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Coherent states of a free electron laser
Volume 32, number 3
OPTICS COMMUNICATIONS
March 1980
COHERENT STATES OF A FREE ELECTRON LASER ~ R. BONIFACIO
Istituto di Fisica, Universit~ di Milano, 20133 Milano, Italy and Optical Science Center, University of Arizona, Tucson, Arizona, USA Received 6 December 1979
We define electron-field coherent quasi-classical states of a free electron laser. In these states both the photon number and the electron momentum are given by a Poisson distribution centered on the classical trajectories.
From a fundamental view-point the theory o f the F.E.L. can be divided into two categories: the classical theories [1 ] and the quantum theories [2]. However the coherence properties and the photon statistics o f the F.E.L. have never been discussed quantum mechanically from first principles. In this paper we define the coherent states o f a F.E.L. as the eigenstates of a properly defined harmonic oscillator operator which acts simultaneously on the electron field Hilbert spaces, and reduces the one particle F.E.L. hamiltonian to that of a single harmonic oscillator driven by a classical field (Wiggler) with a non linear term quadratic in the photon number operator. Assuming small fluctuations the hamiltonian becomes that o f a driven harmonic oscillator with a time dependent frequency. In this approximation the F.E.L. radiates coherently in the Glauber's sense giving a Poisson statistics for the p h o t o n number and the electron's m o m e n t u m centered on the classical values. We start from the B a m b i n i - R e n i e r i hamiltonian [2]
H = p2z/2m + h w (al~a L + a ~ a w ) + ihg(a~awe-ikz _ h . c . ) .
(1)
Here a L and a w represent respectively the Laser and the Wiggler field, Pz and z are the electron's m o m e n t u m and coordinate. The commutation relations are
[aL, a~L] = 1 ;
[aw, a~v] = 1 ;
[Z, pz ] = i h .
¢' Partially supported by Office of Naval Research.
440
(2)
Hamiltonian (1) describes the F.E.L. in a moving frame which has been properly chosen [2] so that the Laser and Wiggler frequency coincide with w = ck/2. The coupling constant is given by:
g = (47folk lOr 0 ,
(3)
r 0 being the classical electron's radius r 0 = e2/ 4neomc2 and V is the interaction volume. The free field hamiltonian can be easily eliminated because it is a constant o f motion. Hence we have
H = p2/2m + ihg(a~awe-ikz
_
h.c.) .
(4)
This hamiltonian has two constants of motion: the total photon number, given by
a~a L + a~vaw = const .
(5a)
and the total momentum:
p + ½(a[a L - a~vaw)= const. = P ,
(5b)
where p = pz/hK. These can be combined to give
p + a~a L =-N.
(5c)
It has been already noticed [3] that using the previous constant o f motion hamiltonian (1) can be written in terms of a single angular m o m e n t u m operator, defined as
J+ = a~aw e-ikz , Jz = ½(a~aL - a~vaw)"
(6a)
The angular m o m e n t u m commutation relations can be easily verified using (2). In this way H can be written as
OPTICS COMMUNICATIONS
Volume 32, number 3 H = ti~2(P - J z ) 2 + ihg(J + - J - ) ,
(6b)
where ~ = ~ k 2 / 2 m . For our purpose it is convenient to define the Harmonic oscillator operators (7a)
A = aLeikz .
functions Im) = (1/L)e im kz which satisfy to periodic boundary conditions on L = 2 n ~ / k . In this way the electron-field states In, m) can be labeled only by [In) defined as: Hn) - In, N - n ) .
(12)
Note that since e+-ikzlm) = Im +- l ) w e have
Note that: [A,A f ] = l ,
March 1980
AtA=a~a
L .
(7b)
Hence, using (5c), (7) and (8), hamiltonian (4) can be written as
Atlln)=x/n
+ l[In + l) ,
A t A l l n ) = nlln).
Go
(8a)
Ila)= e - l a l 2 / 2 ~
(8b)
Af A = p
is the normalized electron's momentum. This is the hamiltonian of two coupled harmonic oscillators with a quadratic non linearity for one of them. Hence the F.E.L. dynamics can be discussed in terms of the "coherent states" [4] la w, a) defined as the simultaneous eigenstates of a w and A awla w, a) = a w l a w, a ) ,
Alaw, a) = ala w, a ) .
