Higher-order constraints on precision of
the time-frequency metrology of atoms
in optical lattices
V. D. Ovsiannikov
Physics Department, Voronezh State University, Universitetskaya pl. 1, Voronezh, 394006, Russia
V. G. Pal'chikov
Institute of Metrology for Time and Space at National Research Institute for Physical--Technical and
Radiotechnical Measurements, Mendeleevo, Moscow Region 141579, Russia
FFK-14, Dubna, December 3,
2014
1
Contents
1.
2.
a)
b)
c)
d)
e)
f)
g)
h)
3.
a)
b)
c)
Principal goal: to determine irremovable clock-frequency shifts induced by multipole, nonlinear
and anharmonic interaction of neutral Sr, Yb and Hg atoms with an optical lattice of a magic
wavelength (MWL) .
Attractive lattice of a Red-detuned MWL:
Spatial distribution of atom-lattice interaction.
Lattice potential wells.
Lattice-induced clock-frequency shift.
Numerical estimates of electromagnetic susceptibilities and clock-frequency shifts of neutral Sr,
Yb and Hg atoms in a lattice of a red-detuned MWL.
MWL for an atom in a traveling wave (TW).
MWL for an atom in a standing wave (SW).
MWL for equal dipole polarizabilities (EDP) in ground and excited clock states
MWL precision.
Elimination of nonlinear effects in a lattice of Sr blue-detuned MWL of λm=389.889 nm.
Spatial distribution of interaction between atom and a repulsive lattice.
Motion-insensitive standing-wave MWL (SW MWL).
Numerical estimates of the blue-detuned-lattice-induced shifts
FFK-14, Dubna, December 3,
2014
2
Typical structure of energy levels in alkaline-earth and alkaline-earth-like
atoms (Mg, Ca, Sr, Zn, Cd, Yb, Hg)
1
P1
Radiation transitions between metastable
and ground states, stimulated in odd
3
3
isotopes by the hyperfine interaction, is
P2
This prohibition makes extremely narrow
3
1
the line of the clock transition,
0
0
which may be stimulated by an external
D1
3
3
S1
P1
M2
3
«Clock»
transition
P0
strictly forbidden in even isotopes.
PS
magnetic field or by the circularly polarized
lattice wave. This transition may be used as
an oscillator with extremely high quality
Q   cl /   1017
2ω(M1+E1)
E1
1
S0
The width
(ΔS=1)

of the oscillator depends on
(and may be regulated by) the intensity of
the lattice wave or a static magnetic field.
FFK-14, Dubna, December 3,
2014
3
Natural isotope composition
Even isotopes
(J=0) abundance
24,26Mg:
90%
40→48Ca:
98.7%
84,86,88Sr:
93%
168→176Yb: 73%
196→204Hg: 69.8%
106→116Cd: 75%
64→70Zn:
95.9%
Odd isotopes
abundance
(J≠0)
25Mg: 10% (J=5/2)
43Ca: 1.3% (J=7/2)
87Sr: 7%
(J=9/2)
171,173Yb: 27% (J=1/2, 5/2)
199,201Hg: 30.2% (J=1/2,3/2)
111,113Cd: 25%
(J=1/2)
67Zn: 4.1%
(J=5/2)
FFK-14, Dubna, December 3,
2014
4
2. Red-detuned MWL
2.a) Spatial distribution of atom-lattice interaction
E( X , t )  2E0 cos(kX )cos(t ),

k

c

2


Vˆ ( X , t )  Re Vˆ ( X ) exp(it )
Vˆ ( X )  VˆE1 cos(kX )  (VˆE 2  VˆM1 )sin(kX )
VˆE1  (r  E0 );
 2
VˆE 2 
r E0  n2  C2 ( ,  )  ;
6
FFK-14, Dubna, December 3,
2014

