Cold three-level atom micromaser in the far-detuning regime

Cold three-level atom micromaser in the far-detuning regime

John Martin and Thierry Bastin*

Institut de Physique Nucléaire, Atomique et de Spectroscopie, Université de Liège au Sart Tilman, Bâtiment B15, B-4000 Liège, Belgium 共Received 13 December 2006; published 24 May 2007兲

We analyze the quantum theory of the two-mode micromaser pumped by cold three-level atoms with a ⌳-configuration under the two-photon resonance condition. We focus more specifically on the large detuning limit where the detuning between the two cavity mode frequencies and the two atomic transition frequencies is large in comparison with the atom-cavity coupling constants. We show that this regime can mimic a virtual two-level atom micromaser without spontaneous emission where the cavity acts for the atoms as a potential barrier or a potential well共depending on the sign of the detuning兲 and a zero potential but no more both as a potential barrier and a potential well as it is the case for the usual cold two-level atom micromaser关M. O. Scully, G. M. Meyer, and H. Walther, Phys. Rev. Lett. 76, 4144共1996兲兴. This introduces interesting options for engineering the interaction between the cold atoms and the cavity. The elimination of the spontaneous emission solves a major issue of the conventional cold two-level atom micromaser as the transit time of cold atoms inside and in the vicinity of the cavity is usually much larger than the atomic level lifetimes. It is also shown that the cold atom regime is very sensitive to the sign of the detuning共in contrast to the hot atom regime兲 and that, according to this sign, the cavity may speed up or slow down the incident three-level atoms after their interaction with the field.

DOI:10.1103/PhysRevA.75.053820 PACS number共s兲: 42.50.Pq, 42.50.Vk, 32.80.Lg

I. INTRODUCTION

Today, laser cooling of atoms has become a very broad topic in atomic physics and quantum optics. Cold and ultra-cold atoms introduce new regimes often not considered in the past. Particularly, the study of the micromaser exhibits important new effects when cold atoms are sent through the maser cavity rather than hot ones关1兴. In this case, Englert et al.关2兴 have shown that the cavity acts simultaneously for the incident atoms as a potential barrier and a potential well, even if the cavity contains no photon. This effect arises from the sudden change of the atom-field interaction when the atoms enter the cavity. It requires the quantization of the center-of-mass motion to be explained. For this reason, the system has been called mazer by Scully and co-workers关1兴 to insist on the fact that the microwave amplification is due to the z motion axis induced emission of radiation.

In the last decade, a lot of studies were dedicated to this subject共see, e.g., references in 关3兴兲. Initially, two-level atoms interacting with the cavity were considered关4–6兴. Later on, models using three-level atoms have been developed关7–15兴. Zhang et al.关7–10兴 focused on three-level atoms in ⌳ and ⌶ configurations interacting with monomodal cavities. Liang et al. 关11兴 investigated the transmission probability of V-type three-level atoms. Arun et al.关12,13兴 considered the case of three-level atoms in⌳ and ⌶ configurations interacting reso-nantly with a bimodal cavity共each transition frequency co-incides with a mode frequency兲. Abdel-Aty and Obada 关14,15兴 studied transient phenomena during the passage of the atoms through the cavity.

In this paper, we consider⌳-type three-level atoms inter-acting with a bimodal cavity in the general two-photon reso-nance condition共see Fig.1兲. We focus more specifically on

the large detuning limit where the detuning ⌬ between the two cavity mode frequencies and the two atomic transition frequencies is large in comparison with the atom-cavity cou-pling constants. In the hot atom regime where the quantiza-tion of the atomic moquantiza-tion is not required, it is well known that the large-detuning limit simplifies the atom-cavity inter-action to the interinter-action of a virtual two-level atom with a monomodal cavity 关16,17兴. In the cold atom regime, we show here that this is more subtle. Still the atom behaves like a virtual two-level one. However, the two-level atom mazer properties are not entirely recovered. The large-detuning limit of the three-level atom mazer mimics a two-level atom mazer where the cavity acts for the atoms initially in one of their two lower states as a potential barrier or a potential well 共depending on the sign of the detuning兲 and a zero potential but no more both as a potential barrier and a potential well as it is the case for the usual two-level atom mazer关1,2,4兴. This changes drastically the properties of the system and intro-duces new interesting options for engineering the interaction between the atoms and the cavity as the sole detuning pa-rameter determines now what kind of potential the atoms interact with and also directly the potential height or depth. In the conventional mazer, no such degree of freedom exists since both mechanical potentials are always present. Also, similarly to three-level atoms interacting with two traveling waves 关18兴, the large-detuning limit allows a system where the spontaneous emission is virtually eliminated when the two lower levels do not define an allowed dipolar transition. This is particularly important in the cold atom regime where the transit times inside and in the vicinity of the cavity can be much larger than the atomic level lifetimes.

