Photodetection using Bose Einstein condensed atoms in a microtrap

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(1)PHYSICAL REVIEW A 78, 043822 共2008兲. Photodetection using Bose-Einstein-condensed atoms in a microtrap S. Wallentowitz1 and A. B. Klimov2. 1. Facultad de Física, Pontificia Universidad Católica de Chile, Casilla 306, Santiago 22, Chile Departamento de Física, Universidad de Guadalajara, Revolución 1500, 44410 Guadalajara, Jal., Mexico 共Received 24 May 2008; published 24 October 2008兲. 2. A model of photodetection using a Bose-Einstein condensate in an atom-chip-based microtrap is analyzed. Atoms absorb photons from the incident light field, receive part of the photon momentum, and leave the trap potential. Upon counting escaped atoms within predetermined time intervals, the photon statistics of the incident light is mapped onto the atom-count statistics. Whereas traditional photodetection theory treats the emission centers of photoelectrons as distinguishable, here the centers of escaping atoms are condensed and thus indistinguishable atoms. From this, an enhancement of the photon-number resolution as compared to the commonly known counting formula is derived for the case of low saturation. DOI: 10.1103/PhysRevA.78.043822. PACS number共s兲: 42.50.Ar, 37.10.Gh. I. INTRODUCTION. The quantum theory of photodetection based on the absorption of photons and emission of photoelectrons represents one of the cornerstones of quantum optics. It serves to obtain the statistics of emitted photoelectrons given the quantum statistics of the incident optical field. Various approximations, as described by this theory, lead eventually to the famous photocounting formula of Mandel 关1,2兴, a quantum version of a previously known semiclassical Poissonian formula 关3,4兴, Pn共t,t0兲 =. 冓. :. 冔. 关␩DÎ共t,t0兲兴n −␩ Iˆ共t,t 兲 e D 0: . n!. 共1兲. Here ⬋ denotes normal operator ordering, ␩D is the quantum efficiency of the detector, and Î共t , t0兲 is the time-integrated light intensity incident on the detector’s entrance plane. This formula has the well-known limit of a purely Poissonian photoelectron statistics, if the integration time ␶ = t − t0 is larger than the coherence time of the incident optical field. This integration time represents the response time of the detector system including the connected electronics to amplify the generated photocurrents. Thus to observe the statistics of the optical field, one must ideally have a fast detector and a large coherence time of the optical field under study. As already mentioned, to derive the Mandel formula, some approximations have to be made. These approximations are perfectly justifiable for a solid-state detector device that operates at not too low temperatures. One crucial assumption is the distinguishability of the atoms emitting the observable photoelectrons. Another approximation is found to consist in the perturbative calculus used to obtain joint probabilities of photoelectron emissions. Together they lead to the Poissonian operator form, rather independent of the underlying absorption dynamics. Consider now a device that operates in a rather different regime, that is, it may be cooled down to ultracold temperatures in order to behave more quantum than a typical solidstate photodetector. For example, let us consider a cloud of magnetically trapped Bose-Einstein-condensed alkali-metal 1050-2947/2008/78共4兲/043822共11兲. atoms 关5兴 floating on the surface of a so-called atom chip 关6,7兴. Atoms can now absorb incident photons to receive part of the photon momentum, giving them sufficient kinetic energy to escape from the trap, to subsequently be detected, for instance by ionization. As such a system is highly degenerate, the emission centers of escaping atoms, i.e., the condensed atoms themselves, are not distinguishable. Furthermore, a perturbative approach to calculate the emission probabilities is hardly appropriate, as we may deal with Rabi cycles, where atoms absorb and stimulatedly emit photons, thereby returning to the condensate. Thus, the crucial approximations that led to the Mandel formula cannot be applied and thus one may expect a rather different counting formula. Such a counting formula connects the statistics of escaping atoms to the statistics of the incident optical field. For the purpose of unveiling the different counting formula, we study in the following a model detector system using a Bose-Einstein-condensed gas. In particular, we focus here on the case of low saturation, when the atom-light coupling is rather weak. Although it serves here merely to demonstrate the differences in the resulting counting formula, we suppose that this system also possibly may be realizable in current experiments. The paper is organized as follows: In Sec. II, the model of the photodetector is introduced. The atom-counting statistics is then derived in Sec. III and evaluated for the case of low saturation in Sec. IV. It is followed by a discussion of the features of the atom count statistics in Sec. V. Finally, a summary and conclusions are given in Sec. VI.. II. PHOTODETECTOR MODEL A. Mechanism of photodetection. Let us assume that a cloud of bosonic atoms is magnetically trapped in a microtrap implemented on an atom chip and being cooled well below the condensation temperature, T Ⰶ Tc. We suppose that the trapping potential is highly elongated into one direction and therefore approximate the system as being effectively one dimensional. Furthermore, the atoms shall interact nearly resonant with a collimated light field with incidence parallel to the elongated trap axis; see. 043822-1. ©2008 The American Physical Society.

