- Email: [email protected]
Quantum theory of the micromaser with ultra-cold Λ-type three-level atoms
1 December 1998
Optics Communications 157 Ž1998. 77–82
Quantum theory of the micromaser with ultra-cold L-type three-level atoms Zhi-Ming Zhang b
a,b,)
, Lin-Sheng He
b
a Department of Applied Physics, Shanghai Jiao Tong UniÕersity, Shanghai 200030, China Laser Spectroscopy Laboratory, Anhui Institute of Optics and Fine Mechanics, Academia Sinica, Hefei 230031, China
Received 15 July 1998; revised 22 September 1998; accepted 23 September 1998
Abstract We present the general quantum theory for the micromaser with ultra-cold L-type three-level atoms. We first derive the dressed states for the interaction of a L-type three-level atom with a quantum cavity field, then treat the interaction between the injection atom with the cavity field as a scattering process, find the reflection and transmission coefficients in this process, and calculate the atomic emission probability. q 1998 Elsevier Science B.V. All rights reserved. PACS: 42.50.Ct; 84.40.Ik; 32.80.Wr; 32.80.Pj Keywords: Micromaser; Ultra-cold atom; Three-level atom; Atomic emission probability
A micromaser is a device in which at most one atom at a time is inside the cavity. Many quantum electrodynamics effects were found in micromasers 1. On the other hand, with the development of laser cooling techniques 2 , one has obtained very cold atoms. Recently, Scully et al. w3x suggested to use the ultra-cold atoms in micromaser studies and proposed the concept of the mazer Žmicrowave amplification via z-motion-induced emission of radiation.. They and their coworkers have studied various properties, for example, the emission probability, the photon statistics and the spectrum, of the mazer in a series of papers w4–6x. Loffler et al. w7x proposed to use the mazer as a velocity filter. However, the two-level atom model was ¨ assumed in these papers. It is well-known that there are plentiful phenomena and effects in the interaction between three-level atoms with radiation fields 3. Especially, the atomic coherence and interference effects, which are related to the electromagnetically induced transparency 4 , the lasing without inversion 5, the laser cooling of atoms 6 and many other effects 7, exist in the interaction between three-level atoms with radiation fields. This motivates us to consider three-level atoms in the study of mazers. In this Communication, we propose to use ultra-cold L-type three-level atoms in a micromaser, and establish the corresponding quantum theory. We derive the dressed states for the interaction of a L-type three-level atom with a quantum cavity field, treat the interaction between the injection atom with the cavity field as a
)
Corresponding author. Department of Physics, Xi’an Jiaotong University, Xi’an 710049, China. E-mail: [email protected] For a review, see Ref. w1x. 2 For laser-cooling techniques, see Ref. w2x. 3 See, for example, Ref. w8x. 4 For a review, see Ref. w9x. 5 For a review, see Ref. w10x. 6 See, for example, Ref. w11x. 7 See, for example, Ref. w12x. 1
0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 5 2 4 - 0
Z.-M. Zhang, L.-S. He r Optics Communications 157 (1998) 77–82
78
Fig. 1. Schematic diagram of the L-type three-level atom interacting with a single-mode radiation field.
scattering process, find the reflection and transmission coefficients of the scattering process, and calculate the emission probability. Suppose that the ultra-cold atom has three levels ŽFig. 1., moves along the z-direction, and enters into a cavity which contains a single-mode quantum radiation field, then the total Hamiltonian operator of the atom-field system is
HT s
p z2 2M
qH
Ž1.
where M is the atomic mass, p z is the atomic momentum operator for the CM Žcenter of mass. motion, and H s H F q HA q u Ž z . V ,
Ž2.
in which uŽ z . is the mode function of the cavity field, and H F q HA s v aqa q
v x < x :² x < ,
Ý
Ž3.
xs a , b ,c X
V s g 1Ž a < b :² a < q aq < a:² b < . q g 2 Ž a < b :² c < q aq < c :² b < . ,
Ž4.
here we have taken " s 1. v is the frequency of the cavity field. aq and a are the photon creation and annihilation operators, < x : Ž x s a,b,c . is the atomic state with the energy v x , g 1 and g 2 are the atom-field coupling constants. We suppose that the states < a: and < c : are degenerate, and let v a s v c s v g and v b s v e , then the states < a,n q 1: and < c,n q 1: are also degenerate and have the energy v g q Ž n q 1. v , and the state < b,n: has the energy v e q n v s w v g q Ž n q 1. v x q D, here the state < x,m: means that the atom is in the state < x : and the field is in the Fock state < m:, and D s Ž v e y v g . y v is the one-photon detuning between the cavity field frequency and the atomic transition frequency. The matrix of H, based on the triplet manifolds < a,n q 1:, < b,n:, and < c,n q 1:, and measured from the energy, v g q Ž n q 1. v , is 0
d 1Ž n .