The paper is organized as follows. In Sec. II, the Hamil-tonian modeling a⌳-type three-level atom interacting with a bimodal cavity in the general two-photon resonance condi-tion is presented and the wave funccondi-tions of the system are described. General expressions for the transition probabilities between the different atomic levels are given. We prove there *Electronic address: t.bastin@ulg.ac.be

that, similarly to two-level atoms interacting nonresonantly with a monomodal cavity关3兴, the cavity may slow down or speed up the atoms according to the sign of the detuning⌬. We then focus on the large detuning limit to show that the interaction of the three-level atoms with the bimodal cavity leads to a one-dimensional scattering process upon a poten-tial barrier or a potenpoten-tial well. Simplified expressions for the transition probabilities are given. In Sec. III, we discuss more specifically the properties of these transition probabili-ties and make the connection to the hot atom regime where the quantization of the atomic motion is not needed. Experi-mental considerations are given in Sec. IV. A brief summary of our results is finally given in Sec. V.

II. MODEL A. The Hamiltonian

1. General expression

We consider a⌳-type three-level atom 共with levels 兩b1典, 兩b2典, and 兩a典 of energy ប⍀b₁, ប⍀b₂, and ប⍀a, respectively兲 moving along the z direction on the way to a bimodal cavity defined to be located in the range 0⬍z⬍L 共see Fig.1兲. We denote by␻1and␻2 the two mode frequencies. We assume no coupling between the two lower levels兩b1典 and 兩b2典 and the two-photon resonance condition⌬⬅␻1−共⍀a−⍀b₁兲=␻2 −共⍀a−⍀b₂兲 is fulfilled. The atomic center-of-mass motion is described quantum mechanically and the usual rotating-wave approximation is made. The Hamiltonian of this system is given by

H = HA+ HF+ HAF, 共1兲

where HA 共HF兲 is the Hamiltonian of the free atom 共field兲 and HAFis the interaction Hamiltonian describing the atom-field interaction, with

HA= p2 2m+ប⍀a兩a典具a兩 +_␣=1

兺

2 ប⍀b_␣兩b␣典具b␣兩, HF=

兺

␣=1 2 ប␻␣a␣†a␣, HAF=

兺

␣=1 2

បg␣u␣共z兲共a␣兩a典具b␣兩 + 兩b␣典具a兩a␣†兲. 共2兲 In Eq.共2兲, p is the atomic center-of-mass momentum along the z axis, m is the atomic mass,兩j典具j兩 共j=a,b1, b2兲 are the projection operators onto the states 兩j典, 兩i典具j兩 共i, j=a,b1, b2; i⫽ j兲 are the transition operators from level j to level i, a␣ and a_␣† 共␣= 1 , 2兲 are, respectively, the annihilation and cre-ation operators of a photon in mode␣, g_␣are the atom-field coupling strengths at the two mode frequencies␻_␣, and u_␣共z兲 are the cavity field mode functions, which are assumed here to be identical, i.e., u1共z兲=u2共z兲⬅u共z兲. In the following, we denote the cavity field eigenstates of the two modes by 兩n1, n2典 and the global state of the atom-field system at any time t by兩␺共t兲典.

2. Effective two-level Hamiltonian

If the atomic upper level兩a典 is initially not populated and if the detuning ⌬ is large in comparison with the Rabi fre-quencies⍀n_␣⬅2g␣

冑n

␣+ 1 associated to the occupation num-bers n_␣of the field state, then the three-level system reduces to a virtual two-level one共comprising the 兩b1典 and 兩b2典 states兲 described in the interaction picture by the effective Hamil-tonian关19兴 Heff= p2 2m+_␣=1

兺

2 បg␣ 2 u_␣2共z兲 ⌬ a␣†a␣兩b␣典具b␣兩 +បg1u1共z兲g2u2共z兲 ⌬ 共a1 † a2兩b1典具b2兩 + a₂†a1兩b2典具b1兩兲. 共3兲 B. Wave functions 1. Arbitrary detuning

Outside the cavity, the mode function u共z兲 vanishes and the Schrödinger equation iបd/dt兩␺共t兲典=H兩␺共t兲典 reads in the z representation and in the noncoupled state basis

iប⳵ ⳵t␺n1,n2 a _{共z,t兲 =}

冉

− ប 2 2m ⳵2 ⳵z2−ប⌬

冊

␺n1,n2 a _共z,t兲, iប⳵ ⳵t␺n1+1,n2 b_{1 共z,t兲 = −} ប2 2m ⳵2 ⳵z2␺n1+1,n2 b_{1 共z,t兲,} iប⳵ ⳵t␺n1,n2+1 b_{2 共z,t兲 = −} ប2 2m ⳵2 ⳵z2␺n1,n2+1 b_{2 共z,t兲, 共4兲} with ␺n₁,n2 a _{共z,t兲 = e}iE_bt/ប_{具z,a,n1,n2兩␺共t兲典,} ␺n₁+1,n₂ b_{1 共z,t兲 = e}iE_bt/ប_具z,b1,n 1+ 1,n2兩␺共t兲典, ␺n₁,n₂+1 b_{2 共z,t兲 = e}iE_bt/ប_具z,b2,n 1,n2+ 1兩␺共t兲典. 共5兲 where

0

L

z

ω

1

ω

2 ω1 ω2 ∆ |a
|b2
|b1

FIG. 1.共Color online兲 System investigated: A ⌳-type three-level atom共with levels 兩b₁典, 兩b₂典, and 兩a典 of energy ប⍀_b

1,ប⍀b2, andប⍀a, respectively兲 moving along the z direction on the way to a bimodal cavity of length L 共with mode frequencies ␻₁ and ␻₂兲 in the general two-photon resonance condition ⌬⬅␻1−共⍀a−⍀b₁兲

=␻₂−共⍀_a−⍀_b 2兲.

