**Detuning effects in the one-photon mazer**

Thierry Bastin*and John Martin†
*Institut de Physique Nucle´aire, Atomique et de Spectroscopie, Universite´ de Lie`ge au Sart Tilman, Baˆtiment B15, B-4000 Lie`ge, Belgium*

共Received 26 December 2002; published 27 May 2003兲

The quantum theory of the mazer in the nonresonant case共a detuning between the cavity mode and the atomic transition frequencies is present兲 is described. The generalization from the resonant case is far from being direct. Interesting effects of the mazer physics are pointed out. In particular, it is shown that the cavity may slow down or speed up the atoms according to the sign of the detuning and that the induced emission process may be completely blocked by use of a positive detuning. It is also shown that the detuning adds a potential step effect not present at resonance and that the use of positive detunings defines a well-controlled cooling mechanism. In the special case of a mesa cavity mode function, generalized expressions for the reflection and transmission coefficients have been obtained. The general properties of the induced emission probability are finally discussed in the hot, intermediate, and cold atom regimes. Comparison with the resonant case is given.

DOI: 10.1103/PhysRevA.67.053804 PACS number共s兲: 42.50.Pq, 42.50.Ct, 42.50.Vk, 32.80.Lg

**I. INTRODUCTION**

*The interaction of cold atoms with microwave high-Q*
cavities共a cold atom micromaser兲 has recently attracted
*in-creasing interest since it was demonstrated by Scully et al.*
关1兴 that this interaction leads to a new type of induced
emis-sion inside the cavity. The new emisemis-sion properties arise
from the necessity to treat quantum mechanically the
center-of-mass motion of the atoms interacting with the cavity. To
insist on the importance of this quantization, usually defined
*along the z axis, the system was called a mazer* 共for
*micro-wave amplification via z-motion-induced emission of *
radia-tion兲. The complete quantum theory of the mazer was
de-scribed in a series of three papers by Scully and co-workers
关2–4兴. The theory was written for two-level atoms
*interact-ing with a sinteract-ingle mode of the high-Q cavity via a one-photon*
transition. In particular, it was shown that the induced
emis-sion properties are strongly dependent on the cavity mode
profile. Results were presented for the mesa, sech2, and
*sinu-soidal modes. Retamal et al.*关5兴 later refined these results in
the special case of the sinusoidal mode, and a numerical
method was proposed by Bastin and Solano 关6兴 for
effi-ciently computing the mazer properties with arbitrary cavity
*field modes. Lo¨ffler et al.* 关7兴 showed also that the mazer
may be used as a velocity selection device for an atomic
*beam. The mazer concept was extended by Zhang et al.*
关8–10兴, who considered two-photon transitions 关8兴, and
three-level atoms interacting with a single cavity 关9兴 and
with two cavities关10兴. Collapse and revival patterns with a
*mazer have been computed by Du et al.* *关11兴. Arun et al.*
关12,13兴 studied the mazer with bimodal cavities and Agarwal
and Arun 关14兴 demonstrated resonant tunneling of cold
at-oms through two mazer cavities.

In all these previous studies, the mazer properties were always presented in the resonant case where the cavity mode frequency is equal to the atomic transition frequency0.

When three-level atoms were considered, a generalized reso-nant condition was assumed 关9,10,12兴. In this paper, we re-move this restriction and establish the theory of the mazer in the nonresonant case (⫽0) for two-level atoms.

The paper is organized as follows. In Sec. II, the Hamil-tonian modeling the mazer in the nonresonant case is pre-sented. The wave functions of the system are described and generalized expressions for the reflection and transmission coefficients in the special case of the mesa mode function are derived. The properties of the induced emission probability when a detuning is present are then discussed in Sec. III. Three regimes of the mazer are considered 共hot, intermedi-ate, and cold兲. A brief summary of our results is finally given in Sec. IV.

**II. MODEL**
**A. The Hamiltonian**

*We consider a two-level atom moving along the z *
*direc-tion on the way to a cavity of length L. The atom is coupled*
unresonantly to a single mode of the quantized field present
in the cavity. The atomic center-of-mass motion is described
quantum mechanically and the usual rotating-wave
approxi-mation is made. We thus consider the Hamiltonian

*H*⫽ប0†⫹ប*a*†*a*⫹ *p*
2

*2m⫹បgu共z兲共a*

†_{}_{⫹a}_{}†_{兲, 共1兲}

*where p is the atomic center-of-mass momentum along the z*
*axis, m is the atomic mass,* 0 is the atomic transition
fre-quency, is the cavity field mode frequency,*⫽兩b*

### 典具

*a兩 (兩a*

### 典

and*兩b*

### 典

are, respectively, the upper and lower levels of the two-level atom*兲, a and a*† are, respectively, the annihilation

*and creation operators of the cavity radiation field, g is the*

*atom-field coupling strength, and u(z) is the cavity field*mode function. We denote in the following the detuning ⫺0by␦, the cavity field eigenstates by