(2) PHYSICAL REVIEW A 78, 043822 共2008兲. S. WALLENTOWITZ AND A. B. KLIMOV Incident Light. Atomic Cloud. Atom Detection. tector integration time and the atom-counting statistics Pa will then be related, in a yet unknown way, to the statistics of the incident optical field.. Atom Chip. B. Interaction with the incident optical field. FIG. 1. 共Color online兲 Outline of a photodetector using trapped Bose-Einstein-condensed atoms on an atom chip. Atoms in the condensate absorb photons from the incident light and escape from the trap, being then counted in the atom detector.. Fig. 1. The overlap of the transverse mode structure of the light with the transverse mode structure of the atomic cloud shall be considered as a constant mode-matching parameter, which will determine the coupling strength and thereby the efficiency of the detector. The resonant electronic transition of the atoms shall be formed by two levels, the lower level 共ground state兲 being subject to the magnetic trapping and the upper level 共excited state兲 being unaffected by the trap. Thus, if all atoms start from the ground state, those being excited by absorption of an incident photon may leave the trap potential to be detected, e.g., by subsequent ionization. Some of the excited atoms, however, will be deexcited by stimulated emission and therefore will be subject to the trapping potential again. The electronic level scheme, including the sidebands generated by the trap potential and the loss of atoms from the trap, is depicted in Fig. 2. Given the atoms being Bose-Einstein-condensed at the initial time t0, one may ask for the probability Pa共t , t0兲 to observe a atoms escaping from the trap in the time interval 关t0 , t兴, during which light is incident on the atomic probe. After this interval the detector is reset, i.e., all atoms are cooled again into the condensate mode 共i.e., the lowest trap level兲, to start the counting again for an identical time interval ␶ = t − t0. Thus ␶ will be the analog to the usual photode-. The Hamilton operator of the complete system including the detector and the incident optical field can be decomposed into free and interaction parts as. where the free evolution is governed by. with Ĥem being the Hamiltonian of the electromagnetic field and Ĥat being the Hamiltonian of the atoms. The electric field of the incident optical beam can be written as a decomposition of monochromatic modes of wave vector k and frequency ck 共H.a. denoted Hermitean adjoint兲, Ê共x兲 =. 冕. 共4兲. dkEkb̂keikx + H.a.,. where Ek denote the rms vacuum fluctuations of the electricfield modes, and b̂k and b̂†k are the bosonic photon annihilation and creation operators, respectively. As the field polarization is selected by the resonant atomic transition, only one type of polarization is considered here. Using this expansion, the free Hamiltonian of the electromagnetic field thus becomes Ĥem =. 冕. 共5兲. dkបckb̂†k b̂k .. The atomic system is described by the bosonic atom-field operators ⌽̂i共x兲, where i = g , e denotes the electronic state, and which satisfy the commutation relations 关⌽̂i共x兲,⌽̂†j 共x⬘兲兴 = ␦ij␦共x − x⬘兲.. γ. 0. The atomic Hamilton operator reads Ĥat =. frequency. 共3兲. Ĥ0 = Ĥem + Ĥat ,. |e. 兺. i=g,e. 冕. 冋. dx⌽̂†i 共x兲 −. 共6兲. 册. ប2⳵2x + ␦i,gU共x兲 ⌽̂i共x兲, 2m. 共7兲. where the trap potential only acts in the electronic ground state and reads. ω. U共x兲 = ν −ω0. 共2兲. Ĥ = Ĥ0 + V̂,. |g. FIG. 2. Energy-level scheme of the atoms constituting the photodetector. The zero-energy level is taken as the threshold, where the continuum of unbound excited states starts. The ground-state level at frequency −␻0 is superimposed by the vibrational sidebands of frequency ␯.. m ␯ 2x 2 − ប␻eg , 2. 共8兲. with ␯ and ␻eg being the trap and electronic-transition frequency, respectively, and m is the atomic mass. For the purpose of diagonalizing the free atomic Hamiltonian, we define the Schrödinger eigen-modes,. 冋. −. 册. ប2⳵2x + ␦i,gU共x兲 ␾i,n共x兲 = ប␻i,n␾i,n共x兲. 2m. 共9兲. Thus ␾g,n共x兲 with discrete n = 0 , 1 , . . . and eigenfrequencies. 043822-2.

(3) PHYSICAL REVIEW A 78, 043822 共2008兲. PHOTODETECTION USING BOSE-EINSTEIN-CONDENSED…. 冉. ␻g,n = n␯ − ␻0. ␻0 = ␻eg −. ␯ 2. 冊. are the harmonic-oscillator eigenmodes corresponding to a trapped atom in its electronic ground state. The modes ␾e,k共x兲 ⬀ exp共ikx兲 with continuous k 苸关−⬁ , ⬁兴 and eigenfrequencies. ␻e,k =. បk2 2m. 共11兲. are plane waves corresponding to a free atom in its electronic excited state. These modes form two independent orthonormal sets and obey the standard completeness relations * 共x兲␾ 共x⬘兲 = ␦共x − x⬘兲, 兺n ␾g,n g,n. 1 ␾គ g,n共k兲 = 冑2␲. 共10兲. 共12兲. 冕. dxe−ikx␾g,n共x兲.. 共20兲. Thus the absorption of a photon of wave vector k transforms a ground-state atom in trap level n into an excitedstate atom with a superposition of wave vectors k⬘ given by ␾គ g,n共k⬘ − k兲. We may therefore define the annihilation operator of an excited wave packet, created from trap level n by absorption of a photon of wave vector k, êk,n =. 冕. * dk⬘êk⬘␾ គ g,n共k⬘ − k兲.. 共21兲. This relation can be inverted to obtain all operators êk from the set of operators êk0,n 共n = 0 , 1 , 2 , . . . 兲 for a specific wave vector k0, ⬁. 冕. * 共x兲␾ 共x⬘兲 = ␦共x − x⬘兲. dk␾e,k e,k. Each electronic component of the quantized atomic field can now be expanded as ⌽̂g共x兲 = 兺 ĝn␾g,n共x兲,. ⌽̂e共x兲 =. n. 冕. 关êk,êk⬘兴 = ␦共k − k⬘兲. †. 共22兲. n=0. The excited-wave-packet operators satisfy the following commutation relations: †. dkêk␾e,k共x兲,. 关êk,n,êk⬘,n⬘兴 =. 共14兲. where the operators ĝn and êk each satisfy again the bosonic commutation relations, † 关ĝn,ĝm 兴 = ␦nm,. êk = 兺 ␾ គ g,n共k − k0兲êk0,n .. 共13兲. 冕. * 共x兲ei共k⬘−k兲x␾ 共x兲, dx␾g,n g,n⬘. 共23兲. † and in particular 关êk,n , êk,n 兴 = 1. Using these wave-packet operators, the interaction Hamiltonian can be simplified to. V̂ = − 兺. 共15兲. n. 冕. † dkប⍀kêk,n ĝnb̂k + H.a.. 共24兲. Using the expansion 共14兲, the free atomic Hamiltonian 共7兲 reduces to Ĥat = 兺. ប␻g,nĝ†nĝn. +. n. 冕. C. Single-mode approximation. dkប␻e,kê†k êk .. 共16兲. The interaction between the atoms and the optical field reads in dipole approximation V̂ = − d␬⬜. 冕. dx 兺 ⌽̂†i 共x兲Ê共x兲⌽̂ j共x兲,. 共17兲. i,j. where d is the transition dipole moment of the atoms and ␬⬜ is the matching between the transverse modes of electromagnetic and atomic field. Using the expansions of electric and atomic fields, cf. Eqs. 共4兲 and 共14兲, respectively, this interaction can be rewritten in optical rotating-wave approximation as V̂ = − 兺 n. 冕 ⬘冕 dk. †. dkប⍀kêk⬘ĝnb̂k␾ គ g,n共k⬘ − k兲 + H.a., 共18兲. where the 共vacuum兲 Rabi frequency has been defined as ⍀k =. dEk␬⬜ , ប. If we start from a Bose-Einstein-condensed gas with all atoms being in the lowest trap level n = 0, a cycled electronic transition will preferably lead again to the lowest trap level by bosonic enhancement. We may thus approximate the ground-state levels by a single mode, corresponding to the lowest trap level, and we may simplify the interaction Hamiltonian 共24兲 to V̂ = −. † dkប⍀kêk,0 ĝ0b̂k + H.a.. 共25兲. Furthermore, we suppose that the incident optical field is quasimonochromatic with wave vector k0, so that only the photon operator b̂k0 has to be kept. Using thus the definitions b̂k0 → b̂, ĝ0 → ĝ, êk0,0 → ê, and ⍀k0 → ⍀, the interaction further simplifies to V̂ = − ប⍀ê†ĝb̂ + H.a.. 共26兲. The free atomic Hamiltonian, on the other hand, can be written in this single-mode approximation as ¯ eê†ê − ប␻0ĝ†ĝ Ĥat = ប␻. 共19兲. and the Fourier transforms of the trap modes are defined as. 冕. 共27兲. with the average frequency of the excited wave packet being determined by the wave vector of the absorbed photon and. 043822-3.