H s d 1Ž n . 0
0
0
D
d2 Ž n. , 0
d2 Ž n.
Ž5.
where d i Ž n. s g i'n q 1 Ž i s 1,2.. The eigenvalues and the eigenstates of H can be derived to be
VnŽ0. s 0,
VnŽ " . s " V n "
ž
D 2
/
Ž6.
Z.-M. Zhang, L.-S. He r Optics Communications 157 (1998) 77–82
79
and < cnŽ0. : s f nŽ0. < a, n q 1: y
d1 d2
< c, n q 1: V Žn" .
< cnŽ " . : s f nŽ " . < a, n q 1: q
d1
Ž7.
< b,n: q
d2
< c, n q 1:
Ž8.
,
Ž9.
q Gn2 .
Ž 10 .
d1
where f nŽ0. s
d2 Ž n. Gn
,
f nŽ " . s
d1 Ž n .
(
D
ž
2 Vn Vn "
(
Gn s d12 Ž n . q d12 Ž n . ,
Vn s
(ž
D 2
2
/
2
/
The states < cnŽ0. : and < cnŽ" . : are the dressed states for the interaction of a L-type three-level atom with a quantum cavity field. They are different from the ones for the interaction of a two-level atom with a quantum cavity field w3,4x. It is expedient to expand the atom-field states in terms of these states. Now we consider the problem in which an atom in the upper state < b : and with a CM momentum k is incident upon a cavity field in the number state < n:. As discussed in Refs. w3,4x, this problem can be treated as a scattering process. The atom-field system before the scattering process is characterized by the state
s u Ž yz . e i k z < b,n:
½
s u Ž yz . e i k z f Žnq .
VnŽq. d1
< cnŽq . : q f nŽy .
VnŽy . d1
Ž 11.
5
< cnŽy . : .
Here, the Heaviside’s unit step function u merely indicates on which side of the cavity the atom can be found. According to the idea of Refs. w3,4x, each component associated with < cnŽ j. : Ž j s q,y . obeys the time-dependent Schrodinger equation ¨
E i
Et
ž
E2
1
2 M E z2
/
q u Ž z . V nŽ j .
Ž 12.
For an arbitrary mode function uŽ z ., the components associated with < cnŽ" . : encounter non-zero-potentials since VnŽ" . / 0. Therefore, they will be partially reflected and partially transmitted by the potentials. Then, the state of the atom-field system after the atom has left the interaction region can be written as
ž
k2 2M
t
/½
yi k z i k Ž zyl . rq u Ž yz . q tq u Ž z y l . f nŽ q . n Žk. e n Žk. e
yi k z i k Ž zyl . q ry u Ž yz . q ty u Ž z y l . f nŽy . n Žk. e n Žk. e
VnŽy. d1
< cnŽy . :
V nŽq .
5
d1
< c nŽ q . :
Ž 13.
where t is the interaction time, l is the cavity length in the z-direction rn"and tn" are the reflection and transmission coefficients of the components < cnŽ" . :, respectively. Substituting Eqs. Ž7. and Ž8. into Eq. Ž13. we obtain
ž
k2 2M
t
/
R b, n Ž k . eyi k z u Ž yz . q Tb , n Ž k . e i k Ž zyl . u Ž z y l . < b,n:
q R a, nq1 Ž k . eyi k z u Ž yz . q Ta , nq1 Ž k . e i k Ž zyl . u Ž z y l . < a, n q 1: q R c, nq1 Ž k . eyi k z u Ž yz . q Tc , nq1 Ž k . e i k Ž zyl . u Ž z y l . < c, n q 1: 4
Ž 14.
Z.-M. Zhang, L.-S. He r Optics Communications 157 (1998) 77–82
80
in which d1
R a, nq1 Ž k . s
d1
Ta, nq1 Ž k . s
Tb, n Ž k . s
2 Vn 1 2 Vn
R c, nq1 Ž k . s Tc, nq1 Ž k . s
ž ž
d2 2 Vn d2
2 Vn
Ž 15.
y w tq n Ž k . y tn Ž k . x ,
2 Vn 1
R b, n Ž k . s
w rqn Ž k . y ryn Ž k . x ,
2 Vn
D Vn q
2
D Vn q
2
/ /
Ž 16.