*兩n*

### 典

, and the global state of the atom-field system by兩*(t)*

### 典

.*Electronic address: T.Bastin@ulg.ac.be

**B. The wave functions**
We introduce the orthonormal basis

兩⌫*n*⫹共兲

### 典

⫽cos*兩a,n*

### 典

⫹sin*兩b,n⫹1*

### 典

,兩⌫*n*⫺共兲

### 典

⫽⫺sin*兩a,n*

### 典

⫹cos*兩b,n⫹1*

### 典

, 共2兲 with an arbitrary parameter. The兩⌫*⫾()*

_{n}### 典

states coincide with the noncoupled states*兩a,n*

### 典

and*兩b,n⫹1*

### 典

when ⫽0 and with the dressed states when⫽n given bycot 2*n*⫽⫺
␦
⍀*n*
, 共3兲
with
⍀*n⫽2g*

### 冑

*n*⫹1. 共4兲

We denote as 兩⫾,n

### 典

the dressed states 兩⌫*⫾(*

_{n}*n*)

### 典

. The*Schro¨dinger equation reads in the z representation and in the*basis共2兲

*i*ប

*t*n, ⫹

_{共z,t兲⫽}### 冋

_{⫺}ប2

*2m*2

*z*2

*⫹共n⫹1兲ប*⫺ប␦cos 2

_{}

*⫹បgu共z兲*

### 冑

*n*⫹1 sin 2

### 册

n,⫹_{}

*共z,t兲*⫹

### 冋

*បgu共z兲*

### 冑

*n*⫹1 cos 2 ⫹1 2ប␦sin 2

### 册

*n,* ⫺

_{共z,t兲,}_{共5a兲}

*i*ប

*t*n, ⫺

_{共z,t兲⫽}### 冋

_{⫺}ប2

*2m*2

*z*2

*⫹共n⫹1兲ប*⫺ប␦ sin 2

_{}

*⫺បgu共z兲*

### 冑

*n*⫹1 sin 2

### 册

n,⫺_{}

*共z,t兲*⫹

### 冋

*បgu共z兲*

### 冑

*n*⫹1 cos 2 ⫹1 2ប␦ sin 2

### 册

*n,* ⫹

_{共z,t兲,}_{共5b兲}with

*n,*⫾

*共z,t兲⫽*

### 具

*z,*⌫⫾

*n*共兲兩

*共t兲*

### 典

. 共6兲*We get for each n two coupled partial differential*equa-tions. In the resonant case (␦⫽0), these equations may be

*decoupled over the entire z axis when working in the dressed*state basis and the atom-field interaction reduces to an el-ementary scattering problem over a potential barrier and a potential well defined by the cavity 共see Ref. 关2兴兲. In the presence of detuning, this is no longer the case: there is no basis where Eqs.

*共5兲 would separate over the entire z axis and*

the interpretation of the atomic interaction with the cavity as a scattering problem over two potentials is less evident.

Outside the cavity 共which we define to be located in the
range 0*⬍z⬍L), the mode function u(z) vanishes and Eqs.*
共5兲 become in the noncoupled state basis (⫽0)

*i*ប
*t*n
*a _{共z,t兲⫽}*

### 冋

_{⫺}ប 2

*2m*2

*z*2

### 册

n*a*

_{共z,t兲,}_{共7a兲}

*i*ប

*t*

*n*⫹1

*b*

_{共z,t兲⫽}### 冋

_{⫺}ប 2

*2m*2

*z*2⫹ប␦

### 册

n⫹1*b*

_{共z,t兲,}_{共7b兲}with

*n*

*a _{共z,t兲⫽e}i(*

_{0}

*⫹n)t*

_{具}

_{z,a,n}_{兩}

_{}

_{共t兲}_{典}

_{,}

_{共8a兲}n

*b*⫹1

*共z,t兲⫽ei(*0

*⫹n)t*

### 具

*z,b,n*⫹1兩

*共t兲*

### 典

. 共8b兲 In Eq. 共8兲, we have introduced the exponential factor*ei(*0

*⫹n)t*

_{in order to define the energy scale origin at the}

*兩a,n*

### 典

level. The solutions to Eqs.共7兲 are obviously given by linear combinations of plane wave functions. If we assume initially a monokinetic atom*共with momentum បk) coming*upon the cavity from the left side

*共negative z values兲 in the*excited state

*兩a*

### 典

and the cavity field in the number state*兩n*

### 典

, the atom-field system is described outside the cavity by the wave function components 共which correspond to the eigen-state 兩

_{k}### 典

*of energy E*⫽ប2

_{k}*2*

_{k}

_{/2m)}n*a _{共z,t兲⫽e}_{⫺i(បk}*2

*/2m)t*

_{n}

*a*

_{共z兲,}_{共9a兲}n

*b*

_{⫹1}

*2*

_{共z,t兲⫽e}_{⫺i(បk}*/2m)t*

_{n}⫹1

*b*

_{共z兲,}_{共9b兲}with

*n*

*a*

_{共z兲⫽}### 再

*e*

*ikz*

_{⫹}

_{}

*n*

*a*

*e⫺ikz*,

*z*⬍0, n

*a*

_{e}ik(z⫺L)_{,}

*共10兲 n*

_{z}_{⬎L,}*b*⫹1

*共z兲⫽*

### 再

n*b*

_{⫹1}

*e⫺ikbz*

_{,}

*⬍0, n*

_{z}*b*⫹1

*eikb(z⫺L)*,

*z⬎L,*共11兲 and

*k*2

_{b}*⫽k*2⫺

*2m*␦ ប . 共12兲 Introducing 2

_{⫽}

*2mg*ប 共13兲 we may write

*k*2

_{b}*⫽k*2⫺2␦

*g*. 共14兲

The solutions共9兲 must be interpreted as follows: The
ex-cited atom coming upon the cavity will be found reflected in
the upper state or in the lower state with the amplitude n*a*
and*n*_{⫹1}

