Abstract

We demonstrate the feasibility of a laser induced complete population transfer to and from a continuum of states. We study the two-photon dissociation of υ = 28, J = 1,…, 10 sodium dimers. We demonstrate that using just a pair of “counter intuitively” ordered pulses we can dissociate 100% of the molecules in an ensemble. The scheme is shown to be stable with respect to the initial choice of rotational level and to fluctuations in the laser frequency and intensity. We also study the reverse phenomenon of complete population transfer from the continuum. We perform calculations on the radiative association of Na atoms to form the Na2 molecule in specific vib-rotational states. It is shown that two pulses of 20 nsec duration and as little as 6 MW/cm2 peak power can photoassociate more than 98% of the atoms within a (pulse and velocity determined) relative effective distance, to yield Na2 molecules in the chosen υ = 28, J = 10 vib-rotational state. This means that given a density of 1016 atoms/cm3 and a temperature of 7K, a 10Hz pulsed laser source of the above parameters can convert half of all the Na atoms in the ensemble to υ = 28, J = 10 Na2 molecules within 15 seconds of operation.

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  1. J. M. Doyle, B. Friedrich, J. Kim, and D. Patterson, "Buffer gas loading of atoms and molecules into a magnetic trap," Phys. Rev. A 52, R2515 (1995).
    [CrossRef] [PubMed]
  2. J. T. Bahns, W. C. Stwalley, and P. L. Gould, "Laser cooling of molecules: A sequential scheme for rotation, translation, and vibration," J. Chem. Phys. 104, 9689 (1996).
    [CrossRef]
  3. A. Bartana, R. Kosloff and D. J. Tannor, "Laser cooling of molecular internal degrees of freedom by a series of shaped pulses," J. Chem. Phys. 99, 196 (1993).
    [CrossRef]
  4. B. Friedrich and D. R. Herschbach, "Alignment and trapping of molecules in intense laser fields", Phys. Rev. Lett. 74, 4623 (1995).
    [CrossRef] [PubMed]
  5. H. R. Thorsheim, J. Weiner, and P. S. Julienne, "Laser-induced photoassociation of ultracold sodium atoms," Phys. Rev. Lett. 58, 2420 (1987).
    [CrossRef] [PubMed]
  6. Y. B. Band and P. S. Julienne, "Ultracold-molecule production by laser-cooled atom photoassociation," Phys. Rev. A 51, R4317 (1995).
    [CrossRef] [PubMed]
  7. K. M. Jones, S. Maleki, L. P. Ratliff, and P. D. Lett, "Two-color photoassociation spectroscopy of ultracold sodium," J. Phys. B, 30, 289 (1997).
    [CrossRef]
  8. R. Cote and A. Dalgarno, "Mechanism for the production of vibrationally excited ultracold molecules of Li2," Chem. Phys. Lett. 279, 50 (1997).
    [CrossRef]
  9. R. Cote and A. Dalgarno, "Photoassociation intensities and radiative trap loss in lithium," Phys. Rev. A 58, 498 (1998).
    [CrossRef]
  10. A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws, and P. Pillet, "Formation of cold Cs2 molecules through photoassociation," Phys. Rev. Lett. 80, 4402 (1998).
    [CrossRef]
  11. A. Vardi, D. Abrashkevich, E. Frishman, and M. Shapiro, "Theory of radiative recombination with strong laser pulses and the formation of ultracold molecules via stimulated photorecombination of cold atoms," J. Chem. Phys. 107, 6166 (1997).
    [CrossRef]
  12. P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, "Stimulated Raman molecule production in Bose-Einstein condensates," Phys. Rev. A 58, R797 (1998).
    [CrossRef]
  13. M. Shapiro, "Theory of one- and two-photon dissociation with strong laser pulses," J. Chem. Phys. 101, 3844 (1994).
    [CrossRef]
  14. E. Frishman and M. Shapiro, "Reversibility of bound-to-continuum transitions induced by a strong short laser pulse and the semiclassical uniform approximation," Phys. Rev. A 54, 3310 (1996).
    [CrossRef] [PubMed]
  15. A. Vardi and M. Shapiro, "Two-photon dissociation/ionization beyond the adiabatic approximation," J. Chem. Phys. 104, 5490 (1996).
    [CrossRef]
  16. U. Gaubatz, P. Rudecki, M. Becker, S. Schiemann, M. Kulz, and K. Bergmann, "Population switching between vibrational levels in molecular beams," Chem. Phys. Lett. 149, 463 (1988).
    [CrossRef]
  17. U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, "Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laser fields. A new concept and experimental results," J. Chem. Phys. 92, 5363 (1990).
    [CrossRef]
  18. J. R. Kuklinski, U. Gaubatz, F. T. Hioe, and K. Bergmann, "Adiabatic population transfer in a three-level system driven by delayed laser pulses," Phys. Rev. A 40, 6741 (1989).
    [CrossRef] [PubMed]
  19. B. W. Shore, K. Bergmann, J. Oreg and S. Rosenwaks, "Multilevel adiabatic population transfer", Phys. Rev. A 44, 7442 (1991).
    [CrossRef] [PubMed]
  20. S. Schiemann, A. Kuhn, S. Steuerwald, and K. Bergmann, "Efficient coherent population transfer in NO molecules using pulsed lasers," Phys. Rev. Lett. 71, 3637 (1993).
    [CrossRef] [PubMed]
  21. T. Halfmann and K. Bergmann, "Coherent population transfer and dark resonances in SO2," J. Chem. Phys. 104, 7068 (1996).
    [CrossRef]
  22. K. Bergmann, H. Theuer, and B. W. Shore, "Coherent population transfer among quantum states of atoms and molecules," Rev. Mod. Phys. 70, 1003 (1998).
    [CrossRef]
  23. The Na-Na potential curves and the relevant electronic dipole moments are from I. Schmidt, Ph.D. Thesis, Kaiserslautern University, 1987.
  24. B. R. Johnson, "New numerical methods applied to solving the one-dimensional eigenvalue problem," J. Chem. Phys. 67, 4086 (1977).
    [CrossRef]
  25. R. E. Langer, "On the connection formulas and the solutions of the wave equation," Phys. Rev. 51, 669 (1937).
    [CrossRef]
  26. R. E. Langer, "On the asymptotic solutions of differential equations, with an application to the Bessel functions of large complex order," Trans. Am. Math. Soc. 34, 447 (1932).
    [CrossRef]
  27. R. E. Langer, Trans. Am. Math. Soc. 37, 937 (1935).
  28. R. E. Langer, Bull. Am. Math. Soc. 40, 545 (1934).
    [CrossRef]
  29. W. H. Miller, "Uniform semiclassical approximations for elastic scattering and eigenvalue problems", J. Chem. Phys. 48, 464 (1968).
    [CrossRef]
  30. J. Javanainen and M. Mackie "Probability of photoassociation from a quasicontinuum approach," Phys. Rev. A 58, R789 (1998).
    [CrossRef]

