Abstract

Atom and nanoparticle arrays trapped in optical lattices are shown to be capable of sustaining collective oscillations of frequency proportional to the strength of the external light field. The spectrum of these oscillations determines the mechanical stability of the arrays. This phenomenon is studied for dimers, strings, and two-dimensional planar arrays. Laterally confined particles free to move along an optical channel are also considered as an example of collective motion in partially-confined systems. The fundamental concepts of dynamical response in optical matter introduced here constitute the basis for potential applications to quantum information technology and signal processing. Experimental realizations of these systems are proposed.

© 2007 Optical Society of America

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References

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  1. M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, "Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms," Nature 415, 39-44 (2002).
    [CrossRef] [PubMed]
  2. M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical binding," Phys. Rev. Lett. 63, 1233-1236 (1989).
    [CrossRef] [PubMed]
  3. M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical matter: Crystallization and binding in intense optical fields," Science 249, 749-754 (1990).
    [CrossRef] [PubMed]
  4. P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, and G. Rempe, "Observation of cavity-mediated longrange light forces between strongly coupled atoms," Phys. Rev. Lett. 84, 4068-4071 (2000).
    [CrossRef] [PubMed]
  5. B. Nagorny, T. Elsässer, and A. Hemmerich, "Collective atomic motion in an optical lattice formed inside a high finesse cavity," Phys. Rev. Lett. 91, 153003 (2003).
    [CrossRef] [PubMed]
  6. D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, "Fast quantum gates for neutral atoms," Phys. Rev. Lett. 85, 2208-2211 (2000).
    [CrossRef] [PubMed]
  7. A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
    [CrossRef]
  8. C. A. Ashley and S. Doniach, "Theory of extended x-ray absorption edge fine structure (EXAFS) in crystalline solids," Phys. Rev. B 11, 1279-1288 (1975).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  16. F. J. García de Abajo, T. Brixner, and W. Pfeiffer, "Nanoscale force manipulation in the vicinity of a metal nanostructure," J. Phys. B 40, S249-S258 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
  18. S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, "One-dimensional optically bound arrays of microscopic particles," Phys. Rev. Lett. 89, 283901 (2002).
    [CrossRef]
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    [CrossRef]
  20. M. Polin, D. G. Grier, and S. R. Quake, "Anomalous vibrational dispersion in holographically trapped colloidal arrays," Phys. Rev. Lett. 96, 088101 (2006).
    [CrossRef] [PubMed]
  21. J. P. Gordon and A. Ashkin, "Motion of atoms in a radiation trap," Phys. Rev. A 21, 1606-1617 (1980).
    [CrossRef]
  22. P. C. Chaumet and M. Nieto-Vesperinas, "Time-averaged total force on a dipolar sphere in an electromagnetic field," Opt. Lett. 25, 1065-1067 (2000).
    [CrossRef]
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    [CrossRef]
  25. M. Guillon, "Field enhancement in a chain of optically bound dipoles," Opt. Express 14, 3045-3055 (2006).
    [CrossRef] [PubMed]
  26. F. J. García de Abajo, "Electromagnetic forces and torques in nanoparticles irradiated by plane waves," J. Quant. Spectrosc. Radiat. Transfer 89, 3-9 (2004).
    [CrossRef]
  27. T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, "Stable optical trapping based on optical binding forces," Phys. Rev. Lett. 96, 113903 (2006).
    [CrossRef] [PubMed]
  28. T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, "Trapping and binding of an arbitrary number of cylindrical particles in an in-plane electromagnetic field," J. Opt. Soc. Am. A 23, 2324-2330 (2006).
    [CrossRef]
  29. F. Depasse and J.-M. Vigoureux, "Optical binding force between two Rayleigh particles," J. Phys. D 27, 914-919 (1994).
    [CrossRef]
  30. M. S. Safronova, C. J. Williams, and C.W. Clark, "Optimizing the fast Rydberg quantum gate," Phys. Rev. A 67, 040303(R) (2003).
    [CrossRef]
  31. It should be noted that Rb has another resonance of width Γ = 2 π×2 MHz at 795 nm (frequency ω1), which combined with the 780 nm resonance (frequency ω0) gives an effective value of Γ = 2 π×6 MHz for light tuned far from this region (|ω. ω0|>> ω0. ω1). However, we are discussing here light tuned very close to the 780 nm resonance (|ω. ω0|<< ω0. ω1), for which the lower-frequency resonance can be overlooked.
  32. F. J. García de Abajo, "Interaction of radiation and fast electrons with clusters of dielectrics: A multiple scattering approach," Phys. Rev. Lett. 82, 2776-2779 (1999).
    [CrossRef]
  33. F. J. García de Abajo, "Momentum transfer to small particles by passing electron beams," Phys. Rev. B 70, 115422 (2004).
    [CrossRef]
  34. A. Ashkin and J. M. Dziedzic, "Optical levitation in high vacuum," Appl. Phys. Lett. 28, 333-335 (1976).
    [CrossRef]
  35. B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
    [CrossRef]
  36. W. H. Weber and G. W. Ford, "Propagation of optical excitations by dipolar interactions in metal nanoparticle chains," Phys. Rev. B 70, 125429 (2004).
    [CrossRef]