(4) PHYSICAL REVIEW A 78, 043822 共2008兲. S. WALLENTOWITZ AND A. B. KLIMOV. 冉. the momentum spread of the ground-state trap level, ¯e = ␻. 冕. ␯ បk20 2 dk␻e,k兩␾ 共k − k 兲兩 = + . គ g,0 0 4 2m. 共28兲. We note that the performed single-mode approximation neglects the dispersion of the excited wave packet, being now considered as a propagating plane wave. The effective transition frequency between the relevant two atomic levels becomes now ¯ e + ␻0 = ␻eg + ⬘ =␻ ␻eg. បk20 ␯ − . 2m 4. ⳵tˆ =. + ĝ†êb̂†e−i␸兲 −. ␲បk20 បk20 ␲ 关1 + 共2␩兲−2兴 ⬇ . m ␩ m␩. E. Atomic pseudospin. Ŝ+ = ê†ĝei␸,. 共32兲. 共33兲. 1 关Ĥ,ˆ 兴 + 兩⳵tˆ 兩esc , iប. 共34兲. where the atom loss is modeled by the Lindblad-form part. Ŝ− = 共Ŝ+兲† ,. 1 Ŝz = 共ê†ê − ĝ†ĝ兲, 2. 共31兲. The escape of atoms and their subsequent detection may be modeled as an incoherent loss of atoms, described by the master equation. ⳵tˆ =. 共37兲. For the atomic quantum fields, we may define the pseudospin operator Ŝ with components 共Ŝ⫾ = Ŝx ⫾ iŜy兲,. where ␩ = k0␦x0 is the Lamb-Dicke parameter with ␦x0 = 冑ប / 共2m␯兲 being the rms position spread of the trap ground level. For a typical magnetic trap potential, the weak binding regime applies, where the Lamb-Dicke parameter is ␩ Ⰷ 1. Thus the conservation of momentum is approximately granted and the excited atom compensates for the momentum of the absorbed photon. If the excited wave packet has moved over a distance ⬃␦x0, it may no longer be recycled into the electronic ground state by stimulated emission of a photon, as the corresponding spatial overlap will be close to zero. It thus has escaped from the trap. The time of flight for this to happen is given by ␶ = ␦x0 / vg, and the corresponding rate for this to happen, ␥ = 2␲␶−1, is therefore obtained as. ␥=. iប␥ † ê ê, 2. with ⍀ = 兩⍀ 兩 ei␸.. 共30兲. It corresponds to a motion into the direction of the wave vector of the previously absorbed photon with the group ve¯ e / k0 given by locity vg = ␻ បk0 关1 + 共2␩兲−2兴, 2m. 共36兲. ¯ eê†ê − ប␻0ĝ†ĝ − ប兩⍀兩共ê†ĝb̂ei␸ Ĥeff = បck0b̂†b̂ + ប␻. The spatio-temporal mode of the excited wave packet in the single-mode approximation is obtained as. vg =. 1 † 兲 + ␥êˆ ê† , 共Ĥeffˆ − ˆ Ĥeff iប. where according to Eqs. 共5兲, 共26兲, and 共27兲, the nonHermitean effective Hamilton operator reads. D. Loss mechanism. * 共x兲. ␾e共x,t兲 = ei共k0x−␻¯ et兲␾g,0. 共35兲. The master equation 共34兲 can be written in the form. 共29兲. In resonance, this transition frequency is compensated for by the frequency ck0 of the optical field, from which we obtain the resonance condition. ␯ ck0 ⬇ ␻eg − . 4. 冊. 1 兩⳵tˆ 兩esc = ␥ êˆ ê† − 兵ê†ê,ˆ 其 . 2. and Ŝ2 =. 冉冊. Â Â +1 , 2 2. 共38兲共39兲. 共40兲. where the atom-number operator is defined as Â = ê†ê + ĝ†ĝ.. 共41兲. The operators Ŝ⫾,z satisfy the standard su共2兲 commutation relations, 关Ŝ+ , Ŝ−兴 = 2Ŝz and 关Ŝz , Ŝ⫾兴 = ⫾ Ŝ⫾. The total number of excitations—excited atoms plus photons—is given by the operator N̂ = ê†ê + b̂†b̂ = Ŝz +. Â + b̂†b̂. 2. 共42兲. The atomic system can now be described in the basis of Dicke states 关8兴, 兩A,Ae典at = 兩A − Ae典g 丢兩Ae典e ,. 共43兲. where A is the total number of atoms, i.e., Â兩A , Ae典at = A兩A , Ae典at, and Ae is the number of excited atoms. The corresponding basis states for the total system can then be written as 兩A,N,n典 = 兩A,N − n典at 丢兩n典em ,. 共44兲. where N is the total number of excitations, i.e., N̂兩A , N , n典 = N兩A , N , n典, and 兩n典em is a photon-number state. Using the definitions of the spin operators 共38兲–共42兲, the effective Hamiltonian 共37兲 can be rewritten as. 043822-4.