Other (30)

J. M. Doyle, B. Friedrich, J. Kim, and D. Patterson, "Buffer gas loading of atoms and molecules into a magnetic trap," Phys. Rev. A 52, R2515 (1995).
[CrossRef] [PubMed]

J. T. Bahns, W. C. Stwalley, and P. L. Gould, "Laser cooling of molecules: A sequential scheme for rotation, translation, and vibration," J. Chem. Phys. 104, 9689 (1996).
[CrossRef]

A. Bartana, R. Kosloff and D. J. Tannor, "Laser cooling of molecular internal degrees of freedom by a series of shaped pulses," J. Chem. Phys. 99, 196 (1993).
[CrossRef]

B. Friedrich and D. R. Herschbach, "Alignment and trapping of molecules in intense laser fields", Phys. Rev. Lett. 74, 4623 (1995).
[CrossRef] [PubMed]

H. R. Thorsheim, J. Weiner, and P. S. Julienne, "Laser-induced photoassociation of ultracold sodium atoms," Phys. Rev. Lett. 58, 2420 (1987).
[CrossRef] [PubMed]

Y. B. Band and P. S. Julienne, "Ultracold-molecule production by laser-cooled atom photoassociation," Phys. Rev. A 51, R4317 (1995).
[CrossRef] [PubMed]

K. M. Jones, S. Maleki, L. P. Ratliff, and P. D. Lett, "Two-color photoassociation spectroscopy of ultracold sodium," J. Phys. B, 30, 289 (1997).
[CrossRef]

