THE INTERACTION OF RADIATION AND MATTER: Q UANTUM THEORY PAGE 19 R. Victor Jones, May 2, 2000 19 III. REPRESENTATIONS OF PHOTON STATES 1. Fock or “Number” States : . 11 As we have seen, the Fock or number states n r k { } ≡∏ s σ n r k s σ [ III-1 ] are complete set eigenstates of an important group of commuting observables -- viz. H rad , N and r M . 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 : n N n = n [ III-2a ] ∆n = "uncertainty" [ ] = n N 2 n - n N n 2 { } = 0 [ III-2b ] b. Expectation value of the fields associated with a single mode : For one mode Equations [ II-24a ] and [ II-24b ] reduce to r E r r , t ( 29 = i ˆ e E a exp i r k ⋅ r r - i ϖ t [ ] - a † exp -i r k ⋅ r r - i ϖ t [ ] [ III-3a ] r H r r , t ( 29 = i ε 0 μ 0 E ˆ k × ˆ e [ ] a t (29 exp i r k ⋅ r r - i ϖ t [ ] - a † t (29 exp - i r k ⋅ r r - i ϖ t [ ] [ III-3b ] 11 In what follows, for simplicity we drop the r k subscripts on the operators and state vectors with the obvious meaning that n r k { } ⇒ n , a r k ⇒ a , etc.. .
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THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 19
R. Victor Jones, May 2, 200019
III. REPRESENTATIONS OF PHOTON STATES
1. Fock or “Number” States : .11
As we have seen, the Fock or number states
n r k { } ≡ ∏
s σn r
k s σ [ III-1 ]
are complete set eigenstates of an important group of commuting observables -- viz.H rad , N and
r M .
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:
n N n = n [ III-2a ]
∆n = "uncertainty" [ ] = n N 2 n − n N n2{ } = 0 [ III-2b ]
b. Expectation value of the fields associated with a single mode:
For one mode Equations [ II-24a ] and [ II-24b ] reduce to
r E
r r , t( ) = i ˆ e E a exp i
r k ⋅
r r − i ω t[ ] − a † exp −i
r k ⋅
r r − i ω t[ ]
[ III-3a ]
r H
r r , t( ) = i
ε0
µ 0
E ˆ k × ˆ e [ ] a t( ) exp ir k ⋅
r r − i ω t[ ] − a † t( ) exp − i
r k ⋅
r r − i ω t[ ]
[ III-3b ]
11 In what follows, for simplicity we drop the r k subscripts on the operators and state vectors with the obvious
meaning that n r k { } ⇒ n , a r
k ⇒ a , etc...
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 20
R. Victor Jones, May 2, 200020
where E=hω
2ε0 V
nr E n = 0
nr H n = 0 [ III-4a ]
∆E = nr E ⋅
r E n − n
r E n
2{ } = hωε0 V
n + 12( )
12 = 2 E n + 1
2( )1
2
∆H = nr H ⋅
r H n − n
r H n
2{ } =hω
µ 0 Vn + 1
2( )1
2 =ε0
µ 0
2 E n + 12( )
12
∆E ∆H = chωV
n + 12( ) =
ε0
µ 0
2E2 n + 12( )
[ 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,12 we define a phase operator , Φ such that
a ≡ N + 1( )12
exp i Φ( )
a† ≡ exp −i Φ( ) N +1( )1
2 [ 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
12 Susskind, L. and Glogower, J., Physics, 1 , 49 (1964)
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 21
R. Victor Jones, May 2, 200021
exp i Φ( ) ≡ N +1( )− 12 a
exp −i Φ( ) ≡ a† N +1( )− 12 [ III-6 ]
and since a a† =N + 1, it follows that
exp i Φ( ) exp −i Φ( ) = 1 [ III-7 ]
but only in this order! Operating on number states with the phase operators,
we obtain from Equation [ I-26 ]
exp i Φ( ) n = N +1( )−12 a n = N +1( )−1
2 n( ) 12 n −1 = n −1
exp − i Φ( ) n = a † N +1( )−12 n = a † n +1( )−1
2 n = n +1[ III-8 ]
Consequently, the only nonvanishing matrix elements of the phase operator
are
n −1 exp i Φ( ) n = 1
n + 1 exp −i Φ( ) n = 1[ 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.
