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Quantum revivals in the motion of an electron in a magnetic field
Volume 86A, number 6,7
PHYSICS LETTERS
30 November 1981
QUANTUM REVIVALS IN THE MOTION OF AN ELECTRON IN A MAGNETIC FIELD Piotr FILIPOWICZ Institute of Physics, Polish Academy of Sciences, 02668 Warsaw, Poland
and Jan MOSTOWSKI1
Research Institute for Theoretical Physics, University of Helsinki, SF-001 70 Helsinki 1 7, Finland Received 23 September 1981
We show that the motion of a relativistic electron in a constant homogeneous magnetic field exhibits quasiperiodic behaviour (quantum revivals) and discuss the possibility of their observation.
In this letter we examine the quantum motion of a relativistic electron in a constant homogeneous magnetic fIeld from the point of view of’ quantum revivals”. As has been shown by Eberly and co-workers [1, 2] (see also ref. [3]) a simple quantum system con-
~,
1
Ia,6,k ,p)”exp[——(Ial 2
ue of the electron position in a constant homogeneous magnetic field exhibits similar behaviour if the initial state is a wavepacket. We will discuss the possibility of observing such phenomena. As is well known (see e.g. ref. [4]) the quantum numbers describing a relativistic electron in a magnetic field are n = 1, 2, 3 (principal quantum number), s = 0, 1, 2 (radial quantum number), k 3(wave vector parallel to the B field), and p = ±1(projection of spin on the direction of B). The energy of the electron is 2 [mc2 +1~2k~/m +phw +2hw(n ~ }1/2(~) E’~{mc where w = —eB/mc > 0. Denote eigenfunctions corresponding to n, s, k, p by In, s, k, p). Coherent states Ia, 6, k 3, p), where 1 Permanent address: Institute of Physics, Polish Academy of Sciences, 02668 Warsaw, Poland.
356
+
161 )] (2)
00
X ~
sisting of an interacting fermion and bosons (the Jaynes—Cummings model) exhibits unexpected quasiperiodical features known as revivals. Following similar lines of reasoning we will show that the average val-
...
6 are arbitrary complex numbers, are defined as
a~l65(n!s!)_hi2In,s, k
3, p) n,s=0 As time develops the coherent states change according to —
r
1
2
2
ln,6,k3,ii) —expi—~(IaI + 161 ) 1E(n)t] ah165(n!s!Yh/2 (3) X n,s=0 exp[—ili ~< ~ s k p Because E(n) is not a linear function of n the wave
~ ‘
‘
~‘
function Ia, 6, k3, ~>t is no longer a coherent state, according to the definition (2). Thus if a coherent state has been formed at time t = 0 it will not be a coherent state after(3) an isinfinitesimally time. Nevertheless function a useful objectshort describing the time development of the wavepacket (2). Projection of the position of the electron on the plane perpendicular to the magnetic field (xy plane) can be most conveniently described by the average value of x + iy. If we use the Foldy—Wouthuysen representation~for the position operator we find
0031-9163/81/0000—0000/S 02.75 © 1981 North-Holland
Volume 86A, number 6,7
PHYSICS LETTERS
t(a, 6, k3,,ulx +iyla, 6, k3, ~>t =(2h/mw)1/2 X
~~~a*exp(_IaI2)
E —
8ir(a +b~3)3/2b—2hk (k = 0,1,2...)