R. Cote and A. Dalgarno, "Mechanism for the production of vibrationally excited ultracold molecules of Li2," Chem. Phys. Lett. 279, 50 (1997).
[CrossRef]

R. Cote and A. Dalgarno, "Photoassociation intensities and radiative trap loss in lithium," Phys. Rev. A 58, 498 (1998).
[CrossRef]

A. Fioretti, D. Comparat, A. Crubellier, O. Dulieu, F. Masnou-Seeuws, and P. Pillet, "Formation of cold Cs2 molecules through photoassociation," Phys. Rev. Lett. 80, 4402 (1998).
[CrossRef]

A. Vardi, D. Abrashkevich, E. Frishman, and M. Shapiro, "Theory of radiative recombination with strong laser pulses and the formation of ultracold molecules via stimulated photorecombination of cold atoms," J. Chem. Phys. 107, 6166 (1997).
[CrossRef]

P. S. Julienne, K. Burnett, Y. B. Band, and W. C. Stwalley, "Stimulated Raman molecule production in Bose-Einstein condensates," Phys. Rev. A 58, R797 (1998).
[CrossRef]

M. Shapiro, "Theory of one- and two-photon dissociation with strong laser pulses," J. Chem. Phys. 101, 3844 (1994).
[CrossRef]

E. Frishman and M. Shapiro, "Reversibility of bound-to-continuum transitions induced by a strong short laser pulse and the semiclassical uniform approximation," Phys. Rev. A 54, 3310 (1996).
[CrossRef] [PubMed]

A. Vardi and M. Shapiro, "Two-photon dissociation/ionization beyond the adiabatic approximation," J. Chem. Phys. 104, 5490 (1996).
[CrossRef]

U. Gaubatz, P. Rudecki, M. Becker, S. Schiemann, M. Kulz, and K. Bergmann, "Population switching between vibrational levels in molecular beams," Chem. Phys. Lett. 149, 463 (1988).
[CrossRef]

U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, "Population transfer between molecular vibrational levels by stimulated Raman scattering with partially overlapping laser fields. A new concept and experimental results," J. Chem. Phys. 92, 5363 (1990).
[CrossRef]

J. R. Kuklinski, U. Gaubatz, F. T. Hioe, and K. Bergmann, "Adiabatic population transfer in a three-level system driven by delayed laser pulses," Phys. Rev. A 40, 6741 (1989).
[CrossRef] [PubMed]

B. W. Shore, K. Bergmann, J. Oreg and S. Rosenwaks, "Multilevel adiabatic population transfer", Phys. Rev. A 44, 7442 (1991).
[CrossRef] [PubMed]

S. Schiemann, A. Kuhn, S. Steuerwald, and K. Bergmann, "Efficient coherent population transfer in NO molecules using pulsed lasers," Phys. Rev. Lett. 71, 3637 (1993).
[CrossRef] [PubMed]

T. Halfmann and K. Bergmann, "Coherent population transfer and dark resonances in SO2," J. Chem. Phys. 104, 7068 (1996).
[CrossRef]

K. Bergmann, H. Theuer, and B. W. Shore, "Coherent population transfer among quantum states of atoms and molecules," Rev. Mod. Phys. 70, 1003 (1998).
[CrossRef]

The Na-Na potential curves and the relevant electronic dipole moments are from I. Schmidt, Ph.D. Thesis, Kaiserslautern University, 1987.

B. R. Johnson, "New numerical methods applied to solving the one-dimensional eigenvalue problem," J. Chem. Phys. 67, 4086 (1977).
[CrossRef]

R. E. Langer, "On the connection formulas and the solutions of the wave equation," Phys. Rev. 51, 669 (1937).
[CrossRef]

R. E. Langer, "On the asymptotic solutions of differential equations, with an application to the Bessel functions of large complex order," Trans. Am. Math. Soc. 34, 447 (1932).
[CrossRef]

R. E. Langer, Trans. Am. Math. Soc. 37, 937 (1935).

R. E. Langer, Bull. Am. Math. Soc. 40, 545 (1934).
[CrossRef]

W. H. Miller, "Uniform semiclassical approximations for elastic scattering and eigenvalue problems", J. Chem. Phys. 48, 464 (1968).
[CrossRef]

J. Javanainen and M. Mackie "Probability of photoassociation from a quasicontinuum approach," Phys. Rev. A 58, R789 (1998).
[CrossRef]

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Figures (10)

Figure 1.
Figure 1.

