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Coherence in Stimulated Emission

Introduction

In 1917 Albert Einstein published an extraordinary piece of analysis which is generally accepted as the foundation of laser physics. This article, "Zur Quantentheorie der Strahlung" (On the Quantum Theory of Radiation), Physika Zeitschrift, Volume 18 (1917), pp 121-128, is also notable for first introducing the concept (but not the name) of the photon. In this article Einstein argues that in the interaction of matter and radiation there must be, in addition to the processes of absorption and spontaneous emission, a third process of stimulated emission. If stimulated emission exists then he can derive the Planck distribution for blackbody radiation and without it the same argument implies the empirically invalid Wien distribution.

But, in addition to establishing the existence of the process of stimulated emission, Einstein also asserts that the radiation produced in stimulated emission is identical in all relevant aspects to the incident radiation. This is a truly remarkable result. It seems to represent some very deep property of the physical world. The sense of marvel this result elicits is captured in a remark made by Einstein himself in a letter to Michael Angelo Besso in November 1916,

A splendid light has dawned on me about the absorption and emission of radiation.

The purpose of this paper is to investigate the identity or near identity of the stimulated emission with the stimulating radiation. The qualification of near identity is included because there is a theoretical reason for believing the stimulated radiation could differ in phase from the stimulating radiation. But the theory upon which this phase difference is based is semiclassical; i.e., the atom is formulated quantum mechanically but the radiation field is given classically. It is possible that a full quantum mechanical treatment of the field would not lead to the same conclusion. And the ultimate confirmation has to be empirical.

There is surprisingly little empirical results on this matter. Even in theoretical analysis there is surprisingly little on the topic of the coherence of the stimulated and stimulating radiation. Almost all of the analysis of stimulated emission in the physical science literature assumes that the stimulated emission is identical to the stimulating radiation in terms of direction and frequency.

It should be noted that the phenomenon of stimulated emission does not apply only to atoms. Electrons confined by crossed electrical and magnetic fields in wave cavities can under stimulation produce microwase radiation. See Riyoupoulos, 1995.

Einstein's Analysis

Einstein asserts early in his article,

[..] for the case of incident radiation, the magnitude of the transferred momentum is the same [as in the case of absorption], but it is in the opposite direction.

but at that point he offers no proof. There is analysis later in the paper on a number of topics and then Einstein states

Most important, however, appears to me the result about the momentum transferred to the molecule by incoming and outgoing radiation. If one of our hypotheses were altered, the result would be a violation of equation (12); it appears hardly possible, except by way of our hypotheses, to be in agreement with this relationship which is demanded by thermodynamics. We may therefore consider the following as pretty much proven. If, through an emission process, the molcule suffers a radiant loss of energy of magnitude hν without the action of an outside agency, then this process, too, is a directed one.

Thus Einstein reaches his conclusion concerning the microscopic characteristics of the stimulated emission process through an argument about the macroscopic properties of a gas. The argument is valid and the conclusions definitely right but one would expect that there would be an alternate microscopic analysis to justify the coherence of stimulated emission.

A search of the physics literature of recent years turned up only one article dealing with the coherence of stimulated emission and that one article was a review of Einstein's 1917 article. The article in the American Journal of Physics(January 1994) is by R. Friedberg of Barnard College and Columbia University is entitled, "Einstein and stimulated emission: A completely corpuscular treatment of momentum balance." Friedberg asserts that there are two parts to Einstein's analysis, the energy balance and the momentum balance, and that what is generally known about Einstein's article concerns the energy balance part of the analysis but most of the article concerns the momentum balance.

Friedberg (and Einstein's) anlysis of momentum transfer between atoms and a radiation field have to do with an effective viscosity experienced by atoms moving in a radiation field. Atoms at rest in an isotropic radiation field will on average experience no force from the interactions. But an atom moving, even in an isotropic radiation field, will experience a net force that slows it. Friedberg refers to the difference between the number of encounters at the front of a moving atom compared to the number at the rear as being like a "head wind." The result is that if p is the momentum of an atom and v is its velocity then


dp/dt = -λv.
 

The analysis concerns the "drag coefficient" λ.

Spontaneous emission is random in direction and cannot produce any drag effect. On the other hand aborption is directional and could have an effect on the drag coefficient. If stimulated emission were random in direction the average effect on momentum transfer would be nil, as is the case of spontaneous emission. But an assumption of directionality for stimulated emission leads to an acceptable formula for the drag coefficient λ. Stimulated emission must not only be directional, the momentum transfer to the atom must be opposite in direction to that involved in absorption.

It is from this type of argument that Einstein concluded that in stimulated emission is directional and the atom recoils in the direction opposite to the incoming photon and thus the emitted photon must leave the atom in the same direction as the incoming photon.

Energy and Momentum

In absorption the conservation of momentum dictates that the recoil of the absorbing atom be in the same direction as the incident photon. Some of the energy of the photon will go into the kinetic energy of the recoiling atom and some will raise the energy state of the atom.

If δ is the energy difference of the states of the atom, m the mass of the atom, v its recoil velocity and νR the resonant frequency then energy and momentum conservation requires (in nonrelativitic approximation):


R = δE + (1/2)mv2
νR/c = mv
 

Since the energy states of the atom are quantized the resonant freguency for absorption will be equal to the sum of the energy difference of the states plus the kinetic energy of an atom with the same momentum all divided by Planck's constant.

