Squeezed States

Besides the coherent states there exist other, more general types of
states with minimum uncertainty product (A.45). Among
these, the coherent states are distinguished by having the same
uncertainties
with respect to the position and momentum operators for
, as expressed by equation (A.46). Therefore
it seems reasonable to consider a broader class of states that satisfy
equation (A.45) but not equation
(A.46). These *squeezed states*
[Ken27,Yue76,HG88]
facilitate a very
compact
definition of the HUSIMI distribution
function in
subsection A.3.3 below.
The following exposition of the subject does not begin with a discussion
of standard deviations, but is organized
on the analogy of
subsection A.3.1.
The above observations concerning
and
can then be
concluded from the definitions.

In subsection A.3.1 the coherent states have been introduced
as the eigenstates of the annihilation operator . The essential steps
of that definition can
be followed just as well
with respect to the
*generalized annihilation operator*

The motivation for referring to these states as

Again, an explicit formula for
may be found
by expanding it with respect to the eigenstates
of the harmonic oscillator. From the ansatz

(A.40) |

where the case has to be excluded; I do not discuss this particular case here any further, since for normalizable eigenstates of do not exist anyway.

If , satisfy then can be normalized: For any one can choose constants and such that for it follows from that . For one can then show by induction and application of the majorant criterion that the series converges, thus giving the finite norm of . In the following, is always chosen in such a way that is normalized. As an interim result, analogous to the findings concerning in the previous subsection, one notes that every is an eigenvalue of , as long as holds.

For the above considerations any values and satisfying
can be chosen.
Moreover,
the
recurrence
relation
(A.57) shows that
depends on the
quotients
and
only, such that
without loss of generality
either
or can be chosen
without any further restriction.
In order to achieve as close an analogy between the operators and
as possible, I require and to satisfy

(A.41) |

which form an orthonormal set and behave in the same way with respect to as the ordinary do with respect to :

The generalized number states can be utilized to give an explicit expression for the squeezed states that is analogous to formula (A.35) for the coherent states:

Here, the expansion coefficients are explicitly known, as opposed to the merely implicit relation (A.57). I do not make any further use of the generalized number states in the present discussion.

Using
the eigenvalue equation (A.55) and
the -representation of and that can be derived from
equations (A.24) and (A.54),

the uncertainties of
the position and momentum operators for the squeezed state
are obtained as

thus giving
the uncertainty product

The use of the term *squeezed states* for the eigenstates
of ,
including those with uncertainty product larger than
,
is motivated by the comparison of equation (A.64) with
the
uncertainties
(A.44c, A.44d)
of the coherent states
: depending on and , the
,
for
can be made smaller
than those for
; in other words, the former
can be
``squeezed'' [KS95].
In addition, in general
and
are not
equal
even if ; this is also contrasted by the coherent
states for which equation (A.46) holds, expressing just
this equality.
More on squeezing -- with respect to the HUSIMI distribution --
can be found in section A.6.

For the time evolution of the squeezed states with respect to the
harmonic oscillator
Hamiltonian (A.47) one has -- in close analogy to
equation (A.48) --

(A.40) |

In the literature (e.g. in [HG88,Lee95]), the most
frequently studied special case of squeezed states is that one that
finally leads to the definition of the HUSIMI distribution function.
The starting point for this discussion is the attempt to rewrite the
generalized annihilation operator

(A.41) |

From this it follows that is a necessary condition for a formula (A.68) to hold. With equation (A.65) one can conclude that all squeezed states corresponding to such a are minimum uncertainty states. And since according to equation (A.58), and must be real numbers. This in turn means that is real as well:

The reverse is also true: squeezed states that correspond to real values of and have the minimal uncertainty product, and their associated generalized annihilation operator can always be written as in equation (A.68). This special case of squeezed states is essential for the considerations in the next subsection.

- ... states
^{A.11} -
In a further generalizing step
*higher order squeezed states*have been introduced as the eigenstates of powers of the generalized ladder operators and with . See [Mar97,Nie97b] and references therein, where some applications are discussed as well.