ž ž
rq n Ž k . q Vn y tq n Ž k . q Vn y
D 2
D 2
/ /
ry n Žk. , ty n Žk. ,
w rqn Ž k . y ryn Ž k . x ,
Ž 17. Ž 18. Ž 19.
y w tq n Ž k . y tn Ž k . x ,
Ž 20.
where Eq. Ž9. for f nŽ" . has been used. When an atom initially in the state < b : enters into a cavity containing n photons, it will be reflected or transmitted while remaining in the state < b : with the amplitudes R b, n and Tb, n , and be reflected or transmitted while making a transition to the state < x : Ž x s a,c . and emitting one photon with the amplitudes R x, nq1 and Tx, nq1 , respectively. The above formalism is very general. It is suitable for arbitrary mode functions uŽ z .. In the following, we consider a special case, the mesa mode function, and calculate rn" and tn" explicitly. The mesa mode function has the form uŽ z . s
½
0-z-l , elsewhere
1, 0,
Ž 21.
it describes the case in which the atom-field coupling inside the cavity is constant along the propagation axis of the atoms. In this case, the component < cnŽq. : encounters a square potential-barrier and the component < cnŽy. : encounters a square potential-well, the reflection and the transmission coefficients can be calculated analytically and have the following forms
rn" s i Dn"sin Ž K n"l . tn" ,
Ž 22. y1
tn" s cos Ž K n" l . y i Sn" sin Ž K n" l .
,
Ž 23.
where
(
2
K n" s k 2 . Ž k n" . ,
Dn" s
1 2
ž
K n"
k y
k
K n"
k n" s
/
,
(
D
ž
2 M Vn "
Sn" s
1 2
ž
K n" k
2
/
Ž 24. k
q
K n"
/
.
Ž 25 .
q Ž . Ž . Ž . Ž . Ž . Note that in the case of k s kq n , i.e., K n s 0, Eqs. 22 and 23 with Eqs. 24 and 25 are not valid for the plus q component. In this case we find
rq n s
yikq n l 2 y ikq n l
,
tq n s
2 2 y ikq n l
.
Ž 26.
The above theory can be used to study many properties of the micromaser in various cases. In this Communication we restrict ourselves to the calculation of the emission probability Pemission Ž n., Pemission Ž n .
s < R a , nq1 Ž k . < 2 q < Ta , nq1 Ž k . < 2 q < R c , nq1 Ž k . < 2 q < Tc , nq1 Ž k . < 2 s
Gn2 2 V n2
1 y Re
)
y q y rq n Ž r n . q tn Ž tn .
)
4.
Ž 27.
Eq. Ž27. differs from the same quantity in Ref. w4x by two points, one is the multiplicative constant Gn2rV n2 , the other is the definitions of k n" Žsee Eq. Ž24.. appearing in rn" and tn". We see that the emission probability is completely determined by
Z.-M. Zhang, L.-S. He r Optics Communications 157 (1998) 77–82
81
the reflective and the transmission coefficients rn" and tn". For l s 0 one has rn"s 0, tn"s 1, and Pemission Ž n. s 0. This is of course reasonable, since l s 0 means that there is no interaction between the atom and the field. In the following, we consider two limit cases of D, i.e., the zero-detuning Ž D s 0. case and the large-detuning case wŽ Dr2. 2 4 Gn2 x Žthis is the condition, under which the L-type three-level atom interacting with a cavity field can be reduced to an effective two-level atom.. We also consider two limits of k, i.e., the thermal-atom Žthe fast-atom. limit and the ultra-cold atom Žthe slow-atom. limit. In the zero-detuning case, D s 0, V n s Gn , we have Pemission Ž n . s
1 2
1 y Re
)
y q y rq n Ž r n . q tn Ž tn .
)
4
Ž 28.
Now we consider the expressions of Eq. Ž28. in the two limit cases of k. 1. For the thermal-atom limit: k 4 k n", K n"s k . Ž1r2 k .Ž k n". 2 , Dn"f 0, Sn"f 1, therefore, rn"f 0, tn"s exp Ž iK n"l .. This means that, corresponding to the incident energy, the potential-barrier is very low and the potential-well is very shallow. Under this condition, the atom transmits through the cavity without reflection, and we have Pemission Ž n . s sin2 Ž Gnt . ,
Ž 29.
where t s Mlrk is the time in which the atom transmits through the cavity Žnote that we have assumed " s 1, otherwise we would have t s Mlr"k .. This is the well-known Rabi oscillation. 2 .. y 2 .. q q y Ž Ž 2 Ž Ž 2 2. For the ultra-cold atom limit, k < k n , Kq n f ik n 1 y k r2 k n , K n f k n 1 q k r2 k n , D n f Sn f ik nr2 k, D n f y Ý n f k nr2 k, and
rq n s yi ry n si
kn 2k
kn 2k
sinh Ž k n l . tq n ,
sin Ž k n l . ty n ,
tq n s cosh Ž k n l . q i
ty n s cos Ž k n l . y i
kn 2k
kn 2k
y1
sinh Ž k n l .