Eb=ប⍀b₁+共n1+ 1兲ប␻1+ n2ប␻2 =ប⍀b₂+ n1ប␻1+共n2+ 1兲ប␻2

is the energy of the atom-field states 兩b1, n₁+ 1 , n2典 and 兩b2, n1, n2+ 1典. In Eq. 共5兲, we merely shift the energy scale origin at the兩b1, n1+ 1 , n2典 and 兩b2, n1, n2+ 1典 level.

The solutions to Eqs. 共4兲 are obviously given by linear combinations of plane wave functions. If we assume initially a monokinetic atom共with momentum បk兲 coming upon the cavity from the left-hand side共negative z values兲 in the state 兩b1典 and the cavity field in the number state 兩n1+ 1 , n2典, the atom-field system is described outside the cavity by the wave function components 共which correspond to the eigenstate 兩␾k典 of energy Ek=ប2k2/ 2m兲 ␺n₁,n₂ a _{共z,t兲 = e}−i共បk2_/2m兲t ␸n₁,n₂ a _{共z兲, 共6a兲} ␺n₁+1,n₂ b_{1 共z,t兲 = e−i共បk}2_/2m兲t ␸n₁+1,n₂ b_{1 共z兲, 共6b兲} ␺n₁,n₂+1 b_{2 共z,t兲 = e}−i共បk2_/2m兲t ␸n₁,n₂+1 b_{2 共z兲, 共6c兲} with ␸n₁,n₂ a _{共z兲 =}

再

␳n1,n2 a e−ikaz_{, z}⬍ 0 ␶n₁,n₂ a eikaz_{, z}⬎ L, 共7兲 ␸n₁+1,n2 b_{1 共z兲 =}

再

e ikz_+␳ n₁+1,n2 b₁ e−ikz_, z⬍ 0 ␶n₁+1,n2 b₁ eikz, z⬎ L, 共8兲 ␸n₁,n₂+1 b_{2 共z兲 =}

再

␳n1,n2+1 b₂ e−ikz, z⬍ 0 ␶n₁,n₂+1 b₂ eikz, z⬎ L, 共9兲 and ka 2 = k2+2m ប ⌬ ⬅ k2+␬2⌬˜, 共10兲 where ␬2₌2m ប

冑

g1 2 + g₂2, ⌬˜ =

_冑

⌬ g₁2+ g₂2. 共11兲 The solutions共6兲 must be interpreted as follows: the atom initially in the state 兩b1典 coming upon the cavity will be found reflected in the state 兩i典 共i=a,b₁, b₂兲 with the ampli-tude␳n₁,n₂ a ,␳n₁+1,n₂ b₁ , and␳n₁,n₂+1 b₂ , respectively, or transmitted with the amplitude␶n₁,n₂

a ,␶n₁+1,n₂ b₁ or␶n₁,n₂+1 b₂ . However, in con-trast to the resonant case共⌬=0, see 关12兴兲, the atom reflected or transmitted in the excited state兩a典 will be found to propa-gate with a momentumបkadifferent from its initial valueបk. The atomic transition兩b₁典→兩a典 induced by the cavity is re-sponsible for a change of the atomic kinetic energy. Accord-ing to the sign of the detunAccord-ing关see Eq. 共10兲兴, the cavity will either speed up the atom 共for ⌬⬎0兲 or slow it down 共for ⌬⬍0兲. This results merely from energy conservation. When, after leaving the cavity region, the atom is passed from the lower state兩b1典 to the upper state 兩a典, the photon number in

mode 1 has decreased by one unit in the cavity and the in-ternal energy of the atom-field system has varied by the quantityប共⍀a−⍀b₁兲−ប␻1= −ប⌬. This variation needs to be exactly counterbalanced by the external energy of the sys-tem, i.e., the atomic kinetic energy. After the process, this energy has varied accordingly by the amount ប⌬ 共this is exactly the same effect that is observed for the nonresonant mazer with two-level atoms关3兴兲. If the initial atomic kinetic energyប2_k2_{/ 2m is lower thanប兩⌬兩, and ⌬⬍0, the transition} 兩b1, n1+ 1 , n2典→兩a,n1, n2典 cannot take place and this process is completely blocked.