tk
(7)
all linear terms in n j3 are multiples of 2iri. Thus the terms in series (5) become phased and we find that the average position takes large values. These are the revivals. In order to find the time evolution of the series (5) around each revival we put ~ tk + r with r ~ 8ir(a + bj3)312hb—2. The series (5) becomes —
al2n(n!)_l (4)
n0
X exp{ih~[E(n + 1)
30 November 1981
E(n)]t})
-
In what follows we will restrict ourselves to the case 1a12 ~ 1 and approximate the difference E(n + 1) E(n) by the differential aE(n)/an. Thus the time evolution of the average position is determined by the series —
00
E
exp(—13)(JY~/n!)exp[4irik(a + b13)b~
n0
2irikl3+3irikb(n —I3)2~(a+b13)—’] X exp{ibr[2h(a+b13)112]’ x [1—b(n—13)~(a+bf3)~ +
~ n0
exp(—IaI2)laI2’~(n!)’exp{i1v1 [aE(n)/an]t} (5)
Similar series, but with a different n dependence in the exponent have been studied in refs. [1—3,5]in connection with the Jaynes—Cummings model, and lead to revivals. We will find an approximate formula for the series (5), but we will avoid a mistake in the original derivation of ref. [21. (A similar calculation for the Jaynes—Cummings model has been performed by Yoo and Eberly [6].) Since the weighting factor exp(_IaI2)IaI2~l(n!)~ peaks at a value n al2 with a narrow dispersion lal we may expand aE/an around n = lcd2: aE/an ~~b(a +b~3)—1I2{J.—b(n —f3)[2(a +b13)]—1 (6)
-
This sum is approximated by an integral and calculated with the saddle point method. The result is Rk(r) exp [iøk where ~k
4irk(a
=
+
—
~i ~k
+
—
ekr2]
,
(8)
b13)b—’ + 2irkf3, b13)11~]—1,
+
=
~k =b413(a +b13+3irikbl3) X ~ + b!3)2 [(a + b13)2
+3b2(n—13)2[8(a+b(3)2]~}, where 13 al2, a = m2c4 + h2k~c2+ (p + 1)hwmc2 and b 2mc2hw. The time behaviour of the series (5) can be studied separately for various time scales. If t ~ 4h(a +b13)312 X b213—1/2, linear and quadratic terms inn —13 in (6) can be neglected. The series can be summed and we find that the average position performs oscillations with a frequency b[2h(a + b(3)1I2]~.This is the quantum counterpart of the classical synchrotron motion. For times of the order of h(a + b13)312b213112, linear and quadratic terms cannot be neglected. We may expect that because of dephasing of the terms the average position will be damped to zero (the wavepacket spreads along its orbit). If t is still larger revivals may occur. Observe that if we take the time t to be
3)2! (a +bj3)—2 ] }
+ 3b2(n
+9ir2k2/32b2]h2}1, ~k
=
tan—1(M/D),
Rk(T) =
(M2 +D2)—114,
M~’b2r(a + bf3){4h(a + bj3)1/2 x [(a + b13)2 + 9ir2k2b2j3] }_1 —
D=1
3~1thf3(t’i+ b13Y1 —3b313r~1r’(a+ b13Y5I2, 3~kb313r{4h(a+ bi3)112 —
X [(a + bj3)2 + 9ir2k2b2!3] }_1 Thus each revival is characterized by its own phase 0k ~ amplitude Rk and characteristic width (Re e)h/Pwhich grows with k. Eventually revivals will overlap, which should lead to “interference” effects, similar to that of ref. [5]. We will not discuss these questions here, hence our discussion is limited to the first few revivals. —
357
Volume 86A, number 6,7
PHYSICS LETTERS
It can be shown that the average value of r = (x2 +y2)l/2 as well as r2 are constant so the spreading of the wave packet takes place in the phase variable only. Note that the formula (9) for (x + iy) is quite similar to the formula for the average value of inversion in the Jaynes—Cummings model (see ref. [2]). Let us now examine more closely the revival time tR = 8irh(a + b13)312b2. It can be expressed in terms of the magnetic field B and the electron energy E: tR 2~E3~1(eBc)2. For the observation of revivals to be possible it is imperative to make tR small. This requires a strong field B and small energy E. If E is in the non-relativistic regime (E~_mc2)andB is 100 kG we find that tR l0~ s. This seems to be the smallest realistic value of the revival time. It is instructive to compare tR with the characteristic dephasing time caused by spontaneous emission. (Dephasing due to inhomogeneities of the magnetic field can be in principle eliminated, but the spontaneous emission cannot.) Since electrons are assumed to be in a highly excited state (ii 1) the radiation rate can be found from the classical formula for synchrotron radiation [4]: ~‘
358
30 November 1981
4c8E[2e4cB2(E2 m2c4)]1. 5 = E(dE/dtY~=3m Condition t 5 ~ tR should make the observation of 2. revivals possible. The last toE ~ 2mc Thus unless the motion ofcondition electrons leads is ultrarelativistic the dephasing time due to spontaneous emission is larger than the revival time, which is a necessary condition for the possibility of observation of the revivals. If a wavepacket has been formed at time t ‘~0it should first spread and later, after time tR given by (7), reproduce itself [up to details given by (9)]. This should lead to the formation of macroscopic polarisation and hence the synchrotron radiation after the revival time should be enhanced. t
—
References [1] J.H.
Eberly, N.B. Naxozhny and J.J. Sanchez-Mondragon, Phys. Rev. Lett. 44(1980) 1323. [2] N.B. Narozhny, J.J. Sanchez-Mondragon and J.H. Eberly, Phys. Rev. A23 (1981) 236. [3] P.L. Knight and P.M. Radmore, to be published. [4] A.A. Sokolov and I.M. Temov, Synchrotron radiation
(Pergamon Press, 1968). [5] H.I. Yoo, J.J. Sanchez-Mondragon and J.H. Eberly, J. Phys. A14 (1981) 1383. [6] H.I. Yoo and J.H. Eberly, private communication.
