22 The Stokes Parameters for Quantum Systems

The Stokes Parameters for Quantum Systems In previous chapters we saw that classical radiating systems could be represented in terms of the Stokes parameters and the Stokes vector.. In a

The Stokes Parameters for

Quantum Systems

In previous chapters we saw that classical radiating systems could be represented

in terms of the Stokes parameters and the Stokes vector In addition, we saw thatthe representation of spectral lines in terms of the Stokes vector enabled us toarrive at a formulation of spectral lines which corresponds exactly to spectroscopicobservations, namely, the frequency, intensity, and polarization Specifically, whenthis formulation was applied to describing the motion of a bound electron moving in

a constant magnetic field, there was a complete agreement between the Maxwell–Lorentz theory and Zeeman’s experimental observations Thus, by the end of thenineteenth century the combination of Maxwell’s theory of radiation (Maxwell’sequations) and the Lorentz theory of the electron appeared to be completelytriumphant The triumph was short-lived, however

The simple fact was that while the electrodynamic theory explained the ance of spectral lines in terms of frequency, intensity, and polarization there was still

appear-a very serious problem Spectroscopic observappear-ations appear-actuappear-ally showed thappear-at even for thesimplest element, ionized hydrogen gas, there was a multiplicity of spectral lines.Furthermore, as the elements increased in atomic number the number of spectrallines for each element greatly increased For example, the spectrum of iron showedhundreds of lines whose intensities and frequencies appeared to be totally irregular

In spite of the best efforts of nineteenth-century theoreticians, no theory was everdevised within classical concepts, e.g., nonlinear oscillators, which could account forthe number and position of the spectral lines

Nevertheless, the fact that the Lorentz–Zeeman effect was completelyexplained by the electrodynamic theory clearly showed that in many ways thetheory was on the right track One must not forget that Lorentz’s theory not onlypredicted the polarizations and the frequencies of the spectral lines, but even showedthat the intensity of the central line in the ‘‘three line linear spectrum (
¼ 90
)’’

would be twice as bright as the outer lines It was this quasi-success that was sopuzzling for such a long time.

Intense efforts were carried on for the first 25 years of the twentieth century

on this problem of the multiplicity of spectral lines The first real breakthroughwas by Niels Bohr in a paper published in 1913 Using Planck’s quantum ideas(1900) and the Rutherford model of an atom (1911) in which an electron rotatedaround a nucleus, Bohr was able to predict with great accuracy the spectrum

of ionized hydrogen gas A shortcoming of this model, however, was that eventhough the electron rotated in a circular orbit it did not appear to radiate, inviolation of classical electrodynamics; we saw earlier that a charged particlemoving in a circular orbit radiates According to Bohr’s model the ‘‘atomicsystem’’ radiated only when the electron dropped to a lower orbit; the phenomenon

of absorption corresponded to the electron moving to a higher orbit In spite of thedifficulty with the Bohr model of hydrogen, it worked successfully It was natural totry to treat the next element, the two-electron helium atom, in the same way Theattempt was unsuccessful

Finally, in 1925, Werner Heisenberg published a new theory of the atom, whichhas since come to be known as quantum mechanics This theory was a radicaldeparture from classical physics In this theory Heisenberg avoided all attempts tointroduce those quantities that are not subject to experimental observation, e.g., themotion of an electron moving in an orbit In its simplest form he constructed atheory in which only observables appeared In the case of spectral lines this was,

of course, the frequency, intensity, and polarization This approach was consideredeven then to be extremely novel By now, however, physicists had long forgotten that

a similar approach had been taken nearly 75 years earlier by Stokes The reader willrecall that to describe unpolarized light Stokes had abandoned a model based onamplitudes (nonobservables) and succeeded by using an intensity formulation(observables) Heisenberg applied his new theory to determining the energy levels

of the harmonic oscillator and was delighted when he arrived at the formula

En¼h!ðn þh 1=2Þ The significance of this result was that for the first time thefactor of 1/2 arose directly out of the theory and not as a factor to be added toobtain the right result Heisenberg noted at the end of his paper, however, that hisformulation ‘‘might’’ be difficult to apply even to the ‘‘simplest’’ of problems such asthe hydrogen atom because of the very formidable mathematical complexities

