AMPLIFICATION OF ATOMIC WAVES BY STIMULATED EMISSION OF ATOMS
Christian J. Borde
Laboratoire de Physique des Lasers, CNRS/URA 282, Universite Paris-Nord, Villetaneuse, France
1. INTRODUCTION:
The recent development of atom optics and interferometry and especially of atomic resonators has demonstrated that one could often exchange the roles of atoms and photons in optical devices in order to design an atomic wave device. This is part of a general view of the world of photons and atoms as elementary eld particles which can mutually scatter each other in a reciprocal way. For example, one could imagine to achieve a device equivalent to optical lasers where photons are replaced by integer spin atoms and where the atoms are re ected or trapped by light (\atom laser", \atomaser" or \ataser").
A similar goal is Bose-Einstein condensation within a collection of cold atoms in thermal equilibrium. We present a di erent approach, closer to the optical case, in which atoms are produced in the same phase space cell by the induced version of a spontaneous process, very far from thermal equilibrium. This process should be the analog of the spontaneous emission of photons: A !A + photon. If we replace photons by bosonic atoms B we are led to consider the more general dissociation process:
C!A+B (1)
which includes the emission of a photonA as a special case.
In the case of an ordinary laser, the photon energy is provided to the emitter by an external energy source, which excites the atoms. Similarly, in the case of the atomic wave, the energy represented by the mass of the atomic constituents needs also to be provided to the emitters. If we do not want to have to recreate each of these masses individually from other particles, they must also be constituents of the emitter. The simplest solution is therefore to use for the emitter C a composite object (molecule, excited atom or ion) which includes the required bosonB as a constituent.
2. THEORETICAL FRAMEWORK:
Alike stimulated emission of photons, the enhancement factor in the emission of atoms in the same phase cell by an incoming atomic wave, comes only from the bosonic character and the exchange properties of indistinguishable particles. Following Feyn-man [1], for nincident atoms B, the statistical weight of the initial state is 1=n! , the
statistical weight of the nal state is 1=(n+ 1)!, and the amplitude for the process is (n+1)! times the amplitude for spontaneous emission. The probability (per second) of emission is then [(n+ 1)!]2=[(n+ 1)!n!] = n+ 1 times the probability of spontaneous dissociation. This can be demonstrated rigorously in a quantized eld description of the stimulated dissociation process. In such a description, atomic systems in a given internal state are described as elementary particles by quantum elds operators [2] satisfying commutation or anti-commutation rules to introduce their bosonic or fermionic character. Here for simplicity, we shall assume scalar (pseudo-scalar) elds which satisfy the Klein-Gordon equation, corresponding to spin zero particles:
(x) = 1 (2 h)32 Z d3p q 2E (~p) c (~p) exp[i h(~p ~r E (~p)t)] (2) where E (~p) = ( (E )2 +p2c2)12 (3) is the relativistic energy of the atom in state
annihilated by the operator c (~p) and where E =c2 = M is the rest mass. The
previous expression corresponds to an expansion in plane waves. For many-particle states it is more convenient to introduce sets of discrete modesm: The starting point is then simply an e ective Hamiltonian density:
fH(x) =gR y
A(x) yB(x) C(x) +h:c: (4) where the quantum elds yA(x), yB(x) and C(x) respectively create atoms A and B and annihilate molecule C at the space-time pointx.
Our purpose will be to derive general formulas for the spontaneous dissociation rate and for the gain (A and B Einstein coe cients and line shape) which are ap-plicable both to massive particles and to photons, in order to keep the parallelism between light and matter wave ampli ers and be guided by this analogy. The deriva-tion of such formulas requires a uni ed approach, in which the rest mass of the various partners may be arbitrarily set to zero. A relativistically covariant descrip-tion of Quantum Mechanics is obtained if the Schr•odinger equation is replaced by the Tomonaga-Schwinger equation. For the density operator e( ), the correspond-ing equation [2] leads to the Liouville-von Neumann equation in a given frame, from which one obtains an integral form for the derivative of the density operator:
@e(t) @t = 1 ih Z d3x[Hf(x);e(t0)] 1 h2 Z d3x Z d3x0 Z t t0 dt0[Hf(x);[Hf(x0);e(t0)]] (5) This equation can be used to derive a master equation for a reduced density matrix T rBe(t) with the Markov approximation and to obtain relaxation operators or to derive @e(t)=@t from the zero-order density operator e(0) in a second-order
3. DERIVATION OF THE SPONTANEOUS DISSOCIATION RATE: With the Hamiltonian density (4), the trace over the empty modes ofByields the following matrix element of the relaxation operator:
<0Aj<1~pC j j1p~0C >j0A>= 2g2 R h2 Re R d 3xR d 3x0Rt1dt0 <0B j B(x) yB(x0)j0B ><0Aj A(x) yA(x0)j0A > <1p~C j y C(x) C(x0)j1~p0 C >= C (~pC ~p 0 C) (6) which gives the spontaneous dissociation rate :
C = 2g2 R 2h4E C(pC) ZZ d 3p A EA(pA) d3pB EB(pB) (~pC ~pA ~pB) (EC(pC) EA(pA) EB(pB)) = 1 2 3g2 R h3c3E2 C pB0c h = C0= (7) where = 1=q1 v2
C=c2 is the time dilation factor, and where we have introduced the momentumpB0 of the emerging particle B in the center-of-mass frame:
pB0 =c
q
(MA+MB+MC)( MA+MB+MC)(MA MB+MC)( MA MB+MC) 2MC
(8) From formula (7), we can recover the spontaneous emission rate of photons by set-ting either MAorMB = 0 (pB0c'h!0 ): C0 ' 2!30=(2 " 0hc3) (an additional
fac-tor 2/3 would come from spin 1 elds). We have used the correspondence between our coupling constant gR and the transition dipole moment : gR2 = 4 2MC2c4(h!)2="0.