*b*

, respectively, or transmitted with the amplituden*a*
orn*b*_{⫹1}. However, in contrast to the resonant case, the atom
reflected or transmitted in the lower state*兩b*

### 典

will be found to propagate with a momentum*បkb*different from its initial value

*បk. The atomic transition 兩a*

### 典

*→兩b*

### 典

induced by the cavity is responsible for a change of the atomic kinetic en-ergy. According to the sign of the detuning关see Eq. 共14兲兴, the cavity will either speed up the atom 共for ␦⬍0) or slow it down共for␦⬎0). This results merely from energy conserva-tion. When, after leaving the cavity region, the atom is passed from the excited state*兩a*

### 典

to the lower state*兩b*

### 典

, the photon number has increased by one unit in the cavity and the internal energy of the atom-field system has varied by the quantity ប⫺ប0⫽ប␦. This variation needs to be exactly counterbalanced by the external energy of the system, i.e., the atomic kinetic energy. In this sense, when a photon is emitted inside the cavity by the atom, the cavity acts as a potential step ប␦ 共see Fig. 1兲, and the atom experiences an attractive or a repulsive force according to the sign of the detuning. Similar共although not identical兲 mechanical effects are obtained under the adiabatic approximation关15兴 共requir-ing no quantum treatment of the atomic center-of-mass mo-tion兲. In this case the dressed levels may also decelerate or accelerate the atoms. However, in this regime and contrary to what is described here, the atom always leaves the cavity with the same kinetic energy 共if no dissipation process is considered兲 and the mechanical effects are not related to the emission of a photon inside the cavity.Presently the use of positive detunings in the atom-field
interaction defines a well-controlled cooling mechanism. A
single excitation exchange between the atom and the field
inside the cavity is sufficient to cool the atom to a desired
*temperature T*⫽ប2*k _{b}*2

*/2mkB*

*(kB*is the Boltzmann constant兲 which may in principle be as low as imaginable. However, if the initial atomic kinetic energy ប2

*k*2

*/2m is lower than*ប␦

*共i.e., if k/*⬍

### 冑

␦*/g), the transition*

*兩a,n*

### 典

*→兩b,n⫹1*

### 典

cannot take place共as it would remove ប␦ from the kinetic energy兲 and no photon can be emitted inside the cavity. In this case the emission process is completely blocked.Due to this change in the kinetic energy when a photon is
emitted, the reflection and transmission probabilities of the
atom in the lower state *兩b*

### 典

are given, respectively, by 共for ប2*2*

_{k}

_{/2m}_{⬎ប}

_{␦}

_{)}

*R*

_{n}b_{⫹1}⫽

*kb*

*k*兩

*n*⫹1

*b*

_{兩}2

_{,}

_{共15a兲}

*T*

_{n}b_{⫹1}⫽

*kb*

*k*兩n⫹1

*b*

_{兩}2

_{.}

_{共15b兲}

These probabilities vanish for ប2*k*2*/2m*⭐ប␦. When the
atom remains in the excited state*兩a*

### 典

after having interacted with the cavity, there is no change in the atomic kinetic en-ergy and the reflection and transmission probabilities are di-rectly given by*R _{n}a*⫽兩n

*a*兩2, 共16a兲

*T*⫽兩n

_{n}a*a*兩2

_{.}

_{共16b兲}To calculate any quantity related to the reflection and transmission coefficients, we must solve the Schro¨dinger

*equation over the entire z axis. Inside the cavity, the problem*is much more complex since we have two coupled partial differential equations. In the special case of the mesa mode function

*关u(z)⫽1 inside the cavity, 0 elsewhere兴, the*prob-lem is, however, greatly simplified. In the dressed state basis (⫽

*n*), the Schro¨dinger equations 共5兲 take the following form inside the cavity:

*i*ប
*t*n
⫾_{共z,t兲⫽}

### 冋

_{⫺}ប2

*2m*2

*z*2

*⫹Vn*⫾

### 册

_{}

*n*⫾

_{共z,t兲}_{共17兲}with n⫾

**

_{共z,t兲⫽e}i(_{0}

*⫹n)t*

_{具}

_{z,}_{⫾,n兩}_{}

_{共t兲}_{典}

_{共18兲}and

*Vn*⫹⫽sin2*n*ប␦*⫹បg*

### 冑

*n*⫹1 sin 2

*n*, 共19a兲

*V _{n}*⫺⫽ប␦

*⫺V*⫹. 共19b兲

_{n}*V _{n}*⫹

*and V*⫺ represent the internal energies of the 兩⫹,n

_{n}### 典

and兩⫺,n### 典

components, respectively. Except for the resonant case, they cannot be strictly interpreted as a potential barrier and a potential well as Eq.共17兲 holds only inside the cavity.Using Eq. 共3兲, we have the well-known relations

sinn⫽

### 冑

⌳*n*⫹␦

### 冑

2⌳*n*, tan

*n*⫽

### 冑

⌳*n*⫹␦ ⌳

*n*⫺␦ , cos

*⫽*

_{n}### 冑

⌳*n*⫺␦

### 冑

2⌳*n*, cot

*⫽*

_{n}### 冑

⌳*n*⫺␦ ⌳

*n*⫹␦ , 共20兲 with ⌳

*n*⫽

### 冑

␦2⫹⍀*n*2 . 共21兲

FIG. 1. Potential step effect of the cavity when a photon is
*emitted by the atom. E represents the total energy of the atom-field*
system.

We thus have

*V _{n}*⫹

*⫽បg*

### 冑

*n*⫹1 tann, 共22a兲

*V*⫺

_{n}*⫽⫺បg*

### 冑

*n*⫹1 cot

*n*. 共22b兲

*The exponential factor ei(*0

*⫹n)t*

_{has been introduced as}

well in Eq. 共18兲 in order to define the same energy scale
inside and outside the cavity. Figure 2 illustrates the internal
*energies of the atom-field system over the whole z axis. The*
*positive internal energy V _{n}*⫹increases with positive detunings
and vice versa with negative ones. For a fixed value of the

*incident kinetic energy Ek*⫽ប2

*k*2

*/2m, Vn*⫹ may be switched

*in this way from a higher value than Ek*to a lower one. For large positive

*共negative兲 detunings, V*⫹ tends to the

_{n}*兩b,n⫹1*

### 典

(*兩a,n*

### 典

) state energy.The most general solution of Eqs. 共17兲 corresponding to the eigenstate兩k

### 典

is given by*n*⫾*共z,t兲⫽e⫺i(បk*
2* _{/2m)t}*

*n*⫾

*共z兲*共23兲 with n⫾

_{共z兲⫽A}*n*⫾

*⫾*

_{e}ik_{n}*z*

_{⫹B}*n*⫾

*⫾*

_{e}⫺ikn*z*

_{,}共24兲

*where A _{n}*⫾

*and B*⫾are complex coefficients and

_{n}*kn*⫾2*⫽k*2⫺
*2m*
ប2 *Vn*
⫾_{.} _{共25兲}
Defining
*n*⫽

### 冑

4*n*⫹1, 共26兲 we have

*k*⫹2

_{n}*⫽k*2⫺

*2 tann, 共27a兲*

_{n}*k*⫺2

_{n}*⫽k*2⫹

*2 cotn. 共27b兲*

_{n}Using Eq. 共19b兲, we may also write

*k _{n}*⫺2

*⫽k*2⫹

_{b}*2 tann. 共28兲*

_{n}From Eq.共2兲, we may express the wave function compo-nents of the atom-field state inside the cavity over the non-coupled state basis. We have

*n*

*a _{共z,t兲⫽cos}*

_{n}

_{n}⫹

_{共z,t兲⫺sin}_{}

*⫺*

_{nn}

_{共z,t兲,}_{共29a兲}n

*b*

_{⫹1}

_{共z,t兲⫽sin}_{}

*⫹*

_{nn}

_{共z,t兲⫹cos}_{}

*⫺*

_{nn}

_{共z,t兲.}_{共29b兲}This allows us to find the wave function components of the eigenstate兩k

### 典

*over the entire z axis. The relations*共9兲 hold with n

*a*

_{共z兲⫽}## 再

*eikz*

_{⫹}

_{n}

*a*

_{e}_{⫺ikz}_{,}

_{z}_{⬍0,}n

*a*

_{共z兲兩}*C*, 0

*⬍z⬍L,*n

*a*

_{e}ik(z_{⫺L)}_{,}

*共30兲 n*

_{z}_{⬎L,}*b*⫹1

*共z兲⫽*

## 再

*n*⫹1

*b*

*e⫺ikbz*

_{,}

*⬍0, n*

_{z}*b*⫹1

*共z兲兩C*, 0

*⬍z⬍L,*n

*b*

_{⫹1}

*eikb(z⫺L)*

_{,}

*共31兲 and n*

_{z}⬎L,*a*

_{共z兲兩}*C*⫽cos

*n共An*⫹

*e*

*ik*

*n*⫹

_{z}*⫹Bn*⫹

*e⫺ikn*⫹

*兲 ⫺sinn*

_{z}*共An*⫺

*e*

*ik*⫺

_{n}*z*

_{⫹B}*n*⫺

*⫺*

_{e}⫺ikn*z*兲, 共32a兲

*n*⫹1

*b*

_{共z兲兩}*C*⫽sin

*n共An*⫹

*e*

*ik*⫹

_{n}*z*

_{⫹B}*n*⫹

*⫹*

_{e}⫺ikn*z*兲 ⫹cos

*n共An*⫺

*eikn*⫺

_{z}*⫹Bn*⫺

*e⫺ikn*⫺

*兲. 共32b兲 The coefficients*

_{z}*,*

_{n}a

_{n}b_{⫹1},n

*a*,n

*b*

_{⫹1}

*, A*⫹

_{n}*, A*⫺

_{n}*, B*⫹, and

_{n}*B*⫺in expressions共30兲–共32兲 are found by imposing the con-tinuity conditions on the wave function and its first deriva-tive at the cavity interfaces. A tedious calculation yields

_{n}FIG. 2. Schematic energy diagram of the atom-field system in-side and outin-side the cavity, for␦⬎0 共a兲 and␦⬍0 共b兲.

