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Feasibility of the optical fiber clock Ekaterina Ilinova1 , James F. Babb2 , and Andrei Derevianko1 1 Department of Physics, University of Nevada, Reno, Nevada 89557, USA and Harvard-Smithsonian Center for Astrophysics, 60 Garden St., MS 14, Cambridge, Massachusetts 02138, USA We explore the feasibility of compact high-precision Hg atomic clock based on the hollow core optical fiber. We evaluate the sensitivity of the 1 S0 -3 P0 clock transition in Hg and other divalent atoms to the fiber inner core surface at non-zero temperatures. The Casimir-Polder interaction induced 1 S0 -3 P0 transition frequency shift is calculated for the atom inside the hollow capillary as a function of atomic position, capillary material, and geometric parameters. For 199 Hg atoms on the axis of silica capillary with inner radius ≥ 15 µm and optimally chosen thickness d ∼ 1 µm, the atom-surface interaction induced 1 S0 -3 P0 clock transition frequency shift can be kept on the level δν/νHg ∼ 10−19 . PACS numbers: 37.10.De, 37.10.Gh, 42.50.Wk INTRODUCTION 𝜺1 !"# 𝜺2 !" !"#$ 𝜺=1 !)* !" %& !! ($in (out % !$# ! %!& "'# $% $ $ & & %! # #*% &' $% !"!#$%&' %! (* , ! '! $' (,.#$%&' $" $' ()*#+%&' $ -+!#)%&' arXiv:1705.05945v1 [physics.atom-ph] 16 May 2017 2 FIG. 1: (a) Cold atoms optically trapped in the ground vibrational state inside the hollow core dielectric cylinder. (b) Energy level diagram for Hg atoms. The clock transition is shown with the dashed straight arrow. Dashed lines show the virtual transitions contributing to the resonant part of the Casimir-Polder interaction potential. The rapid progress in micro- and nano-fabrication technologies and advances in material science together with the growing demand for precision metrology and quantum information processing gave rise to a new branch of research engineering aiming to develop miniaturized quantum devices. Examples include atomic clocks [1–4], atomic sensors and interferometers [5–9], and quantum logic gates [10]. In particular, development of miniaturized atomic clocks is anticipated to enable important applications requiring portability and low power consumption: secure telecommunications [11, 12], mobile timing, navigation [13–15], and deep space atomic clocks [16]. The main platforms for compact quantum devices are micro-electromechanical systems [17, 18], miniaturized wire traps [19], and, more recently, hollow core optical fibers [20]. Here we analyze the feasibility of a “fiber clock”—a device that holds interrogated clock atoms inside a hollow core optical fiber. The atoms are optically trapped, avoiding the collisions with the wall surface [20]. The “fiber clock” is a natural extension of optical lattice clocks [21, 22], which recently reached record levels of fractional inaccuracy at the 10−18 level [23]. The clock transition is the narrow 1 S0 -3 P0 transition present in divalent atoms (e.g. Sr, Yb, Ca, Mg, and Hg). Unlike in the free space configuration, where the maximum interaction length does not exceed the Rayleigh length zR = πw02 /λ (here w0 is the characteristic Gaussian beam waist radius and λ is the wavelength of the laser field ), the fiber clock setup does not suffer from this limitation [20]. Large ensembles of cold atoms trapped in 1D optical lattice can be thus realized within the compact transverse region ∼ 5 − 100 µm, avoiding high atomic density per lattice site. In pursuing a super-precise miniature clock one also needs to keep in mind miniaturization of other clock elements. In this context it is important to note that the “fiber clock” can be easily integrated into the all-fiber framework. Moreover one could envision the enchanced superradiance [24] of 1D trapped atomic ensemble into the fiber guided laser mode. Optical trapping of large ensembles (Na ≫ 1) of cold divalent atoms uniformly distributed over the length L ∼ N λlattice /2 inside the hollow core fiber and suppressing the associated sources of atomic dipole relaxation (other than the radiative decay) will be the next step toward the ultra-narrow linewidth radiation source at 1 S0 -3 P0 clock transition frequency. 