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

1.Fock or "Number" States:[1]Aswe have seen, theFock ornumberstates

[ III-1 ] are complete set eigenstates of an important group of commuting observables --viz., and .Reprise of Characteristics and Properties of Fock States:2.a.The expectation value of the number operator and the:fractional uncertaintyassociated with a single Fock state

[ III-2a ]

[ III-2b ] b.

Expectation value of the fields associated with a single mode:

For one mode Equations [ II-24a ] and [ II-24b ] reduce to

[ III-3a ]

[ III-3b ] where

[ III-4a ]

[ III-4b ]

c.

Phase of field associated with single mode:

To obtain something analogous to the classical theory we would like to separate the creation and destruction operators (and, thus, the electric and magnetic field operators) into a product of amplitude and phase operators. Following Susskind and Glogower,[2] we define asuch thatphase operator,

[ III-5 ] Defined in this way, the basic properties of the phase operator may be evaluated from known properties of the creation, destruction and number operators. Inverting, we obtain

[ III-6 ] and since , it follows that

[ III-7 ] but only in this order!Operating on number states with the phase operators, we obtain from Equation [ I-26 ]

[ III-8 ] Consequently, theonly nonvanishing matrix elementsof the phase operator are

[ III-9 ] The phase operators defined by Equation [ III-36 ] do have the felicitous orclassically analogousproperty of revealinginformation, but unfortunately they are nonHermitian operators --magnitude independenti.e. -- and, hence,cannot represent observables. However, they may bepairedinto operators that are observables --viz.

[ III-10 ] which have the following nonvanishing matrix elements:

[ III-11 ]

Thesenearly commutingoperators [3] may be adopted as the quantum mechanical operators which represent (as we will demonstrate anon) the observable phase properties of the electromagnetic field.For the Fock state:

[ III-12a ]

[ III-12b ]

[ III-12c ] c.

The coordinate orSchrödinger representation of state:

Recall from Equations [ I-10a ] and [I-31] that

[ III-13 ] Therefore, the probability of eigenvalues q for a given Fock state is give by

[ III-14 ] d.

Approximate "localization" of a photon:[4]Of course a plane wave is distributed or "de-localized" in both time and space. Defining the "wave function for a photon" is a task fraught with danger,[5] but the simpler task of defining a wave function approximately localized at a given instant is relatively straight forward --viz.

[ III-14 ] Photon States ofWell-definedPhase:Consider the state defined by

[ III-15 ] Clearly, given the orthonormal properties of the number states. Essential question: Is this state an eigenstate of the phase operators? To answer the question we need to consider the following

potential eigenvalue equation:

[ III-16a ] Using Equations [ III-10 ] and [ III-10 ], we obtain

[ III-16b ] so that the state fails to be a strict eigenket of by terms that diminish faster than as . Similarly, we can see that diagonal matrix elements of and are given by

[ III-17a ]

[ III-17b ]

Reprise of Characteristics and Properties of Phase States:3.a.The expectation value of the number operator and the:fractional uncertaintyassociated with a state of well-defined phase

[ III-18a ]

[ III-18b ] b.

Expectation value of the fields associated with a single mode:From Equation [ III-3a ]

[ III-19 ] c.

Phase of field associated with single mode:

[ III-20a ]

[ III-20b ] d.

Probability of photon number:Finally, we may easily deduce the probability of findingnphotons (i.e.the photon statistics) in a particular state of well defined phase --viz.

[ III-50 ] We see that there is a equal, but small probability of any number: this agrees with the intuition that the magnitude of the field is completely undetermined if the phase is precisely known![6]:CoherentPhoton StatesIt would, indeed, be useful to have eigenstates of the(electric or magnetic field) --destruction operatorviz.

[ III-51 ] Reprise of Characteristics and Properties of Coherent States:a.The Fock state representation of the coherent state:Since.and , then and we are able to write arepresentativeof the sought state in the number state basis --viz.

[ III-52a ] or

[ III-52b ] Using the expansion of the identity operator, the eigenket becomes

[ III-53 ] To normalize the eigenket write

[ III-54 ] so that . Finally, we see that

[ III-55 ] is a normalize representation of the eigenkets of the destruction operator.

b.The expectation value of the number operator and the:fractional uncertaintyassociated with a coherent state

[ III-56a ]

[ III-56b ] c.Thus, we see that the fractional uncertainty diminishes with mean photon number!Expectation value of the electric field associated with a single mode:From Equation [ III-3a ]

[ III-57a ] where .[7]

[ III-57b ]

Electric Field Associated with a Coherent StateState with largest aState with smallest ad.

Probability of photon number:From the representation of the coherent state given in Equation [ III-55 ] we may easily deduce the probability of findingnphotons (the photon statistics) in a particular coherent state is given by aPoisson distributioncharacterized by the mean value . --viz.

[ III-58 ]

SAMPLE POISSON DISTRIBUTIONS - COHERENT STATE PHOTON STATISTICSe.

Phase of field associated with single mode:

[ III-59a] Unfortunately, it is not possible to evaluate this summation analytically. However, Carruthers [8] has given an asymptotic expansion which is valid for a large mean number of photons --viz.