Energy levels of the resonantly-enhanced 2-photon dissociation and association schemes.

Figure 2.
Figure 2.

Potentials and vibrational wave functions used in the simulation of the Na2 2-photon dissociation and the Na+Na 2-photon recombination.

Figure 3.
Figure 3.

Bound-bound (below) and bound-continuum (above) transition-dipole matrix elements for various intermediate vibrational levels. The initial bound state is taken at υ = 28, J = 10 and the kinetic energy of the dissociated atoms is E = 5K.

Figure 4.
Figure 4.

Bound-bound (below) and bound-continuum (above) transition-dipole matrix elements for various rotational transitions. The initial vibrational level is υ = 28, the intermediate level is υ′ = 37 and the kinetic energy of the dissociated atoms is E = 5K.

Figure 5.
Figure 5.

Results of a “counter-intuitive” dissociative pulse sequence. Shown are the population of the υ = 28, J = 10 initial state, the population of the υ = 37, J = 11 intermediate state, and the final integrated population of continuum states, vs. time. Dotted lines are the intensity profiles of the two Gaussian pulses whose maximum intensities are 5 × 104 W/cm2 and 5 × 106 W/cm2 for ϵ 1 and ϵ 2 respectively. Both pulses last 8.5 nsec. The central frequency of the pump pulse is chosen in resonance with the bound-bound transition (Δ1 = 0).

Figure 6.
Figure 6.

Calculated dissociative lineshapes at I 1 = 5 × 104 W/cm2, I 2 = 5 × 106 W/cm2 (—), I 1 = 2 × 105 W/cm2, I 2 = 2 × 107 W/cm2 (┄), and I 1 = 5 × 105 W/cm2, I 2 = 5 × 107 W/cm2 (╴∙╴∙╴), where I 1 and I 2 stand for the peak intensity of ϵ 1 and ϵ 2 respectively. Pulse durations are as in Fig. 5.

Figure 7.
Figure 7.

Bound-continuum PA transition-dipole matrix elements as a function of the initial collision temperature, for the J = 10 radial wave.

Figure 8.
Figure 8.

Results of the “counter-intuitive” associative pulse sequence. Symbols are the same as in Fig. 5. The maximum intensity of the dump pulse is 5 × 104 W/cm2 and that of the pump pulse is 6 × 106 W/cm2. Both pulses last 8.5 nsec. The pump pulse peaks at the peak of the Na+Na wave packet (t 0 = 20 nsec) and the dump pulse peaks at 5 nsec before that time. Central frequencies are chosen so that Δ1 = Δ E 0 , = 0. The initial kinetic energy of the coherent wave packet is E 0 = 5K and its bandwidth is δE = 10-3 cm-1.

Figure 9.
Figure 9.

Cylinder of collisions with a given J, occurring when the PA pump pulse is on.

Figure 10.
Figure 10.

Results of a “counter-intuitive” associative pulse sequence. Symbols are the same as in Fig. 8. The maximum intensity of the dump pulse is 500 W/cm2 and that of the pump pulse is 6 × 105 W/cm2. Both pulses last 85 nsec. The pump pulse peaks at the peak of the Na+Na wave packet (t 0 = 200 nsec) and the dump pulse peaks at 50 nsec before that time. Central frequencies are chosen so that Δ1 = Δ E 0 = 0. The initial kinetic energy of the coherent wave packet is E 0 = 5K and its bandwidth is δE = 10-4 cm-1.