,
Ž 30.
y1
sin Ž k n l .
,
Ž 31.
in this case we have
° Pemission Ž n . s
1 2
1y
~1 y
¢
ChC y
kn
2
ž / ž / 2k
kn
2k
Sh S 2
2
Sh S
q
kn
ž / ž /w
ChC y kn
2k
2k
¶
2
Sh S
•
Ž 32.
2
Ch S q ShC x
ß
2
where C s cosŽ k n l ., S s sinŽ k n l ., Ch s coshŽ k n l ., Sh s sinhŽ k n l .. When k n l s mp Ž m s 1,2, . . . ., S s 0, C s Žy1. m, and Pemission Ž n. f 1r2; when k n l s Ž2 m q 1.pr2 Ž m s 0,1,2, . . . ., C s 0, S s Žy1. m, and Pemission Ž n. f 0. The features of
(
(
Pemission Ž n. can be seen more clearly from Fig. 2, in which we have introduced k n s km n , k s 2 M Ž g 12 q g 12 . , m n s Ž n q 1.1r4 , r s krk and L s k l. We see that the emission probability for the ultra-cold atom case is very different from that for the thermal-atom case described by Eq. Ž29.. We also see that the resonance peaks around L s mp become sharper for slower atoms Žsmaller r .. We can also obtain some interesting results from Eqs. Ž30. and Ž31. directly. When
Fig. 2. The emission probability E.P. versus the dimensionless interaction length L for the ultra-cold atom in the zero-detuning case with ns 0, and r s 0.1 Ždashed., r s 0.01 Žsolid..
82
Z.-M. Zhang, L.-S. He r Optics Communications 157 (1998) 77–82
q expŽ k n l . 4 1, so that coshŽ k n l . f sinhŽ k n l . 4 1, we have rq n f y1, tn f 0, this means that, corresponding to the incident energy, the potential-barrier is very high, so that the plus component is almost fully reflected. For the minus component we y y y Ž Ž . . m for k n l s mp . The condition k n l s mp means have ry n f y1, tn f 0 for k n l / mp m s 1,2, . . . and r n s 0, tn s y1 that the relation between the cavity length l and the de Broglie wavelength ldB of the atom in the cavity is l s m Ž ldBr2.. Then we get the following picture: corresponding to the incident energy, if the potential-well is very deep, then when the cavity length is an integer multiple of half the de Broglie wavelength, the minus component will almost fully transmit through the cavity, and otherwise it will be fully reflected. With the increase of D, V n increases, and therefore, the amplitude of PemissionŽ n. decreases. In the very large-detuning limit, Ž Dr2. 2 4 Gn2 , V n2 4 Gn2 , therefore, PemissionŽ n. is much smaller than 1. Recalling the initial condition, we conclude that when an atom is initially in the state < b : and enters into a cavity containing n photons, it will almost remain in the state < b : and will not emit photons if the detuning between the field frequency and the atomic transition frequency is very large. In summary, we have derived the dressed states for the interaction of a L-type three-level atom with a quantum cavity field, established the general quantum theory of the micromaser with ultra-cold L-type three-level atoms, calculated its emission probability, and discussed the effects of the detuning D and the atomic CM momentum k.
Acknowledgements This work was supported by the Senior Visiting Scholar Project of the Chinese Academy of Sciences and the National Natural Science Foundation of China.