Likewise in stimulated emission there will have to be a recoil of the atom to balance the momentum of the emitted photon. But momentum conservation in this case only requires that the recoil be in the direction opposite to the emitted photon.

For emission the equations for enery and momentum conservation are:


-hνR = -δE + (1/2)mv2
νR/c = mv
which are equivalent to
R = δE - (1/2)mv2
νR/c = mv
 

An interesting complication to the determination of the resonant frequencies for absorption and emission is that the equations have two solutions. This ambiguity can be resolved with a few computations. The energy terms of the equations can be conveniently expressed as electron-volts. For a hyrogen atom δE is on the order of 13.6 eV and mc2=938.8 meV. One solution for the energy of the photons of resonant frequency is 13.6(1-1.45x10-8). The other solution is for a photon with twice the rest-mass energy of the atom. For this latter case the nonrelativistic formula is not sufficiently accurate. But the computation reveals that the solution for photon energy approximately equal to the quantum change in the energy state of the atom is the only physically relevant solution.

Energy and momentum conservation do not require that the stimulated emission duplicate the incident radiation.

Symmetry Principles

There is a high degree of symmetry involved in the coherence of stimulated emission and in its relationship to absorption. It is is worthwhile to look at some principles of symmetry that might apply in this case. The following is a very general such principle as formulated in solid state physics.

Neumann's Principle:
(three versions)

The last version is associated with Pierre Curie and it is the one that is the most convenient for looking at the processes involved in the interaction of matter and radiation.

There is another symmetry principle that is of relavance, the principle of invariance under time reversal. This means that if time runs backward the processes should be physically possible.

Now consider the process of absorption as is illustrated in Figure 1.

An atom in a ground state at rest in laboratory coordinates is hit by a photon. The atom absorbs the photon's momentum and energy, moving to a higher internal energy state as well as recoiling.

Under time reversal, as is shown at the right in Figure 1, the process appears to be an atom with momentum which spontaneously emits a photon.

If in a stimulated emission we considered the possibility of the emitted photon going off at an angle of say 30o we find there is a violation of the Curie version of Neumann's Principle because the 30o trajectory would be an asymmetry which is not in the initial conditions. The only direction that would not vilolate Curie's version of Neumann's Principle is an emitted direction identical to the incident photon or exactly opposite to the incident photon.

In Figure 2 is shown the case of an atom at rest in laboratory coordinates hit by a incident photon. Under the correct version of stimulated emission the emitted photon moves in the same direction as the incident photon.

The atom recoils in a direction exactly opposite to the incident photon. Under time reversal this process would appear to be a case of absorption in which two photons impinge upon a moving atom. One of the photons is absorbed and cancels the momentum. The other continues on its way.

In Figure 3 we have the emitted photon leaving the atom in a direction opposite to that of the incident photon. This means the atom would recoil in the same direction at the incident photon.

There appears to be no violation of any physical principle here. But under time reversal the process appears to be one in which two equal but oppositely directed photons hit an atom and one photon continues. It seems that there is an asymmetry in the matter of which photon continues. In the time reversal picture there does not seems to be any reason for one of the incident photons continuing on its way and the other being absorbed. Thus this case would violate the Curie version of Neumann's Principle under time reversal.

Thus we can conclude that only the case of the emitted photon continuing in the same direction as the incident photon is allowed.

Phase Shifts in Stimulated Emission

Semiclassical analysis of the interaction of an atom with a field predicts that the stimulated radiation lags in phase from the incident radiation by 270o. It would be quite significant if the stimulated emission differed from the incident photon in any way. This is because one way of rationalizing the duplication of the characteristics of the incident photon in the emitted photon is that the field is quantized and the number of photons is a proper quantum number for the quantized field. Thus the effect of a field-atom interaction would be to increase or decrease the quantum number of the field depending upon whether the interaction is an absorption or an emission. This approach would be in terms of the framework of second quantization and its annihilation and creation operators. But if the emitted photon differs from the incident photon in any way the second quantization framework would have to be considerably modified.

The semiclassical analysis from which the phase shift is derived involves a quantum mechanical formulation of the atom and a classical characterization of the field. The oscillating electrical and magnetic field create a dipole moment in the atom which interacts with the electromagnetic fields. The strongest torque on the dipole occurs when the direction of the dipole is perpendicular to the direction of the electrical field. Thus the 270o lag (or a 90o lead) of the emitted radiation results from this effect.

The problem with this result is that any prediction from a semiclassical analysis has to be verified empirically or from a full quantum treatment of the field and the atom.

In the literature of physics there does not seem to be an empirical verification of the semiclassically predicted phase shift for stimulated emission. This is quite surprising but may be indicative of the difficulty of detecting phase shifts.

There are full quantum mechanical treatments of the interaction of a field and an atom, but these treatments use a second quantization characterization of the field which precludes any possible difference between the photons of the incident radiation and the emitted radiation. Thus the full quantum mechanical treatment of field-atom interaction is not sufficiently sophisticated enough to analyze matters of phase shifts between incident and emitted radiation.

Conclusions

The proof of the very important result that a photon produced by stimulated emission is identical in at least most ways with the incident photon is still, in 2007, essentially that given by Albert Einstein in 1917. Einstein's proof is essentially thermodynamic and depends upon averages over time whereas the result is microscopic and not dependent upon what happens in other interactions. This amazing and very beautiful result can be justified on the basis of symmetry principles but a fully satisfactory method of proof would have to be quantum mechanical. Apparently such a proof has not yet been found.

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