2 Super-radiant lasing on the clock transition [25, 26] is a potential alternative to space-consuming bulky reference cavities used in optical clocks. Here we theoretically evaluate the possibility of building the optical lattice clock based on the narrow 1 S0 -3 P0 transition in Hg and other alkaline-earth-like atoms (Cd, Mg, Yb, Sr) optically trapped inside a hollow core fiber. One of the first experimental efforts towards compact optical clocks involved the 3D trapping of an ensemble of Sr atoms in a micron-sized structure [18]. More recently precision spectroscopy of the 1 S0 - 3 P1 transition of Sr atoms optically trapped inside the Kagome fiber has been performed [20]. To the best of our knowledge no 1 S0 - 3 P0 transition based optical lattice clock has been realized with cold atoms inside a hollow core fiber. A mercury clock [27, 28] is of particular interest in applications to the precision tests of fundamental symmetries and probing physics beyond the standard model due to the large value of Hg charge. The relatively low static polarizability of Hg and Cd [27] makes them least sensitive to black body radiation (BBR), as compared to the other atoms Sr, Yb, Ca. This pecularity makes Hg and Cd good candidates for optical clock applications where the ambient temperature is difficult to control, making the BBR effects main constraint on the clock accuracy [29]. The development of a fiber-based atomic clock requires detailed understanding of the effects of surrounding core surface on the shift of clock transition frequency. The atom-surface interactio generally depends on the geometry and material of the fiber. At micron scale inner core dimensions, the role of its shape (i.e., multi-layered cladding structure, mean radius, thickness of the inner interfaces) may become important. In [30] the CasimirPolder (CP) interaction of Rydberg atoms with a cylindrical cavity was analyzed. It was shown that at certain cavity radi there is an enhancement of modes resonant with atomic transitions may occur, leading to the increase of the resonant part of CP interaction potential and to the modification of the atomic radiative decay rate [30, 31]. Considering the high precision required for the optical clock, we study these resonant effects on the ground 1 S0 and metastable 3 P0 clock states of Hg and other alkaline-earth-like atoms. We calculate the general form of the long range atom-surface interaction potential at non-zero temperatures for the hollow core cylindrical geometry and analyze the resulting 1 S0 -3 P0 clock frequency shift and broadening as a function of parameters of the surface interface. We find that for an ensemble of Hg atoms optically trapped near the axis of hollow silica capillary waveguide with inner radius Rin ≥ 15µm, the Casimir Polder interaction induced fractional frequency shift can be suppressed down to the level of δν/ν ∼ 10−19 . The frequency shift due to the nonresonant part of CP interaction in this case is dominant compared to the resonant. It decreases with the growth of the inner core radius. At high relative permittivity of the inner core surface material εr ≫ 1 the contribution of resonant atom-surface interaction as well as the radiative decay rate enhancement (Purcell effect) may become dominant at certain choice of the geometric parameters of the waveguide. Both clock transition fequency shift caused by the resonant part of CP interaction potential and the Purcell effect can be suppressed by slight adjustment of the thickness d at given inner core radius Rin . The adjustment has to be done in order to avoid the resonant waveguide modes at the frequencies of the decay channels. For more complex waveguide geometries the resonances of the atom-surface interaction as a function of the waveguide parameters can be more difficult to predict. 