Equations (62)

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H tot = H 2 μ 1 · ϵ ̂ 1 ϵ 1 ( t ) cos ( ω 1 t ) 2 μ 2 · ϵ ̂ 2 ϵ 2 ( t ) cos ( ω 2 t ) ,
Ψ ( t ) = b 1 1 exp ( i E 1 t ħ ) + b 2 2 exp ( i E 2 t ħ )
+ n dE b E , n ( t ) E , n ± exp ( iEt ħ ) ,
[ E 1 H ] 1 = [ E 2 H ] 2 = [ E H ] E , n ± = 0 .
d b 1 dt = i Ω 1 ( t ) exp ( i Δ 1 t ) b 2 ( t ) ,
d b 2 dt = i Ω 1 ( t ) exp ( i Δ 1 t ) b 1 ( t ) + i dE n Ω 2 , E , n ( t ) exp ( i Δ E t ) b E , n ( t ) ,
d b E , m dt = i Ω 2 , E , m * ( t ) exp ( i Δ E t ) b 2 ( t ) , m = 1 , , N ,
Ω 1 ( t ) 2 μ 1 1 ϵ 1 ( t ) ħ , Ω 2 , E , m ( t ) 2 μ 2 E , m ± ϵ 2 ( t ) ħ ,
Δ 1 ( E 2 E 1 ) ħ ω 1 , Δ E ω 2 ( E 2 E ) ħ ,
b 1 ( t = 0 ) = b 1 0 ; b 2 ( t = 0 ) = b 2 0 ; b E , m ( t = 0 ) = b E , m 0 .
b E , n ( t ) = b E , n 0 + i 0 t dt Ω 2 , E , n * ( t ) exp ( i Δ E t ) b 2 ( t ) ,
d b 2 dt = i Ω 1 ( t ) exp ( i Δ 1 t ) b 1 ( t ) + i n dE Ω 2 , E , n ( t ) exp ( i Δ E t ) b E , n 0
n dE 0 t dt Ω 2 , E . n ( t ) Ω 2 , E , n * ( t ) exp [ i Δ E ( t t ) ] b 2 ( t ) ,
n 2 μ 2 E , n ± 2 n 2 μ 2 E L , n ± 2 .
d b 2 dt = i Ω 1 ( t ) exp ( i Δ 1 t ) b 1 ( t ) Ω 2 ( t ) b 2 ( t ) + i F 2 ( t ) ,
Ω 2 ( t ) π n 2 μ 2 E L , n ± ϵ 2 ( t ) 2 ħ .
F 2 ( t ) = ϵ 2 ( t ) μ ¯ 2 ( t ) ħ ,
μ ̄ 2 ( t ) = n dE 2 μ 2 E , n ± exp ( i Δ E t ) b E , n 0 .
d dt 𝖻 = i { 𝖧 · 𝖻 ( t ) + 𝖿 } ,
𝖻 [ ( exp ( i Δ 1 t ) b 1 b 2 ] ,
𝖧 = [ Δ 1 Ω 1 Ω 1 i Ω 2 ] ,
𝖿 ( t ) = [ 0 F 2 ( t ) ] ,
𝖻 0 𝖻 ( t = 0 ) = [ b 1 0 b 2 0 ] .
𝖴 · 𝖧 = ε ̂ · 𝖴 ,
ε 1,2 = 1 2 { Δ 1 + i Ω 2 ± [ ( Δ 1 i Ω 2 ) 2 + 4 Ω 1 2 ] 1 2 } .
𝖴 ( t ) · 𝖴 𝖳 ( t ) = 𝖨 ,
𝖴 = [ cos θ sin θ sin θ cos θ ]
θ ( t ) = 1 2 arctan ( 2 Ω 1 i Ω 2 Δ 1 ) .
𝖺 ( t ) = 𝖴 ( t ) · 𝖻 ( t )
d dt 𝖺 = { i ε ̂ ( t ) + 𝖠 } · 𝖺 + i 𝐠 .
𝖠 d𝖴 ( t ) dt · 𝖴 𝖳 = [ 0 θ ˙ θ ˙ 0 ] ,
𝐠 ( t ) = [ F 2 ( t ) U 1,2 ( t ) F 2 ( t ) U 2,2 ( t ) ] = [ F 2 ( t ) sin θ ( t ) F 2 ( t ) cos θ ( t ) ]
d dt 𝖺 = i ε ̂ ( t ) · 𝖺 ( t ) + i 𝐠 ( t ) ,
𝖺 0 𝖺 ( t = 0 ) = 𝖴 ( t = 0 ) · 𝖻 0 .