Due to this change in the kinetic energy when the atom undergoes a transition from state兩b1典 to state 兩a典, the reflec-tion and transmission probabilities of the atom in the excited state兩a典 are, respectively, given by

Rn₁,n2 a =ka k兩␳n1,n2 a _兩2_{, 共12a兲} Tn₁,n₂ a =ka k兩␶n1,n2 a _兩2_{, 共12b兲}

provided ប2_k2_{/ 2m⬎ប兩⌬兩 if ⌬⬍0. Otherwise, these} prob-abilities vanish. The total atomic transition probabilityPb₁→a from state兩b1典 to state 兩a典 follows immediately:

Pb_1→a= Rn₁,n₂ a + Tn₁,n₂ a _共13兲 that is Pb₁→a=

冦

ka k共兩␳n1,n2 a _兩2_+兩␶ n₁,n2 a _兩2_{兲 if}

冉

k ␬

冊

2 +⌬˜ ⬎ 0 0 otherwise.

冣

共14兲 When the atom remains in the lower state兩b1典 or under-goes a transition to state兩b2典 after having interacted with the cavity, there is no change in the atomic kinetic energy and the corresponding reflection and transmission probabilities are directly given by

Rn₁+1,n₂ b₁ =兩␳n₁+1,n₂ b_{1 兩}2_{, 共15a兲} Tn₁+1,n₂ b₁ =兩␶n₁+1,n₂ b_{1 兩}2_{, 共15b兲} and Rn₁,n₂+1 b₂ =兩␳n₁,n₂+1 b_{2 兩}2_{, 共16a兲} T_n 1,n2+1 b₂ =兩␶_n 1,n2+1 b_{2 兩}2_{, 共16b兲}

respectively. The total probabilities to find the atom in the state 兩b1典 or 兩b2典 after its interaction with the cavity are ac-cordingly given by Pb_1→b1=兩␳n₁+1,n₂ b_{1 兩}2_+兩␶ n₁+1,n₂ b_{1 兩}2 _共17兲 and Pb₁→b₂=兩␳n₁,n2+1 b₂ 兩2_+兩␶ n₁,n2+1 b₂ 兩2_{, 共18兲} respectively.

To determine any of the probabilities 共12兲–共18兲 the Schrödinger equation must be solved 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 a mesa mode function 关u共z兲=1 inside the cavity, 0 elsewhere兴, the problem is, however, greatly simpli-fied. In this case, inside the cavity, the operator H − p2_{/ 2m} 关see Eq. 共1兲兴 has normalized eigenstates 兩␥n␧₁,n₂典 共␧=0, ±兲 given by关16兴 兩␥n0₁,n₂典 = cb0₁兩b1,n1+ 1,n2典 + cb0₂兩b2,n1,n2+ 1典, 兩␥n₁,n₂ ± _{典 = c} a ±_兩a,n1 ,n2典 + cb₁ ±_兩b1 ,n1+ 1,n2典 + cb₂ ±_兩b2 ,n1,n2+ 1典, 共19兲 with cb0₁= − ⍀n₂

冑

⍀n₁ 2 +⍀n₂ 2 , cb2 0 ₌ ⍀n1

冑

⍀n₁ 2 +⍀n₂ 2 , 共20兲 and ca±= −⌬ ±

冑

⌬2+⍀n2₁+⍀n2₂ ␤± , cb±_␣= ⍀n_␣ ␤± , 共21兲 where ␤±₌

冑

₂

冑

_⍀ n₁ 2 _+⍀ n₂ 2 _{+⌬共⌬ ⫿}

冑

_⌬2_+⍀ n₁ 2 _+⍀ n₂ 2_{兲. 共22兲} The corresponding eigenvalues␥_n

1,n2 ␧ _read ␥n₁,n₂ 0 = Eb, ␥n±₁,n₂= −ប⌬ ± ប

冑

⌬2+⍀n2₁+⍀n2₂ 2 + Eb. 共23兲

In the basis 兩␥n␧₁,n₂典 and in the z representation, the Schrödinger equation takes the following form inside the cavity: iប⳵ ⳵t␺n1,n2 ␧ _{共z,t兲 =}

冉

₋ ប2 2m ⳵2 ⳵z2+ Vn1,n2 ␧

冊

_␺ n₁,n₂ ␧ _{共z,t兲, 共24兲} with ␺n␧₁,n₂共z,t兲 = e iE_bt/ប_具z,␥ n₁,n₂ ␧ _{兩␺共t兲典 共25兲} and Vn₁,n2 ␧ _=␥ n₁,n2 ␧ _{− E} b. 共26兲

The most general solution of Eqs.共24兲 corresponding to the eigenstate兩␾k典 is given by

␺n␧₁,n₂共z,t兲 = e−i共បk 2_/2m兲t ␸n␧₁,n₂共z兲, 共27兲 with ␸␧n₁,n₂共z兲 = A␧n₁,n₂eikn1,n2 ␧ _z + Bn␧₁,n₂e−ikn1,n2 ␧ _z , 共28兲 where A_n 1,n2 ␧ _{and B} n₁,n2

␧ _{are complex coefficients and}

共kn␧₁,n₂兲2= k2− 2m

ប2Vn␧₁,n₂. 共29兲 We can write the wave function components of the atom-field state inside the cavity over the noncoupled state basis. The coefficients ␳n₁,n2 a , ␳n₁+1,n₂ b₁ , ␳n₁,n₂+1 b₂ , ␶n₁,n2 a , ␶n₁+1,n₂ b₁ , ␶n₁,n₂+1 b₂

, An␧₁,n₂, and Bn␧₁,n₂ are then found via the continuity conditions of the wave function and its first derivative at the cavity interfaces. For arbitrary values of the detuning⌬ and in the mesa mode case, the obtained analytical expressions are too huge to be displayed here.

