v ∼ − , L ( u = 5) ∼ km (for ρ ∼ cm − ). Then, for two existing cavities withachievable effective optical pathlnegths:1) l = 1 km, according to Eq. (22), the ratio ∼ ;2) l = 700 km, according to Eq. (22), the ratio ∼ .It follows that large detuned case has a great advantage over the resonance one. IV. ICAS-BEAM EXPERIMENT WITH THE NUMBER OF PHOTONS LARGER THAN THENUMBER OF MOLECULES
For large detunings in an ICAS-beam experiment one can have the number of photons much larger than the numberof molecules (as long as
I/I sat (cid:46) l = 2 L ( ω ) [1]) are no longer valid. To zerothorder, we can set N ev ≈ N in Eq. (4) where N is the initial number of photons injected into the cavity. Accordingto Eq. (4) and Eq. (5), the expression for the P , T -odd Faraday rotation angle can be presented as: ψ ( ω ) = ( ρl ) (cid:2) cm − (cid:3) K (cid:104) cm e (cid:105) d e [ e cm] . (23)For the X1 Π / → X2 Π / transition in the PbF molecule, K ≈ × cm /e . In the scenario employed in thispaper there is no optimal condition l = 2 L ( ω ). In principle, the optical pathlength is limited only by the quality ofthe mirrors in a cavity. For ρ ∼ cm − , d e ≈ . × − e cm and optical pathlength l = 1 km (corresponding tocavity [39] intersected by a molecular beam of 1 cm in diameter), according to Eq. (23), one obtains ψ ∼ × − rad.Under the same conditions but with optical pathlength l = 700 km (corresponding to cavity [40] intersected by amolecular beam of 1 cm in diameter), according to Eq. (23), one obtains ψ ∼ − rad. Another thing one shouldworry about is that the experiment cannot last more than the saturation time. However, in our scenario not the timeof experiment but the transit time of a molecule through the laser beam plays a key role.Let us estimate the saturation intensity for the transition under investigation in the PbF molecule. According toEq. (21), for ω = 1 . × s − , Γ D = 4 . × s − , Γ nat = 2 . × s − and u = 5 one gets I sat ( u = 5) = 5 . × W/cm . (24)Such an intensity corresponds to the injection of N ∼ × photons per second through a laser beam cross-sectionof 1 mm . Taking into account Eq. (18), such a saturation intensity corresponds to the case when W st if = W fi ≈ Γ nat = 2 . × s − .The next question is: how many PbF molecules inside the crossing volume are in the excited (X2 Π / ) state if thelaser intensity is equal to the saturation one? For simplicity and without loss of generality we consider the followingstatement of the problem and do not consider any technical issues. The PbF molecular beam of 1 cm in diametertravels through a cavity of 1 m length in a transverse direction with the speed v mol ≈
300 m/s. Continuous laser lightof 1 mm in diameter of the saturation intensity is coupled to the cavity. Then the transit time of the PbF moleculeto pass through the laser beam is τ tr ≈ − s. The fraction of the molecules in the excited state for the case when W fi τ tr (cid:28) − e − W fi τ tr ) ≈ W fi τ tr = Γ nat τ tr ≈ . . (25)It means that if the saturation intensity is coupled to the cavity then only 3% of the total number of PbF moleculesin the crossing volume will be in the excited state.Alternatively, one can define the saturation parameter κ =excitation rate( u ) / relaxation rate. The excitation rate( u )is proportional to the intensity I . At the detuning the excitation rate ( u = 5) / excitation rate ( u = 0) scalesas ∼ f ( u, v ) /f (0 , v ) ∼ v/u (at the considered conditions v/u is a small number). The choice of the saturationintensity I sat corresponds to κ = 1. In this case one has ∼
33% of molecules in the excited state and ∼
67% ofmolecules in the ground state. It means that one does not “bleach” the molecules. However, in our proposal, since1 /τ tr > relaxation rate, we should define the saturation parameter as κ =excitation rate( u ) · τ tr . As a result, in sucha beam-based ICAS experiment, the number of detected photons can be increased by several orders of magnitude.For the ICAS-beam experiment with the ThO molecules, the coefficient K in Eq. (23) is K ≈ × cm /e .Substituting the adopted parameters of ThO ( ω = 1 . × s − , Γ D = 2 . × s − , Γ nat = 5 × s − and u = 5)in Eq. (21), one obtains I sat ( u = 5) = 3 . × W/cm . (26)This value corresponds to the injection of N ∼ × photons per second through the laser cross-section of 1 mm .Estimating, similarly to the the PbF case, the fraction of the molecules in excited state Γ nat τ tr ≈ . N ∼ × photons per second through the laser cross-section ∼ and5% of the total number of ThO molecules in the excited state in