2007 (2)

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, "Parallel and selective trapping in a patterned plasmonic landscape," Nat. Phys. 3, 477-480 (2007).
[CrossRef]

F. J. García de Abajo, T. Brixner, and W. Pfeiffer, "Nanoscale force manipulation in the vicinity of a metal nanostructure," J. Phys. B 40, S249-S258 (2007).
[CrossRef]

2006 (7)

2005 (1)

P.M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, "Expanding the optical trapping range of gold nanoparticles," Nano Lett. 5, 1937-1942 (2005).
[CrossRef] [PubMed]

2004 (4)

W. H. Weber and G. W. Ford, "Propagation of optical excitations by dipolar interactions in metal nanoparticle chains," Phys. Rev. B 70, 125429 (2004).
[CrossRef]

F. J. García de Abajo, "Electromagnetic forces and torques in nanoparticles irradiated by plane waves," J. Quant. Spectrosc. Radiat. Transfer 89, 3-9 (2004).
[CrossRef]

P. Zemánek, V. Karásek, and A. Sasso, "Optical forces acting on Rayleigh particle placed into interference field," Opt. Commun. 240, 401-415 (2004).
[CrossRef]

F. J. García de Abajo, "Momentum transfer to small particles by passing electron beams," Phys. Rev. B 70, 115422 (2004).
[CrossRef]

2003 (3)

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

M. S. Safronova, C. J. Williams, and C.W. Clark, "Optimizing the fast Rydberg quantum gate," Phys. Rev. A 67, 040303(R) (2003).
[CrossRef]

B. Nagorny, T. Elsässer, and A. Hemmerich, "Collective atomic motion in an optical lattice formed inside a high finesse cavity," Phys. Rev. Lett. 91, 153003 (2003).
[CrossRef] [PubMed]

2002 (2)

M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, "Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms," Nature 415, 39-44 (2002).
[CrossRef] [PubMed]

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, "One-dimensional optically bound arrays of microscopic particles," Phys. Rev. Lett. 89, 283901 (2002).
[CrossRef]

2000 (3)

P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, and G. Rempe, "Observation of cavity-mediated longrange light forces between strongly coupled atoms," Phys. Rev. Lett. 84, 4068-4071 (2000).
[CrossRef] [PubMed]

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, "Fast quantum gates for neutral atoms," Phys. Rev. Lett. 85, 2208-2211 (2000).
[CrossRef] [PubMed]

P. C. Chaumet and M. Nieto-Vesperinas, "Time-averaged total force on a dipolar sphere in an electromagnetic field," Opt. Lett. 25, 1065-1067 (2000).
[CrossRef]

1999 (1)

F. J. García de Abajo, "Interaction of radiation and fast electrons with clusters of dielectrics: A multiple scattering approach," Phys. Rev. Lett. 82, 2776-2779 (1999).
[CrossRef]

1998 (1)

M. Hoppenbrouwers and W. van de Water, "Modes of motion of a colloidal crystal," Phys. Rev. Lett. 80, 3871-3874 (1998).
[CrossRef]

1994 (1)

F. Depasse and J.-M. Vigoureux, "Optical binding force between two Rayleigh particles," J. Phys. D 27, 914-919 (1994).
[CrossRef]

1990 (1)

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical matter: Crystallization and binding in intense optical fields," Science 249, 749-754 (1990).
[CrossRef] [PubMed]

1989 (1)

