The Interaction of Radiation and Matter: Quantum Theory (cont.)

V. Photon Absorption and Emission (pdf)
"POOR MAN'S" SECOND QUANTIZATION OF MATERIAL SYSTEM:
In treating the complete quantum mechanical problem, it is useful to recast the material (atomic) Hamiltonian in terms of an appropriate set of creation and destruction operators. To that end we make the following definition
 
     [ V-1 ]

Using the ubiquitous identity operation  , we may write the material Hamiltonian in second quantized form -- viz.
 

     [ V-2 ]

In general, the operator  applied to any state  yields
 

     [ V-3 ]

-- i.e. the operator changes a state z to a state x if the state is y otherwise it produces zero. In other words, the operator destroys the state y and creates a state x. The second quantization viewpoint is particularly useful in treating the interaction of a two-level material system with the radiation field. This case, is most conveniently formulate in two-vector notation with the use of Pauli spin matrices -- viz.
 

and       [ V-4a ]
and       [ V-4b ]
     [ V-4c ]
     [ V-4d ]

Consequently, the atomic Hamiltonian may be written
 

     [ V-5a ]

and if we neglect the mean energy of the states

     [ V-5b ]

and the electric dipole interaction Hamiltonian becomes
 

      [ V-6 ]

From Equation [ II-24a ] in this lecture set we can write
 

     [ V-7 ]

where  is the so called the electric field per photonand  is the location of the center of the atom under consideration. Thus Equation [ V-6 ] may be written quite generally for a two level atom as
 

     [ V-8 ]

where the coupling constant is given by

     [ V-9 ]

Interaction of a Two-level Atom and a Single Mode Field -- Rabi Flopping:

Let us first consider the interaction of a two-level system with a single photon state to make contact with the discussion in Section III of the lecture set entitled The Interaction of Radiation and Matter: Semiclassical Theory. Equation [ V-8 ] then reduces to
 
     [ V-10a ]

where

.

If we neglect any inter-atomic interference effects by taking , we may simplify Equation [ V-10a ] to
 

     [ V-10b ]

Thus, we may then write the complete effective Hamiltonian of the composite system as
 

     [ V-11a ]

In the rotating wave approximation this reduces to
 

     [ V-11b ]

In the spirit of the discussion in the Section VII, Semiconductor Photonics of the lecture set entitled The Interaction of Radiation and Matter: Semiclassical Theory,this effective Hamiltonian may be adapted to provide a fully quantum mechanical treatment of optical interactions in semiconductors.[1]

"DRESSED" ATOMIC STATES:

We know that the unperturbed Hamiltonian satisfies the following eigenvalue equations
 

     [ V-12 ]

-- where  and  -- and the electric dipole perturbation couples the states  and . It is useful to resolve the complete Hamiltonian into a sum of component Hamiltonians  where the component 's act only within the {,} coupled manifold of states and can be written [2]
 

     [ V-13 ]

where . The second term in this equation is of the same form as the coupling matrix in Equation [ III-8c ] of the lecture set entitled The Interaction of Radiation and Matter: Semiclassical Theory where the semiclassical Rabi frequency  is replaced by its quantum equivalent -- viz. .   Diagonalizing this matrix we find the eigenvalues of the so call dressed atomic states-- viz.[3]
 

      [ V-14 ]

(see energy level diagram below) where  is the quantized field generalization of the Rabi flopping frequency.

The dressed eigenstates are given by
 

     [ V-15a ]

where

     [ V-15b ]

and

     [ V-15c ]

 

DRESSEDENERGY LEVELS OF TWO-STATE ATOM

The time evolution of a state is directly represented in terms of these dressed states -- viz.
 

     [ V-16a ]

or more explicitly

     [ V-16b ]

Therefore, the flopping of the undressed states is given by
 

     [ V-17 ]

Perhaps the most revealing application of the this result is for the case of a resonant coupled system -- i.e.  -- which is prepared so that .   In this instance, Equation [ V-17 ] yields
 

      [ V-18 ]

which clearly exhibits the simplest manifestation of  spontaneous emission -- i.e.Rabi flopping in the absence of an applied field!!!
 
