Quantum Optics Course
Weizmann Institute of Science
Fall 2015
Nabla / Del :
Grad : The vector derivative of a scalar field
is called the gradient, and it can be represented as:
Divergence: The divergence of a vector field is a scalar function that can be represented as:
The divergence is roughly a measure of a vector field's increase in the direction it points; but more accurately, it
is a measure of that field's tendency to converge toward or repel from a point.
Curl
Rot(ation) or Curl describes the infintisimal rotation of a vector field:
The curl at a point is proportional to the on-axis torque to which a tiny pinwheel would be subjected if it were
centered at that point.
The vector product operation can be visualized as a pseudo-determinant:
Laplacian
The Laplace operator is a scalar operator that can be applied to either vector or scalar fields; for cartesian
coordinate systems it is defined as:
∇ ∙ (∇ × 𝐹 ) = 0
∇ × (∇𝑓) = 0
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Quantum Optics Course
Weizmann Institute of Science
Fall 2015
Maxwell Equations
Maxwell’s equations for the electromagnetic field are:
1.3
ε0 ∇ ∙ 𝐸 = 𝜌
1.4
∇∙𝐵 =0
1.1
1.2
∇×𝐸 =0−
1
μ0
𝜕𝐵
𝜕𝑡
𝜕𝐸
∇ × 𝐵 = 𝑗 + ε0 𝜕𝑡
Where  0 is the vacuum permittivity, 0 the vacuum permeability, j and  are the current and charge distributions.
In order to satisfy equation (1.4) automatically, we can always define the vector potential by
B=∇×𝐴
(2)
Substitute into 1.1:
∇×𝐸 =−
𝜕(∇×𝐴)
𝜕𝑡
𝜕𝐴
𝜕𝐴

𝐸 = − 𝜕𝑡 − ∇𝜙
𝜕𝐴

= −∇ × 𝜕𝑡
∇ × (𝐸 + 𝜕𝑡 ) = 0

𝜕𝐴
𝐸 + 𝜕𝑡 = −∇𝜙
(3)
We can then rewrite the Maxwell equations (1.2) and (1.3) using the identity    A      A    A as
2
1
μ0
𝜕2 𝐴
∇ × (∇ × 𝐴) = 𝑗 − ε0 𝜕𝑡 2 − ε0

and
Barak
1
μ0
1
𝜕∇ϕ
𝜕𝑡
𝜕2 𝐴
∇(∇ ∙ 𝐴) − μ ∇2 𝐴 + ε0 𝜕𝑡 2 = 𝑗 − ε0
0
𝜕
∇2 ϕ = 𝜕𝑡 (∇ ∙ 𝐴) − 𝜌/ε0
𝜕∇ϕ
𝜕𝑡
Quantum Optics Course
GAUGE
Weizmann Institute of Science
Fall 2015
These equations are still coupled. But we have an additional degree of freedom! Note that the transformation
A′ = 𝐴 − ∇χ

∇𝜙 ′ = −𝐸 −

𝜙 ′ = 𝜙 + 𝜕𝑡
𝜕𝐴′
𝜕𝑡
= ∇𝜙 +
𝜕∇χ
𝜕𝑡
𝜕χ
Leaves E and B unchanged. This is called gauge invariance. We can choose the Coulomb gauge, such that  A  0
(=transverse). This will allow us to partly disentangle A,  .
∇2 ϕ = −𝜌/ε0
This equation is already the familiar Poisson equation of electrostatics, an ‘instantaneous equation’.
∇2 𝐴 − ε0 μ0
𝜕 2𝐴
𝜕∇ϕ
= ε0 μ0
− μ0 𝑗
2
𝜕𝑡
𝜕𝑡
Now, let’s define longitudinal and transverse fields (by the Helmholtz decomposition) such that D  DT  DL and
  DT  0
  DL  0
Note that In the coulomb gauge, A is transverse because how we chose it, and  is longitudinal because of the rot
grad=0.
So we will now separate the equations to L and T:
𝜕 2𝐴
∇ 𝐴 − ε0 μ0 2 = −μ0 𝑗𝑇
𝜕𝑡
2
ε0 μ0
𝜕∇ϕ
𝜕𝑡
= μ0 𝑗𝐿 , which with the electrostatic equation combines to give the trivial continuity of charge:
𝜕
𝜌 = −∇𝑗𝐿
𝜕𝑡
Hence Maxwell’s equations can be reduced to two independent equations of motion.
AND THE OTHER without charges IS A WAVE EQUATION !!!! SPEED OF LIGHT !!
Note that E    t A
Should be END of 1st Hour
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 ET  t A,
EL  
 8
Quantum Optics Course
Barak
Weizmann Institute of Science
Fall 2015
Quantum Optics Course
Weizmann Institute of Science
Fall 2015
Now, assume a radiation field in a box with periodic boundary conditions. Consider the Fourier decomposition of A.
A
1
 0V