(9)
Using the A-operator the angular m o m e n t u m (6a) can be rewritten ~ la Schwinger [5] as J + = A t a w and Jz = 1( A'~A - a ~ a w ) . Hence the law, a) states are "angular m o m e n t u m coherent states" as deemed and discussed in ref. [6]. Let us now concentrate on the coherent state let) of the operator A defined by (7) giving their explicit expression in the electron-field Hilbert space. The field Hilbert space is spanned by the eigenstates In) of a~LaL a[aLIn)=nln),
n = 0, 1 . . . . .
(10)
Furthermore thanks to the constant of motion (5c) the electron m o m e n t u m Hilbert space can be restricted to the following "accessible" states plm) = m l m ) ,
(13)
an ~ . v IIn).
(14)
n=0
where N-
1),
We can now easily define the electron-field coherent states as
H = I~2(N - A t A ) 2 + i h g ( a w A t - h.c.) = h~p 2 + ihg(awAt - h.c.),
Alln)=x/~[In-
(mira') = 5 re,m,
(11)
withm + n = N. In other words we can assume that the state of the system belongs at all times to the reduced Hilbert space spanned by In, m) = In, N - n) provided it is so at t = 0. Note that the Irn) states correspond to wave
Thanks to eqs. (13) it is easily verified that they satisfy the eigenvalue relation AIla) = allt0. In these states both the electron's m o m e n t u m and the laser photon number are well defined since they obey a Poisson statistics given by P(n) = e - Ic~12[al 2n /n! ,
P(m) = e -la12 l a l 2 ( N - m ) / ( g - m)! ,
(15)
with (n) = (a[a L) = lal 2 ,
(m) = (p) = N -
[al 2 ,
(16)
and (Sn 2) = (Sp 2) = (n) = [al 2 ,
(16a)
where 8n = A t A - (n), 8p = p - (p). Note that, by definition (allaLlla) = (alle± ikz lla) = O, whereas (allAlla) = (allaLe+ikzlla) = a. Hence these states are not strictly speaking phase coherent states for the field and the electron separately, but represent correlated electron-field states in which only the relative phase is well def'med and the photon number and electron m o m e n t u m obey a Poisson distribution with the same fluctuation. Furthermore we note that in general (all(A't ) m A n l l a ) = lal2nSm, n = (all(dr)manila).
(17)
Hence these coherent states have the well known factorization property for all the normally ordered products of equal numbers of creation and annihilation operators. Hamiltonian (8) does not preserve coherent states 441
Volume 32, number 3
OPTICS COMMUNICATIONS
since the nonlinear term introduce fluctuations. However let us define the classical limit of the F.E.L. so that the quantum fluctuations of the photon number and of the electron momentum are negligible, i.e. we assume: ((8n)2) ~ (n)2,
((8p)2) ,~ (p)2.
(18)
Thanks to eqs. (16) we see that inequalities (18) are verified in a coherent state provided (p)2 - - ( N -
]s[2) 2 >> [s[ 2 >> 1 .
(19a)
Note that the condition (p)2 >> 1 is the classical limit defined in ref. [2] assuming that the field is classical from very beginning. However inequality (19) shows that for N >> 1 momentum fluctuations are always relevant in the region I(p)l ~ x/~. In the classical limit (18) we can approximate the operator p2 in hamiltonian (8) as follows
March 1980
the classical equation for the one-particle F.E.L. provided one neglects the depletion of the Wiggler i.e. assuming Sw(t ) = SW(0 ) = s w in (21b). This can be easily seen defining the classical momentum Pc(t), the classical position Zc(t ) and the field's amplitude ~c(t) as follows Pc( t ) =
2c(t ) -
Pc(mr)
'
/3c(t) - s ( t ) e - i k z c (t)= (sllaL e i k z l l s ) e - i k z c ( t )
.
(22)
Note/3c(t ) does not coincide with (a t ) which is zero in an electron-field coherent state. With these definitions eq. (21a) reduces to the classical well known pendulum like equations:
ic(O = Pclm ; Pc = -ghk(swJ3ce- ikz c (t) + c.c.),
p2 = ((p) +8p)2 ... (p)2 + 2(p)~p ~c = gcrwe- ikz c(t) . = 2(p)p -- (p)2,
(19b)
where we have neglected the square of the "fluctuation" operator 6p = p - (p). Inserting (19b) in hamiltonian (8) we have H = 2hg2(p)p + i h g ( a w A ' f - h.c.)