(5) PHYSICAL REVIEW A 78, 043822 共2008兲. PHOTODETECTION USING BOSE-EINSTEIN-CONDENSED…. Ĥeff = Ĥ0⬘ − ប⌬⬘Ŝz −. iប␥ Â − ប兩⍀兩共Ŝ+b̂ + Ŝ−b̂†兲, 4. 共45兲. where for notational simplicity we defined ⌬⬘ = ⌬ + i␥ / 2, with ⌬ = ck0 − ␻eg ⬘ being the detuning from resonance, and the free Hamiltonian is identified as Ĥ0⬘ = បck0N̂ +. ¯ e − ␻ 0兲 ប共␻ Â. 2. 共46兲. As this free part commutes with the remainder of the effective Hamiltonian, we may transform into the interaction picture with respect to Ĥ0⬘, to obtain the master equation in the interaction picture, 1 † ⳵tˆ = 共Ĥeffˆ − ˆ Ĥeff 兲 + ␥êˆ ê† , iប iប␥ Â − ប兩⍀兩共Ŝ+b̂ + Ŝ−b̂†兲. 4. 兺冑N − n兩A − 1,N − 1,n典具A,N,n兩.. Ĥeff = − ប⌬⬘. 共49兲. 共50兲. L̂−. 共51兲. L̂+兩A,N,n典 = 冑共N − n兲共n + 1兲兩A,N,n + 1典,. 共52兲. L̂−兩A,N,n典 = 冑共N − n + 1兲n兩A,N,n − 1典,. 共53兲. 冉冊. Â N̂ − L̂z + ប⌬ 2 2 M̂ + L̂z −. 册. 1 + H.a. , 2. 共57兲. 共54兲. and obeying the usual angular-momentum commutation relations, Eqs. 共50兲 and 共51兲 can be rewritten as. N̂ 1 + 2 2. 冑. M̂ + L̂z −. 冢冑冢. 冑. 1 = L̂− M̂ 1 + 2. = L̂− M̂ 1 +. 共58兲. L̂z −. 1 2. M̂ L̂z − 2M̂. 1 2. 冣. 1/2. 冣. + ¯ ,. 共59兲. where the expansion parameter 共L̂z − 21 兲 / M̂ is chosen to cancel the first-order contribution in Eq. 共57兲, obtaining the zero-order Hamiltonian as Ĥeff共t兲 ⬇ − ប⌬⬘. Defining a new spin operator L̂ with its components being defined by the actions. N 兩A,N,n典, L̂z兩A,N,n典 = n − 2. 共56兲. has been introduced. For a proper functioning of the detector, we assume that the number of atoms in the gas is much larger than the maximum number of photons of the incident optical field. Thus the occupied eigenvalues of the operator M̂ are very large, and consequently we may perform an expansion over a small parameter being proportional to the inverse atom number 关9,10兴. Thus the interaction part of the effective Hamiltonian is expanded as. Ŝ+b̂兩A,N,n典 = 冑共N − n + 1兲共A − N + n兲n兩A,N,n − 1典,. 冊. 冉冊冋冑 M̂ = Â −. Let us now consider the action of the operators appearing in the effective Hamiltonian 共48兲 on the basis states 共44兲. First, the actions of the operators Ŝ+b̂ and Ŝz on a state 兩A , N , n典 are. A 兩A,N,n典. 2. 共55兲. N̂ − Â − L̂z 兩A,N,n典. 2. − ប兩⍀0兩 L̂−. F. Limit of large number of atoms. 冉. 兩A,N,n典,. Thus accordingly we may replace the operators in the effective Hamiltonian 共48兲 to obtain. 共48兲. A,N,n. Ŝz兩A,N,n典 = N − n −. 冊冊. 1/2. where the operator. For notational convenience, we omitted here any indication of being in the interaction picture. In the master equation 共47兲, the last term describes the escape of an excited-wave-packet atom from the trap. The responsible operator ê, which annihilates one such excited atom from the system, can be written in the basis of the states 共44兲 as ê =. Ŝz兩A,N,n典 =. 共47兲. where the transformed effective Hamiltonian becomes Ĥeff = − ប⌬⬘Ŝz −. 冉冉. Ŝ+b̂兩A,N,n典 = L̂−. N̂ Â − + L̂z 2. 冉冊. 冑. Â N̂ − L̂z + ប⌬ − 2ប兩⍀兩 M̂L̂x . 共60兲 2 2. III. ATOM-COUNTING STATISTICS A. Counting statistics. From the solution of the master equation, we need to extract the probability for a atoms having escaped in the time interval 关t0 , t兴, starting at the initial time t0 with a perfect Bose-Einstein-condensed gas with A atoms in the trap ground state. Thus the initial state at time t0 can be written in the form. 043822-5.

(6) PHYSICAL REVIEW A 78, 043822 共2008兲. S. WALLENTOWITZ AND A. B. KLIMOV. ˆ 共t0兲 = 兩A,0典at具A,0兩丢 ␳ˆ em共t0兲,. 共61兲. where ␳ˆ em共t0兲 is the density operator of the incident optical field at the initial time t0. The latter may be expanded in photon-number states as. ␳ˆ em共t0兲 =. 兺 ␳n,n⬘共t0兲兩n典em具n⬘兩,. the trap. The superoperator of the nonunitary evolution is defined as † N̂共t兲ˆ = Ûeff共t兲ˆ Ûeff 共t兲,. the effective nonunitary evolution operator being. 冉. 共62兲. ˆ 共t0兲 =. 兺 ␳n,n⬘共t0兲兩A,n,n典具A,n⬘,n⬘兩.. Given the initial density operator 共63兲, the formal solution of the density operator at time t 艌 t0 can be obtained from the master equation in the form. Ĵˆ = ␥êˆ ê† .. 共72兲. Thus the joint probability density 共67兲 can be written using Eqs. 共69兲–共72兲 and the initial state 共63兲 as pa共t,t0 ;ta, . . . ,t1兲 =. ⬁. ˆ 共t,t0兲 = 兺 ˆ a共t,t0兲,. 共71兲. and the escape of an atom is described by the superoperator. 共63兲. n,n⬘. 冊. it Ûeff共t兲 = exp − Ĥeff , ប. n,n⬘. so that the complete initial density operator can be written in the basis states 共44兲 as. 共70兲. 兺 ␳n,n⬘共t0兲具⌽共t,t0 ;ta, . . . ,t1兩n⬘兲兩⌽共t,t0 ;ta, . . . ,t1兩n兲典,. n,n⬘. 共64兲. 共73兲. a=0. where ˆ a共t , t0兲 is the 共unnormalized兲 conditional density operator corresponding to the history of the detector system where in total a atoms have escaped in the time interval 关t0 , t兴. The norm of this conditional density operator is the probability for this history to occur, which is the desired probability to count a atoms escaping from the trap, Pa共t,t0兲 = Tr关ˆ a共t,t0兲兴.. 共65兲. where the 共unnormalized兲 quantum-trajectory states starting with the initial state 兩A , n , n典 are 兩⌽共t,t0 ;ta, . . . ,t1兩n兲典 = ␥a/2Ûeff共t − ta兲ê ¯ êÛeff共t1 − t0兲 ⫻兩A,n,n典.. Using the representation of the operator ê in the basis states, Eq. 共49兲, this state vector can be rewritten as. The conditional density operator ˆ a共t , t0兲 is itself a sum of all possible histories where a atoms escape in such a way that the jth atom escapes at time t j 苸关t0 , t兴, where j = 1 , . . . , a, ˆ a共t,t0兲 =. 冕. t. t0. dta ¯. 冕. t2. dt1ˆ a共t,t0 ;ta, . . . ,t1兲.. 共67兲. Thus the required counting statistics is obtained as. 冕. t. t0. dta ¯. =. 冕. t2. dt1 pa共t,t0 ;ta, . . . ,t1兲.. 共68兲. 兺. Ûeff共t − ta兲兩A − a,n − a,na典. na,. . .,n1. ⫻⌿nA−a+1,n−a+1 共ta − ta−1兲 ¯ ⌿nA−1,n−1 共t2 − t1兲 ,n ,n a a−1. ⫻⌿nA,n,n共t1 1. t0. pa共t,t0 ;ta, . . . ,t1兲 = Tr关ˆ a共t,t0 ;ta, . . . ,t1兲兴.. Pa共t,t0兲 =. 兩⌽共t,t0 ;ta, . . . ,t1兩n兲典. 共66兲. The norm of the conditional density operator on the righthand side 共rhs兲 is the joint probability density for a atoms to escape from the trap at times t1 , . . . , ta,. 共74兲. 2 1. − t0兲,. 共75兲. where the transition amplitudes are A,N 共t兲 = 冑␥共N − m兲具A,N,m兩Ûeff共t兲兩A,N,n典. ⌿m,n. 共76兲. Here we made use of the fact that the effective Hamiltonian preserves both the atom number and the total number of excitations. As the trajectory 共75兲 has exactly A − a atoms and n − a remaining excitations, in the sum of Eq. 共73兲 only terms with n = n⬘ contribute,. t0. pa共t,t0 ;ta, . . . ,t1兲 = 兺 Pn pa共t,t0 ;ta, . . . ,t1兩n兲,. 共77兲. n. B. Quantum trajectories. Given the initial density operator 共63兲, the conditional density operator ˆ a共t , t0 ; ta , . . . , t1兲 is given by the quantum trajectory 关11–13兴. where Pn = ␳n,n共t0兲 is the initial photon statistics and the probability density conditioned on initially n photons is defined as pa共t,t0 ;ta, . . . ,t1兩n兲 = 储兩⌽共t,t0 ;ta, . . . ,t1兩n兲典储2 . Thus the atom-counting statistics can be written as. ˆ a共t,t0 ;ta, . . . ,t1兲 = N̂共t − ta兲ĴN̂共ta − ta−1兲Ĵ ¯ ĴN̂共t1 − t0兲ˆ 共t0兲.. 共78兲. Pa共t,t0兲 = 兺 Pa共t,t0兩n兲Pn ,. 共69兲. It is a nonunitary evolution N̂ intermittent by so-called jump operators Ĵ that describe the escape of a single atom from. 共79兲. n. where the conditional probability for a atoms to escape in the time interval 关t0 , t兴 given that n photons are present is. 043822-6.

