258
J. Moran and V. Hussin
The states |n, m in configuration space have the following wavefunction:
x, y|n, m = ψ n (x)ψ m (y) =
1
√
2 n+m n!m!
1
π
e
−
x 2
2 −
y 2
2 H n (x) H m (y) ,
(8)
where ψ n (x) =
1
√
2 n n!
1
π
1
4 e
−
x 2
2 H n (x) is the wavefunction of the 1D oscillator
and H n (x) are the Hermite polynomials. For the physical position and momentum
operators, ˆ
X i =
1
√
2
(a
+
i + a
−
i ), ˆ
P i =
1
√
2i
(a
−
i − a
+
i ), respectively in the i direction,
the states |n, m have the following dispersions:
(Δ ˆ
X)
2
|n,m = (Δ ˆ
P x )
2
|n,m =
1
2
+ n;
(9)
(Δ ˆ
Y )
2
|n,m = (Δ ˆ
P y )
2
|n,m =
1
2
+ m.
(10)
They satisfy the Heisenberg uncertainty relation (Δ ˆ
X) |n,m (Δ ˆ
P x ) |n,m =
1
2 + n
which grows linearly in n in the x direction. Similarly for the Y quadratures, we
obtain (Δ ˆ
Y ) |n,m (Δ ˆ
P y ) |n,m =
1
2 + m.
In what follows we will construct two new ladder operators as linear combinations of the operators in (7) and proceed to define a single indexed Fock state for the
2D system which yields the SU (2) coherent states. The new ladder operators and
SU (2) coherent states are used to extend the definitions of the 1D squeezed states
in Sect. 2 to the 2D oscillator.
4 SU (2) Coherent States
We use the ladder operators presented in Sect. 3 to construct a single set of creation
and annihilation operators for the 2D oscillator. Introducing a set of states {|ν}, and
defining a new set of ladder operators through their action on the set,
A
−
|ν =
√
ν |ν − 1 , A
+
|ν =
√
ν + 1 |ν + 1 , ν|ν = 1, ν = 0, 1, 2, . . .
(11)
These states have a linear increasing spectrum E ν = ν + 1. We may build the states
by hand starting with the only non-degenerate state, the ground state, |0 ≡ |0, 0
and we take simple linear combinations of the 1D ladder operators
A
+
α,β = αa
+
x ⊗ I y + I x ⊗ βa
+
y ;
A
−
α,β = ¯
αa
−
x ⊗ I y + I x ⊗ ¯
βa
−
y ;
[A
−
α,β , A
+
α,β ] = (|α|
2
+ |β|
2 )I x ⊗ I y ≡ I,
(12)