𝖺 ( t ) = 𝗏 ( t ) · ϕ ( t ) + 𝖺 0 · 𝗏 ( t ) ,
𝗏 ( t ) = exp { i 0 t ε ̂ ( t ) dt }
ϕ ( t ) = i 0 t 𝗏 1 ( t ) · 𝐠 ( t ) dt .
𝖺 ( t ) = 𝖺 0 · 𝗏 ( t ) .
b 1 ( t ) = exp ( i Δ 1 t ) { cos θ ( t ) exp { i 0 t ε 1 ( t ) dt } ( cos θ ( 0 ) b 1 0 + sin θ ( 0 ) b 2 0 )
sin θ ( t ) exp { i 0 t ε 2 ( t ) dt } ( sin θ ( 0 ) b 1 0 + cos θ ( 0 ) b 2 0 ) }
b 2 ( t ) = sin θ ( t ) exp { 0 t ε 1 ( t ) dt } ( cos θ ( 0 ) b 1 0 + sin θ ( 0 ) b 2 0 )
+ cos θ ( t ) exp { i 0 t ε 2 ( t ) dt } ( sin θ ( 0 ) b 1 0 + cos θ ( θ ) b 2 0 ) .
b 1 ( t ) = exp ( i Δ 1 t ) { cos θ ( t ) exp { i 0 t ε 1 ( t ) dt } cos θ ( 0 )
+ sin θ ( t ) exp { i 0 t ε 2 ( t ) dt } sin θ ( 0 ) }
b 2 ( t ) = sin θ ( t ) exp { i 0 t ε 1 ( t ) dt } ( cos θ ( 0 )
cos θ ( t ) exp { i 0 t ε 2 ( t ) dt } sin θ ( 0 ) .
𝖺 ( t ) = 𝗏 ( t ) · ϕ ( t ) ,
b 1 ( t ) = i exp ( i Δ 1 t ) { cos θ ( t ) 0 t exp { i t t ε 1 ( t ) dt } F 2 ( t ) sin θ ( t ) dt }
sin θ ( t ) 0 t exp { i t t ε 2 ( t ) dt } F 2 ( t ) cos θ ( t ) dt } ,
b 2 ( t ) = i { sin θ ( t ) 0 t exp { i t t ε 1 ( t ) dt } } F 2 ( t ) sin θ ( t ) dt
+ cos θ ( t ) 0 t exp { i t t ε 2 ( t ) dt } F 2 ( t ) cos θ ( t ) dt } .
Ψ ( t = 0 ) = dE b E 0 E , 3 s + 3 s ,
b E 0 b E ( t = 0 ) = ( δ E 2 π ) 1 4 exp { ( E E 0 ) 2 2 δ E 2 + i Δ E t 0 } . ,
F 2 ( t ) = Ω 2 , E 0 ( t ) dE exp [ i Δ E t ] b E 0 ,
F 2 ( t ) = ( 4 δ E 2 π ) 1 4 Ω 2 , E 0 ( t ) exp { δ E 2 ( t t 0 ) 2 2 ħ 2 i Δ E 0 ( t t 0 ) } .
ϵ 1,2 ( t ) = ϵ 1,2 0 exp { ( t t 1,2 ) 2 Δ t 1,2 2 } ,
F 2 ( t ) = ( 4 δ E 2 ) 1 4 ħ 2 μ 2 E 0 , m ± ϵ 2 0 exp { ( t t 2 ) 2 Δ t 2 2 i Δ E 0 ( t t 0 ) δ E 2 ( t t 0 ) 2 2 ħ 2 } .
η J ( T ) = π ( b J 2 b J 1 2 ) Δ t 2 = πn ħ 2 m 2 υ ( T ) ( 2 J + δ J , 0 4 ) Δ t 2
δ E δ E s , ϵ 1 0 ϵ 1 0 s , ϵ 2 0 ϵ 2 0 s ,
Δ t 1,2 Δ t 1,2 s ,
F 2 ( t ) F ̄ 2 ( t ) = F 2 ( t s ) s ; Ω 1,2 ( t ) Ω ̄ 1,2 ( t ) = Ω 1,2 ( t s ) s
d dt s 𝖻 ̄ = i { 𝖧 ( t s ) · 𝖻 ̅ + 𝖿 ( t s ) } ,

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