(7) PHYSICAL REVIEW A 78, 043822 共2008兲. PHOTODETECTION USING BOSE-EINSTEIN-CONDENSED…. Pa共t,t0兩n兲 =. 冕. t. t0. dta ¯. 冕. t2. N. t0. IV. OVERDAMPED RESONANT REGIME. The features of the dynamics of the absorption and stimulated emission of photons and the loss of atoms from the trap, given a state with A atoms and N excitations, depend on the saturation parameter 共cf. Appendix A兲 SA,N =. 4兩⍀兩2M A,N , ⌬2 + 共␥/2兲2. 共81兲. where M̂兩A , N , n典 = M A,N兩A , N , n典 with M A,N = A − 共N − 1兲 / 2. Given perfect resonance of the incident monochromatic light field, ⌬ = 0, for SA,N ⬎ 1, Rabi oscillations occur that involve cycles of absorption and 共stimulated兲 reemission of photons until an atom is lost from the trap. In the opposite overdamped case, i.e., for low saturation SA,N ⬍ 1, however, no cycling transition is observed but photons are absorbed and their excitation is removed from the system by an atom leaving the trap. In other words, in this case the recoil frequency ␻rec = បk20 / 2m is larger than the effective coupling strength of the atoms with the photon field,. ␻rec ⬎. 兺 m=0. dt1 pa共t,t0 ;ta, . . . ,t1兩n兲. 共80兲. 2␩ 兩⍀兩冑M A,N . ␲. 共82兲. For an experiment with A ⬃ 105 trapped rubidium atoms and comparably low photon numbers, typical parameters are M A,N ⬃ 105 and ␻rec / 2␲ ⬃ 3 kHz, which corresponds to a recoil temperature of 150 nK. Thus, for obtaining low saturation, the atom-light coupling is limited to ␩兩⍀兩 / 2␲ ⱗ 20 Hz. This can be accomplished either by a weak atom-light coupling 兩⍀兩 or by a large trap frequency, which correspondingly produces a small Lamb-Dicke parameter ␩. It should be noted that for species with lower atomic mass than rubidium, the condition for low saturation 共82兲 is less stringent. In the following, we will focus on the case of low saturation, i.e., the case in which an overdamped behavior is observed, as mentioned above. As then cycling transitions are suppressed, the number of absorbed photons equals approximately the number of lost atoms, which means that only the transition amplitudes A,n 共t兲 = 冑␥共n − m兲具A,n,m兩Ûeff共t兲兩A,n,n典 ⌿m,n. 共83兲. have to be considered, since only they start from initial states with all excitation being photonic. From Eq. 共75兲, it becomes clear then that among these transition amplitudes we may further consider only those where one photon has been absorbed, i.e., A,n 共t兲 = 冑␥具A,n,n − 1兩Ûeff共t兲兩A,n,n典. ⌿n−1,n. ⬁ A,N dt兩⌿m,n 共t兲兩2 = 1,. 共85兲. 0. meaning that starting from a state 兩A , N , n典 the system eventually will end up with certainty in one of the states 兩A , N , m典 with m = 0 , . . . , N. Taking into account now only the transition amplitudes 共84兲, we must use the renormalized transition amplitude A,n A,n A,n ⌿n−1,n 共t兲 → ⌿A,n共t兲 = ⌿n−1,n 共t兲/冑Pn−1,n ,. 共86兲. where the probability for the considered transition is defined as A,n = Pn−1,n. 冕. ⬁. A,n dt兩⌿n−1,n 共t兲兩2 ⯝ 1.. 共87兲. 0. In this way, we obtain an atom waiting-time distribution wA,n共t兲 = 兩⌿A,n共t兲兩2. 共88兲. that is properly normalized,. 冕. ⬁. dtwA,n共t兲 = 1.. 共89兲. 0. The quantum trajectory in the overdamped regime is thus simplified to 兩⌽共t,t0 ;ta, . . . ,t1兩n兲典 = Ûeff共t − ta兲兩A − a,n − a,n − a典 a−1. ⫻ 兿 ⌿A−k,n−k共tk+1 − tk兲,. 共90兲. k=0. so that the joint probability density 共78兲 becomes a−1. pa共t,t0 ;ta, . . . ,t1兩n兲 = WA−a,n−a共t − ta兲兿 wA−k,n−k共tk+1 − tk兲, k=0. 共91兲 where † 共t兲Ûeff共t兲兩A,n,n典 WA,n共t兲 = 具A,n,n兩Ûeff. 共92兲. is the probability that no atom leaves the trap within the time interval t starting from the state 兩A , n , n典. When approximating the transition amplitudes, cf. Eq. 共86兲, to obtain statistically correct quantum trajectories that lead to a normalized atom-count statistics Pa, also the probability 共92兲 must be consistently approximated. This can be done by using the general relation WA,n共t兲 = 1 −. 共84兲. However, as these are approximations, we must renormalize correctly the transition amplitudes to obtain statistically correct quantum trajectories. Originally the normalization read. 冕. 冕. t. dt⬘wA,n共t⬘兲,. 共93兲. 0. employing on the rhs the approximation 共88兲. Using the above results, the conditional probability for a atoms to leave the trap given that n photons are present, Eq. 共80兲, becomes now. 043822-7.