References w1x G. Raithel, C. Wagner, H. Walther, L.M. Harducci, M.O. Scully, in: P.R. Berman ŽEd.., Cavity Quantum Electrodynamics, Academic Press, Boston, 1994, p. 57. w2x E. Arimondo, W.D. Phillips, F. Strumia ŽEds.., Laser Manipulation of Atoms and Ions, Amsterdam, North-Holland, 1992. w3x M.O. Scully, G.M. Meyer, H. Walther, Phys. Rev. Lett. 76 Ž1996. 4144. w4x G.M. Meyer, M.O. Scully, H. Walther, Phys. Rev. A 56 Ž1997. 4142. w5x M. Loffler, G.M. Meyer, M. Schroder, M.O. Scully, H. Walther, Phys. Rev. A 56 Ž1997. 4153. ¨ ¨ w6x M. Schroder, K. Vogel, W.P. Schleich, M.O. Scully, H. Walther, Phys. Rev. A 56 Ž1997. 4164. ¨ w7x M. Loffler, G.M. Meyer, H. Walther, Europhys. Lett. 41 Ž1998. 593. ¨ w8x H.-I. Yoo, J.H. Eberly, Phys. Rep. 118 Ž1985. 239. w9x S.E. Harris, Physics Today 50 Ž7. Ž1997. 36. w10x O. Kocharovskaya, Phys. Rep. 219 Ž1992. 175. w11x C. Cohen-Tannoudji, Phys. Rep. 219 Ž1992. 153. w12x M.O. Scully, Phys. Rep. 219 Ž1992. 191.
Optics Communications 157 Ž1998. 77–82
Quantum theory of the micromaser with ultra-cold L-type three-level atoms Zhi-Ming Zhang b
a,b,)
, Lin-Sheng He
b
a Department of Applied Physics, Shanghai Jiao Tong UniÕersity, Shanghai 200030, China Laser Spectroscopy Laboratory, Anhui Institute of Optics and Fine Mechanics, Academia Sinica, Hefei 230031, China
Received 15 July 1998; revised 22 September 1998; accepted 23 September 1998
Abstract We present the general quantum theory for the micromaser with ultra-cold L-type three-level atoms. We first derive the dressed states for the interaction of a L-type three-level atom with a quantum cavity field, then treat the interaction between the injection atom with the cavity field as a scattering process, find the reflection and transmission coefficients in this process, and calculate the atomic emission probability. q 1998 Elsevier Science B.V. All rights reserved. PACS: 42.50.Ct; 84.40.Ik; 32.80.Wr; 32.80.Pj Keywords: Micromaser; Ultra-cold atom; Three-level atom; Atomic emission probability
A micromaser is a device in which at most one atom at a time is inside the cavity. Many quantum electrodynamics effects were found in micromasers 1. On the other hand, with the development of laser cooling techniques 2 , one has obtained very cold atoms. Recently, Scully et al. w3x suggested to use the ultra-cold atoms in micromaser studies and proposed the concept of the mazer Žmicrowave amplification via z-motion-induced emission of radiation.. They and their coworkers have studied various properties, for example, the emission probability, the photon statistics and the spectrum, of the mazer in a series of papers w4–6x. Loffler et al. w7x proposed to use the mazer as a velocity filter. However, the two-level atom model was ¨ assumed in these papers. It is well-known that there are plentiful phenomena and effects in the interaction between three-level atoms with radiation fields 3. Especially, the atomic coherence and interference effects, which are related to the electromagnetically induced transparency 4 , the lasing without inversion 5, the laser cooling of atoms 6 and many other effects 7, exist in the interaction between three-level atoms with radiation fields. This motivates us to consider three-level atoms in the study of mazers. In this Communication, we propose to use ultra-cold L-type three-level atoms in a micromaser, and establish the corresponding quantum theory. We derive the dressed states for the interaction of a L-type three-level atom with a quantum cavity field, treat the interaction between the injection atom with the cavity field as a
)
Corresponding author. Department of Physics, Xi’an Jiaotong University, Xi’an 710049, China. E-mail: [email protected] For a review, see Ref. w1x. 2 For laser-cooling techniques, see Ref. w2x. 3 See, for example, Ref. w8x. 4 For a review, see Ref. w9x. 5 For a review, see Ref. w10x. 6 See, for example, Ref. w11x. 7 See, for example, Ref. w12x. 1
0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 5 2 4 - 0
Z.-M. Zhang, L.-S. He r Optics Communications 157 (1998) 77–82
78
Fig. 1. Schematic diagram of the L-type three-level atom interacting with a single-mode radiation field.
scattering process, find the reflection and transmission coefficients of the scattering process, and calculate the emission probability. Suppose that the ultra-cold atom has three levels ŽFig. 1., moves along the z-direction, and enters into a cavity which contains a single-mode quantum radiation field, then the total Hamiltonian operator of the atom-field system is
HT s
p z2 2M
qH
Ž1.
where M is the atomic mass, p z is the atomic momentum operator for the CM Žcenter of mass. motion, and H s H F q HA q u Ž z . V ,
Ž2.
in which uŽ z . is the mode function of the cavity field, and H F q HA s v aqa q
v x < x :² x < ,
Ý
Ž3.