the crossing volume.The next question concerns fundamental noises which determine the statistical error of the experiment. The figure-of-merit for the fundamental noise-limited experiment on the molecular spin-precession observation (ACME-style) isas follows: δd e ∼ E eff τ coh (cid:112) ˙ N mol T , (27)where τ coh is the coherence time (a few ms), ˙ N mol is the number of molecules supplied to the experiment by themolecular beam per unit time in the desired initial molecular state and T is the time of the experiment. In the ACMEexperiment the statistics is determined by the number of molecules (molecular spin-noise). The experiment is carriedout on the excited state of a molecule with nonzero total angular momentum (spin).Contrary to this, in our proposed experiment we do not need to prepare molecules in the excited state. Ourexperiment is carried out with molecules in the ground (zero spin) states. In the ground state of the ThO molecule,we don’t have any spin. In turn, excited by a laser state is optically inactive to this laser (provided decoherence effectsare negligible). So there is no molecular spin noise in this case. Note, that taking into account the nuclear spin,the PbF molecule can also formally be considered as a molecule with zero total angular momentum in the groundstate [58]. The excited states are produced in small amounts ( ∼ δd e ∼ E eff τ coh Γ nat L ( u ) c (cid:113) ˙ N phot T , (28)where ˙ N phot is the number of detected photons per unit time and τ coh = l/c is the coherence time in the opticalrotation experiment. The factor N phot is the key difference between our proposal and the ACME-style experiments. V. CAVITY TRANSMISSION IN THE ICAS-BEAM EXPERIMENT
In this section we consider the cavity transmission, T cav , i. e. the transmission of the light determined by theproperties of the mirrors. In principle, the sensitivity of the ICAS experiments strongly depends on the parametersof mirrors. In this paper we only briefly consider the problem of the cavity transmission in the simplest model.Let us consider two identical mirrors with high reflectivity R = 1 − δ , δ ≈ (10 − − − ). Transmission of such aninterferometer can be described as [59]: T cav = I tr I in ≈ − R ) (1 − A/ − R + A ) , (29)where I tr is the transmitted intensity, I in is the initial laser intensity and A is the light intensity loss in the absorberduring one round trip. For the case of PbF beam for one pass ( l ∼ u = 5, A < ρσl ∼ − .Then, A (cid:28) δ and, in principle, can be negligible. In this case, I tr ≈ I in . Note also, that it is possible to choose suchan initial laser intensity, that the coupled intracavity intensity I int = I tr /δ will reach the saturation intensity. Thismeans that the transmitted intensity is now I tr = I sat × δ . The photon shot-noise limit for an ideal polarimeter (see,e.g., the review [38]) is δψ ≈ (cid:112) N phot , (30)where N phot is the number of detected photons.Consider two cases of the existing cavities:1a) The cavity [39] with δ ∼ − , according to Eq. (24), for the PbF case gives I tr = I sat × δ ≈ . × − W/cm .It corresponds to the detection of N phot ∼ × photons per second. Then, in such experiments with the integrationtime of the order of two weeks ∼ s (such an observation time was in the ACME experiments), the number ofdetected photons is N phot ∼ × . According to Eq. (30), in this case δψ ∼ − rad. According to Eq. (23) forsuch a cavity and the recent ACME experimental bound on the e EDM value, ψ ∼ − rad. As a result, for suchparameters PbF is a candidate to verify the recent ACME results via the alternative method.1b) The cavity [39] with δ ∼ − , for the ThO case, gives N × δ ≈ × × − ≈ × detected photonsper second. Then, in such experiments with the integration time on the order of two weeks ∼ s, the number ofdetected photons is N phot ∼ × . According to Eq. (30), in this case δψ ∼ × − rad. According to Eq. (23)for such a cavity and the recent ACME experimental bound on the e EDM value, ψ ∼ ∼ − rad. As a result,ThO is a good candidate for improving the e EDM bound by 1 order of magnitude.2a) The cavity [40] with δ ∼ − , according to Eq. (24), for the PbF case gives I tr = I sat × δ ≈ . × − W/cm .It corresponds to the detection of N phot ∼ × photons per second. Then, in such experiments with the integrationtime on the order of two weeks ∼ s, the number of detected photons is N phot ∼ × . According to Eq. (30),in this case δψ ∼ − rad. According to Eq. (23) for such a cavity and the recent ACME experimental bound onthe e EDM value, ψ ∼ − rad. As a