n −1 exp i Φ( ) n ≠ n exp i Φ( ) n − 1∗
-- and, hence, cannot represent observables. However, they may be paired
into operators that are observables -- viz.
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 22
R. Victor Jones, May 2, 200022
cosΦ = 1
2exp i Φ( ) + exp −i Φ( ){ }
sinΦ = 1
2 iexp i Φ( ) − exp −i Φ( ){ }
[ III-10 ]
which have the following nonvanishing matrix elements:
n −1 cosΦ n = n cosΦ n −1 = 1
2
n −1 sin Φ n = − n sin Φ n −1 = 1
2 i
[ III-11 ]
These nearly commuting operators13 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:
n cos Φ n = n sinΦ n = 0 [ III-12a ]
∆ cos Φ = ∆sin Φ= n cos2 Φ n − n cosΦ n2{ } = 1
2 [ III-12b ]
∆ cosΦ ∆ sin Φ = 12
[ III-12c ]
c. The coordinate or Schrödinger representation of state:
Recall from Equations [ I-10a ] and [I-31] that
13 Also, it may be easily established that the matrix elements of their commutator are given by
n cos Φ,sinΦ[ ] ′ n =i
2δn ′ n δn0
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 23
R. Victor Jones, May 2, 200023
q n =1
n !
m
2 hω
n
ω q−hm
d
dq
n
q 0
=1
2n n!
ωπ h
H n
m ωh
q
exp −
m ω2 h
q2
[ III-13 ]
Therefore, the probability P q( ) of eigenvalues q for a given Fock state n is give
by
P q( ) = q n =1
2n n!
ωπ h
H n2 m ω
hq
exp −
m ωh
q2
[ III-14 ]
d. Approximate “localization” of a photon: 14
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,15 but the
simpler task of defining a wave function approximately localized at a given instant
is relatively straight forward -- viz.
ψr r 0( ) r
k 0 ∆r k
2 = C exp
r k −
r k 0
2
2 ∆r k
2
exp ir k ⋅
r r 0[ ] 0,0,0,Knr
k =1,K ,0,0,0r k
∑ [ III-14 ]
2. Photon States of Well-defined Phase :
Consider the state defined by
ϕ ≡ lims →∞
s + 1( )− 12 exp in ϕ[ ] n
n= 0
s
∑ [ III-15 ]
14 See Section 10.4.2 in Leonard Mandel and Emil Wolf, Optical Coherence and Quantum Optics, Cambridge
Press (1995), ISBN 0-521-417112.