n*a*_{⫽} 共cos
2_{}
*n*兲n⫺*共k兲/*n⫺*共kb*兲n⫹*共kb*兲⫹sin2*nn*⫺*共k兲*
关共cos2_{n}_{兲共k⫺k}*b兲/kn*
*c*
*共L兲⫺1兴关共cos*2_{n}_{兲共k⫺k}*b兲/kn*
*t*
*共L兲⫺1兴*, 共33兲
*n*
*a*_{⫽}
共cos2_{n}_{兲} n
⫺* _{共k兲}*
n

### ⬘

⫺

_{共k,k}*b*兲 ⌬

*n*⫹

*共k兲*⌬

*n*⫹

*共kb*兲 n⫹

_{共k}*b*兲⫹

### 冉

1⫺共cos2*n*兲 n⫹

_{共k}*b*兲

*n*

### ⬙

⫹*共k,kb*兲

### 冊

n⫺*cos2n 4*

_{共k兲⫹}### 冉

*kb*

*k*⫺

*k*

*kb*

### 冊

*u*

_{n}共k兲*n*⫺

_{共k兲}_{}

*n*⫹

_{共k}*b*兲 ⌬

*n*⫺

*共k兲⌬n*⫹

*共kb*兲

### 冉

cos2_{}

*n*

*k⫺kb*

*kn*

*c*⫺1

_{共L兲}### 冊冉

cos2*n*

*k⫺kb*

*kn*

*t*⫺1

_{共L兲}### 冊

, 共34兲 n*b*

_{⫹1}

_{⫽}sin 2n 4

### 冉

1⫹*k*

*kb*

### 冊

关n⫺

_{共k兲/}˜_{}

*n*⫺

_{共k,k}*b*兲兴n⫹

*共kb*兲⫺关n⫹

*共kb*兲/

*˜*

*n*⫹

*共k,kb*兲兴n⫺

*共k兲*关共cos2

_{}

*n兲共k⫺kb兲/kn*

*c*2

_{共L兲⫺1兴关共cos}_{}

*n兲共k⫺kb兲/kn*

*t*, 共35兲

_{共L兲⫺1兴}*n*⫹1

*b*

_{⫽sin 2}

_{}

*n*1 2关n⫺

*共k兲/¯*

*n*⫺

*共k,kb*兲兴关⌬

*n*⫹

*共k兲/⌬n*⫹

*共kb*兲兴n⫹

*共kb*兲⫺ 1 2关n⫹

*共kb*兲/

*¯*

*n*⫹

*共k,kb*兲兴

*n*⫺

*共k兲⫹*1 4

*共k/kb⫺1兲vn共k兲*n⫺

*共k兲*n⫹

*共kb*兲 关共cos2

_{}

*n兲共k⫺kb兲/kn*

*c*2

_{共L兲⫺1兴关共cos}_{n}

_{兲共k⫺k}*b兲/kn*

*t*, 共36兲 with

_{共L兲⫺1兴}*un共k兲⫽sin*2

*n*

*Sn*⫹⫺ sin

*共k*⫹

_{n}*L兲sin共k*⫺

_{n}*L*兲⫹

### 冉

*kn*⫺

*kn*⫹

*k*2 ⫺

*k*2

*k*⫺

_{n}*k*⫹

_{n}### 冊

, 共37兲*vn共k兲⫽i*

### 冉

*k*⫹

_{n}*k*sin

*共kn*⫹

*L兲cos共kn*⫺

*L*兲 ⫺

*kn*⫺

*k*sin

*共kn*⫺

_{L}_{兲cos共k}*n*⫹

_{L}_{兲}

### 冊

_{⫺}cos 2

*n*2

*Sn*⫹⫺

_{,}共38兲

*S*⫹⫺

_{n}*⫽sin共k*⫹

_{n}*L兲sin共k*⫺

_{n}*L*兲

### 冉

*kn*⫺

*k*⫹⫹

_{n}*k*⫹

_{n}*k*⫺

_{n}### 冊

*⫹2关cos共kn*⫺

*L兲cos共kn*⫹

*L*兲⫺1兴, 共39兲 and n⫾

_{共k兲⫽i⌬}*n*⫾

_{共k兲sin共k}*n*⫾

_{L}_{兲}

_{n}⫾

_{共k兲,}_{共40兲}n⫾

_{共k兲⫽关cos共k}*n*⫾

_{L}_{兲⫺i⌺}*n*⫾

_{共k兲sin共k}*n*⫾

_{L}_{兲兴}⫺1

_{,}

_{共41兲}⌺

*n*⫾

*共k兲⫽*1 2

### 冉

*k*⫾

_{n}*k*⫹

*k*

*kn*⫾

### 冊

, 共42兲 ⌬*n*⫾

*共k兲⫽*1 2

### 冉

*k*⫾

_{n}*k*⫺

*k*

*k*⫾

_{n}### 冊

, 共43兲 n### ⬘

⫾

_{共k,k}*b*兲⫽

### 冋

cos*共kn*⫾

*L兲⫺i*

*kb*

*k*⌺

*n*⫾

_{共k兲sin共k}*n*⫾

_{L}_{兲}

### 册

⫺1_{,}共44兲 n

### ⬙

⫾

_{共k,k}*b*兲⫽

### 冋

cos*共kn*⫾

*L兲⫺i*

*k*

*k*⌺

_{b}*n*⫾

*共k兲sin共kn*⫾

*L*兲

### 册

⫺1 , 共45兲 *˜*

*n*⫾

_{共k,k}*b兲⫽关cos共kn*⫾*L兲⫺i⌺˜n*⫾*共k,kb兲sin共kn*⫾*L*兲兴⫺1, 共46兲

*¯*
*n*
⫾_{共k,k}*b*兲⫽

### 冋

cos*共kn*⫾

*L兲⫺i*

*k⫹kb*

*2k*⌺

_{b}*n*⫾

*共k兲sin共kn*⫾

*L*兲

### 册

⫺1 , 共47兲*⌺˜n*⫾

*共k,kb*兲⫽

### 冉

*k*⫾

_{n}*k⫹kb*⫹

*k*

_{b}*k⫹kb*

*k*

*kn*⫾

### 冊

, 共48兲*k _{n}c共L兲⫽i„k⫹i cot共kn*
⫺

_{L/2}_{兲k}*n*⫺

_{…„k}*b⫹i cot共kn*⫹

*L/2兲kn*⫹… cot

*共k*⫺

_{n}*L/2兲k*⫺

_{n}*⫺cot共k*⫹

_{n}*L/2兲k*⫹ , 共49兲