1 S0 -3 P0 CLOCK TRANSITION INSIDE THE HOLLOW CORE FIBER The probability of BBR induced transitions from the clock levels to the other atomic states is small over the typical clock operation time and can be neglected. That is the clock atoms are not in thermal equilibrium with the BBR bath [32]. The CP interaction induced shift of the clock frequency has to be found as the difference of free energy shifts of the clock levels [30, 32]: δFa (r) = 1X (n(|ωna |) − Θ(ωan )(1 + 2n(|ωan |))|µan |2 × 3 n Tr[ReG(r, r, |ωna |)] − kB T ∞ ′ X Tr[G(r, r, iξk )]αan (iξk ),(1) k=0 where ξk = 2πkB T · k/h is the k-th Matsubara frequency, µan is the dipole matrix element of transition a → n, G(r, r, ω) is the classical Green tensor [33–35] for a given waveguide geometry, ρ is the radial coordinate of an atom inside the cylinder and the prime on the Matsubara sum indicates that the j = 0 term is to be taken with half weight. We approximate the geometry of the fiber inner interface by a dielectric capillary of a given thickness d = Rout − Rin , see Fig. 1(a), where Rin and Rout are the inner and outer radii of the capillary. We consider the surface in thermal equilibrium with black body radiation at given temperature TS . The mean occupation number n of photons with energy ~ωna is given by the Bose-Einstein distribution, n(ωna ) = (e~ωna /kB TS − 1)−1 . Cold atoms prepared in a given state (either 1 S0 or 3 P0 ) are trapped by the red-detuned standing wave forming the 1D optical lattice inside the fiber, see Fig. 1. The “magic” wavelength of the laser field forming the lattice λL = 360 nm ([27]) is chosen in order to cancel the differential ac Stark shift for the clock transition. The atomic ensemble temperature can be reduced down to ∼ few nK[27], decreasing the trapping potential depth to U ∼ 10ER , where ER is the recoil energy. Atoms 3 trapped in the ground vibrational state are located near the lattice nodes along the axis of a cylinder (ρ = 0), where they are most distant from the dielectric walls. 0.7 10 2 0.3 -1 0.1 8 -𝛿𝜈/𝜈×10 0 -𝛿𝜈/𝜈 ×1018 1 !"#$%$&'!(#)%$&*+ 18 0.5 d, 𝜇m resonant Casimir-Polder interaction induced clock transition frequency shift in Hg atoms on the capillary axis, at surface temperature TS = 293 K and inner core radius Rin = 15 µm is δν/νHg ∼ −2 × 10−18 . It can be further reduced by choosing the optimal values of waveguide inner radius and thickness. The Green function is given #"! 6 2 !"! 8 10 12 14 #$ 0 !!"' 105 6 !" 4 !"' (a) 110 115 '#)125 120 !"!!!#! !"!!!#%!"!!!#&!"!!!#' d, nm ! !"!!!##!"!!!#$ !"# 16 ,%!- Rin , 𝜇m #"! !"!"#$ FIG. 2: (a) The 199 Hg 1 S0 -3 P0 clock transition frequency shift due to the resonant part of atom interaction with the inner surface of a silica capillary as a fuction of its thickness d and inner radius Rin on the capillary axis ρ = 0. !"!"#$ !"* !"( !"!""#$ !"& !"$ !"! !"! '() !"# !"$ !"% !"& !"' !"( !") 