(8) PHYSICAL REVIEW A 78, 043822 共2008兲. S. WALLENTOWITZ AND A. B. KLIMOV. Pa共t,t0兩n兲 =. 冕. 冕. t. t0. dta ¯. t2. W គ A−a,n−a共z兲. dt1WA−a,n−a共t − ta兲. t0. =. a−1. ⫻ 兿 wA−k,n−k共tk+1 − tk兲.. 共94兲. k=0. +. This convolution integral is expressed as the inverse Laplace transform. 冋. 册. a−1. គ A−a,n−a共z兲兿 w គ A−k,n−k共z兲 , Pa共␶兩n兲 = L−1 W k=0. 共95兲. 冊. ␥t 冑冑 1 − SA e−共␥t/2兲关n−共n−1兲 1−SA兴 , wA,n共t兲 ⬀ sinh2 4. Thus the complete conditional probability 共95兲 becomes the sum of the three inverse Laplace transforms. Pa共␶兩n兲 =. 共96兲. 共n − k兲共n − k + q兲共n − k + 2q兲 , 共n − k + ␶0z兲共n − k + ␶0z + q兲共n − k + ␶0z + 2q兲. 共98兲. w គ A−k,n−k共z兲 = 兿 k=0. n+q. n + ␶ 0z. a n + q + ␶ 0z. a. a. 30 ⫻G33. a. W គ A−a,n−a共z兲 =. 1−w គ A−a,n−a共z兲 , z. which can be written as the sum. 共101兲. a 2q + ␶0z a. a 2q + ␶0z a+1. n + 2q a. 1−p. 共1 − p兲n. 冥. 共103兲. .. 1. n a+1. 1−p. q+1. n+q a. 2q + 1. n+q a. 1. q+1. 2q + 1. − a q − a 2q − a + 1. n a+1. 1−p. n a. − a q − a + 1 2q − a + 1. 1. n+q a+1. q + 1 2q + 1. − a q − a 2q − a. p = 1 − e −␶/␶0 .. 共100兲. The Laplace transform of the no-escape probability is, according to Eq. 共93兲,. a. ,. 共104兲. where we have introduced the parameter. where the binomial coefficients are defined by means of the Gamma function, x ⌫共x + 1兲 . = y ⌫共x − y + 1兲⌫共y + 1兲. a. 冉冊冋冉冊冉冊冉冏冏冊冉冊冉冊冉冏冏冊冉冊冉冊冉冏冏冊册. + 共a + 1兲2. a , n + 2q + ␶0z. 冉冊. a 2q + ␶0z. n + 2q. 30 ⫻G33. n + 2q. 共99兲. n + 2q. a q + ␶ 0z. n+q n a+1 a+1 + q + ␶ 0z ␶ 0z a+1 a+1. + 共a + 1兲. 冉冊冉冊冉冊冉冊冉冊冉冊 n a. 冊冉冊冊冉冊冊冊冊冊. n+q. n + 2q. 30 ⫻G33. As we deal with low saturation, SA Ⰶ 1, the introduced parameters behave as q Ⰷ 1 and ␶−1 0 ⬇ ␥SA / 4. Thus the product of transformed waiting-time distributions in Eq. 共95兲 becomes a−1. 冤冉. n a ␶ 0z a+1. n n+q a+1 a + q + ␶ 0z ␶ 0z a+1 a+1. Pa共q,p兩n兲 = 共a!兲3. where we defined. ␥ 冑 ␶−1 0 = 共1 − 1 − SA兲. 2. ␶ 0e L−1 a+1. 冉冊冉冊冉冊冉冊冉冊冉冊冉. Each Laplace transform results as a Meijer G function 关14兴,. 共97兲. 冑1 − SA , q= 1 − 冑1 − SA. −n␶/␶0. 冉冊冉冉冊冉冉冊冉冉冊冉. where the saturation parameter 共A1兲 has been approximated by SA = 4⍀2A / 共␥ / 2兲2. This form shows a behavior quite similar to overdamped Rabi oscillations of a two-level system interacting with a resonant laser field, with the saturation being now dependent on the number of atoms. The individual Laplace transforms become then w គ A−k,n−k共z兲 =. ␶0共n − a兲共n − a + q兲 . 共n − a + ␶0z兲共n − a + ␶0z + q兲共n − a + ␶0z + 2q兲共102兲. where ␶ = t − t0. In the resonant case 共⌬ = 0兲, low saturation 共SA Ⰶ 1兲, and assuming numbers of photons much lower than the atom number, n Ⰶ A, we obtain the atom waiting-time distribution as 关cf. Eq. 共A2兲兴. 冉. ␶0共n − a兲 ␶0 + n − a + ␶0z 共n − a + ␶0z兲共n − a + ␶0z + q兲. 共105兲. V. DISCUSSION. The effective rate at which an atom leaves the trap is 2 ␶−1 0 ⬇ 4兩⍀兩 A / ␥ so that the introduced parameter p, cf. Eq. 共105兲, can be identified as the probability for an atom to escape from the trap by the absorption of a photon. However, there is additionally the large saturation-dependent parameter q ⬇ 2 / SA, cf. Eq. 共98兲, that may change the shape of the conditional probability 共104兲. It is thus not obvious how. 043822-8.