result, PbF is a good candidate for improving the e EDM bound by 2 orders ofmagnitude in this case.2b) The cavity [40] with δ ∼ − , for the ThO case, gives N × δ ≈ × × − ≈ × detected photonsper second. Then, in such experiments with the integration time on the order of two weeks ∼ s, the number ofdetected photons is N phot ∼ × . According to Eq. (30), in this case δψ ∼ × − rad. According to Eq. (23)for such a cavity and the recent ACME experimental bound on the e EDM value, ψ ∼ × − rad. As a result, ThOis a good candidate for improving the e EDM bound by 3 orders of magnitude in this case.In conclusion of the section, we comment on the possible sources of improving the P , T -odd Faraday signal-to-noiseratio. Note, that according to Eq. (5), the rotation angle is proportional to ψ ∼ h ( u, v ) / Γ D . For large dimensionlessdetunings u (e.g., u = 5), ψ ∼ / ( u Γ D ) . For the case where the number of photons is much larger than the number ofmolecules (near the saturation threshold), one can neglect the absorption of photons. Then, it is no longer necessaryto make such a large detuning. However, even at the detuning u = 1 . h ( u, v ) function)where h ( u, v ) ≈ . ψ ∼ (1 / × / Γ D . Thus, the rotation angle enhances by a factor of 5, but the shot-noise(connected in our case with the saturation intensity Eq. (21) which depends on u ) drops down by a factor of ∼ √ P , T -odd Faraday rotation is determined,among other things, by the largest of the widths (natural, collisional, transit-time, Doppler, etc.). For instance, forthe PbF molecular beam case, the largest width is the Doppler one (Γ nat = 2 . × s − , the transit-time widthΓ tr ∼ / (2 πτ tr ) ≈ . × s − , Γ D = 4 . × s − ). Finally, we would like to mention that with squeezed statesof light the photon shot-noise limit can be surpassed which would be favorable for the P , T -odd Faraday effectobservation. However, these squeezed states of light have not yet found their application in polarimetry. VI. CONCLUSIONS
The recent most advanced e EDM constraint obtained in the experiment with ThO is | d e | < . × − e cm.In this experiment, electron spin precession in an external electric field is employed and the effect is proportionalto the time spent by a particular molecule in an electric field. In the present paper we suggest another method forobservation of such effects – a beam-ICAS P , T -odd Faraday experiment with molecules. A theoretical simulation ofthe proposed experiment is based on the recently available ICAS parameters. In this experiment it is not necessaryto keep a separate molecule in an electric field since the effect is accumulated in the laser beam which encountersmany molecules. According to our estimates for the PbF molecule, the current e EDM sensitivity can be improvedby 1-2 orders of magnitude. In its turn, for the ThO molecule the current e EDM sensitivity can be improved by 1-3orders of magnitude. This implies testing of new particles at energy 1-2 order of magnitude larger than the currentbest constraint.Making these predictions we understand that some technical problems, not mentioned here, may arise. In this paperwe did not discuss the possible systematic errors among which the stray magnetic fields, the uncontrolled ellipticityof the laser beam and uncontrolled drift of mirrors are the most evident. A problem of avoiding the P -odd opticalrotation, much stronger than the P , T -odd rotation also should be resolved. All these problems we hope to addressin the future studies. Acknowledgments
Preparing the paper and the calculations of the P , T -odd Faraday signals, as well as finding optimal parameters forexperiment were supported by the Russian Science Foundation grant 17-12-01035. L.V.S. acknowledges the support ofthe Foundation for the advancement of theoretical physics and mathematics “BASIS” grant according to the researchprojects No. 18-1-3-55-1. D.V.C. acknowledges the support of the President of Russian Federation Stipend No. SP-1213.2021.2. The authors would like to thank Dr. Dmitry Budker, Dr. Mikhail G. Kozlov, Dr. L. Bougas, Dr. Timur0A. Isaev and Dr. Peter Rakitzis for helpful discussions. References [1] I.B. Khriplovich, Parity Nonconservation in Atomic Phenomena, Gordon and Breach, London, 1991[2] J.S. Ginges and V.V. Flambaum, Phys. Rep. , 63 (2004)[3] M.S. Safronova, D. Budker, D. DeMille, D.F.J. Kimball, A. Derevianko, C.W. Clark, Rev. Mod. Phys. , 025008 (2018)[4] V. Andreev et al. (ACME collaboration), Nature , 355 (2018)[5] B.C. Regan, E.D. 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