VˆM 1  [n  E0 ]  (Jˆ  Sˆ )
2

5

Eglatt(e) ( X )  Eg(2)(e) ( X )  Eg(4)(e) ( X )  ...
E
(2)
g (e)
†
 ˆ
 ˆ †
ˆ
ˆ
( X )   g (e) V ( X )G V ( X )  V ( X )G V ( X ) g (e)
   gE(1e ) ( ) cos 2 (kX )   gqm( e ) ( ) sin 2 ( kX )  I
 gqm( e ) ( )   gE(2e ) ( )   gM( e1) ( ); I is the laser intensity:
2 I is the mean intensity of a standing wave,
0 is the intensity of the node,
4 I is the intensity of antinode.
Eg(4)(e) ( X )  g (e) ()cos4 (kX ) I 2 .
FFK-14, Dubna, December 3,
2014
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2b) Lattice potential wells.
Clock-level shift is the Lattice-trap potential energy
E
latt
g ( e)
(X ) U
latt
g ( e)
Dg ( e ) ( ,  , I )  
( X )  Dg (e)  U
E1
g (e)
( harm)
g (e)
X U
2
( anh)
g ( e)
4
X ;
( ) I   g ( e ) ( ,  ) I ,  depth
2
Mat ( ,  , I )
U
  ( ) I  2 g ( e ) ( ,  ) I  k 
,
2
4
k
( anh )
dqm
2

U g ( e ) ( ,  , I )   g ( e ) ( ) I  5 g ( e) ( ,  ) I 
3
2
( harm )
g (e)
dqm
g (e)
2
FFK-14, Dubna, December 3,
2014
2
7
X / L
5
4
3
2
1
n=0
U glatt(e) ( X ) / Dg (e)
L / 2
Stark-trap potential and vibration-state energies of an atom in a standing
wave of a lattice field
FFK-14, Dubna, December 3,
2014
8
2
ˆ
P

at
latt
at
Hˆ g ( e)(X )=
 U g ( e) ( X ), Pˆat  i
;
2Mat
X
ˆ atg (e)(X)n(X)= Evib
H
g ( e) n(X)
1
 1  anh
 2
E (,  , I , n)   Dg (e) (,  , I )  g (e) (,  , I )  n   -Eg (e) (,  , I )  n  n  
2
2


anharmonic energy
depth
harmonic oscillations
vib
g (e)
rec
E
anh
Eg ( e ) ( ,  , I ) 
2
rec
t
E

2
2Matc
2
 3 g ( e ) ( ,  ) I 
1 
;
dqm
 g ( e ) ( ) 

is the recoil energy of a lattice photon
FFK-14, Dubna, December 3,
2014
9
latt
g (e)
E
E
(0)
g ( e)
E
vib
g ( e)
(,  , I , n)
The strict magic-wavelength condition should imply the equality
vib
Evib
(

,

,
I
,
n
)

E
e
mag
g (mag ,  , I , n)
To hold this condition, the equality should hold for the susceptibilities:
E1
qm
 gdqm
(

)


(

)


(e)
g (e)
g ( e ) ( );