2. Large-detuning limit

For large detunings, the various probabilities 共15兲–共18兲 can be obtained in a much simpler way by use of the effec-tive two-level Hamiltonian 共3兲. In basis 兩␥_n

1,n2 共i兲 _{典 共i=1,2兲} given by 兩␥n共1兲₁,n₂典 = cb0₁兩b1,n1+ 1,n2典 + cb0₂兩b2,n1,n2+ 1典, 兩␥n共2兲₁,n₂典 = cb₂ 0_兩b1 ,n1+ 1,n2典 − cb₁ 0_兩b2 ,n1,n2+ 1典, 共30兲 and in the z representation, the Schrödinger equation for the effective two-level Hamiltonian共3兲 takes the following form on the whole z axis:

iប⳵ ⳵t␺n1,n2 共i兲 _{共z,t兲 =}

冉

₋ ប2 2m ⳵2 ⳵z2+ Vn1,n2 共i兲 _u2_共z兲

冊

_␺ n₁,n₂ 共i兲 _共z,t兲, 共31兲 with ␺共i兲n₁,n₂共z,t兲 = 具z,␥共i兲n₁,n₂兩␺共t兲典, 共32兲 and Vn共1兲₁,n₂= 0, Vn共2兲₁,n₂=ប ⍀n2₁+⍀n2₂ 4⌬ . 共33兲

Equation 共31兲 describes an elementary one-dimensional scattering problem for each component ␺_n

1,n2

共i兲 _{共z,t兲. The}

␺n₁,n₂

共1兲 _{共z,t兲 components evolve like a free particle wave} func-tion whereas the ␺_n

1,n2

共2兲 _{共z,t兲 ones are similar to the wave} function of a particle encountering the potential V_n

1,n2

共2兲 _u2_共z兲 which is either a potential barrier or a potential well accord-ing to the sign of⌬ 共positive or negative, respectively兲 共see Fig. 2兲. This leads to the conclusion that, in the large-detuning limit, the three-level atom mazer does not mimic perfectly a two-level atom mazer for which the atom-cavity interaction reduces to the scattering of the atom upon a po-tential barrier and a popo-tential well 关2兴. In this respect, the three-level atom mazer introduces new interesting options for engineering the interaction between the atoms and the cavity. It is also interesting to note that the scattering poten-tial amplitude can be very easily tuned through the value of the detuning ⌬. With a two-level atom mazer, the potential

amplitudes are fixed by the atom-cavity coupling constants and the cavity photon numbers.

If we again assume initially a monokinetic atom 共with momentum បk兲 coming upon the cavity from the left-hand side in the state兩b1典 and the cavity field in the number state 兩n1+ 1 , n2典, the atom-field system is described by the two wave function components共which correspond to the eigen-state兩␾k典 of energy Ek=ប2k2/ 2m兲 ␺n共i兲₁,n₂共z,t兲 = e−i共បk 2_/2m兲t ␸n共i兲₁,n₂共z兲 共i = 1,2兲 共34兲 with ␸n共1兲₁,n₂共z兲 = e ikz _共35兲 and ␸n共2兲₁,n₂共z兲 =

再

eikz+␳e−ikz, z⬍ 0 ␶eikz, z⬎ L, 共36兲

where␳ and␶are, respectively, the reflection and transmis-sion coefficients of a particle of momentumបk coming upon the potential V_n

1,n2

共2兲 _u2_{共z兲 关see Eq. 共31兲兴. Except for very} par-ticular cases, these coefficients are obtained numerically by solving the time-independent version of the Schrödinger equation共31兲. For a mesa mode, they are elementarily given by

␳= i⌶ sin共k

⬘

L兲␶, 共37兲

␶=关cos共k

⬘

L兲 − i⌺ sin共k

⬘

L兲兴−1_{, 共38兲} with k

⬘

=

冑

k2₋ ␩ ⌬˜␬ 2_{, 共39兲} ⌶ =1 2

冉

k

⬘

k − k k

⬘

冊

, 共40兲 ⌺ =1 2

冉

k

⬘

k + k k

⬘

冊

, 共41兲 and ␩=共n1+ 1兲 + 共g1/g2兲 −2_{共n2+ 1兲} 1 +共g1/g2兲−2 . 共42兲 The initial wave vector reads

具z兩␺共0兲典 = eikz_兩b1 ,n1+ 1,n2典 = eikz共cb₁ 0_兩␥ n₁,n₂ 共1兲 _{典 + c} b₂ 0_兩␥ n₁,n₂ 共2兲 _典兲 共43兲 and after the atom has left the interaction region, the initial state has evolved into

