256
J. Moran and V. Hussin
Work on degeneracy in coherent state theory has been done, Klauder described
coherent states of the hydrogen atom [2] which preserved many of the usual
properties required by coherent state analysis [3]. Fox and Choi proposed the
Gaussian Klauder states [4], an alternative method for producing coherent states
for more general systems with degenerate spectra. An analysis of the connection
between the two definitions was studied in [5].
In the first part of the paper we address the degeneracy in the energy spectrum
by constructing non-degenerate states, the SU (2) coherent states, and we define
a generalized ladder operator formed from a linear combination of the 1D ladder
operators with complex coefficients.
In the last part of the paper we use a generalized squeezing operator and Fock
space expansion to define squeezed states for the 2D system. In both cases we use
the same definitions as for the 1D squeezed states, but replacing the Fock states with
the SU (2) coherent states and the 1D ladder operators with the new generalized
ladder operators. We discuss the spatial probability distributions of the 2D squeezed
states, as well as their dispersions.
2 Squeezed States of the 1D Harmonic Oscillator
Squeezed states, or squeezed coherent states, are a generalization of the standard
coherent states first studied by Schrödinger [6], and then formalized in the context of
quantum optics by Glauber and Sudarshan [7, 8]. In terms of the displacement and
squeezing operators D(ψ) = e ψa † − ¯
ψa , S(ξ ) = e
1
2 (ξ a † 2 − ¯
ξa 2 ) respectively, where
a, a † are the annihilation and creation operators, squeezed states are expressed as
|ψ, ξ = D(ψ)S(ξ ) |0 ,
(1)
ψ, ξ ∈ C. Writing ξ = re iθ , in terms of Fock states, {|n}, the squeezed states are
given by
|z, γ =
1
N (z, γ )
∞
n=0
1
√
n!
γ
2
n
2
H n
z
√
2γ
|n ,
(2)
where
1
N (z,γ ) =
1
√
cosh r
e
−
|z| 2
2 e
tanh r
2 Re(e iθ ¯
z 2 ) . The states in Eq. (2) are solutions to the
eigenvalue equation
(a + γ a
† ) |z, γ = z |z, γ .
(3)
Equivalence between definitions (1) and (2) is understood through the following
relationships between the parameters [9]:
J. Moran and V. Hussin
Work on degeneracy in coherent state theory has been done, Klauder described
coherent states of the hydrogen atom [2] which preserved many of the usual
properties required by coherent state analysis [3]. Fox and Choi proposed the
Gaussian Klauder states [4], an alternative method for producing coherent states
for more general systems with degenerate spectra. An analysis of the connection
between the two definitions was studied in [5].
In the first part of the paper we address the degeneracy in the energy spectrum
by constructing non-degenerate states, the SU (2) coherent states, and we define
a generalized ladder operator formed from a linear combination of the 1D ladder
operators with complex coefficients.
In the last part of the paper we use a generalized squeezing operator and Fock
space expansion to define squeezed states for the 2D system. In both cases we use
the same definitions as for the 1D squeezed states, but replacing the Fock states with
the SU (2) coherent states and the 1D ladder operators with the new generalized
ladder operators. We discuss the spatial probability distributions of the 2D squeezed
states, as well as their dispersions.
2 Squeezed States of the 1D Harmonic Oscillator
Squeezed states, or squeezed coherent states, are a generalization of the standard
coherent states first studied by Schrödinger [6], and then formalized in the context of
quantum optics by Glauber and Sudarshan [7, 8]. In terms of the displacement and
squeezing operators D(ψ) = e ψa † − ¯
ψa , S(ξ ) = e
1
2 (ξ a † 2 − ¯
ξa 2 ) respectively, where
a, a † are the annihilation and creation operators, squeezed states are expressed as
|ψ, ξ = D(ψ)S(ξ ) |0 ,
(1)
ψ, ξ ∈ C. Writing ξ = re iθ , in terms of Fock states, {|n}, the squeezed states are
given by
|z, γ =
1
N (z, γ )
∞
n=0
1
√
n!
γ
2
n
2
H n
z
√
2γ
|n ,
(2)
where
1
N (z,γ ) =
1
√
cosh r
e
−
|z| 2
2 e
tanh r
2 Re(e iθ ¯
z 2 ) . The states in Eq. (2) are solutions to the
eigenvalue equation
(a + γ a
† ) |z, γ = z |z, γ .
(3)
Equivalence between definitions (1) and (2) is understood through the following
relationships between the parameters [9]: