FFK-14, Dubna, December 3, 2014 11 Higher-order constraints on precision of the time-frequency metrology of atoms in optical lattices V. D. Ovsiannikov

FFK-14, Dubna, December 3, 2014

11

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


22

Contents1. 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) .

2. Attractive lattice of a Red-detuned MWL:

a) Spatial distribution of atom-lattice interaction.b) Lattice potential wells.c) Lattice-induced clock-frequency shift.d) Numerical estimates of electromagnetic susceptibilities and clock-frequency shifts of neutral Sr,

Yb and Hg atoms in a lattice of a red-detuned MWL.

e) MWL for an atom in a traveling wave (TW).f) MWL for an atom in a standing wave (SW).g) MWL for equal dipole polarizabilities (EDP) in ground and excited clock statesh) MWL precision.

3. Elimination of nonlinear effects in a lattice of Sr blue-detuned MWL of λm=389.889 nm.

a) Spatial distribution of interaction between atom and a repulsive lattice.b) Motion-insensitive standing-wave MWL (SW MWL).c) Numerical estimates of the blue-detuned-lattice-induced shifts


3

«Clock» transition

Typical structure of energy levels in alkaline-earth and alkaline-earth-like atoms (Mg, Ca, Sr, Zn, Cd, Yb, Hg)

Radiation transitions between metastable

and ground states, stimulated in odd

isotopes by the hyperfine interaction, is

strictly forbidden in even isotopes.

This prohibition makes extremely narrow

the line of the clock transition,

which may be stimulated by an external

magnetic field or by the circularly polarized

lattice wave. This transition may be used as

an oscillator with extremely high quality

The width of the oscillator depends on

(and may be regulated by) the intensity of

the lattice wave or a static magnetic field.

11P

23P

13S

13P

03P

01S

13D

01

03 SP

17/ 10clQ

M2

E1

2ω(M1+E1)

(ΔS=1)


4

Natural isotope composition Even isotopes Odd isotopes

(J=0) abundance abundance (J≠0)24,26Mg: 90% 25Mg: 10% (J=5/2)40→48Ca: 98.7% 43Ca: 1.3% (J=7/2)84,86,88Sr: 93% 87Sr: 7% (J=9/2)168→176Yb: 73% 171,173Yb: 27% (J=1/2, 5/2)196→204Hg: 69.8% 199,201Hg: 30.2% (J=1/2,3/2)106→116Cd: 75% 111,113Cd: 25% (J=1/2) 64→70Zn: 95.9% 67Zn: 4.1% (J=5/2)


55

0( , ) 2 cos( )cos( ),X t kX tE E2

kc

ˆ ˆ( , ) Re ( )exp( )V X t V X i t

1 2 1ˆ ˆ ˆ ˆ( ) cos( ) ( )sin( )E E MV X V kX V V kX

21 0 2 0 1 02

ˆˆ ˆ ˆ ˆ( ); ( , ) ; [ ] ( )26

E E MV V r V 2r E E n C n E J S

2. Red-detuned MWL

2.a) Spatial distribution of atom-lattice interaction


6

(2) (4)( ) ( ) ( )( ) ( ) ( ) ...latt

g e g e g eE X E X E X

(2) † †( )

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ) ( )g eE X g e V X G V X V X G V X g e

1 2 2( ) ( )( ) cos ( ) ( )sin ( )E qm

g e g ekX kX I 2 1

( ) ( ) ( )( ) ( ) ( ); is the laser intensity:

2 is the mean intensity of a standing wave,

0 is the intensity of the node,

4 is the intensity of antinode.

qm E Mg e g e g e I

I

I

(4) 4 2( ) ( )( ) ( ) cos ( ) .g e g eE X kX I


7

2b) Lattice potential wells. Clock-level shift is the Lattice-trap potential energy

( ) 2 ( ) 4( ) ( ) ( ) ( ) ( )( ) ( ) ;latt latt harm anh

g e g e g e g e g eE X U X D U X U X

1 2( ) ( ) ( )

2( ) 2 2

( ) ( ) ( )

4( ) 2

( ) ( ) ( )

( , , ) ( ) ( , ) , depth

( , , )( ) 2 ( , ) ,

2

( , , ) ( ) 5 ( , )3

Eg e g e g e

harm dqmg e g e g e

anh dqmg e g e g e

D I I I

IU I I k

kU I I I

atM


8

/ LX

n=0

1

2

3

4

( ) ( )( ) /lattg e g eU X D

Stark-trap potential and vibration-state energies of an atom in a standing wave of a lattice field

/ 2L

5


99

( ) ( )

ˆˆ ˆ( ), ;

2at lattg e g eX U X i

X

2at

atat

PH ( )= P

M

( ) ( )ˆ at vib

g e n g e nX X X H ( ) ( )=E ( )

2( ) ( ) ( ) ( )

1 1( , , , ) ( , , ) ( , , ) ( , , )

2 2vib anhg e g e g e g eI n D I I n I n n

E -E

depth harmonic oscillations anharmonic energy

( )( )

( )

3 ( , )( , , ) 1 ;

2 ( )

recg eanh

g e dqmg e

II

EE

2

22rec

t c

atME is the recoil energy of a lattice photon


10

1( ) ( ) ( )

2 1( ) ( ) ( )

2( ) ( ) ( ) ( )

( ) ( ) ( );

( ) ( ) ( );

( , ) ( ) ( ) ( ) ; | | 1.

dqm E qmg e g e g e

qm E Mg e g e g e

lin c ling e g e g e g e

(0)( ) ( ) ( ) ( , , , )latt vib

g e g e g e I n E E E

( , , , ) ( , , , )vib vibe mag g magI n I n E E

The strict magic-wavelength condition should imply the equality

To hold this condition, the equality should hold for the susceptibilities:

The most important of which is the E1 polarizability, so the primitive MWL condition implied

1 1( ) ( )E Ee mag g mag


11

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 10kW/cm2.

λmag =762.3 nm (theory) λmag =759.3537 nm (experiment)

I

nm

kHz


12

Wavelength dependence of the linear in the lattice-laser intensity I=25 kW/cm2

Stark shifts ΔE/kHz of Hg atoms in their upper 6s6p3P0 (e) and lower 6s2 1S0 (g) clock states.

λmag =364 nm (theory) λmag =362.53 nm (experiment)

nm

ΔE/kHz


13

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.

nm

kHz


14

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.

kHz

nm


15

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

nm

kHz


1616

(0) (0) (0) (0); ; ;latt latt latt vib vibcl cl cl cl e g cl e g E E E E

21 1;

2 2latt anhcl D n n n

E

1 1 2( ) ( ) ( ) ( ) ;E Ee g e gD I I

2 ( ) 2 ( ) ( ) 2 ( ) ;rec dqm dqme g e e g gI I I E

( , )( , )3

2 ( ) ( )ganh rec e

dqm dqme g

I

E E

2c) Lattice-induced clock-frequency shift.


17

1/2 3/2 21 21/2 3/2( , , ) ( ) ( , ) ( , ) ( )latt

cl n I n n nc I c I c I c I

If sign ( , ) sign ( , ) ,l cmag mag

then 1/ 1 /c lmag

( , ) 0mag mag


1818

2.d) Numerical estimates of electromagnetic susceptibilities and clock-frequency uncertainties

mag

2

1 kHz

kW/cm

Em

2

mHz

kW/cm

qmm

22

Hz

kW/cmRe l

m

22

ImkW/cm

lm

Hz

22

Hz

kW/cmRe c

m

22kW/cm

Im cm

Hz

2

kHz

kW/cmm I

1

19 210 kW/cmEm

kHzrecE

Atom Sr Yb Hg

/nm 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

0.254 10.3 0.720 0.134

3.47 15.1 2.00 7.57

Table 1


19

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

nm

μHz/(kW/cm2)2

3P23P0

3F2


20

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

nm

μHz/(kW/cm2)2

3P03P2

3F2


2121

2.e) MWL for an atom in a traveling wave

( ) ( ) .t tg m e m t

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

1( ) ( ) ( )( ) ( ) ( )E qm

g e g e g e

So, the MWL 2 /t tm mc

is determined from the equality


22

At this condition,

2 (0)

(0)

/ ,

where ; ( ) ( );

( ) ( );

qm qmt t t t t t t

qm qm t qm tt t t e m g m

t tt e m g m

D I I D I

D I

and coefficients for the intensity dependence of the shift

are

211/2

23/2

3 1( ) 2 1 ; ( , ) ( ) ;

2 2

( , ) ( ) 2 1 ; ( ) ( ),

rec rect tqm qm

t t tt t

rect

t tt

t t

t t

n n n n n

n n

c c

c c

E E

E

( , , )tcl n I


23

(a) Sr TW MWL (n=0) (b) Yb TW MWL (n=0)

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

kW/cm2

mHz mHz

kW/cm2

mHz


24

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 – clock-frequency broadening for linear (black solid) and circular (red dashed) polarizations are negative values (thin curves at the plot bottom).

( , , ) Im ( , , )n I n I

Hg TW MWL (n=0)

( , , )n I

( , , )n I mHz

kW/cm2


2525

2.f) MWL for an atom in a standing wave of an optical lattice (motion-insensitive MWL)

( ) ( ) .dqm s dqm s dqmg m e m s

At this condition,

1/ 2

21 1

3/ 2 2

( , ) 0;

3 1( , ) ( ) ( , ),

2 2

( , ) ( ) 2 1 , ( ) ( ).

s

recs qm ts

s sdqms

recs ss

s sdqms

с n

с n n n с n

с n n с

E

E


26

(a) Sr SW MWL (n=0) (b) Yb SW MWL (n=0)

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

kW/cm2 kW/cm2

mHz mHz


27

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 – clock-frequency broadening for linear (red solid) and circular (black dashed) polarizations are negative values (thin curves at the plot top).

( , , ) Im ( , , )n I n I

Hg SW MWL (n=0)

( , , )n I

( , , )n I

kW/cm2

mHz


2828

2.g) MWL for equal dipole polarizabilities in ground and excited clock states

1 1 1 1 1( ) ( ) .E E E E Eg m e m mag

At this condition,

1 1 21 11/ 2 1 11 1

1 113/ 2 1 2 11

31 1( ) , ( , ) ( ) ,

2 2 2

( , ) ( ) 2 1 , ( ) ( ).

rec recE qm EE E

m EE Em m

recE EE

E EEm

c n n c n n n

c n n c

E E

E


29

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.

(a) Sr EDP MWL (n=0) (b) Yb EDP MWL (n=0)mHz

mHz

kW/cm2

kW/cm2


30

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 for linear (red solid) and circular (black dashed) polarizations are negative values.

( , , ) Im ( , , )n I n I

Hg EDP MWL (n=0)

0( , , )n I

( , , )mag

n I

| 1(| , , )n I

0( , , )n I

(| | 1, , )n I

kW/cm2

mHz


3131

Dependence of the lattice-induced clock-frequency shift on the lattice intensity , circular polarization degree ξ and on the

vibration quantum number n in Yb1. For the TW MWL ( ):( ) ( )t t

g m e m t 1/ 2 2 2

2 3/ 2 2 2

1.79(2 1) [8.06 (0.0136 0.0225 )(2 2 1)]

(0.0814 0.1348 )(2 1) (0.366 0.606 ) ;

TWcl n I n n I

n I I

2. For the SW MWL ( ):( ) ( )dqm s dqm s dqmg m e m s

2 2

2 3/ 2 2 2

[8.06 (0.0136 0.0225 )(2 2 1)]

(0.0814 0.1348 )(2 1) (0.366 0.606 ) ;

SWcl n n I

n I I

3. For the ED MWL ( ):1 1 1( ) ( )E ED E ED Eg m e m m

1/ 2 2 2

2 3/ 2 2 2

0.895(2 1) (0.0136 0.0225 )(2 2 1)

(0.0814 0.1348 )(2 1) (0.366 0.606 ) .

EDcl n I n n I

n I I

I


3232

2.h) MWL precision

Uncertainties of the clock frequency are directly proportional to the uncertainties of the MWL:

clcl 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:

1( ) 1

2

Ecl m m

m m m

I n


3333

A 15% precision estimate of frequency derivatives for polarizabilities in Sr atoms gives:

1 2 11 0

3 31 0

1

(0) (0)

5 5 ( ) 5 ( )

(0) (0)

5 6 ( ) 5 5 ( )

10

1 1 1;

2 2

281.95 THz,

72.778 THz;

110 6.57 1.52 .

2

Ecl mmag res res

m e g

resg ms p P s S

rese ms s S s p P

cl

m

I n

E E

E E

I I n

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

18| | / 10cl cl


3434

Conclusions 1 (Red-detuned MWL)

1. At least 3 different methods may be used for determining MWL for the red-detuned optical lattice, providing MWL, and their mean value

(in Sr, ). 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.

,t sm m

1 ( ) / 2E t sm m m 20.5 MHzs t

m m


3535

3. Elimination of nonlinear effects in a lattice of Sr blue-detuned MWL of

λm=389.889 nm3.1. Spatial distribution of interaction between

atom and a repulsive lattice.

Trapped atoms locate near nodes of the lattice field:

0( , ) 2 sin( )sin( ),X t kX tE E

Atom-lattice interaction:

1 2 1ˆ ˆ ˆ ˆ( ) sin( ) ( ) cos( )E E MV X V kX V V kX


3636

(2) † †( )

ˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( ) ( )g eE X g e V X G V X V X G V X g e

2( ) ( )( )sin ( ) ( )dqm qm

g e g ekX I

The second-order term is linear in the laser intensity I and is determined by the E1 and multipole polarizabilities ( E2, M1…) :

1( ) ( )

1 2( ) ( ) ( )

1( )

1( ) ( ) ( ) .

where the dipole polarizability is ; ( ) >> ( ) ;

( ) ( ) ( )

( ) 0

( ) ( ) ( ) 0;

E qmg e g e

qm M Eg e g e g e

Eg e

dqm qmEg e g e g e

(4) 4 2( ) ( )( ) ( )sin ( ) .g e g eE X kX I

The fourth-order term is quadratic in the laser intensity I and is determined by the dipole hyperpolarizability:


37

(2) (4)( ) ( ) ( ) ( )

2( ) ( )

42

( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( )

( )( ) 3 ( ) ;

3

latt lattg e g e g e g e

qm dqmg e g e

dqmg e g e

E X E X E X U X

I I kX

kXI I

The Stark-effect energy determines the trap potential energy for excited and ground-state atom:

2( ) ( ) ( ) ( ) ( )(0) ( ) ( )

4latt latt latt dqmg e g e g e g e g eD U X U I I

is the depth of the lattice well, quite similar to the red-detuned lattice, but the position-independent energy shift involves only the E2-M1 polarizability , in contrast to the red-detuned MWL, where both E1 polarizability and hyperpolarizability were involved.

The difference between top (X=λ/4) and bottom (X=0) of the trap potential

( ) ( )qmg e


3838

(0) 2( ) ( ) ( ) ( )

1 1( , , ) ( , )

2 2vib anhg e g e g e g eI n U n I n n

E -E

bottom harmonic oscillations anharmonic energy

( )( )

( )

3 ( , )3( , ) 1 ;

2 ( )g eanh rec

g e dqmg e

II

E E

( ) ( ) ( )2 ( ) 2rec dqm rec lattg e g e g eI D -E E

with the energy

2

( )

3For k , ,

2 2atom is trapped into an eigenstate of the vibrational Hamiltonian

latt rec therm recg e B

kD T E

ME E

(0)( ) ( ) ( )(0) ( )latt qm

g e g e g eU U I


3939

Lattice-induced clock-frequency shift is

1/ 21/ 2 1( , , ) ( ) ( , )latt vib vib

cl m e gI n c n I c n I E E

where

1/ 2 ( ) ( ) ( ) (2 1);rec dqm dqme m g mc n n E

21

( , )( , )3 1( , ) ( ) ;

2 ( ) ( ) 2

recg mqm e m

e m dqm dqme m g m

c n n n

E


4040

3.2. Motion-insensitive standing-wave MWL (SW MWL)

( ) ( ),dqm dqme m g m

the lattice-induced clock-frequency shift is

1( , , , ) ( , ) ,lattcl m I n c n I

21

21

3 1Re ( , ) ( ) Re ( ) ;

2 2

3 1Im ( , ) Im ( ) ,

2 2

is the depth of the lattice well at the MWL frequency

recqme m mlatt

m

rec

mlattm

lattm

c n n n ID

c n n n ID

D

E

E

The hyperpolarizability effects in the shift and broadening (caused by two-photon ionization) are strongly reduced by the factor (as follows from the data of table 1 for the blue MWL, where intensity is in kW/cm2).

/ 1rec lattmD E

/ 1/(6 ),rec lattmD IE389.889 nm,b

m

is determined by the equality


4141

From the data of table 1 for the Sr blue-detuned MWL we have

2 21

2 2 21

Re ( , 0) (13.74 0.05 ) mHz/(kW/cm ),

Im ( , ) (0.606 0.026 )( 1/ 2) Hz/(kW/cm )

c n

c n n n

3.3. Numerical estimates of the blue-detuned-lattice-induced shifts

The hyperpolarizability effects in the shift and broadening (caused by two-photon ionization)

are strongly reduced by the factor (as follows from the data of table 1 for

the blue MWL, where intensity is in

kW/cm2).

/ 1rec lattmD E

389.889 nm,bm / 1/(6 ),rec latt

mD IE

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 lattice-

induced width is about 6 μHz.


4242

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.

3. The intensity of the lattice laser 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%.

24 kW/cmI


4343

Thank you for attention!

Documents

FFK-14, Dubna, December 3, 2014 11 Higher-order constraints on precision of the time-frequency metrology of atoms in optical lattices V. D. Ovsiannikov