(20) = 2hg2(Gt?A) - ~ ) A ? A
+ i h g ( a w A ? - h.c.),
where we have neglected c-numbers since they do not contribute to the equation of motion and we have used (8b). Hamiltonian (20) preserves coherent states [4], i.e. if the initial state is a coherent state Is, s w) the state at time t will be the coherent state Is(t), O~w(t)) correspondent to the complex numbers s(t), Sw(t) which satisfy the equations 8 (t) = -2i~2(1~t 2 _ OV))s + go~w ,
(21 a)
8W (t) = - g s ( t ) ,
(21 b)
which must be solved with the initial condition s(0) = s and O~w(0) = s w. These equations are obtained fromthe Heisenberg equation o f motion for A from hamiltonian (20) replacing the operators A and a w with the c-number
s(t), Sw(t) It is easily recognized that eq. (21) corresponds to 442
(23)
In conclusion, in the classical limit (19) the F.E.L. quantum state is a coherent state correspondent to a complex time dependent eigenvalue which evolves according to the classical trajectory. The validity of the semiclassical approximation (19b) and the effect of quantum fluctuations will be discussed elsewhere.
References [ l ] F.A. Hopf, P. Meystre, M.O. ScuUy and W.H. Louisell, Phys. Rev. Lett. 37 (1976) 1342. [2] A. Bambini and A. Renieri, Lett. Nuovo Cimento 3l (1978) 399; A. Bambini and A. Renieri, Optics Comm. 29 (1978) 244; A. Bambini, A. Renieri and S. Stenholm, Phys. Rev° A19 (1979) 2013. [3] G. Dattoli, Nuovo Cimento Lett., to be published. [4] R.J. Glauber, in: Quantum optics and electronics, eds. D° DeWitt, A. Blandlin and C. Cohen-Tannoudji (Gordon and Breach, Science Publishers, Inc. New York, 1965) p. 65. [5] J. Schwinger, in: Quantum theory of angulax momentum; perspectives in physics, eds. L.C. Biedenharn and H. VanDam (Academic Press Inc., New York, 1965). [6] R. Bonifacio, Dae M. Kim and M.O. ScuUy, Phys. Rev. 187 (1969) 44l.
OPTICS COMMUNICATIONS
March 1980
COHERENT STATES OF A FREE ELECTRON LASER ~ R. BONIFACIO
Istituto di Fisica, Universit~ di Milano, 20133 Milano, Italy and Optical Science Center, University of Arizona, Tucson, Arizona, USA Received 6 December 1979
We define electron-field coherent quasi-classical states of a free electron laser. In these states both the photon number and the electron momentum are given by a Poisson distribution centered on the classical trajectories.
From a fundamental view-point the theory o f the F.E.L. can be divided into two categories: the classical theories [1 ] and the quantum theories [2]. However the coherence properties and the photon statistics o f the F.E.L. have never been discussed quantum mechanically from first principles. In this paper we define the coherent states o f a F.E.L. as the eigenstates of a properly defined harmonic oscillator operator which acts simultaneously on the electron field Hilbert spaces, and reduces the one particle F.E.L. hamiltonian to that of a single harmonic oscillator driven by a classical field (Wiggler) with a non linear term quadratic in the photon number operator. Assuming small fluctuations the hamiltonian becomes that o f a driven harmonic oscillator with a time dependent frequency. In this approximation the F.E.L. radiates coherently in the Glauber's sense giving a Poisson statistics for the p h o t o n number and the electron's m o m e n t u m centered on the classical values. We start from the B a m b i n i - R e n i e r i hamiltonian [2]
H = p2z/2m + h w (al~a L + a ~ a w ) + ihg(a~awe-ikz _ h . c . ) .
(1)
Here a L and a w represent respectively the Laser and the Wiggler field, Pz and z are the electron's m o m e n t u m and coordinate. The commutation relations are
[aL, a~L] = 1 ;
[aw, a~v] = 1 ;
[Z, pz ] = i h .
¢' Partially supported by Office of Naval Research.
440
(2)
Hamiltonian (1) describes the F.E.L. in a moving frame which has been properly chosen [2] so that the Laser and Wiggler frequency coincide with w = ck/2. The coupling constant is given by:
g = (47folk lOr 0 ,
(3)
r 0 being the classical electron's radius r 0 = e2/ 4neomc2 and V is the interaction volume. The free field hamiltonian can be easily eliminated because it is a constant o f motion. Hence we have
H = p2/2m + ihg(a~awe-ikz
_
h.c.) .
