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

"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 ofandcreationoperators. To that end we make the following definitiondestruction

[ V-1 ] Using the ubiquitous identity operation , we may write the material Hamiltonian in

form --second quantizedviz.

[ V-2 ] In general, the operator applied to any state yields

[ V-3 ] --

i.e.the operator changes a statezto a statexif the state isyotherwise it produces zero. In other words, the operatordestroysthe stateyandcreatesa statex. 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 matricesviz.

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

and 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 aselectric field per photon

[ 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 entitledThe 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 approximationthis reduces to

[ V-11b ] In the spirit of the discussion in the Section VII,

Semiconductor Photonicsof the lecture set entitledThe 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 Theorywhere the semiclassical Rabi frequency is replaced by its quantum equivalent --viz.. Diagonalizing this matrix we find the eigenvalues of the so call--dressed atomic statesviz.[3]

[ V-14 ] (see energy level diagram below) where is the

quantized field generalizationof theRabi flopping frequency.The dressed eigenstates are given by

[ V-15a ] where

[ V-15b ] and

[ V-15c ]

DRESSEDENERGY LEVELS OF TWO-STATE ATOMThe time evolution of a state is directly represented in terms of these dressed states --

viz.

[ V-16a ] or more explicitly

[ V-16b ] Therefore, the

floppingof 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 emissioni.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

uncorrelatedwe may, for simplicity, dispense with the factors --i.e.we will neglect, for the present, any possibleand writeinterference effects --

[ 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-conservingterms --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

of the coupled system. If we define the row vector and the column vector then Equation [ V-22 ] becomesdressedstates

[ 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 columnand

[ 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

Finally, we see that

[ V-28e ]

[ 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

for the frequency dependent susceptibility (see Equation [ VA-12 ] becomesstandardized form

[ V-33 ] By eliminating the U's from Equations [ V-26a ] and [ V-26b ] we see that

Again

[ V-34 ]

[ 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 thatand 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

10^{9}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

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.

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 ofRadiation 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).

This page was prepared and is maintained by R. Victor Jones,

Last updated April 18, 2000