 

Interaction of a Two-level Atom and a Multi Mode Field -- Spontaneous Emission:
To broaden (make more realistic) our treatment of spontaneous emission we return to Equation [ V-8 ] to include the interaction with many modes with a two-level atom -- viz.
 
     [ V-8' ]

This equation may be easily generalized to encompass multi-level material systems.[4]

If we are dealing with situation in which the locations of the atoms are uncorrelated we may, for simplicity, dispense with the  factors -- i.e. we will neglect, for the present, any possible interference effects --and write
 

     [ V-19 ]

Transforming to the Schrödinger picture and taking the unperturbed ground state  as the zero energy reference point we may write the complete effective Hamiltonian as
 

     [ V-20a ]

If we include only energy-conserving terms -- i.e. in the rotating wave approximation --
 

     [ V-20b ]

It is important to note that, in general, the interaction terms in this Hamiltonian include contributions from the coupling of the material system (atom) to any externally excited mode(s) (the incident electromagnetic field) and to all available electromagnetic cavity modes. For the present, we ignore the coupling to externally excited modes: we treat the external interaction later as a perturbation. Our goal at this point is to diagonalize the Hamiltonian of the complete unperturbed system which may be written
 

     [ V-21 ]

We are looking for the eigenstates and eigenvalues of this Hamiltonian which we write as
 

     [ V-22 ]

Since the square matrix is Hermitian, it is possible, in principle, to diagonalize it via a unitary transformation of the form
 

     [ V-23 ]

where  is a diagonal matrix whose elements are the  eigenvalues of the dressed states of the coupled system. If we define the row vector  and the column vector  then Equation [ V-22 ] becomes
 

      [ V-24 ]

For consistency,  and, hence,  and  are, respectively, paired creation and destruction operators in the sense of the operators defined in Equation [ V-3 ] above -- i.e.  and .   From Equation [ V-21 ] we may write
 

     [ V-25a ]

which can be written explicitly as
 

     [ V-25b ]

Expanding out the matrix product on the left we see that
 

     [ V-26a ]

for elements in the first column and

     [ V-26b ]

for elements not in the first column. Hence
 

     [ V-26c ]

and all elements of the unitary matrix can be expressed in terms of the first-column elements as
 

     [ V-27 ]

Quite generally the columns of any unitary matrix satisfy an orthonormal condition -- viz.
 

     [ V-28a ]

so the normalization of the first column gives

     [ V-28b ]

and the orthogonality of the first column with any other column yields
 

     [ V-28c ]

for each one of the cavity frequencies .

Further we may multiply Equation [ V-28c ] by a product of factors
 

     [ V-28d ]

It is obvious from this expression the 's are roots of the left side of the equation if the 's are replaced by  .   Thus
 

     [ V-28e ]
Finally, we see that
 
     [ V-29 ]

We can make use of this expression to obtain the time varying polarization induced by an externally excited field. The Hamiltonian associated with this perturbation may be written

     [ V-30 ]

Since  and  we see that
 

     [ V-31 ]

and from Equation [ V-6 ] we may write
 

     [ V-32 ]

In light of Equation [ V-29 ], the standardized form for the frequency dependent susceptibility (see Equation [ VA-12 ] becomes
 

     [ V-33 ]

By eliminating the U's from Equations [ V-26a ] and [ V-26b ] we see that
 

     [ V-34 ]
Again
 
     [ V-35 ]

so that we can write
 

     [ V-36 ]

Therefore
 

     [ V-37 ]

where the sum  gives an explicit, non-phenomenological accounting of interactions with the cavity modes and hence of spontaneous emission!

EVALUATION OF SPONTANEOUS EMISSION RATE:

Recall the discussion of phenomenologically defined damping in semiclassical models of the dielectric response function in the lecture set entitled The Interaction of Radiation and Matter: Semiclassical Theory. Recollect, in particular, Equation [ III-19c ] in those notes. In reconciling that discussion with the content of Equation [ V-37 ] above, we see that the summation  replaces the simple damping parameter . Our task here is evaluate this integral which we write as
.