xs a , b ,c X
V s g 1Ž a < b :² a < q aq < a:² b < . q g 2 Ž a < b :² c < q aq < c :² b < . ,
Ž4.
here we have taken " s 1. v is the frequency of the cavity field. aq and a are the photon creation and annihilation operators, < x : Ž x s a,b,c . is the atomic state with the energy v x , g 1 and g 2 are the atom-field coupling constants. We suppose that the states < a: and < c : are degenerate, and let v a s v c s v g and v b s v e , then the states < a,n q 1: and < c,n q 1: are also degenerate and have the energy v g q Ž n q 1. v , and the state < b,n: has the energy v e q n v s w v g q Ž n q 1. v x q D, here the state < x,m: means that the atom is in the state < x : and the field is in the Fock state < m:, and D s Ž v e y v g . y v is the one-photon detuning between the cavity field frequency and the atomic transition frequency. The matrix of H, based on the triplet manifolds < a,n q 1:, < b,n:, and < c,n q 1:, and measured from the energy, v g q Ž n q 1. v , is 0
d 1Ž n .
H s d 1Ž n . 0
0
0
D
d2 Ž n. , 0
d2 Ž n.
Ž5.
where d i Ž n. s g i'n q 1 Ž i s 1,2.. The eigenvalues and the eigenstates of H can be derived to be
VnŽ0. s 0,
VnŽ " . s " V n "
ž
D 2
/
Ž6.
Z.-M. Zhang, L.-S. He r Optics Communications 157 (1998) 77–82
79
and < cnŽ0. : s f nŽ0. < a, n q 1: y
d1 d2
< c, n q 1: V Žn" .
< cnŽ " . : s f nŽ " . < a, n q 1: q
d1
Ž7.
< b,n: q
d2
< c, n q 1:
Ž8.
,
Ž9.
q Gn2 .
Ž 10 .
d1
where f nŽ0. s
d2 Ž n. Gn
,
f nŽ " . s
d1 Ž n .
(
D
ž
2 Vn Vn "
(
Gn s d12 Ž n . q d12 Ž n . ,
Vn s
(ž
D 2
2
/
2
/
The states < cnŽ0. : and < cnŽ" . : are the dressed states for the interaction of a L-type three-level atom with a quantum cavity field. They are different from the ones for the interaction of a two-level atom with a quantum cavity field w3,4x. It is expedient to expand the atom-field states in terms of these states. Now we consider the problem in which an atom in the upper state < b : and with a CM momentum k is incident upon a cavity field in the number state < n:. As discussed in Refs. w3,4x, this problem can be treated as a scattering process. The atom-field system before the scattering process is characterized by the state
s u Ž yz . e i k z < b,n:
½
s u Ž yz . e i k z f Žnq .
VnŽq. d1
< cnŽq . : q f nŽy .
VnŽy . d1
Ž 11.
5
< cnŽy . : .
Here, the Heaviside’s unit step function u merely indicates on which side of the cavity the atom can be found. According to the idea of Refs. w3,4x, each component associated with < cnŽ j. : Ž j s q,y . obeys the time-dependent Schrodinger equation ¨
E i
Et
ž
E2
1
2 M E z2
/
q u Ž z . V nŽ j .
Ž 12.
For an arbitrary mode function uŽ z ., the components associated with < cnŽ" . : encounter non-zero-potentials since VnŽ" . / 0. Therefore, they will be partially reflected and partially transmitted by the potentials. Then, the state of the atom-field system after the atom has left the interaction region can be written as
ž
k2 2M
t
/½
yi k z i k Ž zyl . rq u Ž yz . q tq u Ž z y l . f nŽ q . n Žk. e n Žk. e
yi k z i k Ž zyl . q ry u Ž yz . q ty u Ž z y l . f nŽy . n Žk. e n Žk. e
VnŽy. d1
< cnŽy . :
V nŽq .
5
d1
< c nŽ q . :
Ž 13.
where t is the interaction time, l is the cavity length in the z-direction rn"and tn" are the reflection and transmission coefficients of the components < cnŽ" . :, respectively. Substituting Eqs. Ž7. and Ž8. into Eq. Ž13. we obtain
ž
k2 2M
t
/
R b, n Ž k . eyi k z u Ž yz . q Tb , n Ž k . e i k Ž zyl . u Ž z y l . < b,n:
q R a, nq1 Ž k . eyi k z u Ž yz . q Ta , nq1 Ž k . e i k Ž zyl . u Ž z y l . < a, n q 1: q R c, nq1 Ž k . eyi k z u Ž yz . q Tc , nq1 Ž k . e i k Ž zyl . u Ž z y l . < c, n q 1: 4
Ž 14.