qm
g (e)
( )  
E2
g (e)
( )  
M1
g ( e)
( );
 g ( e ) ( ,  )   glin( e ) ( )   2   gc( e ) ( )   glin( e) ( )  ; |  | 1.
The most important of which is the E1 polarizability, so the primitive MWL condition implied
 (mag )   (mag )
E1
e
E1
g
FFK-14, Dubna, December 3,
2014
10
kHz
nm
Wavelength dependences of the linear in the lattice-laser intensity Stark shifts for Yb
atoms in their upper 6s6p3P0 (e) and lower 6s2 1S0 (g) clock states at I 10kW/cm2.
λmag =762.3 nm (theory)
λmag =759.3537 nm (experiment)
FFK-14, Dubna, December 3,
2014
11
ΔE/kHz
nm
Wavelength dependence of the linear in the lattice-laser intensity
Stark shifts ΔE/kHz of
clock states.
I=25 kW/cm2
Hg atoms in their upper 6s6p3P0 (e) and lower 6s2 1S0 (g)
λmag =364 nm (theory)
λmag =362.53 nm (experiment)
FFK-14, Dubna, December 3,
2014
12
kHz
nm
The wavelength dependence of Stark shifts ΔE/kHz of Mg clock levels. The shifts
of the ground state 3s2 1S0 (red solid line) and the excited state 3s3p 3P0 (black
dashed curve) in a lattice field of a laser intensity I=40 kW/cm^2 (chosen
provisionally to provide the Stark trapping potential depth of about 40-50 photon
recoil energies). The magic wavelength λmag≈453 nm is determined by the point of
intersection of the lines.
FFK-14, Dubna, December 3,
2014
13
kHz
nm
Stark shifts of magnesium clock levels in case of a right-handed circular polarization of lattice. Red solid
line is for the ground state 3s2 1S0, all the rest for different magnetic sublevels of the excited 3s3p 3P1 state in
a lattice field of a laser intensity I=40 kW/cm^2 (about 40-50 photon recoil energies). The magic wavelengths
(MWL) are 419.5 nm for M=-1 and 448.1 nm for M=0 magnetic substates of the upper clock level 3s3p(3P1),
correspondingly. There is no MWL for the state M=1 in a circularly polarized lattice.
FFK-14, Dubna, December 3,
2014
14
kHz
nm
Stark shifts of magnesium clock levels in case of a linearly polarized lattice wave of the laser intensity
I=40 kW/cm^2. The shifts of states 3s3p 3P1 (M=±1) are identical and completely equivalent to that of
the state M=0 in a circularly polarized lattice beam with the MWL 448.1 nm, which is nearly equal to
the MWL 453.5 nm for an averaged over M, independent of polarization (scalar) shift; the MWL for
the M=0 state is 527 nm. The shifts of upper clock states experience the resonance enhancement on the
3s4s(3S1)-state at 517 nm, except for the state M=0 in the case of linearly polarized lattice and M=1
(M=-1) state in the right-handed (left-handed) case of circular polarization
FFK-14, Dubna, December 3,
2014
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2c) Lattice-induced clock-frequency shift.

latt
cl

(0)
cl

  ; 
latt
cl
latt
cl
(0)
cl
 E  E ; 
(0)
e
(0)
g
latt
cl
 E E ;
vib
e
vib
g
1
1

anh  2
 D    n    E  n  n   ;
2
2


D   eE1 ( )   gE1 ( )  I    e ( )   g ( )  I 2 ;
  e  g  2 Erec I  edqm ()  2e () I   gdqm ()  2 g () I  ;


E
anh
3 rec   e ( ,  )  g ( ,  ) 
 E  dqm
 dqm
I
2
  e ( )  g ( ) 
FFK-14, Dubna, December 3,
2014
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 (n,  , I )  c1/2 (n) I
latt
cl
If
1/2
 c1 (n,  ) I  c3/2 (n,  ) I  c2 ( ) I
3/2
2
sign  l ( , mag )  sign  c ( , mag ) ,
then
 mag  1/ 1   / 
c
l
 (mag , mag )  0
FFK-14, Dubna, December 3,
2014
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2.d) Numerical estimates of electromagnetic
susceptibilities and clock-frequency uncertainties
Table 1
mag
Atom
Sr
/nm