      [ V-38 ]

where  is the number of cavity modes with frequencies between  and .. Following arguments best explicated long ago by Heitler,[5] it may be shown that
..

     [ V-39 ]

which we write as . This is an extremely important result!!   It shows that the interaction between the atom and the cavity modes leads to a frequency shift or correction in the atomic splitting 
 

     [ V-40a ]

and a spontaneous emission decay rate

     [ V-40b ]

If we assume that the cavity modes defined for the blackbody calculation in Section IV of the lecture set entitled The Interaction of Radiation and Matter: Semiclassical Theory (see Equation [ IV-5 ] in those notes) are the appropriate modes, we know that

and from Equation [ V-9 ] we know that

.

Treating the shift , the radiative correction to atomic energy level separation, is a very complex and much studied matter. The simple interpretation of Equation [ V-38a ] is problematic since the integrand is proportional to  at large  and, thus, the correction significantly diverges!!

The divergence in  was for many years an unresolved discrepancy between the quantum theory of radiation and observational spectroscopy. The difficulty was overcome by Bethe in 1947[6] using a technique known as mass renormalization. Bethe point out that the divergence can mainly be associated with the mass of the electron. It is found that the energy of a free electron has an infinite contribution arising from the interaction of the electron with the electromagnetic field. In other words, the apparent mass of the electron is shifted by an infinite amount from the mass of an electron which is not in interaction with the radiation field. However, the former mass is the one measured experimentally, since it is never possible to isolate an electron from the radiation field. Identification of the measured electron mass with the theoretical mass, after renormalization to take account of the energy of interaction with the radiation field, removes most of the divergence from .......

Calculations for the hydrogen atom show that  vanish unless one of the states in the transition is an S state. Even when it does not vanish, the renormalized  is always very small compared with the excitation frequency , and varies slowly with . For example, the magnitude of  for the  state of hydrogen is about
109 Hz, or roughly six orders of magnitude smaller than the state excitation energy..... The existence of level shifts was first demonstrated by Lamb and Retherford in experiments on radiative transition between the  state of hydrogen and the unshifted  state. The splitting between these states is known as the Lamb shift. [7]

However Equation [ V-38b ] is not complicated by divergences and, consequently, we easily obtain the famous Weisskopf-Wigner formul[8] for the spontaneous emission decay rate into the modes of a three-dimensional cavity
 .

      [ V-41 ]


[1] In Semiconductor Photonics we noted that Chow, Koch and Sargent in their Semiconductor-Laser Physics (Springer-Verlag - 1994) treat semiconductor problems in terms of the following semiclassical Hamiltonian for an inhomogeneous two-level system:


where  and  are, respectively, electron and hole operators  is the dipole matrix element between vertical states in the valence and conduction bands. In the fully quantal treatment the effective Hamiltonian becomes

where .

[2] In particular, using the identity operator -- i.e., 


[3] It should be noted that the rotation matrix

diagonalizes the Hamiltonian in Equation [ V-13 ] through the transformation
.
.Further,   relates the dressedand bare robability amplitudes as


where 

[4] For a multilevel atomic system the effective interaction Hamiltonian in second quantized form becomes


[5] W. Heitler, in Chapter II, Section 8 of The Quantum Theory of Radiation (3rd edition), Oxford Press (1954) uses contour integral arguments to shown that
 

.
To quote W. H. Louisell's summary in Chapter 5 of Radiation and Noise in Quantum Electronics,
"...if  is well behaved and has no poles at , then
where means the Cauchy principle part and is defined by
provided the limit on the right side exists."


[6] Bethe, H. A., Phys. Rev. 72, 339 (1947)

[7] From Chapter 8, Rodney Loudon, Quantum Theory of Light (1st edition), Oxford (1973)

[8] V. Weisskopf and E. Wigner, Z. Phys., 63, 54 (1930).

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This page was prepared and is maintained by R. Victor Jones, jones@deas.harvard.edu
Last updated April 18, 2000