(9) PHYSICAL REVIEW A 78, 043822 共2008兲. 1.00. 0.4. 0.96. 0.3. 0.88 (a). 0.2 0.1 0. Deviation from Binomial. 1.00 0.98. 2. 4 6 8 Atom Count. 10. 0.06 (c). 0.04 0.02 0. −0.02. 0.96. 50 100 150 200 250 300 350 400 450 500 q. FIG. 3. Dependence of the scaled quantum efficiency ␩D / p on the saturation-dependent parameter q for n = 1 共solid兲, n = 10 共dashed兲, and n = 20 共dotted兲 photons. Atom-escape probabilities are p = 0.9 共a兲 and p = 0.6 共b兲.. these two parameters can be merged into a possibly existing single parameter, such as the quantum efficiency. However, such a quantum efficiency may be defined phenomenologically, demanding that the average atom count conditioned on the presence of n photons reads ān = ␩Dn.. 共106兲. Thus, only the fraction ␩D of the n photons leads on average to ān escaping atoms. In general, the above relation is not necessarily linear, as the efficiency may be a function of the photon number, revealing a nonlinear relation between incoming photon and escaping atom numbers. For our specific case, such a phenomenological quantum efficiency depends additionally on the atom-escape probability p and the saturation-dependent coefficient q, thus we obtain. ␩D = ␩D共q,p,n兲.. 共107兲. In Fig. 3共a兲, this dependence is shown for an atom-escape probability p = 0.9 in dependence of the parameter q for n = 1 共solid兲, n = 10 共dashed兲, and n = 20 共dotted兲 photons. It can be observed that for large values of q, all these curves converge to the value of the atom-escape probability, i.e., lim ␩D共q,p,n兲 = p.. q→⬁. 共108兲. The same behavior is also observed for a lower escape probability p = 0.6, see Fig. 3共b兲. Thus for sufficiently low saturation SA, i.e., for sufficiently large q, a linear regime is attained, where the quantum efficiency becomes independent. 2. 4 6 8 Atom Count. 10. 4 6 8 Atom Count. 10. 0.06 (d). 0.04 0.02 0. −0.04. 0. (b). 0. −0.02. −0.04. 0.94. 0.2. 0 0. 0.8. 0.3. 0.1. Deviation from Binomial. 0.84. (b) Probability. 0.92. 50 100 150 200 250 300 350 400 450 500 q Efficiency / Escape Probability. 0.4 (a). Probability. Efficiency / Escape Probability. PHOTODETECTION USING BOSE-EINSTEIN-CONDENSED…. 2. 4 6 8 Atom Count. 10. 0. 2. FIG. 4. Conditional atom-count statistics Pa共q , p 兩 n兲 for n = 10 photons, p = 0.9, q = 100 共a兲 and q = 10 共b兲共filled boxes兲. The lower part shows the deviation from the corresponding binomial statistics with quantum efficiency ␩D = 0.8862 for q = 100 共c兲 and ␩D = 0.7730 for q = 10 共d兲; the binomial statistics are shown in parts 共a兲 and 共b兲 as empty boxes.. of the incident photon number n and coincides with the atom-escape probability p. Does the statistics become then identical to that known from the Mandel counting formula? To answer this question, we proceed as follows: Given that by use of Eq. 共106兲 we may identify a quantum efficiency ␩D for each conditional atom-count statistics Pa共q , p 兩 n兲, we may compare it with the corresponding conditional count statistics that would correspond to the Mandel formula of photodetection 共1兲. The latter is given by the binomial statistics, P共M兲 a 共␩D兩n兲 =. 冉冊. n a ␩ 共1 − ␩D兲n−a . a D. 共109兲. In Fig. 4共a兲, the atom-count statistics is shown for an incident light field with n = 10 photons, with an atom-escape probability p = 0.9 and q = 100. From part 共c兲 it can be seen that the deviation from the binomial statistics of corresponding quantum efficiency ␩D = 0.8862 is not too large. However, this changes if the saturation-dependent coefficient is lowered to q = 10; see the atom-count statistics shown in Fig. 4共b兲. Now the deviation from a binomial statistics with corresponding quantum efficiency ␩D = 0.7730 becomes rather large, cf. Fig. 4共d兲. In fact, the statistics becomes narrower as compared to the binomial form. Whereas these two cases used a rather large atom-escape probability of p = 0.9, for a lower value such as p = 0.6 already for q = 100 a somewhat larger deviation from the binomial statistics can be observed in Fig. 5共c兲. However, the general trend of larger deviation for smaller values of q is confirmed.. 043822-9.

(10) PHYSICAL REVIEW A 78, 043822 共2008兲. S. WALLENTOWITZ AND A. B. KLIMOV. (a). 0.2 0.1 0. 0.2 0.1 0. 2. 4 6 8 Atom Count. 10. (c). 0.08. 0 Deviation from Binomial. 0 Deviation from Binomial. (b). 0.3 Probability. Probability. 0.3. 0.04 0. −0.04. 2. 4 6 8 Atom Count. 10. 4 6 8 Atom Count. 10. (d). 0.08 0.04 0. −0.04. 0. 2. 4 6 8 Atom Count. 10. 0. 2. FIG. 5. Conditional atom-count statistics Pa共q , p 兩 n兲 for n = 10 photons, p = 0.6, q = 100 共a兲 and q = 10 共b兲共filled boxes兲. The lower part shows the deviation from the corresponding binomial statistics with quantum efficiency ␩D = 0.5639 for q = 100 共c兲 and ␩D = 0.3968 for q = 10 共d兲; the binomial statistics are shown in parts 共a兲 and 共b兲 as empty boxes.. We may thus conclude that for a given value of p in the limit q → ⬁, we approach the form of a binomial statistics, agreeing with the Mandel formula. This limit corresponds to SA → 0, i.e., to extremely low saturation. For such low saturation, the atomic gas absorbs photons one by one, making the appearance of bunches of simultaneously escaping atoms highly improbable. As in this limit, at a given instant of time, no more than one escaped atom may be detected, the indistinguishability of the bosonic atoms can be safely disregarded. Thus the Mandel formula is reproduced, as given by its perturbative derivation assuming distinguishable photoelectron emission centers. For larger saturation, however, it becomes more probable that several photons are absorbed simultaneously, generating thus bunches of escaping atoms. In this regime, the indistinguishability of the bosonic atoms becomes relevant and large deviations from the Mandel counting formula are observed. They manifest themselves by a reduced fluctuation of atom counts as compared to the binomial form. This noise reduction leads to an enhanced photon-number resolution as compared to a conventional photodetector at equal detection efficiency, which constitutes an important advantage of a microtrap-based detector. It should be noted that the detection efficiency ␩D determines the mapping photon to atom numbers to all orders of moments for a conventional photodetector. In our case, however, it only connects the mean photon and atom-count numbers. The shape and width of the atom-count statistics depends additionally on a second parameter, e.g., the saturation, which allows for a difference from the binomial statistics. The present model assumes an ideal Bose gas in the limit of zero temperature. However, the inclusion of atomic colli-. sions can be consistently made by identifying the lowest trap mode as the solution of the Gross-Pitaevskii equation. In Thomas-Fermi approximation, the only modification will then be the increased 共decreased兲 size of the ground-state mode for repulsive 共attractive兲 scattering, which will modify the time of flight of excited atoms to escape from the trap. Thus repulsive 共attractive兲 scattering will decrease 共increase兲 the atom-loss rate ␥. The treatment of finite temperature 共but below the condensation temperature兲 requires a departure from the singlemode approximation. However, for low saturation as discussed here, the inclusion of noncondensed modes will not lead to a mixing of condensed and noncondensed atoms by photon absorption: Photon absorption will excite both initially condensed as well as noncondensed atoms to immediately leave the trap. Thus they do not return to the electronic ground state, preventing the mixing. The small fraction of noncondensed atoms at finite temperature will represent distinguishable emission centers that will contribute as a small binomial admixture to the atom-count statistics. With increasing temperature, this admixture will deterioate the enhancement of photon-number resolution made by the condensed atoms, as we expect from the transition beyond the condensation temperature to a gas of distinguishable atoms.. VI. SUMMARY AND CONCLUSIONS. In summary, we have introduced a model for a photodetector using a Bose-Einstein-condensed gas trapped on an atom chip. By use of an atomic pseudospin approximation based on large atom numbers, the photon absorption and atom-escape dynamics could be approximated for the overdamped case where Rabi oscillations are suppressed. A quantum-trajectory method then led us to a counting formula involving a sum of three Meijer G functions. The conditional count statistics has been shown to depend on a saturation-dependent coefficient q and an atom-escape probability p, to which the quantum efficiency of the detector converges in the limit of extremely low saturation, q → ⬁. In this limit, the differences from a binomial statistics vanish so that our result coincides with the Mandel counting formula. However, for realistic values of the saturation, substantial deviations can be observed as a reduced atom-count fluctuation. An open question is yet the behavior of our model detector for saturation SA 艌 1, where full Rabi oscillations start to occur. We assume that in this regime we may expect even larger and dramatic deviations from the Mandel formula. Then coherent processes must be included, which describe the temporary storage of photon energy in the atomic gas in the absence of correspondingly escaping atoms. This is left for a future investigation.. ACKNOWLEDGMENTS. The authors acknowledge support by FONDECYT project no. 7070220 and S.W. acknowledges support by FONDECYT project no. 1051072.. 043822-10.