k
Ak  t  eik r
9
Where we require k x , y , z  mx , y , z 2 / L . The wave equation now becomes an algebraic one
k 2 c 2 Ak   t2 A k  0
The general solution is Ak  t   ck eik t  dk eik t
And so A    0V 
1/2

k
ck eik t kr   dk eik t kr  


9
*
where the frequency is defined by k  kc . But since we require that A be real ck  d  k
Ak .
Let’s now write the vectors in some unspecified orthonormal basis of the polarization
Ak   s 1,2 Ak , s ek , s  A    0V 
1/2

k ,s
 Ak , s ek , s eik r k t   Ak*, s e*k , s eik r k t  


10
Note that since we work in the Coulomb gauge   A  0  ek , s  kˆ  0 , explaining the name ‘transversal field’.
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Quantum Optics Course
Writing Ak ,s  t   Ak , s e
ET  i   0V 
1/2

k ,s
Weizmann Institute of Science
ik t
Fall 2015
, and using ET  t A . (We will drop the T later for convenience)
k  Ak ,s  t  ek ,s eikr  Ak*,s  t  e*k ,s eikr 
And from B   A , which is always transverse
B  i  0V 
1/2

k ,s
 Ak , s  t   k  ek , s  eik r  Ak*, s  t   k  e*k , s  eik r 


H  1/ 2   dV  0 ET2  B 2 / 0 
The energy of the radiation field is given by
11
Let’s calculate this expression using the orthonormality condition of the normal modes
0
E
2 
1
2m0
2
òB

2
1
2V
=

c2
2V
e
i  k  k ' x
dx  2  k  k '
k k '  dV  Ak , s ek , s eik r  Ak*, s e*k , s e  ik r   Ak* 'e*k ', s 'e  ik 'r  Ak ', s 'ek ', s 'eik 'r    k , s k2 Ak , s  RE
2
k ,k ', s , s '
òå
k,k '
s,s'
(
(
)
)
(
)
(
)
2
é A k ´ e eik×r - A* k ´ e* e -ik×r ùé A* k '´ e* e -ik '×r - A k '´ e
eik '×r ùû = å wk2 Ak,s + RB
k,s
k,s
-k,s
k ',s'
k ',s'
-k ',s'
ë k,s
ûë k ',s'
k,s
Homework 1.1: prove that the cross terms are opposite: RE   RB
We get a time independent sum of the energies
H   k , s 2k2 Ak , s  t    k , s 2k2 Ak , s   k , s k2  Ak , s Ak*, s  Ak*, s Ak , s 
2
2
We will now define real canonical variables q and p s.t.
So we can now write
H
1
  pk2,s  k2 qk2,s 
2 k ,s 
12 
Ak , s  t  

1
i
pk , s  t 
qk , s  t  
2
k

13
14 
Let’s prove that these are indeed canonical variables
qk , s  t   Ak , s  t   Ak*, s  t 
pk , s  t   i  Ak , s  t   Ak*, s  t  
Since we have Ak , s  t   Ak ,s eik t , we get
So q,p are canonical coordinates, and the Hamiltonian of the field is a system of uncoupled harmonic oscillators
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.
Quantum Optics Course
A
Weizmann Institute of Science
1
2
ET 
Summary:
B

V
k ,s
0



i


pk , s  t   ek , s eik r  c.c.
  qk , s  t  
k





  q  t   ip  t  e
V
i
k ,s
2
k
k ,s
k ,s
k ,s

eik r  c.c.
15.1
15.2 
0
i
2
Fall 2015

V
0
k ,s



i


pk , s  t    k  ek , s  eik r  c.c.
  qk , s  t  
k





15.3
We then treat the position and momentum as operators:  qˆk , s  t  , pˆ k ', s '  t    i kk ' s , s ' . (r and t are just parameters.)
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Quantum Optics Course
Barak
Weizmann Institute of Science
Fall 2015