kHz


 mE1 

kW/cm
 mqm
2
 mHz 

2 
 kW/cm 


 Hz

Re  ml  
  kW/cm2 2 




 Hz

Im  ml  
  kW/cm 2 2 





Hz

Re  mc 
  kW/cm2 2 


  Hz 

Im  mc 
  kW/cm2 2 




kHz
m
I

2
 kW/cm 
10 
9
   mE1 
E

rec
 kW/cm 
kHz
2 1
Yb
Hg
813.42727
389.889
759.35374
362.53
45.2
– 92.7
40.5
5.70
1.38
– 13.6
-8.06
8.25
–200.0
1150
– 366.3
– 2.50
0
2.48
0
4.34
– 311.0
1550
240.2
2.53
0
2.37
0
6.37
25.05
74.8
18.03
13.1
10.3
0.720
0.134
15.1
2.00
7.57
0.254
3.47
FFK-14, Dubna, December 3,
2014
18
μHz/(kW/cm2)2
nm
3P
2
3P
0
3F
2
The wavelength (in nanometers) dependence of the hyperpolarizability (in
μHz/(kW/cm2)2) of clock transition in Yb atoms for the linear (red dashed curve)
and circular (black solid curve) polarization of the lattice-laser wave. The vertical
lines indicate positions of two-photon resonances on 6s8p(3P2) state at 754.226
nm, 6s8p(3P0) state at 759.71 nm (this resonance appears only for linear
polarization) and 6s5f(3F2) state at 764.953 nm
FFK-14, Dubna, December 3,
2014
19
μHz/(kW/cm2)2
nm
3P
2
3P
0
3F
2
The wavelength (in nanometers) dependence of the hyperpolarizability (in
μHz/(kW/cm2)2) of clock transition in Sr atoms for the linear (red dashed curve) and
circular (black solid curve) polarization of the lattice-laser wave. The vertical lines
indicate positions of two-photon resonances on 5s7p(3P2) state at 795.5 nm, 5s7p(3P0)
state at 797 nm (this resonance does not appear for circular polarization) and 5s4f(3F2)
state at 818.6 nm
FFK-14, Dubna, December 3,
2014
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2.e) MWL for an atom in a traveling wave
Due to homogeneous spatial distribution of intensity in a traveling
wave, the second-order shift of clock levels is determined by the
sum of E1, E2 and M1 polarizabilities
g(e) ()  gE(1e) ()  gqm(e) ()
So, the MWL
is determined from
the equality
  2 c / 
t
m
t
m
 ( )   ( )   .

g
t
m

e
FFK-14, Dubna, December 3,
2014
t
m

t
21
At this condition,
Dt   tqm I   t I 2  Dt(0)   tqm   t I  /  t ,
where Dt(0)   t I ;
 tqm   eqm (mt )   gqm (mt );
t   e ( )   g ( );
t
m
t
m
 clt (n,  , I )
and coefficients for the intensity dependence of the shift
are
t
c1/2
(n)    tqm
Etrec

t
c 3/2
( , n)    t ( )

t
 2n  1 ;
Etrec
 t
c1t ( , n)    tqm 
3Etrec
1
 2
  t ( )  n  n   ;

2 t
2

 2n  1 ; c 2t ( )    t ( ),
FFK-14, Dubna, December 3,
2014
22
mHz
(a) Sr TW MWL (n=0)
mHz
kW/cm2
(b) Yb TW MWL (n=0)
kW/cm2
Intensity I/(kW/cm2) dependence of the lattice-induced clock-frequency shift (Δν/mHz)
for (a) Sr and (b) Yb atoms in a linearly (red solid), elliptically (green dotted) and
circularly (black dashed) polarized lattice of a traveling-wave MWL
FFK-14, Dubna, December 3,
2014
23
mHz
Hg TW MWL (n=0)
 ( , n, I )
kW/cm2
( , n, I )
Intensity _I_/(kW/cm2)) dependence of the lattice-induced clock-frequency shift
(Δν/mHz) in Hg atoms in a linearly (red solid), elliptically (green dotted) and circularly
(black dashed) polarized lattice of a traveling-wave MWL. The imaginary part – clockfrequency broadening ( , n, I )  Im ( , n, I ) for linear (black solid) and circular
(red dashed) polarizations are negative values (thin curves at the plot bottom).
FFK-14, Dubna, December 3,
2014
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2.f) MWL for an atom in a standing wave of
an optical lattice (motion-insensitive MWL)
gdqm (ms )  edqm (ms )  sdqm.
At this condition,
с1/s 2 ( , n)  0;
с ( , n)  
s
1
qm
s
rec
s
dqm
s
1
 2
t
 s ( )  n  n    с1 ( , n),
2

rec
s
dqm
s
 2n  1 ,
3E

2
E
с ( , n)   s ( )

s
3/ 2
FFK-14, Dubna, December 3,
2014
с ( )   s ( ).
s
2
25
mHz
(a)
Sr
SW MWL (n=0)
mHz
kW/cm2
(b) Yb SW MWL (n=0)
kW/cm2
Intensity _I_/(kW/cm2)) dependence of the lattice-induced clock-frequency shift
(Δν/mHz) for (a) Sr and (b) Yb atoms in a linearly (red solid) elliptically (green dotted)
and circularly (black dashed) polarized lattice of a standing-wave MWL
FFK-14, Dubna, December 3,
2014
26
Hg
SW MWL
(n=0)
kW/cm2
( , n, I )
 ( , n, I )
mHz
Intensity _I_/(kW/cm2)) dependence of the lattice-induced clock-frequency shift
(Δν/mHz) in Hg atoms in a linearly (red solid), elliptically (green dotted) and circularly
(blue dashed) polarized lattice of a standing-wave MWL. The imaginary part – clockfrequency broadening ( , n, I )  Im ( , n, I ) for linear (red solid) and circular
(black dashed) polarizations are negative values (thin curves at the plot top).
FFK-14, Dubna, December 3,
2014
27
2.g) MWL for equal dipole polarizabilities
in ground and excited clock states
E1
gE1 (mE1 )  eE1 (mE1 )  mag
.
At this condition,
c1/E12 (n)   mqm
rec
EErec
3
E
1
1
 E1
 2
1 
E1
n   , c1 ( , n)   E1  E1 ( )  n  n   ,
E1 
m  2 
2 m
2

rec
E
c3/E12 ( , n)  EE11  E1 ( )  2n  1 , c2E1 ( )   E1 ( ).
m
FFK-14, Dubna, December 3,
2014
28
(a)
Sr
EDP MWL
(b) Yb
(n=0)
mHz
EDP MWL
(n=0)
mHz
kW/cm2
kW/cm2
Intensity I/(kW/cm2)) dependence of the lattice-induced clock-frequency shift
(Δν/mHz) for: (a) Sr and (b) Yb atoms in a linearly (red solid), elliptically (green dotted)
and circularly (black dashed) polarized lattice of an “equal dipole polarizabilities”
MWL.
FFK-14, Dubna, December 3,
2014
29
Hg
EDP MWL
(n=0)
 (  0, n, I )
kW/cm2
 (mag , n, I )
(  0, n, I )
 (|  | 1, n, I )
(|  | 1, n, I )
mHz
Intensity _I_/(kW/cm2)) dependence of the lattice-induced clock-frequency shift,
Re(Δν/mHz) in Hg atoms in a linearly (red solid), elliptically (green dotted) and
circularly (blue dashed) polarized lattice of an “equal dipole polarizabilities” MWL.
The imaginary part – clock-frequency broadening ( , n, I )  Im ( , n, I ) for linear
(red solid) and circular (black dashed) polarizations are negative values.
FFK-14, Dubna, December 3,
2014
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Dependence of the lattice-induced clock-frequency shift on the
lattice intensity I, circular polarization degree ξ and on the
vibration quantum number n in Yb
1. For the TW MWL (  g (mt )  e (mt )  t ):
 clTW  1.79(2n  1) I 1/ 2  [8.06  (0.0136  0.0225 2 )(2n2  2n  1)]I
 (0.0814  0.1348 2 )(2n  1) I 3/ 2  (0.366  0.606 2 ) I 2 ;
2. For the SW MWL (  gdqm (ms )  edqm (ms )  sdqm ):
 clSW  [8.06  (0.0136  0.0225 2 )(2n2  2n  1)]I
 (0.0814  0.1348 2 )(2n  1) I 3/ 2  (0.366  0.606 2 ) I 2 ;
3. For the ED MWL ( g (m )  e (m )  m ):
E1
ED
E1
ED
E1
 clED  0.895(2n  1) I 1/ 2  (0.0136  0.0225 2 )(2n2  2n  1) I
 (0.0814  0.1348 2 )(2n  1) I 3/ 2  (0.366  0.606 2 ) I 2 .
FFK-14, Dubna, December 3,
2014
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2.h) MWL precision
Uncertainties of the clock frequency are directly proportional to the
uncertainties of the MWL:
 cl
 cl 
m
m
The principal contribution to the derivative comes from the E1
polarizability in the lattice well depth and in the frequency of
harmonic vibrations:
 cl
( )
m 
1

I
n 
m
m
m 
2
E1
m
FFK-14, Dubna, December 3,
2014
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A 15% precision estimate of frequency derivatives for
polarizabilities in Sr atoms gives:
 cl
m 
 E1
1   1
1
   mag I 
 n     res  res
m
2 
2   e
g

(0)
(0)
 res

E

E
g
5 s 5 p ( 1P )
5 s 2 ( 1S
1
0

 ;

 m  281.95 THz,
)
(0)
 eres  E5(0)

E
 m  72.778 THz;
s 6 s ( 3S )
5s5 p ( 3P )
1
0
 cl
1 

10 
 10 6.57 I  1.52 I  n    .
m
2 


For I=10 kW/cm2 the departure from the magic frequency Δωm < 100 kHz
provides the fractional uncertainty of the clock frequency at the level
|  cl | / cl  1018
FFK-14, Dubna, December 3,
2014
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Conclusions 1 (Red-detuned MWL)
1. At least 3 different methods may be used for determining MWL for the redt
s
detuned optical lattice, providing MWL, m , m and their mean value
mE1  (mt  ms ) / 2 (in Sr, ms  mt  20.5 MHz ). These MWLs provide
different lattice-induced shifts and uncertainties on the clock frequency, with
different dependencies on the lattice laser intensity.
2. The polarizabilities contribute only to the lattice potential depth and harmonic
oscillation frequencies and never contribute to the anharmonic terms, where the
contributions come from hyperpolarizabilities only.
3. The hyperpolarizability provides quadratic, power 3/2 and linear contributions
to the lattice-potential depth, frequency of vibrations and anharmonic
interaction, correspondingly. At I>10 kW/cm2 the hyperpolarizability
contribution to the lattice-induced shift in Sr and Yb atoms becomes comparable
or exceeding that of polarizability. In Hg atoms the hyperpolarizability terms do
not exceed 10% of polarizability terms at I<100 kW/cm2.
FFK-14, Dubna, December 3,
2014
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3. Elimination of nonlinear effects in a
lattice of Sr blue-detuned MWL of
λm=389.889 nm
3.1. Spatial distribution of interaction between
atom and a repulsive lattice.
Trapped atoms locate near nodes of the lattice field:
E( X , t )  2E0 sin(kX )sin(t ),
Atom-lattice interaction:
Vˆ ( X )  VˆE1 sin(kX )  (VˆE 2  VˆM 1 )cos(kX )
FFK-14, Dubna, December 3,
2014
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The second-order term is linear in the laser intensity I and is determined
by the E1 and multipole polarizabilities ( E2, M1…) :
Eg(2)( e ) ( X )   g (e) Vˆ † ( X )GVˆ ( X )  Vˆ ( X )G Vˆ † ( X ) g (e)
2
qm

   gdqm
(

)
sin
(
kX
)


(e)
g ( e ) ( )  I
where the dipole polarizability is  gE(1e) ( )  0;
E1 ( )   qm ( )  0;
 gdqm
(

)


(e)
g (e)
g (e)
 gE(1e ) ( ) >>  gqm( e ) ( ) ;
 gqm(e) ( )   gM(e1) ( )   gE(2e ) ( ).
The fourth-order term is quadratic in the laser intensity I and is
determined by the dipole hyperpolarizability:
(4)
g (e)
E
( X )  g (e) ()sin (kX )I .
4
FFK-14, Dubna, December 3,
2014
2
36
The Stark-effect energy determines the trap potential
energy for excited and ground-state atom:
Eglatt( e ) ( X )  Eg(2)( e ) ( X )  Eg(4)( e ) ( X )  U glatt( e ) ( X )
2
  gqm( e ) ( ) I   gdqm
(

)
I
(
kX
)
(e)
 
4
dqm
g (e)
(kX )
( ) I  3 g ( e ) ( ) I 
;
3
2
The difference between top (X=λ/4) and bottom (X=0) of the trap potential
latt
g (e)
D
U
latt
g (e)


latt
dqm
2
X


U
(0)



(

)
I


(

)
I
g (e)
g (e)
g (e )


4


is the depth of the lattice well, quite similar to the red-detuned lattice, but the positionindependent energy shift involves only the E2-M1 polarizability  gqm
, in contrast
( e ) ( )
to the red-detuned MWL, where both E1 polarizability and hyperpolarizability were
involved.
FFK-14, Dubna, December 3,
2014
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3
k2
rec
For D  E  E
 k BT , E 
,
2
2M
atom is trapped into an eigenstate of the vibrational Hamiltonian
latt
g (e)
rec
therm
with the energy
vib
g (e)
E
( I ,  , n)  U
(0)
g (e)
bottom
1  anh
1

 2
  g ( e )  n   -Eg (e ) ( , I )  n  n  
2
2


harmonic oscillations
anharmonic energy
Ug(0)(e)  Uglatt(e) (0)  gqm(e) ()I
rec latt
 g ( e )  2 -Erec gdqm
(

)
I

2
E
Dg ( e )
(e)
anh
g (e)
E
3 rec  3 g ( e ) ( ,  ) I 
( , I )  E 1 
;
dqm
2
 g ( e ) ( ) 

FFK-14, Dubna, December 3,
2014
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Lattice-induced clock-frequency shift is
vib
1/ 2
 cllatt (m , I , n)  Evib

E

c
(
n
)
I
 c1 ( , n)I
e
g
1/ 2
where
c1/ 2 (n)  Erec


edqm (m )   gdqm (m ) (2n  1);
rec
3
E
c1 ( , n)   eqm (m ) 
2
  e ( , m )  g ( , m )   2
1
 dqm
 dqm
  n  n  ;
2
  e (m )  g (m )  
FFK-14, Dubna, December 3,
2014
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3.2. Motion-insensitive standing-wave MWL (SW
MWL)
is determined by the equality
edqm (m )  gdqm (m ),
the lattice-induced clock-frequency shift is
latt
cl
m
1

( , I ,  , n)  c ( , n)I ,
rec
3
E
Re c1 ( , n)   eqm (m ) 
2 Dmlatt
3E rec
Im c1 ( , n) 
2 Dmlatt
1
 2
 n  n   Re  m ( ) I ;
2

1
 2
 n  n   Im  m ( ) I ,
2

Dmlatt is the depth of the lattice well at the MWL frequency
The hyperpolarizability effects in the shift and broadening (caused by two-photon
rec
latt
ionization) are strongly reduced by the factor E / Dm  1 (as follows from the
data of table 1 for the blue MWL,  b  389.889 nm, Erec / Dlatt  1/(6I ), where
m
m
intensity is in kW/cm2).
FFK-14, Dubna, December 3,
2014
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3.3. Numerical estimates of the blue-detuned-latticeinduced shifts
From the data of table 1 for the Sr blue-detuned MWL we have
Re c1 ( , n  0)  (13.74  0.05 2 ) mHz/(kW/cm 2 ),
Im c1 ( , n)  (0.606  0.026 2 )( n 2  n  1/ 2)  Hz/(kW/cm 2 )
In the blue-detuned lattice of Sr atoms the shift of the clock frequency is directly proportional
to the lattice-laser intensity and is mainly determined by the difference of E2-M1
polarizabilities of the clock levels. The influence of hyperpolarizability appears only in the
third digit number. The broadening (imaginary part of the shift) is more than 4 orders smaller
than the shift. For I=10 kW/cm2 the lattice-induced shift is about 137 mHz, the latticeinduced width is about 6 μHz.
The hyperpolarizability effects in the shift and broadening (caused by two-photon ionization)
rec
latt
are strongly reduced by the factor E / Dm  1 (as follows from the data of table 1 for
b
the blue MWL, m  389.889 nm, Erec / Dlatt  1/(6I ),where intensity is in kW/cm 2).
m
FFK-14, Dubna, December 3,
2014
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Conclusions 2
1. The motion-insensitive blue-detuned MWL depends on only
the polarizabilities and is not influenced by
hyperpolarizability effects.
2. The hyperpolarizability effects on the clock levels appear
only in anharmonic interaction of atom with lattice.
2
3. The intensity of the lattice laser I  4 kW/cm is sufficient to
trap atoms cooled to 1 μK at the lowest vibrational state.
4. To achieve the clock frequency precision at the 18th decimal
place, the irremovable multipole-interaction-induced shift by
the field of optical lattice should be taken into account with
uncertainty below 1.0%.
FFK-14, Dubna, December 3,
2014
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Thank you for attention!
FFK-14, Dubna, December 3,
2014
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