Z.-M. Zhang, L.-S. He r Optics Communications 157 (1998) 77–82
80
in which d1
R a, nq1 Ž k . s
d1
Ta, nq1 Ž k . s
Tb, n Ž k . s
2 Vn 1 2 Vn
R c, nq1 Ž k . s Tc, nq1 Ž k . s
ž ž
d2 2 Vn d2
2 Vn
Ž 15.
y w tq n Ž k . y tn Ž k . x ,
2 Vn 1
R b, n Ž k . s
w rqn Ž k . y ryn Ž k . x ,
2 Vn
D Vn q
2
D Vn q
2
/ /
Ž 16.
ž ž
rq n Ž k . q Vn y tq n Ž k . q Vn y
D 2
D 2
/ /
ry n Žk. , ty n Žk. ,
w rqn Ž k . y ryn Ž k . x ,
Ž 17. Ž 18. Ž 19.
y w tq n Ž k . y tn Ž k . x ,
Ž 20.
where Eq. Ž9. for f nŽ" . has been used. When an atom initially in the state < b : enters into a cavity containing n photons, it will be reflected or transmitted while remaining in the state < b : with the amplitudes R b, n and Tb, n , and be reflected or transmitted while making a transition to the state < x : Ž x s a,c . and emitting one photon with the amplitudes R x, nq1 and Tx, nq1 , respectively. The above formalism is very general. It is suitable for arbitrary mode functions uŽ z .. In the following, we consider a special case, the mesa mode function, and calculate rn" and tn" explicitly. The mesa mode function has the form uŽ z . s
½
0-z-l , elsewhere
1, 0,
Ž 21.
it describes the case in which the atom-field coupling inside the cavity is constant along the propagation axis of the atoms. In this case, the component < cnŽq. : encounters a square potential-barrier and the component < cnŽy. : encounters a square potential-well, the reflection and the transmission coefficients can be calculated analytically and have the following forms
rn" s i Dn"sin Ž K n"l . tn" ,
Ž 22. y1
tn" s cos Ž K n" l . y i Sn" sin Ž K n" l .
,
Ž 23.
where
(
2
K n" s k 2 . Ž k n" . ,
Dn" s
1 2
ž
K n"
k y
k
K n"
k n" s
/
,
(
D
ž
2 M Vn "
Sn" s
1 2
ž
K n" k
2
/
Ž 24. k
q
K n"
/
.
Ž 25 .
q Ž . Ž . Ž . Ž . Ž . Note that in the case of k s kq n , i.e., K n s 0, Eqs. 22 and 23 with Eqs. 24 and 25 are not valid for the plus q component. In this case we find
rq n s
yikq n l 2 y ikq n l
,
tq n s
2 2 y ikq n l
.
Ž 26.
The above theory can be used to study many properties of the micromaser in various cases. In this Communication we restrict ourselves to the calculation of the emission probability Pemission Ž n., Pemission Ž n .
s < R a , nq1 Ž k . < 2 q < Ta , nq1 Ž k . < 2 q < R c , nq1 Ž k . < 2 q < Tc , nq1 Ž k . < 2 s
Gn2 2 V n2
1 y Re
)
y q y rq n Ž r n . q tn Ž tn .
)
4.
Ž 27.
Eq. Ž27. differs from the same quantity in Ref. w4x by two points, one is the multiplicative constant Gn2rV n2 , the other is the definitions of k n" Žsee Eq. Ž24.. appearing in rn" and tn". We see that the emission probability is completely determined by
Z.-M. Zhang, L.-S. He r Optics Communications 157 (1998) 77–82
81
the reflective and the transmission coefficients rn" and tn". For l s 0 one has rn"s 0, tn"s 1, and Pemission Ž n. s 0. This is of course reasonable, since l s 0 means that there is no interaction between the atom and the field. In the following, we consider two limit cases of D, i.e., the zero-detuning Ž D s 0. case and the large-detuning case wŽ Dr2. 2 4 Gn2 x Žthis is the condition, under which the L-type three-level atom interacting with a cavity field can be reduced to an effective two-level atom.. We also consider two limits of k, i.e., the thermal-atom Žthe fast-atom. limit and the ultra-cold atom Žthe slow-atom. limit. In the zero-detuning case, D s 0, V n s Gn , we have Pemission Ž n . s
1 2
1 y Re
)
y q y rq n Ž r n . q tn Ž tn .
)
4
Ž 28.