具z兩␺共t兲典 = e−i共បk2_/2m兲t

兵c0b₁eikz␪共z − L兲兩␥共1兲n₁,n₂典 + cb0₂关␳e−ikz␪共− z兲 +␶eikz␪共z − L兲兴兩␥n共2兲₁,n₂典其, 共44兲 where␪共z兲 is the Heaviside step function indicating that the atom is coming from the left. The wave function can be rewritten in the form

具z兩␺共t兲典 = e−i共បk2_/2m兲t ⫻兵关rn₁+1,n2 b₁ e−ikz_{␪共− z兲} + t_n 1+1,n2 b₁ eikz_{␪共z − L兲兴兩b1,n} 1+ 1,n2典 +关rn₁,n₂+1 b₂ e−ikz␪共− z兲 + tn₁,n₂+1 b₂ eikz␪共z − L兲兴兩b2,n1,n2+ 1典其, 共45兲 with rn₁+1,n₂ b₁ = ⍀n1 2 ⍀n2₁+⍀n2₂ ␳, 共46兲 tn₁+1,n₂ b₁ = ⍀n2 2 ⍀n2₁+⍀n2₂ + ⍀n1 2 ⍀n2₁+⍀n2₂ ␶, 共47兲 and rn₁,n₂+1 b₂ = ⍀n1⍀n2 ⍀n2₁+⍀n2₂ ␳, 共48兲 tn₁,n₂+1 b₂ = ⍀n1⍀n2 ⍀n₁ 2 +⍀n₂ 2 共␶− 1兲. 共49兲

The probabilities Pb₁→b₁ and Pb₁→b₂ are then, respec-tively, given by Pb₁→b₁=兩rn₁+1,n₂ b_{1 兩}2_+兩t n₁+1,n₂ b_{1 兩}2 _共50兲 and Pb₁→b₂=兩rn₁,n₂+1 b_{2 兩}2_+兩t n₁,n₂+1 b_{2 兩}2_{. 共51兲}

III. PROPERTIES OF THE TRANSITION PROBABILITIES

In the following, we distinguish the hot and the cold atom regime. In the hot共respectively, cold兲 atom regime, the

inci-0 L z 0 ¯ h2_k2 2m ¯ hΩ 2 n1+Ω2n2 4∆ E

FIG. 2. Schematic representation of the energyប2_k2_{/ 2m of an} atom moving upon the micromaser cavity of length L. For large detunings, the cavity acts as a repulsive共or an attractive兲 potential 共dashed curve兲 while one atomic component is freely transmitted 共plain curve兲. The picture illustrates the case where the atom-field coupling is constant inside the cavity and the detuning⌬ is positive 共potential barrier兲.

dent kinetic energy of the atom is much larger共respectively, smaller兲 than the internal energies Vn₁,n₂

± _{共for small ⌬兲 or} V_n

1,n2

共2兲 _{共for large ⌬兲.}

A. Hot atom regime: Rabi limit

In the hot atom regime, the atomic kinetic energy is such that the atoms are always transmitted through the cavity and the transition probabilities simplify to their expression ob-tained when the atomic motion is treated classically. We show this here below explicitly for a mesa mode and in the two extreme cases: at resonance 共⌬=0兲 and in the far-detuning regime共兩⌬兩 ⍀n₁,⍀n₂兲.

For a mesa mode, the atomic momenta inside the cavity associated to the corresponding ␸_n

1,n2

␧ _{共z兲 共␧=0, ±兲 wave} functions are given by

បkn₁,n₂ 0 =បk, 共52a兲 បkn±₁,n₂⯝ បk

冉

1 + ␬2 4k2共⌬˜ ⫿

冑

⌬˜ 2_{+ 4␩兲}

冊

_{, 共52b兲} respectively.

At resonance, after the atom initially in the兩b1典 state has interacted with the cavity field, the global state of the system reduces to 兩␺共t兲典 =

冕

dz␺共z,t兲共cb0₁兩z,␥0n₁,n₂典 + cb+₁e−i共␬ 2_L/2k兲冑_␩ 兩z,␥n+₁,n₂典 + cb−₁e i共␬2L/2k兲冑␩_兩z,␥ n₁,n₂ − _{典兲 共53兲} with ␺共z,t兲 = eikz e−i共បk2/2m兲t. 共54兲 The transition probability from state 兩b1典 to state 兩b2典 is thus given by Pb_1→b2=兩具z,b2,n1,n2+ 1兩␺共t兲典兩2 = 4 ⍀n1 2_⍀ n₂ 2 共⍀n2₁+⍀n2₂兲2 sin4

冉

␬ 2_L 4k

冑

␩

冊

共55兲 which can be rewritten

Pb₁→b₂= 4 ⍀n2₁⍀n2₂ 共⍀n₁ 2 +⍀n₂ 2_兲2sin 4

冉

⍀␶ 2

冊

共56兲 with ⍀ =1 2

冑

⍀n1 2 _+⍀ n₂ 2 _{, 共57兲}

where␶=mL_បk is the classical transit time of the thermal atoms through the cavity. Equation共56兲 is exactly the classical ex-pression of the transition probability from state兩b1典 to state 兩b2典 for a three-level atom interacting with a bimodal cavity during a time␶关16兴.