(4)
This hamiltonian has two constants of motion: the total photon number, given by
a~a L + a~vaw = const .
(5a)
and the total momentum:
p + ½(a[a L - a~vaw)= const. = P ,
(5b)
where p = pz/hK. These can be combined to give
p + a~a L =-N.
(5c)
It has been already noticed [3] that using the previous constant o f motion hamiltonian (1) can be written in terms of a single angular m o m e n t u m operator, defined as
J+ = a~aw e-ikz , Jz = ½(a~aL - a~vaw)"
(6a)
The angular m o m e n t u m commutation relations can be easily verified using (2). In this way H can be written as
OPTICS COMMUNICATIONS
Volume 32, number 3 H = ti~2(P - J z ) 2 + ihg(J + - J - ) ,
(6b)
where ~ = ~ k 2 / 2 m . For our purpose it is convenient to define the Harmonic oscillator operators (7a)
A = aLeikz .
functions Im) = (1/L)e im kz which satisfy to periodic boundary conditions on L = 2 n ~ / k . In this way the electron-field states In, m) can be labeled only by [In) defined as: Hn) - In, N - n ) .
(12)
Note that since e+-ikzlm) = Im +- l ) w e have
Note that: [A,A f ] = l ,
March 1980
AtA=a~a
L .
(7b)
Hence, using (5c), (7) and (8), hamiltonian (4) can be written as
Atlln)=x/n
+ l[In + l) ,
A t A l l n ) = nlln).
Go
(8a)
Ila)= e - l a l 2 / 2 ~
(8b)
Af A = p
is the normalized electron's momentum. This is the hamiltonian of two coupled harmonic oscillators with a quadratic non linearity for one of them. Hence the F.E.L. dynamics can be discussed in terms of the "coherent states" [4] la w, a) defined as the simultaneous eigenstates of a w and A awla w, a) = a w l a w, a ) ,
Alaw, a) = ala w, a ) .
(9)
Using the A-operator the angular m o m e n t u m (6a) can be rewritten ~ la Schwinger [5] as J + = A t a w and Jz = 1( A'~A - a ~ a w ) . Hence the law, a) states are "angular m o m e n t u m coherent states" as deemed and discussed in ref. [6]. Let us now concentrate on the coherent state let) of the operator A defined by (7) giving their explicit expression in the electron-field Hilbert space. The field Hilbert space is spanned by the eigenstates In) of a~LaL a[aLIn)=nln),
n = 0, 1 . . . . .
(10)
Furthermore thanks to the constant of motion (5c) the electron m o m e n t u m Hilbert space can be restricted to the following "accessible" states plm) = m l m ) ,
(13)
an ~ . v IIn).
(14)
n=0
where N-
1),
We can now easily define the electron-field coherent states as
H = I~2(N - A t A ) 2 + i h g ( a w A t - h.c.) = h~p 2 + ihg(awAt - h.c.),
Alln)=x/~[In-
(mira') = 5 re,m,
(11)
withm + n = N. In other words we can assume that the state of the system belongs at all times to the reduced Hilbert space spanned by In, m) = In, N - n) provided it is so at t = 0. Note that the Irn) states correspond to wave
Thanks to eqs. (13) it is easily verified that they satisfy the eigenvalue relation AIla) = allt0. In these states both the electron's m o m e n t u m and the laser photon number are well defined since they obey a Poisson statistics given by P(n) = e - Ic~12[al 2n /n! ,
P(m) = e -la12 l a l 2 ( N - m ) / ( g - m)! ,
(15)
with (n) = (a[a L) = lal 2 ,
(m) = (p) = N -
[al 2 ,
(16)
and (Sn 2) = (Sp 2) = (n) = [al 2 ,
(16a)
where 8n = A t A - (n), 8p = p - (p). Note that, by definition (allaLlla) = (alle± ikz lla) = O, whereas (allAlla) = (allaLe+ikzlla) = a. Hence these states are not strictly speaking phase coherent states for the field and the electron separately, but represent correlated electron-field states in which only the relative phase is well def'med and the photon number and electron m o m e n t u m obey a Poisson distribution with the same fluctuation. Furthermore we note that in general (all(A't ) m A n l l a ) = lal2nSm, n = (all(dr)manila).
