Download Feasibility of the optical fiber clock

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
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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.
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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
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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.
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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
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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. Zeeman shift and vector-light shift cancellation techniques have been developed in [45] based on the
averaging of frequency measurements for two transitions
(1 S0 , F = 92 , mF = ± 29 ) − (3 P0 , F = 29 , mF = ± 92 ).
For the same parameters ∆n, L, U , and (ek · eB ) = 0,
(e · eB ) = 1, δνt 29 = 41.5 mHz.
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