At the same time that Heisenberg was working, an entirely different approachwas being taken by another physicist, Erwin Schro¨dinger Using an idea put forth in

a thesis by Louis de Broglie, he developed a new equation to describe quantumsystems This new equation was a partial differential equation, which has sincecome to be known as Schro¨dinger’s wave equation On applying his equation to anumber of outstanding problems, such as the harmonic oscillator, he also arrived atthe same result for the energy as Heisenberg Remarkably, Schro¨dinger’s formula-tion of quantum mechanics was totally different from Heisenberg’s His formulation,unlike Heisenberg’s, used the pictorial representation of electrons moving in orbits in

a wavelike motion, an idea proposed by de Broglie

The question then arose, how could two seemingly different theories arrive

at the same results? The answer was provided by Schro¨dinger He discoveredthat Heisenberg’s quantum mechanics, which was now being called quantummatrix mechanics, and his wave mechanics were mathematically identical In a

very remarkable result Schro¨dinger showed that Heisenberg’s matrix elements could

be obtained by simply integrating the absolute magnitude squared of his wave tion solution multiplied by the variable over the volume of space This result isextremely important for our present problem because it provides the mechanismfor calculating the variables €x, €yand €zz in our radiation equation

equa-We saw that the radiation equations for E
and E were proportional to theacceleration components €xx €y, and €zz To obtain the corresponding equations forquantum mechanical radiating systems, we must calculate these quantities usingthe rules of quantum mechanics In Section 22.4 we transform the radiation equa-tions so that they also describe the radiation emitted by quantum systems In Section22.5 we determine the Stokes vectors for several quantized systems We therefore seethat we can describe both classical and quantum radiating systems by using theStokes vector

Before we carry this out, however, we describe some relationships betweenclassical and quantum radiation fields

AND QUANTUM MECHANICAL DENSITY MATRIX

In quantum mechanics the treatment of partially polarized light and the polarization

of the radiation emitted by quantum mechanical systems appears to be very differentfrom the classical methods In classical optics the radiation field is described interms of the polarization ellipse and amplitudes On the other hand, in quantumoptics the radiation field is described in terms of density matrices Furthermore, thepolarization of the radiation emitted by quantum systems is described in terms ofintensities and selection rules rather than the familiar amplitude and phase relations

of the optical field Let us examine the descriptions of polarization in classical andquantum mechanical terms We start with a historical review and then present themathematics for the quantum mechanical treatment

It is a remarkable fact that after the appearance of Stokes’ paper (1852) and hisintroduction of his parameters, they were practically forgotten for nearly a century!

It appears that only in France was the significance of his work fully appreciated.After the publication of Stokes’ paper, E Verdet expounded upon them (1862) Itappears that the Stokes parameters were thereafter known to French students ofopitcs, e.g., Henri Poincare´ (ca 1890) and Paul Soleillet (1927) The Stokesparameters did not reappear in any publication in the English-speaking worlduntil 1942, in a paper by Francis Perrin (Perrin was the son of the Nobel laureateJean Perrin Both father and son fled to the United States after the fall of France inJune 1940 Jean Perrin was a scientist of international standing, and he also appears

to have been a very active voice against fascism in prewar France Had both fatherand son remained in France, they would have very probably been killed during theoccupation.)

Perrin’s 1942 paper is very important because he (1) reintroduced the Stokesparameters to the English-speaking world, (2) presented the relation between theStokes parameters for a beam that underwent rotation or was phase shifted, (3)showed the connection between the Stokes parameters and the wave statistics ofJohn von Neumann, and (4) derived conditions on the Mueller matrix elementsfor scattering (the Mueller matrix had not been named at that date) Perrin also

stated that Soleillet (1927) had pointed out that only a linear relation could existbetween the Stokes parameter for an incident beam (Si) and the transmitted (orscattered) beam ðSi0Þ According to Perrin the argument for a linear relation was adirect consequence of the superposition of the Stokes parameters for n independentbeams; only a linear relation would satisfy this requirement This is discussed further

in this section The impact of his paper did not appear for several years, because ofits publication during the Second World War As a result, even by 1945 the Stokesparameters were still not generally known