We get a similar result for the transition probability from state兩b1典 to state 兩a典,

˜ ∆ Pb1 → b2 0.1 0.05 0 −0.05 −0.1 0.75 0.5 0.25 0 ˜ ∆ Pb1 → b2 4 2 0 −2 −4 1 0.75 0.5 0.25 0

FIG. 3. 共Color online兲 Transition probability P_b

1→b2 with re-spect to the detuning ⌬˜ for k/␬=0.01, ␬L=10, n1= n2= 0, and g₁/ g₂= 1.共Inset, k/␬=100 and ␬L=105_.兲

˜

∆

P

b1 → a 4 2 0 −2 −4 0.02 0.015 0.01 0.005 0

FIG. 4.共Color online兲 Transition probability P_b

1→awith respect to the detuning⌬˜ for k/␬=0.01, ␬L=10, n1= n2= 0, and g1/ g2= 1.

˜ ∆ = 0 κL Pb1 → b2 15 10 5 0 0.75 0.5 0.25 0 ˜ ∆ =−0.1 ˜ ∆ = +0.1 κL Pb1 → b2 20 15 10 5 0 1 0.75 0.5 0.25 0

FIG. 5. 共Color online兲 Transition probability P_b

1→b2 with re-spect to the interaction length␬L for k/␬=0.01, ⌬˜= ±0.1, n1= n2 = 0, and g₁/ g₂= 1.共Inset, ⌬˜=0.兲

Pb_1→a=兩具z,a,n1,n2兩␺共t兲典兩2= ⍀n₁ 2 ⍀n₁ 2 _+⍀ n₂ 2 sin 2

冉

␬ 2_L 2k

冑

␩

冊

, 共58兲 which again reduces to the classical expression of this prob-ability关16兴, Pb₁→a= ⍀n2₁ ⍀n₁ 2 +⍀n₂ 2 sin2共⍀␶兲. 共59兲 For large detunings共兩⌬兩 ⍀n₁,⍀n₂兲, the global state of the system can be written after the atom-field interaction

兩␺共t兲典 ⯝

冕

dz␺共z,t兲共cb0₁兩z,␥n0₁,n₂典 + cb+₁e−i共␬ 2 L/2k兲␩/⌬˜_兩z,␥ n₁,n₂ + _典兲 共60兲 since c_b 1

− _{⯝0 关see Eqs. 共21兲 and 共22兲兴. The transition} prob-ability from state兩b1典 to state 兩b2典 is thus given by

Pb_1→b2=兩具z,b2,n1,n2+ 1兩␺共t兲典兩2 = 4 ⍀n1 2_⍀ n₂ 2 共⍀n2₁+⍀n2₂兲2 sin2

冉

␬ 2_L 4k ␩/⌬˜

冊

. 共61兲 It is then enough to notice that Eq. 共61兲 can be cast in the form

Pb_1→b2= ⍀eff

2 ⍀eff2 _+⌬eff2 sin

2

冉

冑

_⍀ eff 2 +⌬_eff2 ␶ 2

冊

共62兲 with ⌬eff=⍀n2 2 −⍀n₁ 2 4⌬ , ⍀eff= ⍀n₁⍀n₂ 2⌬ . 共63兲

Equation 共62兲 is exactly the transition probability of a two-level system 兩b1典 and 兩b2典 interacting during a time ␶ with a monomodal cavity and with a detuning⌬effand a Rabi frequency ⍀_eff. It is well known that this is precisely the

behavior of a three-level system interacting during a time␶ with a bimodal cavity in the far-detuning regime and in the two-photon resonance condition关17,19,20兴.

B. Cold atom regime

In the cold atom regime, the properties of the atomic tran-sition probabilities differ significantly from the hot atom re-gime where the Rabi oscillatory behavior is observed with respect to the cavity length L关see Eqs. 共55兲 and 共61兲兴. The specific properties of a bimodal cavity pumped by cold three-level atoms were already pointed out at resonance by Arun et al. 关12兴. Out of resonance, other specific properties are ex-hibited by the system.

First, and in contrast to the hot atom regime, the transition probabilities are sensitive to the sign of the detuning. This is illustrated in Figs.3and4in the case of a mesa mode func-tion for the probabilitiesPb₁→b₂andPb₁→a, respectively. For negative detunings, as discussed in Sec. II, the transition probabilityPb₁→avanishes very quickly共see Fig.4兲.

Second, the transition probability Pb₁→b₂ exhibits in the cold atom regime very fine resonances with respect to the interaction length ␬L that are amplified by use of a small negative detuning共see Fig.5兲. Rather, for small positive de-tunings, these resonances are significantly decreased. About the transition probabilityPb₁→a, the fine resonances that are observed in absence of detuning decrease significantly by use of a small positive detuning, whereas they are annihi-lated for small negative detunings共see Fig.6兲. In this case, the three-level atom mazer behaves like a two-level atom one even if the system is not running in the far-detuning regime. We then get in the strong coupling regime a virtual two-level atom mazer without spontaneous emission between the two levels兩b1典 and 兩b2典. We eliminate this way a major issue of the conventional two-level atom mazer and parameters like those of Fig. 5 could be advantageously used to test the specific properties of the micromaser in the cold atom re-gime. With these parameters, the resonances are well peaked and reach the value 1 while those of the usual two-level atom mazer are limited to 0.5关1兴.