[ III-59b] f.

Coherent states as a basis:As we will see presently, the coherent states are very useful in describing the quantized electromagnetic field, but, alas, there is a complication --the coherent states are not truly orthogonal!From Equation [ III-6 ] we see that

[ III-60 ] so that

[ III-61 ] That is,g.the eigenkets are approximately orthogonalonly when is large!The "displacement operator:"There are a growing and significant set of applications where it is useful to express the coherent states directly in terms of the vacuum state . If we use the number state generating rule --i.e.Equation [ I-27 ] -- the coherent state may be written in the form

[ III-62 ] If we make us of the Baker-Hausdorff theorem,[9]we may easily show that

[ III-63 ] so that may be interpreted as acreationoperator which generates a coherent state from the vacuum. (Its adjoint operator is adestructionoperator which destroys a state).In some treatments is described as the "displacement operator" (written ) [10] and the coherent states are called the "displaced states of the vacuum." [11] To explore this point of view (and to give some meaning to the phase of the coherent state eigenvalue), we may express in a two-dimensional, dimensionless "phase space" representation. To that end, following Equation [ I-16 ], we write the dimensionless coordinate as

[ III-64a ] and the dimensionless momentum as

[ III-64b ] so that

[ III-64c ]

and since these variables are canonical [12]

[ III-64d ]

Since

[ III-65 ]

the mode field (see Equation [II-24a])

[ III-66a ]

becomes

[ III-66b ]

Since has a coordinate space representation [13] and has a momentum representation Then

[ III-67a ] and

[ III-67b ] Thus, defines or generates a two-dimensional Taylor expansion when it acts on a function of and . In particular, if we take the "phase space" representation of the ground or vacuum state as the product of two Gaussians (see Equations [ I-10a ] and [ I-29 ]), then represents a shift or displacement of this "phase space" representation --i.e.

[ III-68 ] In light of Equation [ II-23b ], we can write

[ III-69 ] where .

h.The diagonal coherent-state representation of the density operator (Glauber-Sudarshan P-representation):It may be easily established that

[ III-70 ]

so that it seems quite reseasonable to look for a representation of the density matrix is the form

[ III-71 ] For a pure coherent state,Pis clearly a two-dimensional delta function

Example 1 -- Coherent state

[ III-72 ] In general, using Equation [ III-60 ] --i.e.

[ III-60 ] we may find a simple procedure for finding the P-representation by writing

[III-73 ] Thus, is the two-dimensional Fourier transform of the function and we may write

[ III-74 ] As a second example, consider a thermal radiation field described by a canonical ensemble

[III-75 ] where . Thus,

[III-76 ] and

[III-77 ] so that

[III-78 ] Thus, we can write

[III-79 ] and

[III-80 ] Finally, we see that

Example 2 -- Thermal radiation - a chaotic state

[III-81 ]

As a third example, consider Fock or number state. From Equation [ III-55 ] we see that

[III-82a ] and

[ III-82b ] so that

Example 3 -- Pure Fock or number state

[ III-82b ] i.

The Glauber-Sudarshan-Klauder "optical equivalence" theorem:Suppose we have some "normally ordered" function

[III-83 ] The expectation value is given by

[III-84 ] Using Equation [ III-71 ] we see that

[III-85a ] or, finally, the"optical equivalence" theorem

[III-85b ] j.

The Uncertainty Relationship for :Since we see from Equation [ III-64a ] that

[ III-86 ] where symbollizes the normally ordered expectation value of the operator . From Equation [III-85b ]

[ III-87 ]

[ III-88 ] Therefore, if we choose (and) such that , then and (squeezed states)!

[1] In what follows, for simplicity we drop the subscripts on the operators and state vectors with the obvious meaning that

, ,etc...

[2] Susskind, L. and Glogower, J., *Physics,***1**,
49 (1964)

[3] Also, it may be easily established that the matrix elements of their commutator are given by

[4] See Section 10.4.2 in Leonard Mandel and Emil Wolf,

[5] See Section 1.5.4 in Marlan O. Scully and M. Suhail
Zubairy, Quantum Optics*,* Cambridge Press (1997), ISBN 0-521-43458.

[6] The coherent state is a **Harvard invention**!
See R. J. Glauber, Phys. Rev. **131**, 2766 (1963).

[7] Similarly for the coherent state, so that .

[8] Carruthers, P. and Nieto, M. M., *Phys. Rev.
Lett.
***14**, 387 (1965)

[9] The Baker-Hausdorff theorem or identity may be stated as

where

For a proof, see, for example, Charles P. Slichter's *Principles
of Magnetic Resonance*, Appendix A or William Louisell's *Radiation
and Noise in Quantum Electronics*.

[10] We can (or rather you will) show that

and

[11] See *Elements of Quantum Optics*, Pierre
Meystre and Murray Sargent III, Spinger-Verlag (1991), ISBN 0-387-54190-X.

where

[13] If this unfamiliar, see Equations [ I-20 ] and
[ I-22 ] in the lecture notes entitled *The Interaction of Radiation
and Matter: Semiclassical Theory.*

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

Last updated April 11, 2000