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical binding," Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef] [PubMed]

1988 (1)

B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

1987 (1)

A. Ashkin and J. M. Dziedzic, "Optical trapping and manipulation of viruses and bacteria," Science 235, 1517-1520 (1987).
[CrossRef] [PubMed]

1980 (2)

J. P. Gordon and A. Ashkin, "Motion of atoms in a radiation trap," Phys. Rev. A 21, 1606-1617 (1980).
[CrossRef]

A. Ashkin, "Applications of laser radiation pressure," Science 210, 1081-1088 (1980).
[CrossRef] [PubMed]

1976 (1)

A. Ashkin and J. M. Dziedzic, "Optical levitation in high vacuum," Appl. Phys. Lett. 28, 333-335 (1976).
[CrossRef]

1975 (1)

C. A. Ashley and S. Doniach, "Theory of extended x-ray absorption edge fine structure (EXAFS) in crystalline solids," Phys. Rev. B 11, 1279-1288 (1975).
[CrossRef]

1970 (1)

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

," Phys. Rev. A (1)

M. S. Safronova, C. J. Williams, and C.W. Clark, "Optimizing the fast Rydberg quantum gate," Phys. Rev. A 67, 040303(R) (2003).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Ashkin and J. M. Dziedzic, "Optical levitation in high vacuum," Appl. Phys. Lett. 28, 333-335 (1976).
[CrossRef]

Astrophys. J. (1)

B. T. Draine, "The discrete-dipole approximation and its application to interstellar graphite grains," Astrophys. J. 333, 848-872 (1988).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. B (1)

F. J. García de Abajo, T. Brixner, and W. Pfeiffer, "Nanoscale force manipulation in the vicinity of a metal nanostructure," J. Phys. B 40, S249-S258 (2007).
[CrossRef]

J. Phys. D (1)

F. Depasse and J.-M. Vigoureux, "Optical binding force between two Rayleigh particles," J. Phys. D 27, 914-919 (1994).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer (1)

F. J. García de Abajo, "Electromagnetic forces and torques in nanoparticles irradiated by plane waves," J. Quant. Spectrosc. Radiat. Transfer 89, 3-9 (2004).
[CrossRef]

Nano Lett. (1)

P.M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, "Expanding the optical trapping range of gold nanoparticles," Nano Lett. 5, 1937-1942 (2005).
[CrossRef] [PubMed]

Nat. Phys. (1)

M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, "Parallel and selective trapping in a patterned plasmonic landscape," Nat. Phys. 3, 477-480 (2007).
[CrossRef]

Nature (2)

M. Greiner, O. Mandel, T. Esslinger, T.W. Hänsch, and I. Bloch, "Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms," Nature 415, 39-44 (2002).
[CrossRef] [PubMed]

D. G. Grier, "A revolution in optical manipulation," Nature 424, 810-816 (2003).
[CrossRef] [PubMed]

Opt. Commun. (1)

P. Zemánek, V. Karásek, and A. Sasso, "Optical forces acting on Rayleigh particle placed into interference field," Opt. Commun. 240, 401-415 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (1)

J. P. Gordon and A. Ashkin, "Motion of atoms in a radiation trap," Phys. Rev. A 21, 1606-1617 (1980).
[CrossRef]

Phys. Rev. B (3)

C. A. Ashley and S. Doniach, "Theory of extended x-ray absorption edge fine structure (EXAFS) in crystalline solids," Phys. Rev. B 11, 1279-1288 (1975).
[CrossRef]

W. H. Weber and G. W. Ford, "Propagation of optical excitations by dipolar interactions in metal nanoparticle chains," Phys. Rev. B 70, 125429 (2004).
[CrossRef]

F. J. García de Abajo, "Momentum transfer to small particles by passing electron beams," Phys. Rev. B 70, 115422 (2004).
[CrossRef]

Phys. Rev. Lett. (11)

F. J. García de Abajo, "Interaction of radiation and fast electrons with clusters of dielectrics: A multiple scattering approach," Phys. Rev. Lett. 82, 2776-2779 (1999).
[CrossRef]

T. M. Grzegorczyk, B. A. Kemp, and J. A. Kong, "Stable optical trapping based on optical binding forces," Phys. Rev. Lett. 96, 113903 (2006).
[CrossRef] [PubMed]