15 See Section 1.5.4 in Marlan O. Scully and M. Suhail Zubairy, Quantum Optics, Cambridge Press (1997),ISBN 0-521-43458.
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 24
R. Victor Jones, May 2, 200024
Clearly, ϕ ϕ =1 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:
cosΦ ϕ =1
2lims→∞
s+1( )−12 exp i n ϕ[ ] exp i Φ[ ] n + exp in ϕ[ ] exp − i Φ[ ] n
n=0
s
∑n=0
s
∑
[ III-16a ]
Using Equations [ III-10 ] and [ III-10 ], we obtain
cosΦ ϕ =1
2lims→∞
s +1( )− 12 exp in ϕ[ ] n − 1 + exp i n ϕ[ ] n +1
n= 0
s
∑n =1
s
∑
=1
2lims→∞
s +1( )− 12 exp iϕ( ) exp i ν ϕ[ ] ν + exp −i ϕ[ ] exp i ν ϕ[ ] ν
ν = 1
s +1
∑ν = 0
s− 1
∑
= cosϕ ϕ
+ 12
lims→∞
s +1( )− 12 exp i s ϕ[ ] s +1 − exp i s + 1( ) ϕ[ ] s − exp −i ϕ[ ] 0{ }
[ III-16b ]
so that the state ϕ fails to be a strict eigenket of cosΦ by terms that diminish faster
than s + 1( )− 12 as s →∞ . Similarly, we can see that diagonal matrix elements of cosΦ
and sinΦ are given by
ϕ cos Φ ϕ = cosϕ 1− lims →∞
s + 1( )−1{ } ⇒ cosϕ [ III-17a ]
ϕ sin Φ ϕ = sinϕ 1− lims→∞
s + 1( )−1{ } ⇒ sinϕ [ III-17b ]
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 25
R. Victor Jones, May 2, 200025
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:
ϕ N ϕ = lim
s →∞s + 1( )−1
nn = 0
s
∑ = lims→∞
s +1( )− 1 s s +1( )2
= lim
s→∞
s
2[ III-18a ]
fractional
uncertainty
=
ϕ N 2 ϕ − ϕ N ϕ 2{ }ϕ N ϕ
=
lims→∞
s+1( )−1n2
n=0
s
∑ − lims→∞
s+1( )−1n
n=0
s
∑
2
lims→∞
s+1( )−1n
n=0
s
∑
=lims→∞
1
62s2 + s( ) −
1
4s2
lims→∞
s2
= 1
3
[ III-18b ]
b. Expectation value of the fields associated with a single mode:
From Equation [ III-3a ]
ϕr E ϕ =− 2
h ω2 ε0 V
ˆ e sinr k ⋅
r r − ω t + ϕ( ) lim
s →∞s + 1( )−1 n + 1( )1
2
n= 0
s
∑⇒ diverges as s for large s !
[ III-19 ]
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 26
That is, the eigenkets are approximately orthogonal only when α−β 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 0 . If we use the number
state generating rule
n =a †
n
n !0
-- i.e. Equation [ I-27 ] -- the coherent state may be written in the form
α = exp −1
2α
2
α a †
n
n !0
n
∑ = exp α a † −1
2α
2
0 [ III-62 ]
If we make us of the Baker-Hausdorff theorem,19 we may easily show that
19 The Baker-Hausdorff theorem or identity may be stated as
exp A +B{ } = exp A{ } exp B{ } exp − 12 A ,B[ ]{ }
when A , A ,B[ ][ ] = B , A ,B[ ][ ] = 0 . 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.
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 32
R. Victor Jones, May 2, 200032
α = A † α( ) 0 = exp α a † − α∗ a
0 [ III-63 ]
so that A † α( ) may be interpreted as a creation operator which generates a
coherent state from the vacuum. (Its adjoint operator A α( ) = A † −α( ) is a
destruction operator which destroys a state). In some treatments A † α( ) is
described as the “displacement operator” (written D α( ) )20 and the coherent states
are called the “displaced states of the vacuum.” 21
To explore this point of view (and to give some meaning to the phase of thecoherent state eigenvalue), we may express α in a two-dimensional,
dimensionless “phase space” representation. To that end, following Equation
[ I-16 ], we write the dimensionless coordinate as
θ =2 mω
h
12
q = a † exp i γ[ ] +a exp − i γ[ ] [ III-64a ]
and the dimensionless momentum as
π = 2
m hω
12
p = a † exp i γ + π 2( )[ ] +a exp − i γ + π 2( )[ ] [ III-64b ]
so that θ , π[ ] = 2 i a ,a †
= 2 i [ III-64c ]