Now we consider the expressions of Eq. Ž28. in the two limit cases of k. 1. For the thermal-atom limit: k 4 k n", K n"s k . Ž1r2 k .Ž k n". 2 , Dn"f 0, Sn"f 1, therefore, rn"f 0, tn"s exp Ž iK n"l .. This means that, corresponding to the incident energy, the potential-barrier is very low and the potential-well is very shallow. Under this condition, the atom transmits through the cavity without reflection, and we have Pemission Ž n . s sin2 Ž Gnt . ,
Ž 29.
where t s Mlrk is the time in which the atom transmits through the cavity Žnote that we have assumed " s 1, otherwise we would have t s Mlr"k .. This is the well-known Rabi oscillation. 2 .. y 2 .. q q y Ž Ž 2 Ž Ž 2 2. For the ultra-cold atom limit, k < k n , Kq n f ik n 1 y k r2 k n , K n f k n 1 q k r2 k n , D n f Sn f ik nr2 k, D n f y Ý n f k nr2 k, and
rq n s yi ry n si
kn 2k
kn 2k
sinh Ž k n l . tq n ,
sin Ž k n l . ty n ,
tq n s cosh Ž k n l . q i
ty n s cos Ž k n l . y i
kn 2k
kn 2k
y1
sinh Ž k n l .
,
Ž 30.
y1
sin Ž k n l .
,
Ž 31.
in this case we have
° Pemission Ž n . s
1 2
1y
~1 y
¢
ChC y
kn
2
ž / ž / 2k
kn
2k
Sh S 2
2
Sh S
q
kn
ž / ž /w
ChC y kn
2k
2k
¶
2
Sh S
•
Ž 32.
2
Ch S q ShC x
ß
2
where C s cosŽ k n l ., S s sinŽ k n l ., Ch s coshŽ k n l ., Sh s sinhŽ k n l .. When k n l s mp Ž m s 1,2, . . . ., S s 0, C s Žy1. m, and Pemission Ž n. f 1r2; when k n l s Ž2 m q 1.pr2 Ž m s 0,1,2, . . . ., C s 0, S s Žy1. m, and Pemission Ž n. f 0. The features of
(
(
Pemission Ž n. can be seen more clearly from Fig. 2, in which we have introduced k n s km n , k s 2 M Ž g 12 q g 12 . , m n s Ž n q 1.1r4 , r s krk and L s k l. We see that the emission probability for the ultra-cold atom case is very different from that for the thermal-atom case described by Eq. Ž29.. We also see that the resonance peaks around L s mp become sharper for slower atoms Žsmaller r .. We can also obtain some interesting results from Eqs. Ž30. and Ž31. directly. When
Fig. 2. The emission probability E.P. versus the dimensionless interaction length L for the ultra-cold atom in the zero-detuning case with ns 0, and r s 0.1 Ždashed., r s 0.01 Žsolid..
82
Z.-M. Zhang, L.-S. He r Optics Communications 157 (1998) 77–82
q expŽ k n l . 4 1, so that coshŽ k n l . f sinhŽ k n l . 4 1, we have rq n f y1, tn f 0, this means that, corresponding to the incident energy, the potential-barrier is very high, so that the plus component is almost fully reflected. For the minus component we y y y Ž Ž . . m for k n l s mp . The condition k n l s mp means have ry n f y1, tn f 0 for k n l / mp m s 1,2, . . . and r n s 0, tn s y1 that the relation between the cavity length l and the de Broglie wavelength ldB of the atom in the cavity is l s m Ž ldBr2.. Then we get the following picture: corresponding to the incident energy, if the potential-well is very deep, then when the cavity length is an integer multiple of half the de Broglie wavelength, the minus component will almost fully transmit through the cavity, and otherwise it will be fully reflected. With the increase of D, V n increases, and therefore, the amplitude of PemissionŽ n. decreases. In the very large-detuning limit, Ž Dr2. 2 4 Gn2 , V n2 4 Gn2 , therefore, PemissionŽ n. is much smaller than 1. Recalling the initial condition, we conclude that when an atom is initially in the state < b : and enters into a cavity containing n photons, it will almost remain in the state < b : and will not emit photons if the detuning between the field frequency and the atomic transition frequency is very large. In summary, we have derived the dressed states for the interaction of a L-type three-level atom with a quantum cavity field, established the general quantum theory of the micromaser with ultra-cold L-type three-level atoms, calculated its emission probability, and discussed the effects of the detuning D and the atomic CM momentum k.
Acknowledgements This work was supported by the Senior Visiting Scholar Project of the Chinese Academy of Sciences and the National Natural Science Foundation of China.
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