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical binding," Phys. Rev. Lett. 63, 1233-1236 (1989).
[CrossRef] [PubMed]

P. Münstermann, T. Fischer, P. Maunz, P. W. H. Pinkse, and G. Rempe, "Observation of cavity-mediated longrange light forces between strongly coupled atoms," Phys. Rev. Lett. 84, 4068-4071 (2000).
[CrossRef] [PubMed]

B. Nagorny, T. Elsässer, and A. Hemmerich, "Collective atomic motion in an optical lattice formed inside a high finesse cavity," Phys. Rev. Lett. 91, 153003 (2003).
[CrossRef] [PubMed]

D. Jaksch, J. I. Cirac, P. Zoller, S. L. Rolston, R. Côté, and M. D. Lukin, "Fast quantum gates for neutral atoms," Phys. Rev. Lett. 85, 2208-2211 (2000).
[CrossRef] [PubMed]

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett. 24, 156-159 (1970).
[CrossRef]

N. K. Metzger, K. Dholakia, and E. M. Wright, "Observation of bistability and hysteresis in optical binding of two dielectric spheres," Phys. Rev. Lett. 96, 068102 (2006).
[CrossRef] [PubMed]

S. A. Tatarkova, A. E. Carruthers, and K. Dholakia, "One-dimensional optically bound arrays of microscopic particles," Phys. Rev. Lett. 89, 283901 (2002).
[CrossRef]

M. Hoppenbrouwers and W. van de Water, "Modes of motion of a colloidal crystal," Phys. Rev. Lett. 80, 3871-3874 (1998).
[CrossRef]

M. Polin, D. G. Grier, and S. R. Quake, "Anomalous vibrational dispersion in holographically trapped colloidal arrays," Phys. Rev. Lett. 96, 088101 (2006).
[CrossRef] [PubMed]

Science (3)

M. M. Burns, J.-M. Fournier, and J. A. Golovchenko, "Optical matter: Crystallization and binding in intense optical fields," Science 249, 749-754 (1990).
[CrossRef] [PubMed]

A. Ashkin, "Applications of laser radiation pressure," Science 210, 1081-1088 (1980).
[CrossRef] [PubMed]

A. Ashkin and J. M. Dziedzic, "Optical trapping and manipulation of viruses and bacteria," Science 235, 1517-1520 (1987).
[CrossRef] [PubMed]

Other (2)

It should be noted that Rb has another resonance of width Γ = 2 π×2 MHz at 795 nm (frequency ω1), which combined with the 780 nm resonance (frequency ω0) gives an effective value of Γ = 2 π×6 MHz for light tuned far from this region (|ω. ω0|>> ω0. ω1). However, we are discussing here light tuned very close to the 780 nm resonance (|ω. ω0|<< ω0. ω1), for which the lower-frequency resonance can be overlooked.

R. Loudon, The Quantum Theory of Light (Oxford University Press, Oxford, 2000).

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Figures (4)

Fig. 1.
Fig. 1.

Oscillation frequencies of two identical atoms trapped in contiguous wells of an optical lattice as a function of light frequency w near an atomic resonance ω0 . The lattice is set up by three pairs of counter-propagating lasers of direction and (linear) polarization as shown in the lower inset. The wells are a distance λ/2 apart, where λ=2πc/w is the light wavelength. The atomic polarizability α (upper inset) is obtained by assuming that radiative decay is the dominant contribution to the resonance width Γ [see Eq. (2)]. The oscillation frequency Ω is normalized using the atom mass M and the intensity of each beam I. The oscillations take place around equilibrium positions corresponding to the vanishing of the electric field strength (yellow regions in lower inset). Six oscillation modes are obtained: two pairs of doubly-degenerate modes (solid curves) and two nondegenerate modes (dashed curves). The relative directions of motion of the atoms in the dimer are indicated by arrows for each of the modes. The oscillation frequency of a single trapped atom is given for reference (dotted curve).

Fig. 2.
Fig. 2.

Dispersion relation of the oscillation modes in an infinite periodic string of atoms located at the minima of electric field strength along the 〈111〉 direction under the same conditions of illumination as in Fig. 1 (see lower inset in that Fig.; the coordinate axes are taken parallel to the laser beams). The nearest-neighbor distance is a=√3λ/2. The light frequency is tuned to ω=ω0 +Γ, using the same notation and atomic polarizability as in Fig. 1. The oscillation frequency is given as a function of momentum q along the string within the one-dimensional first Brillouin zone. Oscillations parallel to the 〈111〉 direction (broken curve) are decoupled from those along perpendicular directions (solid curve).