_{n}*k*

_{n}t共L兲⫽i„k⫺i tan共kn⫺_{L/2}_{兲k}*n*
⫺_{…„k}*b⫺i tan共kn*⫹*L/2兲kn*⫹…
tan*共k _{n}*⫹

*L/2兲k*⫹

_{n}*⫺tan共k*⫺

_{n}*L/2兲k*⫺ . 共50兲 At resonance,␦⫽0,

_{n}*n*⫽

*/4, kb⫽k,*n⫾

*(k)*⫽n

### ⬘

⫾*(k,kb*) ⫽

### ⬙

*n*⫾

*(k,kb*)⫽

*˜*⫾

*n(k,kb*)⫽

*¯*

*n*⫾

*(k,kb*), and the reflection and transmission coefficients reduce to the well-known results 共see Ref. 关2兴兲 n

*a*

_{⫽}1 2共n ⫹

_{⫹}

_{n}⫺

_{兲,}

_{共51a兲}

*n*

*a*

_{⫽}1 2共n ⫹

_{⫹}

_{n}⫺

_{兲,}

_{共51b兲}and

n*b*⫹1⫽
1
2共n
⫹_{⫺}_{n}⫺_{兲,} _{共52a兲}
*n*⫹1
*b* _{⫽}1
2共*n*⫹⫺*n*⫺兲. 共52b兲

**III. INDUCED EMISSION PROBABILITY**

The induced emission probability of a photon inside the
cavity is given by
*P*em*共n兲⫽Rn*⫹1
*b* _{⫹T}*n*⫹1
*b*
. 共53兲

According to Eqs.共15兲, this probability may be written as

*P*em*共n兲⫽*

### 再

*k*

_{b}*k*关兩n⫹1

*b*

_{兩}2

_{⫹兩}

_{n}⫹1

*b*

_{兩}2

_{兴 if}

*k*⬎

### 冑

␦*g*, 0 otherwise. 共54兲

We distinguish three regimes determined by the incident
kinetic energy of the atom compared with the internal energy
*Vn*⫹ *共see Fig. 2兲: the hot atom regime 共when kⰇ**n*

### 冑

tann*and kb*

*is approximated to k), the intermediate regime (k*⬇

*n*

### 冑

tan*n), and the cold atom regime (k*Ⰶ

*n*

### 冑

tann). In comparison with the resonant case, the detuning defines an additional parameter that fixes the working regime of the system.**A. Hot atom regime**

In the hot atom regime, the kinetic energy is much higher
*than the energies V _{n}*⫾ and the atoms are always transmitted
through the cavity. For the mesa mode, the atomic
momen-tum inside the cavity is given by

*បkn*⫹*⯝បk*

### 冉

1⫺ *n*2

*2k*2tan

*n*

### 冊

, 共55a兲*បkn*⫺

*⯝បk*

### 冉

1⫹ *n*2

*2k*2cot

*n*

### 冊

. 共55b兲 After the atom-field interaction, the global state of the system reduces to 兩*共t兲*

### 典

⫽### 冕

*dz*

*共z,t兲共cos*

*2*

_{n}e⫺i(n*L/2k)tan*

_{n}_{兩z,}_{⫹,n}_{典}

⫺sinn*ei(*

*n*2

_{L/2k)cot}_{}

*n兩z,⫺,n*

### 典

_{)}共56兲 with

*共z,t兲⫽eikz _{e}⫺i(បk*2

*/2m)t*

_{.}

_{共57兲}

The induced emission probability is thus given by

*P*em*共n兲⫽円*

### 具

*z,b,n*⫹1兩

*共t兲*

### 典

円2 ⫽sin 2_{共2}

_{}

*n*兲 4

*兩e*

*⫺i(n*2

*L/2k)tan*

_{n}*⫺ei(*

*2*

_{n}*L/2k)cot*

_{n}_{兩}2

_{.}

_{共58兲}

Using Eq. 共20兲 we get straightforwardly

*P*em*共n兲⫽*

### 冋

1⫹### 冉

␦ ⍀*n*

### 冊

2### 册

⫺1 sin2### 冉

2### 冑

⍀*n*2

_{⫹}

_{␦}2

### 冊

_{,}

_{共59兲}where

*⫽mL/បk is the classical transit time of the thermal*atoms through the cavity.