In the case of three non-relativistic massive particles: pB0 '(2 AB E)1=2 where AB is the reduced mass and where E=c2 is the mass defect.
4. DERIVATION OF THE AMPLIFICATION COEFFICIENT:
The gain for the bosonic wave will be simply obtained from the time rate of change of the average number of bosons in the mode m, by:
= 1 vB 1 nB @nB @t (9)
where vB is the group velocity of that wave. This rate is opposite to the time rate of change of particles C under the in uence of stimulated emission of bosonsB in a given mode: =n(0)C h g2 R=h 2 =[8E B(~pB)vBE 2 C] i 2ReR0+1d R d 3p CEC2=[EC(~pC)EA(~pC ~pB)]
expfi[EC(~pC) EA(~pC ~pB) EB(~pB)] =h =2gF(~pC)
where n(0)C F(~pC) is the zero-order population of species C, with a normalized mo-mentum distribution F(~pC) of width MCu: Using (7), the coupling constant can be eliminated in favor of the decay constant:
" g2 R h2 ! 1 8EB(~pB)vBE 2 C # = h 2 pBpB0 C0 4
Like in the case of light ampli cation, the gain may thus be either homogeneously or inhomogeneously broadened, depending on the relative size of h and of the in-homogeneous energy width. This inin-homogeneous width is of the order of the spread in kinetic energies ( MCu2) in the case of massive particles, but it is much larger ( MCuc) whenMA = 0.
We will give expressions for the gain in the limits of a purely homogeneously or inhomogeneously broadened line shape. In the inhomogeneous case the integral gives a function for the energy and if the distribution F(~pC) is a non-relativistic Maxwell-Boltzmann distribution we obtain:
(pB) = n(0)C 4 3=2 h3 p2 BpB0 C0 u 8 < : p c ue c2=u2 2 4 erf h c u[EB(~pB0)EB(~pB) +pB0pBc 2]=E2 B i erfhc u[EB(~pB0)EB(~pB) pB0pBc 2]=E2 B i 3 5 9 = ; (11) which gives back the usual Doppler pro le when MB vanishes. When the boson mass is di erent from zero, the previous formula simpli es in the non-relativistic approximation: (pB) = n(0)C 4 3=2 h3 p2 BpB0 C0 u ( exp " (pB pB0)2 M2 Bu2 # exp " (pB+pB0)2 M2 Bu2 #) (12) Both (11) and (12) exhibit a resonance for pB 'pB0; if pB0is large enough
com-pared toMBu:We can also extract the EinsteinBcoe cient: B =h2c C0=(4 p2BEB) associated with a line shape normalized in frequency =pBc=h:IfMBuis large com-pared to pB0 then: (pB)'n
(0)
C = 3=2(h=(MBu))3 C0=vB:
The homogeneous gain formula is obtained in the limit where F(~pC) is considered as a function in (10) : (pB) = n (0) C h2 pBpB0 EC EA( ~pB) 1 C0=4 ( =2)2+ [E C EA( ~pB) EB(~pB)]2=h2 (13) ForMB = 0;we recover the usual formula for light ampli cation with the natural width for = C0 =A: (!) = n(0)C 20 1 2 (A=2)2 (A=2)2+ (! ! 0)2 (14)
For MA = 0, the homogeneous limit is obtained only in the extreme case h > MCuc: (pB) =n (0) C 2 0 c vB 1 (A=2)2 (A=2)2+ [E C EB(~pB) pBc]2=h2 (15) For MA; MB; MC 6= 0; = C0 (pB) =n (0) C h2 pBpB0 MB AB 1 ( =2)2 ( =2)2 + [(p2 B0 p2B)=(2 AB)]2=h2 (16) We see that, at resonance, the cross section is again proportional to a de Broglie wavelength squared, and therefore one should use atoms as slow as possible. The mass defect E=c2 should thus be as small as possible.
5. CONCLUSION AND POSSIBLE GAIN MEDIA:
We get formulas for atomic waves which are similar to the optical case. This demonstrates that comparable gains could be obtained with the same wavelengths and population inversion densities. It should be noted, however, that a divergent gain occurs for vanishing atomic velocity and this could be in favor of the slowest atomic waves for a given length of ampli er and xed resonator losses per round-trip. In equation (1) species B is an atomic or a molecular boson, but A can be anything from a photon to a solid, i.e., C can be an excited atom, a negative ion, a molecule, a cluster or even a surface (B may in fact be trapped in any external potential). Especially interesting possibilities are the predissociation of electroni-cally excited molecules (e.g. Na2(B1 u)!Na(3p)+Na(3s)) or of vibrationally excited van der Waals molecules (e.g. He-I2) or the dissociative recombination of ions (e.g.
HeH+ +e
!HeH!He+H). Losses by elastic and inelastic collisions of B with A or C need to be investigated in each case. The only requirement, to maintain the population inversion, is that speciesA should escape or be removed as fast as possi-ble through spontaneous decay, optical pumping or any other means. Finally, higher-order processes (such asC!A+A'+B,whereAandA'can again be either photons or massive particles) and simultaneous stimulated emission of light and atoms should also be considered as interesting possibilities.
6. References:
[1] R.P. Feynman, Quantum Electrodynamics, Frontiers in Physics, Addison Wes-ley (1962)
[2] Ch. J. Borde, in Advances in Laser Spectroscopy, Plenum Press, NATO ASI Series B95 (1983) pp. 1-70