(11) PHYSICAL REVIEW A 78, 043822 共2008兲. PHOTODETECTION USING BOSE-EINSTEIN-CONDENSED… APPENDIX A: TRANSITION AMPLITUDE. z=. The transition amplitude can be simplified to A,n 共␶兲 = 冑␥e−␥n␶/4具A,n,n − 1兩ei␶兩共2兩⍀兩 ⌿n−1,n. 冑M A,nLˆx−i␥Lˆz/2兲. 兩A,n,n典,. where M A,n = A − 共n − 1兲 / 2. The above matrix element can be calculated as 具A,n,n − 1兩e. i共␣Lˆz+␤Lˆx兲. 兩A,n,n典 = 冑nz−n/2␨n−1共1 + z兲n−1 ,. SA,n − 1 , SA,n sinh2共␦⬘/2兲. ␦⬘ =. ␥␶ 冑 1 − SA,n , 2. where we defined the saturation parameter, SA,N =. where i␣ sin共␦/2兲 , ␨= ␦ cos共␦/2兲 − i␤ sin共␦/2兲 z=−. ␦2 ␣2 sin2共␦/2兲. A,n ⌿n−1,n 共␶兲 = − i冑n␥e−␥n␶/4共− 1兲n−1关冑1 − SA,n cosh共␦⬘/2兲. ,. + sinh共␦⬘/2兲兴n−1. using ␣ = 2兩⍀兩␶冑M A,n and ␤ = −i␥␶ / 2 for the resonant case 共⌬ = 0兲. Using the explicit values for ␣ and ␤, we obtain the explicit expressions for the above parameters with ␦ = i␦⬘, i冑SA,n sinh共␦⬘/2兲. 冑1 − SA,n cosh共␦⬘/2兲 − sinh共␦⬘/2兲. 冉冊. 共A1兲. In the overdamped regime, SA,n ⬍ 1, the transition amplitude becomes thus. ␦ = 冑␣2 + ␤2 ,. ␨=. 4兩⍀兩2M A,N . ␥ 2 2. 1 共1 − SA,n兲n/2. 册冑. S sinh共␦⬘/2兲.. In the limit SA,n Ⰶ 1, this expression reduces in lowest-order approximation to A,n ⌿n−1,n 共␶兲 = i共− 1兲n冑SA,nn␥e−共␥␶/4兲关n−共n−1兲. 冉. ⫻ sinh. ,. 关1兴 See L. Mandel and E. Wolf, Optical Coherence and Quantum Optics 共Cambridge University Press, Cambridge, UK, 1995兲, and references therein. 关2兴 P. L. Kelley and W. H. Kleiner, Phys. Rev. 136, A316 共1964兲; R. J. Glauber, Quantum Optics and Electronics, Les Houches Summer School of Theoretical Physics, University of Grenoble, edited by C. DeWitt, A. Blandin, and C. CohenTannoudji 共Gordon and Breach, New York, 1965兲, p. 53. 关3兴 L. Mandel, Proc. Phys. Soc. London 72, 1037 共1958兲; 74, 233 共1959兲; Prog. Opt. 2, 181 共1963兲. 关4兴 L. Mandel, E. C. G. Sudarshan, and E. Wolf, Proc. Phys. Soc. London 84, 435 共1964兲; W. E. Lamb Jr., and M. O. Scully, Polarization: Matiere et Rayonnement 共Presses Universitaires de France, Paris, 1969兲. 关5兴 M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 共1995兲; C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 共1995兲; K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, ibid. 75, 3969 共1995兲. 关6兴 H. Ott, J. Fortágh, G. Schlotterbeck, A. Grossmann, and C. Zimmermann, Phys. Rev. Lett. 87, 230401 共2001兲; W. Hänsel, J. Reichel, P. Hommelhoff, and T. W. Hänsch, Phys. Rev. A 64, 063607 共2001兲. 关7兴 See also J. Fortágh and C. Zimmermann, Rev. Mod. Phys. 79, 235 共2007兲, and references therein.. 冋. 冊. ␥␶ 冑 1 − SA,n . 4. 冑1−SA,n兴. 共A2兲. 关8兴 R. H. Dicke, Phys. Rev. 93, 99 共1954兲. 关9兴 M. Kozierowski, A. A. Mamedov, and S. M. Chumakov, Phys. Rev. A 42, 1762 共1990兲. 关10兴 S. M. Chumakov, A. B. Klimov, and M. Kozierowski, From the Jaynes-Cummings Model to Collective Interactions, in Theory of Nonclassical States of Light, edited by V. V. Dodonov and V. I. Man’ko 共Taylor & Francis, London, 2002兲, p. 313. 关11兴 G. C. Hegerfeldt and T. S. Wilser, in Classical and Quantum systems, Proceedings of the II. International Wigner Symposium, 1991 共World Scientific, Singapore, 1992兲, p. 104; C. W. Gardiner, A. S. Parkins, and P. Zoller, Phys. Rev. A 46, 4363 共1992兲; J. Dalibard, Y. Castin, and K. Molmer, Phys. Rev. Lett. 68, 580 共1992兲. 关12兴 H. J. Carmichael, An Open Systems Approach to Quantum Optics, Vol. m18 of Lecture Notes in Physics 共Springer-Verlag, Berlin, 1993兲. 关13兴 K. Molmer, Y. Castin, and J. Dalibard, J. Opt. Soc. Am. B 10, 524 共1993兲; G. C. Hegerfeldt and M. B. Plenio, Quantum Opt. 6, 15 共1994兲; B. M. Garraway and P. L. Knight, Phys. Rev. A 50, 2548 共1994兲; M. B. Plenio and P. L. Knight, Rev. Mod. Phys. 70, 101 共1998兲. 关14兴 A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series 共Gordon and Breach Science Publishers, New York, 1992兲, Vol. 4, p. 578.. 043822-11.

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