The question of the relation between the classical and quantum representation

of the radiation field only appears to have arisen after the ‘‘rediscovery’’ of Stokes’

1852 paper and the Stokes parameters by the Nobel laureate SubrahmanyanChandrasekhar in 1947, while writing his fundamental papers on radiative transfer.Chandrasekhar’s astrophysical research was well known, and consequently, hispapers were immediately read by the scientific community

Shortly after the appearance of Chandrasekhar’s radiative transfer papers,

U Fano (1949) showed that the Stokes parameters are a very suitable analyticaltool for treating problems of polarization in both classical optics and quantummechanics He appears to have been the first to give a quantum mechanicaldescription of the electromagnetic field in terms of the Stokes parameters; he alsoused the formalism of the Stokes parameters to determine the Mueller matrix forCompton scattering Fano also noted that the reason for the successful application

of the Stokes parameters to the quantum theoretical treatment of electromagneticradiation problems is that they are the observable quantities of phenomenologicaloptics

The appearance of the Stokes parameters of classical optics in quantum physicsappears to have come as a surprise at the time The reason for their appearance waspointed out by Falkoff and MacDonald (1951) shortly after the publication ofFano’s paper In classical and quantum optics the representations of completely(i.e., elliptically) polarized light are identical (this was also first pointed out byPerrin) and can be written as

However, the classical and quantum interpretations of this equation are quite ferent In classical optics 1and 2represent perpendicular unit vectors, and theresultant polarization vector for a beam is characterized by the complex ampli-tudes c1and c2 The absolute magnitude squared of these coefficients then yields theintensities jc1j2and jc2j2that one would measure through an analyzer in the direction

dif-of 1and 2 In the quantum interpretation 1and 2represent orthogonal ization states for a photon, but now jc1j2and jc2j2yield the relative probabilities for asingle photon to pass through an analyzer which admits only quanta in the states 1

polar-and 2, respectively

In both interpretations the polarization of the beam (photon) is completelydetermined by the complex amplitudes c1and c2 In terms of these quantities one candefine a 2  2 matrix with elements:

In quantum mechanics an arbitrary wave equation can be expanded into any desiredcomplete set of orthonormal eigenfunctions; that is,

Thus, the expectation value of F, hF i, is determined by taking the trace of the matrixproduct of F and .

In classical statistical mechanics the density function (p, q) in phase space,where p and q are the momentum and the position, respectively, is normalized by thecondition:

The polarization of electromagnetic radiation can be described by the vibration

of the electric vector For a complete description the field may be represented by twoindependent beams of orthogonal polarizations That is, the electric vector can berepresented by

where e1and e2are two orthogonal unit vectors and c1and c2, which are in generalcomplex, describe the amplitude and phase of the two vibrations From the two ex-pansion coefficients in (22-13) we can form a 2  2 density matrix Furthermore,from the viewpoint of quantum mechanics the equation analogous to (22-13) is given

by (22-1), which is rewritten here:

In complex notation, (22-14) is written as

Equation (22-14) can then be expressed as

¼ cos2
cos
sin ei

sin
cos
ei sin2

!ð22-19Þ

Complete polarization can be described by writing (22-1) in terms of a singleeigenfunction for each of the two orthogonal states Thus, we write

where we have set c1c1 and c2c2 equal to 1 to represent a beam of unit intensity

We can use (22-21a) and (22-21b) to obtain the density matrix for unpolarizedlight Since an unpolarized beam may be considered to be the incoherent superposi-tion of two polarized beams with equal intensity, if we add (22-21a) and (22-21b) thedensity matrix is

In general, a beam will have an arbitrary degree of polarization, and wecan characterize such a beam by the incoherent superposition of an unpolarizedbeam and a totally polarized beam From (22-19) the polarized contribution isdescribed by

P¼ c1c1 c1c2

c2c1 c2c2

ð22-23Þ

The density matrix for a beam with arbitrary polarization can then be written in theform:

1 If 0 < P < 1, then the beam is partially polarized

2 If P ¼ 0, then the beam is unpolarized

3 If P ¼ 1, then the beam is totally polarized

For P ¼ 0, we know that

U¼12

Equation (22-27) is the density matrix for a beam of arbitrary polarization

By the proper choice of pure states of polarization i, the part of the densitymatrix representing total polarization can be written in one of the forms given by(22-20) Therefore, we may write the general density matrix as

 ¼12

!