PHYSICS LETTERS
30 November 1981
QUANTUM REVIVALS IN THE MOTION OF AN ELECTRON IN A MAGNETIC FIELD Piotr FILIPOWICZ Institute of Physics, Polish Academy of Sciences, 02668 Warsaw, Poland
and Jan MOSTOWSKI1
Research Institute for Theoretical Physics, University of Helsinki, SF-001 70 Helsinki 1 7, Finland Received 23 September 1981
We show that the motion of a relativistic electron in a constant homogeneous magnetic field exhibits quasiperiodic behaviour (quantum revivals) and discuss the possibility of their observation.
In this letter we examine the quantum motion of a relativistic electron in a constant homogeneous magnetic fIeld from the point of view of’ quantum revivals”. As has been shown by Eberly and co-workers [1, 2] (see also ref. [3]) a simple quantum system con-
~,
1
Ia,6,k ,p)”exp[——(Ial 2
ue of the electron position in a constant homogeneous magnetic field exhibits similar behaviour if the initial state is a wavepacket. We will discuss the possibility of observing such phenomena. As is well known (see e.g. ref. [4]) the quantum numbers describing a relativistic electron in a magnetic field are n = 1, 2, 3 (principal quantum number), s = 0, 1, 2 (radial quantum number), k 3(wave vector parallel to the B field), and p = ±1(projection of spin on the direction of B). The energy of the electron is 2 [mc2 +1~2k~/m +phw +2hw(n ~ }1/2(~) E’~{mc where w = —eB/mc > 0. Denote eigenfunctions corresponding to n, s, k, p by In, s, k, p). Coherent states Ia, 6, k 3, p), where 1 Permanent address: Institute of Physics, Polish Academy of Sciences, 02668 Warsaw, Poland.
356
+
161 )] (2)
00
X ~
sisting of an interacting fermion and bosons (the Jaynes—Cummings model) exhibits unexpected quasiperiodical features known as revivals. Following similar lines of reasoning we will show that the average val-
...
6 are arbitrary complex numbers, are defined as
a~l65(n!s!)_hi2In,s, k
3, p) n,s=0 As time develops the coherent states change according to —
r
1
2
2
ln,6,k3,ii) —expi—~(IaI + 161 ) 1E(n)t] ah165(n!s!Yh/2 (3) X n,s=0 exp[—ili ~< ~ s k p Because E(n) is not a linear function of n the wave
~ ‘
‘
~‘
function Ia, 6, k3, ~>t is no longer a coherent state, according to the definition (2). Thus if a coherent state has been formed at time t = 0 it will not be a coherent state after(3) an isinfinitesimally time. Nevertheless function a useful objectshort describing the time development of the wavepacket (2). Projection of the position of the electron on the plane perpendicular to the magnetic field (xy plane) can be most conveniently described by the average value of x + iy. If we use the Foldy—Wouthuysen representation~for the position operator we find
0031-9163/81/0000—0000/S 02.75 © 1981 North-Holland
Volume 86A, number 6,7
PHYSICS LETTERS
t(a, 6, k3,,ulx +iyla, 6, k3, ~>t =(2h/mw)1/2 X
~~~a*exp(_IaI2)
E —
8ir(a +b~3)3/2b—2hk (k = 0,1,2...)