𝜌, μm Resonant atom-surface interaction The first sum over the Matzubara frequencies in Eq. (1) corresponds to virtual dipole absorption and emission of thermal photons. In presence of surface polaritons [32] or the resonant waveguide modes at the atomic transitions frequencies ωna these terms may become resonant [16]. The dashed lines in Fig. 1(b) indicate the virtual transitions contributing to the resonant part of the CP interaction potential. The term arising from virtual emis1 sion 3 P0 S0 remains finite even at zero temperature, although it is strongly suppressed due to the small values of the corresponding dipole matrix elements for nonzero nuclear spin isotopes (which vanishes for nuclear spin-zero isotopes). The other contributions come from virtual absorption of thermal photons via the 3 P0 -3 S1 , 3 P0 -3 D1 ,1 S0 -1 P1 , and 1 S0 -3 P1,0 transitions. Their amplitudes are suppressed exponentially with decreasing surface temperature. Fig. 2 shows the fractional frequency shift of the clock transition on the capillary axis (ρ = 0) caused by the resonant part of the CP interaction potential at the surface temperature TS = 293 K. One can see the resonant structure in its dependence on the capillary thickness parameter d and inner radius Rin . For the vacuum-silica-air interface the cavity effect is small as compared to the case of highly reflective capillary material. Also, there are no resonances in the relative permitivity εsilica corresponding to transitions from 1 S0 or 3 P0 states in divalent atoms. The upper limit on the %#%&!%#"#$%& FIG. 3: Resonant atom-surface interaction induced 1 S0 -3 P0 transition frequency shift in Hg atoms on the axis of silica capillary with inner radius Rin = 15.5 µm as a function of thickness parameter d at two different values of electric permitivity εr = 2.45 (blue) and εr = 11. (red). (b) The normalized value of A = ρ∆ν(ρ)|ψ(ρ)|2 (purple solid line) as a function of atom position ρ inside the capillary at the inner core radius Rin = 15 µm and the thickness d = 1 µm. The solid red and blue dashed curves show the normalized values of the frequency shift ∆ν/∆νmax and the amplitude of the ground vibrational state wave function for the atom inside the red detuned optical lattice with trapping potential depth U = 180 µK. [35] by: Z∞ X ∞ ′ q2 iω 2 Tr[ReG(r, r, ω)] = r + r × dq M N 2πc2 k2 m=0 0 2 m η2 2 2 ′ 2 J (ηρ) + Jm (ηρ) + rN 2 Jm (ηρ) ,(2) η 2 ρ2 m k where rM,N (n, q) are the functions of the frequency ω and the surface parameters: (Rin , d, ε). The resonances of rM,N coefficients at q = 0 determine the true resonances of the Green function. On the axis ρ = 0 the resonant waveguide mode at the given frequency ω can 4 be suppressed when minimizing the corresponding sum Im[rN (0, 0) + rM (1, 0)/2]. (a) Sr Log[-𝛿𝜈/ν×10 18] 10 8 8 6 4 4 2 0 0 !2 !4 -4 Mg Cd Hg 5 10 15 (b) 2.95 2.53 2.5 25 25 µm 3.52 Rin, 20 20 4 2 5 10 20 15 Rin, -4 .0 2 0.5 -3 .5 -2. 5 Hg -4 -3. 0 -1.5 -2 -2.0 -0.51 0 -1.0 0.47 1.0 -0.003 1.03 µm d, 1.5 Log[-𝛿𝜈/ν×10 18] 1.49 2.01 2.0 25 µm FIG. 4: (a) The frequency shift of the 1 S0 -3 P0 transition in Hg (blue), Cd (green), Mg (orange) and in Sr (red) atoms originating from the nonresonant part of Casimir-Polder interaction potential. The atoms are assumed to be on the capillary axis. The temperature of the capillary is TS = 77 K and its thickness d = 1 µm. (b) The dependence of 1 S0 -3 P0 transition frequency shift (non resonant part) in Hg atom on the waveguide inner radius Rin and thickness d. the capillary thickness and reducing its inner radius also leads to the larger frequency shifts. In summary, we have studied the possibility of creating the optical lattice clock based on the 1 S0 -3 P0 transition in Hg, Cd, Mg and Sr atoms optically trapped inside the micron-scale hollow core waveguide. The effect of the atom-surface interaction on the clock transition frequency stability at non zero surface temperature has been evaluated. For ensemble of cold Sr atoms, with Ta ∼ µK, optically trapped on the axis of a silica capillary (TS = 293 K) the main contribution to the 1 S0 -3 P0 transition frequency shift comes from nonresonant part of Casimir-Polder interaction potential. For Hg atoms it is reduced by the factor ∼ δαHg (0)/δαSr (0). The resonant effects which may reinforce the 1 S0 -3 P0 decay channel and produce additional frequency shifts