Equation 共59兲 is exactly the classical expression of the induced emission probability of an atom interacting with a single mode during a time . We recover the well-known Rabi oscillations in the general case where the field and the atomic frequencies are detuned by the quantity␦.

**B. Intermediate regime**

The frontier between the cold atom and the hot atom
re-gime appears when the atomic kinetic energy becomes equal
*to the positive energy Vn*⫹ inside the cavity, i.e., when the
*ratio k/**n* is equal to the critical value

*k*
*n*

### 冏

*⫽*

_{c}### 冑

tan*⫽*

_{n}### 冉

### 冑

共␦*/g*兲 2

_{⫹4共n⫹1兲⫹}_{␦}

_{/g}### 冑

共␦*/g*兲2

*⫹4共n⫹1兲⫺*␦

*/g*

### 冊

1/4 . 共60兲*At resonance, this frontier occurs for k/*n⫽1. This
condi-tion changes significantly when the cavity and the atomic
transition are detuned. This is well illustrated in Fig. 3,
which shows the induced emission probability as a function
*of k/* at resonance and for a positive value of the detuning.
In the second case, the regime change occurs for a larger
*value of k/* than one. Please notice also that the induced
*emission probability vanishes for k/*⬍

### 冑

␦*/g according to*Eq. 共54兲.

Equation共60兲 may be inverted to yield a critical detuning
*when working with a fixed value of k/*. We get

FIG. 3. Induced emission probability*P*em*(n*⫽0) with respect to

␦
*g*

### 冏

*⫽*

_{c}### 冑

*n*⫹1

### 冋冉

*k*n

### 冊

2 ⫺### 冉

_{n}

*k*

### 冊

⫺2### 册

. 共61兲 Figure 4 illustrates the induced emission probability as a*function of the detuning for k/* slightly greater than 1. At resonance, the system is on the hot atom regime side as the

*atomic kinetic energy is greater than the internal energy V*⫹.

_{n}*When the detuning is increased, V*⫹is increased as well and becomes greater than the kinetic energy 关see Fig. 2共a兲兴, switching the system toward the cold atom regime. This therefore could define a convenient way to switch from one regime to the other rather than by varying the incident atomic momentum.

_{n}A similar effect of the detuning is presented in Fig. 5
which presents the induced emission probability with respect
to the interaction length in the intermediate regime. It
ap-pears clearly there that working with a negative detuning is
similar to working with hotter atoms at resonance. This
*re-sults merely from the level of V _{n}*⫹compared to the incident

kinetic energy, which is identical in both cases considered in Fig. 5.

From all these situations, we conclude that a detuning
variation has an identical effect as a change of the kinetic
energy of the incoming atoms, confirming that the only
im-portant parameter of the system in the intermediate regime is
*the actual value of the V _{n}*⫹ energy in comparison with the
kinetic energy ប2

*k*2

*/2m.*

**C. Cold atom regime**

*In the cold atom regime (k/**n*Ⰶ

### 冑

tan*n*), the induced emission probability exhibits a completely different behav-ior. We have in this regime for兩␦兩/⍀

*n*Ⰶ1, exp(n

*L)*Ⰷ1, and

*nL*Ⰶ(

*n/k)*2,

*P*em*共n兲⫽*

*B共L兲*

2

1⫹关共cot*n*兲/2兴sin共2*n*

### 冑

cot*nL*兲 1⫹共n

*/2k*兲2 cot

*n*sin2共

*n*

### 冑

cot*nL*兲

,
共62兲
with
*B共L兲⫽kb*
*k*
1
兩共cos2_{}
*n兲共k⫺kb兲/kn*
*c _{共L兲⫺1兩}*2

_{兩共cos}2

_{}

*n兲共k⫺kb兲/kn*

*t*2. 共63兲

_{共L兲⫺1兩}At resonance, this expression simplifies to

*P*em*共n兲⫽*
1
2
1⫹ 1
2 sin共2n*L*兲
1⫹共n*/2k*兲2 sin2共n*L*兲 共64兲
*and the results of Meyer et al.*关2兴 are well recovered. Figure
6 illustrates the induced emission probability 共62兲 with
re-spect to the interaction length *L for various values of the*
detuning. The curves present a series of peaks where the
induced emission probability is optimum. The detuning
af-fects the peak position, amplitude, and width. Similarly to
the resonant case, the curves still look like the Airy function
of classical optics *关1⫹F sin*2(⌬/2)兴⫺1 *with finesse F and*
total phase difference⌬, even if the structure of Eq. 共62兲 has

become complicated with the factor*B(L). In fact, this *
equa-tion is extremely well fitted in its domain of validity by the
function
*P*em*共n兲⫽*
*2kb/k*
*共1⫹kb/k*兲2
1⫹12 sin共2*n*

### 冑

cot*nL*兲 1⫹关

*n*/

*共kb⫹k兲兴*2 sin2共

*n*

### 冑

cot*nL*兲 . 共65兲

**1. Peak position**The induced emission probability is optimum when

*n*

### 冑

cot*nL⫽m*

*共m a positive integer兲.*共66兲

FIG. 4. Induced emission probability*P*em*(n*⫽0) with respect to

␦*/g in the intermediate regime (k/*⫽1.01). The interaction length
was fixed to*L⫽100.*

FIG. 5. Induced emission probability*P*em*(n*⫽0) with respect to

the interaction length*L in the intermediate regime. Comparison of*
a detuning variation with a change of the atomic kinetic energy.