(17)
Hence these coherent states have the well known factorization property for all the normally ordered products of equal numbers of creation and annihilation operators. Hamiltonian (8) does not preserve coherent states 441
Volume 32, number 3
OPTICS COMMUNICATIONS
since the nonlinear term introduce fluctuations. However let us define the classical limit of the F.E.L. so that the quantum fluctuations of the photon number and of the electron momentum are negligible, i.e. we assume: ((8n)2) ~ (n)2,
((8p)2) ,~ (p)2.
(18)
Thanks to eqs. (16) we see that inequalities (18) are verified in a coherent state provided (p)2 - - ( N -
]s[2) 2 >> [s[ 2 >> 1 .
(19a)
Note that the condition (p)2 >> 1 is the classical limit defined in ref. [2] assuming that the field is classical from very beginning. However inequality (19) shows that for N >> 1 momentum fluctuations are always relevant in the region I(p)l ~ x/~. In the classical limit (18) we can approximate the operator p2 in hamiltonian (8) as follows
March 1980
the classical equation for the one-particle F.E.L. provided one neglects the depletion of the Wiggler i.e. assuming Sw(t ) = SW(0 ) = s w in (21b). This can be easily seen defining the classical momentum Pc(t), the classical position Zc(t ) and the field's amplitude ~c(t) as follows Pc( t ) =
2c(t ) -
Pc(mr)
'
/3c(t) - s ( t ) e - i k z c (t)= (sllaL e i k z l l s ) e - i k z c ( t )
.
(22)
Note/3c(t ) does not coincide with (a t ) which is zero in an electron-field coherent state. With these definitions eq. (21a) reduces to the classical well known pendulum like equations:
ic(O = Pclm ; Pc = -ghk(swJ3ce- ikz c (t) + c.c.),
p2 = ((p) +8p)2 ... (p)2 + 2(p)~p ~c = gcrwe- ikz c(t) . = 2(p)p -- (p)2,
(19b)
where we have neglected the square of the "fluctuation" operator 6p = p - (p). Inserting (19b) in hamiltonian (8) we have H = 2hg2(p)p + i h g ( a w A ' f - h.c.)
(20) = 2hg2(Gt?A) - ~ ) A ? A
+ i h g ( a w A ? - h.c.),
where we have neglected c-numbers since they do not contribute to the equation of motion and we have used (8b). Hamiltonian (20) preserves coherent states [4], i.e. if the initial state is a coherent state Is, s w) the state at time t will be the coherent state Is(t), O~w(t)) correspondent to the complex numbers s(t), Sw(t) which satisfy the equations 8 (t) = -2i~2(1~t 2 _ OV))s + go~w ,
(21 a)
8W (t) = - g s ( t ) ,
(21 b)
which must be solved with the initial condition s(0) = s and O~w(0) = s w. These equations are obtained fromthe Heisenberg equation o f motion for A from hamiltonian (20) replacing the operators A and a w with the c-number
s(t), Sw(t) It is easily recognized that eq. (21) corresponds to 442
(23)
In conclusion, in the classical limit (19) the F.E.L. quantum state is a coherent state correspondent to a complex time dependent eigenvalue which evolves according to the classical trajectory. The validity of the semiclassical approximation (19b) and the effect of quantum fluctuations will be discussed elsewhere.
References [ l ] F.A. Hopf, P. Meystre, M.O. ScuUy and W.H. Louisell, Phys. Rev. Lett. 37 (1976) 1342. [2] A. Bambini and A. Renieri, Lett. Nuovo Cimento 3l (1978) 399; A. Bambini and A. Renieri, Optics Comm. 29 (1978) 244; A. Bambini, A. Renieri and S. Stenholm, Phys. Rev° A19 (1979) 2013. [3] G. Dattoli, Nuovo Cimento Lett., to be published. [4] R.J. Glauber, in: Quantum optics and electronics, eds. D° DeWitt, A. Blandlin and C. Cohen-Tannoudji (Gordon and Breach, Science Publishers, Inc. New York, 1965) p. 65. [5] J. Schwinger, in: Quantum theory of angulax momentum; perspectives in physics, eds. L.C. Biedenharn and H. VanDam (Academic Press Inc., New York, 1965). [6] R. Bonifacio, Dae M. Kim and M.O. ScuUy, Phys. Rev. 187 (1969) 44l.