Fig. 3.
Fig. 3.

Dispersion relation of oscillation modes in an optical lattice of silica spherical particles (ε=2.1) in vacuum. The lattice is set up by four pairs of counter-propagating linearly-polarized beams with incident magnetic field in the plane of the particles. The inset shows the particles in their equilibrium positions forming a planar square lattice of nearest-neighbor distance λ/sinθ, where θ=50° is the angle formed by the light beams and the normal to the plane of the array. Continuous, dashed, and dashed-dotted thick curves correspond to the three oscillation modes for each value of the parallel momentum along the excursion ΓXMΓ within the first Brillouin zone. The spheres diameter is 240 nm, the light wavelength is λ=550 nm, and the flux of each laser is 100 W/cm2. These results are obtained with inclusion of higher-order multipoles beyond the dipole. Calculations in the dipole approximation (thin dotted curves) are shown for reference, with the polarizability obtained from the electric dipole Mie coefficient as α=(3/2k 3) tE 1 [32].

Fig. 4.
Fig. 4.

Oscillation modes and stability of a linear array of particles trapped in a two-dimensional optical lattice set up by two pairs of counter-propagating beams with polarizations as shown in (a). The array is assumed to be periodic, infinitely long, and surrounded by vacuum. A typical spectrum of oscillations is presented in panel (b) for specific values of the free-space momentum of the trapping light k and the particle polarizability α, normalized to the lattice spacing a (see text insets). The spectrum is given within the first Brillouin zone of momentum along the array q, with normalization of frequency and momentum as in Fig. 2. The contour plot of panel (c) represents squared oscillation frequencies as a function of q and k for α=1/(α-1 0-2ik3 /3), where α0=0.001a 3 is the electrostatic polarizability (this prescription for α is consistent with the optical theorem for non-absorbing particles [35, 36]). White regions in (c) correspond to imaginary oscillation eigenfrequencies, which signal array instabilities.

Equations (19)

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F = 2 { α l E l ext [ E l ext ] * } = V 2 { α } { l E l ext [ E l ext ] * } ,
α ( ω ) = 3 c 3 Γ 2 ω 0 2 ω 0 2 ω 2 i Γ ω 3 ω 0 2 3 c 3 Γ 4 ω 0 3 ω 0 ω i Γ 2 ,
p j = α [ E ext ( r j ) + j j G ( r j r j ) p j ] ,
G ( r ) = ( k 2 + ) e i kr r = A B r 2 r r ,
A = e i kr r 3 [ ( kr ) 2 + i kr 1 ] ,
B = e i kr r 3 [ ( kr ) 2 + 3 i kr 3 ] ,
E ext ( r , t ) = 4 { E 0 e i ω t } [ sin ( k z ) x ˆ + sin ( k x ) y ˆ + sin ( k y ) z ˆ ] ,
Ω 1 ± = E 0 k M 8 { 1 α 1 ± A }
Ω 2 ± = E 0 k M 8 { 1 α 1 ± ( A B ) } ,
{ Σ q } u q = M Ω q 2 u q ,
G q = j 0 e i q · r j G ( r j )
G q = 4 j = 1 cos ( q a j ) e i kaj ( a j ) 3 ( 1 i kaj )
G q = 2 j = 1 cos ( q a j ) e i kaj ( a j ) 3 [ ( kaj ) 2 + i kaj 1 ] ,
E ext ( r , t ) = 8 { E 0 e i ω t } { [ sin ( Q x ) x ˆ + sin ( Q y ) y ˆ ] sin ( k z z ) cos + [ cos ( Q x ) + cos ( Q y ) z ˆ cos ( k z z ) sin θ } ,
E ext ( r , t ) = 4 { E 0 e i ω t } [ cos ( k z ) + cos ( k y ) ] x ˆ ,
E j = E ext + η j ,
η q = j e i q a j η j ,
Ω q = [ 2 α E 2 M { g q 2 1 α G q + H q } ] 1 2
Ω q = [ 2 α E 2 M { g q 2 4 1 α G q + H q } + 4 k 2 M { E 0 * α E } ] 1 2 ,

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