tk
(7)
all linear terms in n j3 are multiples of 2iri. Thus the terms in series (5) become phased and we find that the average position takes large values. These are the revivals. In order to find the time evolution of the series (5) around each revival we put ~ tk + r with r ~ 8ir(a + bj3)312hb—2. The series (5) becomes —
al2n(n!)_l (4)
n0
X exp{ih~[E(n + 1)
30 November 1981
E(n)]t})
-
In what follows we will restrict ourselves to the case 1a12 ~ 1 and approximate the difference E(n + 1) E(n) by the differential aE(n)/an. Thus the time evolution of the average position is determined by the series —
00
E
exp(—13)(JY~/n!)exp[4irik(a + b13)b~
n0
2irikl3+3irikb(n —I3)2~(a+b13)—’] X exp{ibr[2h(a+b13)112]’ x [1—b(n—13)~(a+bf3)~ +
~ n0
exp(—IaI2)laI2’~(n!)’exp{i1v1 [aE(n)/an]t} (5)
Similar series, but with a different n dependence in the exponent have been studied in refs. [1—3,5]in connection with the Jaynes—Cummings model, and lead to revivals. We will find an approximate formula for the series (5), but we will avoid a mistake in the original derivation of ref. [21. (A similar calculation for the Jaynes—Cummings model has been performed by Yoo and Eberly [6].) Since the weighting factor exp(_IaI2)IaI2~l(n!)~ peaks at a value n al2 with a narrow dispersion lal we may expand aE/an around n = lcd2: aE/an ~~b(a +b~3)—1I2{J.—b(n —f3)[2(a +b13)]—1 (6)
-
This sum is approximated by an integral and calculated with the saddle point method. The result is Rk(r) exp [iøk where ~k
4irk(a
=
+
—
~i ~k
+
—
ekr2]
,
(8)
b13)b—’ + 2irkf3, b13)11~]—1,
+
=
~k =b413(a +b13+3irikbl3) X ~ + b!3)2 [(a + b13)2
+3b2(n—13)2[8(a+b(3)2]~}, where 13 al2, a = m2c4 + h2k~c2+ (p + 1)hwmc2 and b 2mc2hw. The time behaviour of the series (5) can be studied separately for various time scales. If t ~ 4h(a +b13)312 X b213—1/2, linear and quadratic terms inn —13 in (6) can be neglected. The series can be summed and we find that the average position performs oscillations with a frequency b[2h(a + b(3)1I2]~.This is the quantum counterpart of the classical synchrotron motion. For times of the order of h(a + b13)312b213112, linear and quadratic terms cannot be neglected. We may expect that because of dephasing of the terms the average position will be damped to zero (the wavepacket spreads along its orbit). If t is still larger revivals may occur. Observe that if we take the time t to be
3)2! (a +bj3)—2 ] }
+ 3b2(n
+9ir2k2/32b2]h2}1, ~k
=
tan—1(M/D),
Rk(T) =
(M2 +D2)—114,
M~’b2r(a + bf3){4h(a + bj3)1/2 x [(a + b13)2 + 9ir2k2b2j3] }_1 —
D=1
3~1thf3(t’i+ b13Y1 —3b313r~1r’(a+ b13Y5I2, 3~kb313r{4h(a+ bi3)112 —
X [(a + bj3)2 + 9ir2k2b2!3] }_1 Thus each revival is characterized by its own phase 0k ~ amplitude Rk and characteristic width (Re e)h/Pwhich grows with k. Eventually revivals will overlap, which should lead to “interference” effects, similar to that of ref. [5]. We will not discuss these questions here, hence our discussion is limited to the first few revivals. —
357
Volume 86A, number 6,7
PHYSICS LETTERS
It can be shown that the average value of r = (x2 +y2)l/2 as well as r2 are constant so the spreading of the wave packet takes place in the phase variable only. Note that the formula (9) for (x + iy) is quite similar to the formula for the average value of inversion in the Jaynes—Cummings model (see ref. [2]). Let us now examine more closely the revival time tR = 8irh(a + b13)312b2. It can be expressed in terms of the magnetic field B and the electron energy E: tR 2~E3~1(eBc)2. For the observation of revivals to be possible it is imperative to make tR small. This requires a strong field B and small energy E. If E is in the non-relativistic regime (E~_mc2)andB is 100 kG we find that tR l0~ s. This seems to be the smallest realistic value of the revival time. It is instructive to compare tR with the characteristic dephasing time caused by spontaneous emission. (Dephasing due to inhomogeneities of the magnetic field can be in principle eliminated, but the spontaneous emission cannot.) Since electrons are assumed to be in a highly excited state (ii 1) the radiation rate can be found from the classical formula for synchrotron radiation [4]: ~‘
358
30 November 1981
4c8E[2e4cB2(E2 m2c4)]1. 5 = E(dE/dtY~=3m Condition t 5 ~ tR should make the observation of 2. revivals possible. The last toE ~ 2mc Thus unless the motion ofcondition electrons leads is ultrarelativistic the dephasing time due to spontaneous emission is larger than the revival time, which is a necessary condition for the possibility of observation of the revivals. If a wavepacket has been formed at time t ‘~0it should first spread and later, after time tR given by (7), reproduce itself [up to details given by (9)]. This should lead to the formation of macroscopic polarisation and hence the synchrotron radiation after the revival time should be enhanced. t
—
References [1] J.H.
Eberly, N.B. Naxozhny and J.J. Sanchez-Mondragon, Phys. Rev. Lett. 44(1980) 1323. [2] N.B. Narozhny, J.J. Sanchez-Mondragon and J.H. Eberly, Phys. Rev. A23 (1981) 236. [3] P.L. Knight and P.M. Radmore, to be published. [4] A.A. Sokolov and I.M. Temov, Synchrotron radiation
(Pergamon Press, 1968). [5] H.I. Yoo, J.J. Sanchez-Mondragon and J.H. Eberly, J. Phys. A14 (1981) 1383. [6] H.I. Yoo and J.H. Eberly, private communication.