can be minimized by proper choice of geometric parameters of the waveguide. For the silica capillary the “resonant” effects are generally small as compared to materials with high electric permittivity εr ≫ 1. In the later case one could consider the possibility to compensate the nonresonant part of Casimir-Polder interaction on the core axis in combination with its resonant part at certain geometries of the waveguide. Additional effects may appear in case of the resonances in dielectric constants of waveguide material. As an applications one could consider the possibility of using the compact Hg or Cd optical fiber clock for detecting the dark matter objects. So, for example, in [36] the method of dark matter detection using a single optical lattice clock has been proposed. The metod is based on the difference of the susceptibilities of the atomic transition frequency and the resonant cavity mode to the variation of the fine structure constant. The resonant part of the Casimir polder interaction will be sensitive to such difference. Thus the “dark matter objects” can be detected by measuring the α -variation induced clock transition frequency variations originating from the resonant part of atom-surface interaction. All the other systematic effects leading to clock transition frequency uncertainty need to be well suppressed in order such variations to be detcetectable. Nonresonant Casimir-Polder interaction potential LIFETIME OF THE ATOMS IN A TRAP The last sum in Eq. (1) corresponds to nonresonant quantum fluctuations of the atomic dipole. Away from resonances and at low temperatures this term is the primary contributor to the atomic energy level shift. Fig. 4 shows the calculated clock transition frequency shift for the Hg, Sr, Cd and Mg atoms inside the silica capillary (on the axis ρ = 0) at temperature TS = 77 K. The least sensitive to the nonresonant atom surface interaction is the Hg atom. The clock shift is larger for Cd, Mg, Sr atoms wich have larger differential static polarizability δα = α3 P0 − α1 S0 between the clock states. Increasing Collisions of cold atoms with the background gas molecules inside the fiber lead to their heating and eventual escape from the optical dipole trap. Considering that the average kinetic energy of the incident buffer gas molecule Ebg is much larger than the one of the trapped atoms and that the scattering process (inside the shallow optical lattice) happens at relatively large internuclear distances R (so that |Vcol (R) = −C6 /R6 | ≪ Ebg ), one can consider the energy exchange between the two particles using the impulse approximation √ [37]. The resulting loss rate is given by γloss = 2 πn · 5 2kB Tb 1/3 M 15πC 6 √ 8 2mU 1/3 Γ 11 6 , where n is the buffer gas concentration, M and m are, respectively, the masses of the buffer gas molecule and trapped atom, U is the trapping potential depth, and Tb is the temperature of the buffer gas. Compared to the macroscopic vacuum chamber, achieving high vacuum in hollow core fiber is more challenging due to the small inner core size. The residual buffer gas pressure in the experiments with cold atoms inside the hollow core fiber [20, 38] is Pbg ∼ 10−6 Pa. The collisions of cold Hg atoms with molecular nitrogen N2 at buffer gas pressure PN2 = 10−6 Pa, temperature TN2 = 293 K and trapping potential depth U = 180 µK results in a Hg loss rate γloss = 0.53 1/s. In our calculations we used the van der Waals interaction constants C6 given in [39]. The collisional heating of the trapped atoms dTHg M2m 3 = 2π kB (M+m) can be estimated as 2 vr n · dt Rπ σ(χ) cos χ2 dχ, where vr is the most probable π−2θmin speed of the buffer gas molecule, σ(χ) = σ(χ)|4 sin χ2 |, σ(χ) [37] is the differential collision cross section, χ = π − 2θ, θmin = cos−1 (1 − µUm 2 v 2 ) is the minimum scatr