20 We can (or rather you will) show that D † α( ) a D α( ) = a + α and D † α( ) a † D α( ) = a † + α∗
21 See Elements of Quantum Optics, Pierre Meystre and Murray Sargent III, Spinger-Verlag (1991), ISBN 0-387-54190-X.
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 33
R. Victor Jones, May 2, 200033
and since these variables are canonical 22
∆ θ( )2∆ π( )2
≥ 1 [ III-64d ]
Since
a † = 1
2θ− i π( ) exp − i γ[ ]
a =12
θ + i π( ) exp i γ[ ][ III-65 ]
the mode field (see Equation [II-24a]) b
r E
r r , t( ) = i ˆ e E a exp i
r k ⋅
r r − i ω t[ ] − a † exp − i
r k ⋅
r r + i ω t[ ]
[ III-66a ]
becomes
r E
r r , t( ) = − ˆ e E π cos
r k ⋅
r r −ω t +γ( ) +θ sin
r k ⋅
r r −ω t +γ( ){ } [ III-66b ]
Since p has a coordinate space representation −i h d dq = − i h ω 2( ) 12 d d θ
and q has a momentum representation i h d d p = i h 2 ω( ) 12 d d π , 23
αa † − α∗ a = αr a † − a
+ i αi a † +a
=− αr d d θ +α i d d π[ ] [ III-67a ]
22 Of course, in general ∆ A( )2∆ B( )2
≥1
2A ,B[ ]
2
where ∆ A( )2= A 2 − A
2
23 If this unfamiliar, see Equations [ I-20 ] and [ I-22 ] in the lecture notes entitled The Interaction of Radiationand Matter: Semiclassical Theory.
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 34
R. Victor Jones, May 2, 200034
and
A † α( ) = exp αa † − α∗ a
= exp − αr d d θ +α i d d π( )[ ] [ III-67b ]
Thus, A† α( ) defines or generates a two-dimensional Taylor expansion when it
acts on a function of θ and π . In particular, if we take the “phase space”representation of the ground or vacuum state θπ as the product of two Gaussians
(see Equations [ I-10a ] and [ I-29 ]), then A † α( ) θπ represents a shift or
displacement of this “phase space” representation -- i.e.
θπ α = θπ A † α( ) 0 = uG θ− αr( ) uG π − αi( ) [ III-68 ]
In light of Equation [ II-23b ], α t( ) = α exp − i ω t( ) we can write
θπ α t( ) = uG θ− α cos ω t +φ( )( ) uG π − α cos ω t +φ( )( ) [ III-69 ]
where α= α exp i φ( ) .
h. The diagonal coherent-state representation of the density operator
(Glauber-Sudarshan P-representation):
It may be easily established that
1 =1
πβ β d2 β∫∫ = β β d Re β( ) d Im β( )∫∫ [ III-70 ]
so that it seems quite reseasonable to look for a representation of the density matrix
is the form
ρ = P β( ) β β d2 β∫∫ [ III-71 ]
For a pure coherent state, P is clearly a two-dimensional delta function
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 35
R. Victor Jones, May 2, 200035
Example 1 -- Coherent state
P β( ) = δ 2( ) β− α( ) =δ 1( ) Re β( ) − Re α( )( ) δ 1( ) Im β( ) − Im α( )( ) [ III-72 ]
In general, using Equation [ III-60 ] -- i.e.
