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

III. Representations of Photon States (pdf)
 
1. Fock or "Number" States: [1]
As we have seen, the Fock or number states
     [ III-1 ]
are complete set eigenstates of an important group of commuting observables -- viz., and .
Reprise of Characteristics and Properties of Fock States:
a. The expectation value of the number operator and the fractional uncertainty associated 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 a phase operator, such that
     [ 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, the only nonvanishing matrix elements of the phase operator are
     [ III-9 ]
The phase operators defined by Equation [ III-36 ] do have the felicitous or classically analogous property of revealing magnitude independent information, but unfortunately they are nonHermitian operators -- i.e.
-- and, hence, cannot represent observables. However, they may be paired into operators that are observables -- viz.
     [ III-10 ]
which have the following nonvanishing matrix elements:
     [ III-11 ]

 
These nearly commuting operators [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 or Schrödinger 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 ]
2. Photon States of Well-defined Phase:
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 followingpotential 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:

a. The expectation value of the number operator and the fractional uncertainty associated 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 finding n photons (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!
3. Coherent Photon States: [6]
It would, indeed, be useful to have eigenstates of the destruction operator (electric or magnetic field) -- viz.
 
     [ 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 a representative of 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 uncertainty associated with a coherent state:
 

      [ III-56a ]

 
     [ III-56b ]
 
Thus, we see that the fractional uncertainty diminishes with mean photon number!
c. 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 State
State with largest a
State with smallest a

d. Probability of photon number:

From the representation of the coherent state given in Equation [ III-55 ] we may easily deduce the probability of finding n photons (the photon statistics) in a particular coherent state is given by a Poisson distribution characterized by the mean value . -- viz.
     [ III-58 ]

SAMPLE POISSON DISTRIBUTIONS - COHERENT STATE PHOTON STATISTICS

e. 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, the eigenkets are approximately orthogonal only when  is large!
g. 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 a creation operator which generates a coherent state from the vacuum. (Its adjoint operator  is a destructionoperator 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, P is 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, Optical Coherence and Quantum Optics, Cambridge Press (1995), ISBN 0-521-417112.

[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.

[12] Of course, in general

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.
 

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This page was prepared and is maintained by R. Victor Jones, jones@deas.harvard.edu
Last updated April 11, 2000