˜ ∆ = 0 κL Pb1 → a 15 10 5 0 0.75 0.5 0.25 0 ˜ ∆ =−0.1 ˜ ∆ = +0.1 κL Pb1 → a 20 15 10 5 0 1 0.75 0.5 0.25 0

FIG. 6.共Color online兲 Transition probability P_b

1→awith respect to the interaction length␬L for k/␬=0.01, n1= n2= 0, g1/ g2= 1, and ⌬˜= ±0.1. 共Inset, ⌬˜=0.兲 ˜ ∆ =−50 ˜ ∆ = +50

κL

P

b1 → b2 200 150 100 50 0 1 0.75 0.5 0.25 0

FIG. 7. 共Color online兲 Transition probability P_b

1→b2 with re-spect to the interaction length ␬L for k/␬=0.01, ⌬˜= ±50, n1= n2 = 0, and g₁/ g₂= 1.

In the large detuning regime, the resonances are much less peaked, especially as the cavity mode gets smoother 共see Figs. 7 and 8 for a mesa and a sinusoidal mode兲. When positive detunings are used, the transition probabilityPb₁→b₂ very rapidly reaches a plateau at a valueP given by

P = 2 ⍀n1 2_⍀ n₂ 2 共⍀n2₁+⍀n2₂兲2 . 共64兲

This is true for all modes. Indeed, for cold atoms and posi-tive detunings,兩␳兩 ⯝1 and␶⯝0. Inserting those values into Eqs.共48兲, 共49兲, and 共51兲 yields Eq. 共64兲.

IV. EXPERIMENTAL CONSIDERATIONS

We now discuss how our proposal could be experimen-tally implemented with present technology. The apparatus of Hood et al.关21,22兴 yield a microlaser cavity with a Gaussian mode waist of 15␮m and a free spectral range共FSR兲 of the order of 7 THz. Sending cold metastable xenon atoms with no nuclear spin关27兴 through a similar cavity would define a ⌳-configuration system with a coupling strength g/2␲of the order of 75 MHz via the two transitions 6s关3/2兴₂o-6p关5/2兴2 and 6s 关3/2兴₁o-6p 关5/2兴2 at 904.54 and 992.32 nm, respec-tively 关23兴. These two transitions are separated by a fre-quency difference of 29.2 THz that can match 4 FSR by tuning properly the cavity length. The far-detuning regime would be easily obtained while remaining far below the free spectral range 共we have ⌬/g=100 for ⌬/2␲⯝7.5 GHz FSR兲. For a detuning ⌬/g of 1 共respectively, 10 and 100兲,

the cold atom regime is reached at a translational motion temperature of 3 mK 共respectively, 0.5 mK and 51␮K兲. With atoms launched horizontally like in our scheme of Fig. 1 共using, e.g., the conveyor belt technique 关24兴兲, such tem-peratures would produce a vertical fall during the atomic transit time through the cavity mode waist of the order of 2.8 nm共respectively, 17 nm and 0.17␮m兲. These falls are so small in comparison with the size of the mode waist that the gravity action during the atom-cavity interaction could be neglected. All these parameters show that an experiment ex-hibiting the effects described in this paper is accessible with present technology in the optical domain. Also, this domain has the advantage of being thermal photon free. We would not recommend working in the microwave domain since the coupling strengths 共and thus the maximal allowed transla-tional temperatures defining the cold atom regime兲 are usu-ally much smaller there. Also the cavity mode sizes are much larger and the gravity action could not be considered any-more as negligible. A model including gravity and thermal photons as shown in Ref.关25兴 for two-level atoms would be necessary in this case.

V. SUMMARY

In this paper, we have analyzed the quantum theory of the two-mode micromaser pumped by cold three-level atoms with a⌳ configuration under the two-photon resonance con-dition. We have focused more specifically on the large-detuning limit and have shown that this regime can mimic a virtual two-level atom micromaser without spontaneous emission where the cavity acts for the atoms as a potential barrier or a potential well 共depending on the sign of the detuning兲 and a zero potential but not as a potential barrier and a potential well as it is the case for the usual cold two-level atom mazer. This introduces interesting options for en-gineering the interaction between the cold atoms and the cavity. The elimination of the spontaneous emission solves a major issue of the conventional two-level atom mazer as the transit times of cold atoms inside and in the vicinity of the cavity is usually much larger than the atomic level lifetimes. We have also shown that the cold atom regime is very sensitive to the sign of the detuning共in contrast to the hot atom regime兲 and that, according to this sign, the cavity may speed up or slow down the incident three-level atoms after their interaction with the field.

ACKNOWLEDGMENT

One of the authors 共J.M.兲 thanks the Belgian FRIA for financial support.

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P

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FIG. 8. 共Color online兲 Transition probability P_b

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