β α = exp −1
2α
2−
1
2β
2+αβ∗
[ III-60 ]
we may find a simple procedure for finding the P-representation by writing
−α ρ α = P β( ) −α β β α d2 β∫∫= exp − α
2( ) P β( ) exp −β2( )
exp αβ∗ −β α∗[ ] d2 β∫∫[III-73 ]
Thus, −α ρ α exp − α2( ) is the two-dimensional Fourier transform 0f the
function P β( ) exp −β2( ) and we may write
P β( ) =1
π 2 exp β2( ) −α ρ α exp α
2( )∫∫ exp −αβ∗ +β α∗[ ] d2 α [ III-74 ]
As a second example, consider a thermal radiation field described by a canonical
ensemble
ρ =exp − H kB T( )
Tr exp − H kB T( )[ ] [III-75 ]
where H = hω a † a +1
2
. Thus,
ρ = 1− exph ωkB T
exp −n hωkB T
n
∑ n n [III-76 ]
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 36
R. Victor Jones, May 2, 200036
and n = Tr ρ a † a
= exp
hωkB T
−1
−1
n
∑ [III-77 ]
so that ρ =n
n
1+ n( )n+1n
∑ n n [III-78 ]
Thus, we can write n ρ n =n
n
1+ n( )n+1 [III-79 ]
and
−α ρ α =n
n
1+ n( )n+1n
∑ −α n n α
=exp −α 2( )
1+ n
−α 2( )n
n!n
∑ n
1+ n( )
n
=exp −α 2( )
1+ nexp −α 2
1+ 1n
[III-80 ]
Finally, we see that
Example 2 -- Thermal radiation - a chaotic state
P α( ) =exp α 2( )π 2 1+ n( ) exp −β 2
1+ 1n
exp −βα * + α β *( )∫∫ d2 β
=1
π nexp − α 2
n( )[III-81 ]
As a third example, consider Fock or number state. From Equation [ III-55 ] we
see that
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 37
R. Victor Jones, May 2, 200037
−α ρ α = −α n n α =exp − α 2( )
n !−α
2( )n
[III-82a ]
and
P β( ) = 1
n!
1
π 2exp β 2( ) −α 2( )n
∫∫ exp −αβ∗ +β α∗[ ] d2 α
=exp β 2( )
n !
∂ 2 n
∂β *n∂β
n
1
π 2exp −αβ∗ +βα∗[ ] d2 α∫∫
[ III-82b ]
so that
Example 3 -- Pure Fock or number state
P β( ) =exp β 2( )
n!
∂ 2 n
∂β *n∂β
n δ 2( ) β( ) [ III-82b ]
i. The Glauber-Sudarshan-Klauder “optical equivalence” theorem:
Suppose we have some “normally ordered” function
f N( ) a ,a †
= c n m a †n a m
m
∑n
∑ [III-83 ]
The expectation value is given by
f N( ) a ,a †
= Tr ρ f N( ) a ,a †
[III-84 ]
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 38
R. Victor Jones, May 2, 200038
Using Equation [ III-71 ] we see that
f N( ) a ,a †
= Tr P α( ) c n m α α a †n a m
m
∑n
∑ d2 α∫∫
= P α( ) c n m α a †n a m αm
∑n
∑ d2 α∫∫= P α( ) c n m α*n
αm d2 αm
∑n
∑∫∫
[III-85a ]
or, finally, the “optical equivalence” theorem
f N( ) a ,a †
= P α( ) f N( ) α, α*( )∫∫ [III-85b ]
j. The Uncertainty Relationship for θ , π{ } :
Since a ,a †
=1 we see from Equation [ III-64a ] that
∆ θ 2 = θ 2 − θ 2
= a † a † exp 2 i γ[ ] + a a exp −2 i γ[ ] + a † a + a a †
− a †2
exp 2 i γ[ ] − a 2exp 2 i γ[ ] − 2 a † a
= : ∆ θ 2 : +1
[ III-86 ]
where : A : symbollizes the normally ordered expectation value of the operator
A . From Equation [III-85b ]
: ∆ θ 2 a ,a †
: = P α( ) ∆ θ 2 α, α*( )∫∫ d2 α [ III-87 ]
THE INTERACTION OF RADIATION AND MATTER: QUANTUM THEORY PAGE 39
R. Victor Jones, May 2, 200039
: ∆ θ 2 a ,a †
: = P α( ) ∆ α* exp i γ( ) + ∆ α exp − i γ( )[ ]2
d2 α∫∫ [ III-88 ]
If we choose γ (and P α( ) ) such that : ∆ θ 2 a ,a †