This occurs when the cavity length fits a multiple of one-half
the de Broglie wavelength_{dB}of the atom inside the cavity:

*L⫽m*dB

2 . 共67兲

Indeed, in the cold atom regime, only the 兩⫺,n

### 典

compo-nent propagates inside the cavity with the de Broglie wave-length dB⫽ 2*k*⫺⯝ 2 n

_{n}### 冑

cotn. 共68兲 Inserting Eq.共68兲 into Eq. 共67兲 gives the condition 共66兲.**2. Peak amplitude**

The peak amplitude of the induced emission probability

*P*em*(n) is given by*
*A⬅B共L⫽m*dB/2兲
2 ⯝
1
2
*4kb/k*
*共1⫹kb/k*兲2
. 共69兲

We illustrate this amplitude in Fig. 7 as a function of the detuning␦. In contrast to the hot atom regime关see Eq. 共59兲兴, the curves present a strong asymmetry with respect to the sign of the detuning. This results from the potential stepប␦ 共see Sec. II兲 experienced by the atoms when they emit a photon. For cold atoms whose energy is similar to or less

than the step height, the sign of the step is a crucial param-eter. According to Eq.共54兲, the induced emission probability drops down very rapidly to zero for positive detunings, in contrast to what happens for negative detunings.

It is also interesting to note that the peak amplitude共69兲 is
equal to the amplitude at resonance (1/2) times the factor
*(4kb/k)/(1⫹kb/k)*2 that corresponds exactly to the
trans-mission factor of a particle of momentum *បk through a *
po-tential stepប␦. This is an additional argument to say that the
use of a detuning adds a potential step effect for atoms
emit-ting a photon inside the cavity共see Fig. 1兲.

**3. Peak width**

The peak width is determined by the finesse 关*n/(kb*
*⫹k)兴*2_{. Positive detunings increase the finesse (k}

*b⬍k) while*
*negative ones decrease it (kb⬎k).*

**4. Large detunings**

For large detunings, Eq. 共62兲 is no longer valid and the
induced emission probability must be computed using the
general relation*共54兲. We present in Fig. 8 P*em(0) as a
func-tion of the detuning and the interacfunc-tion length*L. The *
varia-tion of the resonance posivaria-tions with respect to the detuning is
very clear in this figure. It is interesting to note that one
resonance out of two disappears when increasing the
detun-ing toward negative values. Also, the induced emission
prob-ability does not decrease monotonically with the detuning
共especially for small interaction lengths兲. This effect is
strictly limited to the cold atom regime as for hot atoms the
Rabi oscillation amplitudes always decrease when larger and
larger detunings are used 关see Eq. 共59兲兴.

The use of large negative detunings in the cold atom
re-gime is not limitless. As ⫺␦ increases, the internal energy
*V _{n}*⫹ decreases关see Fig. 2共b兲兴 and may finally become lower
than the incident kinetic energy. To keep the system in the

*cold atom regime, we must have for k/*nⰆ1 关see Eq. 共61兲兴

⫺␦* _{g}*⬍

### 冑

*n*⫹1

### 冉

n*k*

### 冊

2

. 共70兲

FIG. 6. Induced emission probability*P*em*(n*⫽0) with respect to

the interaction length *L in the cold atom regime (k/⫽0.1) and*
for various values of the detuning.

FIG. 7. Amplitude*A of the resonances with respect to*␦*/g for*
*two values of k/* in the cold atom regime.

FIG. 8. Induced emission probability*Pem(n*⫽0) with respect to
the interaction length*L and*␦*/g共for k/*⫽0.1).

This condition is well respected in Fig. 8 as the limiting
lower value of ␦*/g to keep the system in the cold atom*
regime is*⫺100 for k/*⫽0.1.

The use of large positive detunings in the cold atom
re-gime is not possible as the induced emission probability
van-ishes for
␦
*g*⭓

### 冉

*k*

### 冊

2 . 共71兲**IV. SUMMARY**

In this paper we have presented the quantum theory of the mazer in the nonresonant case. Interesting effects have been pointed out. In particular, we have shown that the cavity may slow down or speed up the atoms according to the sign of the detuning and that the induced emission process may be com-pletely blocked by use of a positive detuning. We have also demonstrated that the detuning adds a potential step effect not present at resonance. This defines a well-controlled

cool-ing mechanism for positive detuncool-ings. In the special case of the mesa mode function, generalized expressions for the re-flection and transmission coefficients have been obtained. The properties of the induced emission probability in the presence of a detuning have been discussed. In the cold atom regime, we have obtained a simplified expression for this probability and have been able to describe the detuning ef-fects on the resonance amplitude, width, and position. In contrast to the hot atom regime, we have shown that the mazer properties are not symmetric with respect to the sign of the detuning. In the intermediate regime, the use of detun-ing could be a convenient way to switch from the hot atom regime to the cold atom one.

**ACKNOWLEDGMENTS**

This work has been supported by the Belgian Institut In-teruniversitaire des Sciences Nucle´aires共IISN兲. T.B. wants to thank H. Walther and E. Solano for their hospitality at Max-Planck-Institut fu¨r Quantenoptik, Garching 共Germany兲.

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