tering angle corresponding to the escape of initially mM trapped atom from the optical lattice, and µ = m+M is the reduced mass. Using the impulse approxima 13 5π(m+M)C6 1 . tion [37] one can obtain: σ(χ) = 12 χ 4mEbg sin2 ( ) dT 2 2 mM The corresponding heating rate is dtHg = π5 kB (m+M) 2 · 31 5 6 3 1 − (1 − µUm . At buffer gas panvr3 5π(m+M)C 2 v2 ) 4mEbg r rameters the same as mentioned above, the calculated dT collisional heating rate is dtHg = 5.6 µ K/s. To calculate the heating and loss rates for Sr atom we estimated I I αSr αbg the van der Waals coefficient as C6,Sr = 32 ISrSr+Ibgbg (4πǫ 2, 0) where αSr,bg and ISr,bg are the polarizabilities and ionization potentials of the colliding Sr atom and buffer gas molecule. The ionization energy of molecular nitrogen is IN2 = 1503 kJ mol−1. The polarizability [40] αN2 = 11.74 a.u. The corresponding Sr loss rate is γSr,loss = 1.03 1/s, dTSr dt = 9.87 µK/s. The residual buffer gas density inside the fiber can be further reduced using light induced desorption [41] together with ultra high vacuum pumping techniques. The rate of spontaneous scattering of the photons from the lattice field (λ = 360 nm) by Hg atoms trapped in Ω2Rabi the ground 1 S0 state is given by γscat ∼ 4∆ 2 [42], where ΩRabi is the Rabi frequency of the 1 S0 -3 P1 transition, ∆ = ωres − ωlattice . Considering the large detuning between the frequencies of the lattice field and the resonant frequency of 1 S0 -3 P1 transition, such that ∆ ≫ ΩRabi , the heating due to the spontaneous scattering can be neglected. Another source of heating comes from the lattice intensity noise [43]. The corresponding heating rate is given by dT dt = P n Pn (n + 21 )~Ωtrap · Ω2trap 4 · Sǫ (2νtrap ), where Sǫ (2νtrap ) is the intensity noise power spectrum, n is the vibrational state of the trapped atom inside the lattice, and Pn is the population of the given vibrational state n. Taking the argon-ion laser noise spectrum [43] as an example, we estimate the lattice intendT noise Hg = sity noise produced heating rate of Hg atoms: dt 3 √ −1 2 161I0 Sǫ (2 × 34.29 I0 kHz)K s , where I0 is the ampli2 tude of the lattice field intensity, measured in kW/cm . At lattice potential depth U ∼ 10ER,Hg = 3.67 µK , with the atoms being in the ground vibrational state n = 0, dT noise the corresponding heating rate Hg is reduced down dt to 76 nKs−1 , where ER,Hg is the recoil energy for Hg. For Sr the lattice intensity noise induced heating (in the 3 √ dT noise 1 = 1302I02 Sǫ (2 × 15.45 I0 kHz) Ks−1 . S0 state) is Sr dt At trapping potential U ∼ 10ER,Sr = 1.65 µK, we find noise dTSr dt = 8.7 nKs−1 . The residual birefringence of the fiber causes the nonuniformity of the polarization along the lattice. This results in additional clock transition frequency uncertainty. The 87 Sr clock transition frequency shift due to contribution of the vector and tensor polarizabilities was presented in [44] as ∆νv,t = (0.22Hz · mF ξ(ek · eB ) − 0.0577 mHz · β)U/ER , where U is the lattice potential depth, ER is the recoil energy, ek ,eB are the unitary vectors along the lattice wave vector and the quantization axis, β = (3|e · eB |2 − 1)[3m2F − F (F + 1)], e is the complex polarization vector, and ξ is the degree of the ellipticity of the lattice field. Taking (ek · eB ) = 1, ∆ξ ∼ π∆n λL · L ≪ 1, where ∆n is the difference of the refractive indexes for two orthogonal polarizations and L is the length of the atomic cloud inside the fiber, one can estimate the vector polarizability induced frequency shift uncertainty as δνv,mF ∼ λπL mF ∆nL · 0.22 Hz. For U −7 mF = 9/2, ∆n ∼ 10−7 [20], δνv 92 = 3.1 Hz · L . λ ER · 10 For L = 8.13 cm, U = 10 ER , δνv 92 = 0.3 Hz. The vector light-shifts of the components mF = ± 29 have the opposite signs. 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