ð22-29Þ

Hence, any intensity measurement made in relation to these pure states will yield theeigenvalues:

Comparing c1an c2in (22-1) with E1and E2in (22-15) suggests that we set

We see that (22-33) are exactly the classical Stokes parameters (with a1 and a2

replacing, e.g., E0xand E0y as previously used in this text) Expressing (22-32) interms of the density matrix elements, 11 ¼c1c1 etc., the Stokes parameters arelinearly related to the density matrix elements by

Thus, the Stokes parameters are linear combinations of the elements of the 2  2density matrix.

It will be convenient to express (22-34a) by the symbol I for the intensity andthe remaining parameters of the beam by P1, P2and P3, so

We also recall that the Stokes parameters satisfy the condition:

There is one further point that we wish to make The wave function can beexpanded in a complete set of orthonormal eigenfunctions For electromagneticradiation (optical field) this consists only of the terms:

Using this wave function leads to the following expressions for the expectation values(see 22-9a) of the unit matrix and the Pauli spin matrices:

Further information on the quantum mechanical density matrices and theapplication of the Stokes parameters to quantum problems, e.g., Compton scatter-ing, can be found in the numerous papers cited in the references

PARAMETERS, DENSITY MATRIX, AND LINEARITY OF THE

MUELLER MATRIX ELEMENTS

It is worthwhile to discuss Perrin’s observations further It is rather remarkable that

he discussed the Stokes polarization parameters and their relationship to thePoincare´ sphere without any introduction or background While they appear tohave been known by French optical physicists, the only English-speaking references

to them are in the papers of Lord Rayleigh and a textbook by Walker Walker’stextbook is remarkably well written, but does not appear to have had a wide circula-tion It was in this book, incidentally, that Chandrasekhar found the Stokes polar-ization parameters and recognized that they could be used to incorporate thephenomenon of polarization in the (intensity) radiative transfer equations

As is often the case, because Perrin’s paper was one of the first papers on theStokes parameters, his presentation serves as a very good introduction to the subject.Furthermore, he briefly described their relation to the quantum mechanical densitymatrix

For completely polarized monochromatic light the optical vibrations may berepresented along the two rectangular axes as

where a1and a2are the maximum amplitudes and 1and 2are the phases The phasedifference between these components is

Ea ¼k1a1cosð!t þ 1þ1Þ þk2a2cosð!t þ 2þ2Þ ð22-44Þ

We note that this form is identical to the quantum mechanical form given by (22-1).The mean intensity of (22-44) is then

Ia¼12

h

ðk21þk22Þðha21i þ ha22iÞ: þ ðk21k22Þðha21i  ha22iÞ

þ2k1k2cosð12Þðh2a1a2cos iÞ

þ2k1k2sinð12Þðh2a1a2sin iÞi

h

ðk21þk22ÞS0þ ðk21k22ÞS1þ2k1k2cosð12ÞS2:þ2k1k2sinð12ÞS3i

ð22-47Þ

As we have seen, by choosing different combinations of a1and a2and 1and 2wecan determine S0, S1, S2, and S3 Equation (22-47) is essentially the equation firstderived by Stokes

The method used by Stokes to characterize a state of polarization may begeneralized and connected with the wave statistics of von Neumann Consider a

system of n harmonic oscillations of the same frequency subjected to small randomperturbations This may be represented by the complex expression:

nonhar-we wish to investigate Perrin noted that Soleillet first pointed out that, when a beam

of light passes through some optical arrangement, or, more generally, produces

a secondary beam of light, the intensity and the state of polarization of the emergentbeam are functions of those of the incident beam If two independent incident beamsare superposed, the new emergent beam will be, if the process is linear, the