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 $m_0\omega_0=1$, 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 $\Delta {\hat{x}}$ and $\Delta {\hat{p}}$ 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 ${\hat{a}}$. The essential steps of that definition can be followed just as well with respect to the generalized annihilation operator

$$\b\; := \; \mu{\hat{a}}+\nu{\hat{a}}^\dagger, \quad \mu,\nu\in\mathbb{C}$$ (A.38)

as a replacement for ${\hat{a}}$ -- the results of subsection A.3.1 may then be obtained by restricting to the special case $\mu=1$, $\nu=0$. As in that case I discuss the eigenstates $\left\vert \beta \right>_{\rm s}$ of $\b$ with respect to the eigenvalue $\beta\in\mathbb{C}$:

$$\b\left\vert \beta \right>_{\rm s} \; = \; \beta \left\vert \beta \right>_{\rm s}.$$ (A.39)

The motivation for referring to these states as squeezed states^A.11is outlined below on page .

Again, an explicit formula for $\left\vert \beta \right>_{\rm s}$ may be found by expanding it with respect to the eigenstates $\left\vert n \right>$ of the harmonic oscillator. From the ansatz

$$\left\vert \beta \right>_{\rm s} \; = \; \sum_{n=0}^\infty c_n \left\vert n \right>, \quad c_n\in\mathbb{C}$$ (A.40)

one gets for arbitrary $c_0$ the following recurrence relation for the coefficients $c_n$:

where the case $\mu=0$ has to be excluded; I do not discuss this particular case here any further, since for $\mu=0$ normalizable eigenstates of $\b$ do not exist anyway.

If $\mu$, $\nu$ satisfy $\vert\nu\vert<\vert\mu\vert$ then $\left\vert \beta \right>_{\rm s}$ can be normalized: For any $k>2$ one can choose constants $C\in\mathbb{R}^+$ and $n_0\in\mathbb{N}$ such that for $n>n_0$ it follows from $\left\vert c_n\right\vert^2 \leq Cn^{-k}$ that $\left\vert c_{n+1}\right\vert^2 \leq C(n+1)^{-k} \vert\nu\vert/\vert\mu\vert$. For $\vert\nu\vert<\vert\mu\vert$ one can then show by induction and application of the majorant criterion that the series $\sum_{n=0}^\infty \left\vert c_n\right\vert^2$ converges, thus giving the finite norm of $\left\vert \beta \right>_{\rm s}$. In the following, $c_0$ is always chosen in such a way that $\left\vert \beta \right>_{\rm s}$ is normalized. As an interim result, analogous to the findings concerning $\alpha$ in the previous subsection, one notes that every $\beta\in\mathbb{C}$ is an eigenvalue of $\b$, as long as $\vert\nu\vert<\vert\mu\vert$ holds.

For the above considerations any values $\mu$ and $\nu$ satisfying $\vert\nu\vert<\vert\mu\vert$ can be chosen. Moreover, the recurrence relation (A.57) shows that $\left\vert \beta \right>_{\rm s}$ depends on the quotients $\beta/\mu$ and $\nu/\mu$ only, such that without loss of generality either $\mu$ or $\nu$ can be chosen without any further restriction. In order to achieve as close an analogy between the operators ${\hat{a}}$ and $\b$ as possible, I require $\mu$ and $\nu$ to satisfy

$$\vert\mu\vert^2 - \vert\nu\vert^2 \; = \; 1,$$ (A.40)

whereby the condition for normalizability is automatically met, and the special case of coherent states, $\mu=1$ and $\nu=0$, is included as well. The choice (A.58) leads to

$$[\b,\b^\dagger]\; = \; [{\hat{a}},{\hat{a}}^\dagger] \; = \; 1;$$ (A.41)

this is one example for properties that carry over from the ordinary ladder operators ${\hat{a}},{\hat{a}}^\dagger$ to the generalized ladder operators $\b,\b^\dagger$. The analogy can be carried further by defining generalized number states $\left\vert n \right>_{\rm g}$,

which form an orthonormal set and behave in the same way with respect to $\b$ as the ordinary $\left\vert n \right>$ do with respect to ${\hat{a}}$:

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:

$$\left\vert \beta \right>_{\rm s} \; = \; e^{-\frac{1}{2}\ver... ... \frac{1}{\sqrt{n!}} \, \beta^n \left\vert n \right>_{\rm g}.$$ (A.40)

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 $\b$-representation of ${\hat{x}}$ and ${\hat{p}}$ that can be derived from equations (A.24) and (A.54),

the uncertainties of the position and momentum operators for the squeezed state $\left\vert \beta \right>_{\rm s}$ are obtained as

thus giving the uncertainty product

$$\frac{\hbar}{2} \; \leq \; (\Delta{\hat{x}})(\Delta{\hat{p}}) \; = \; \frac{\hbar}{2} \left\vert \mu^2-\nu^2 \right\vert.$$ (A.39)

Therefore, for states characterized by $\left\vert\mu^2-\nu^2\right\vert=1$ -- one example again being the coherent states with $\mu=1$, $\nu=0$ -- the uncertainty product takes on its minimum value. It is these ``squeezed states in the narrow sense'' that I referred to in the introduction of the present subsection.

The use of the term squeezed states for the eigenstates $\left\vert \beta \right>_{\rm s}$ of $\b$, including those with uncertainty product larger than $\hbar/2$, is motivated by the comparison of equation (A.64) with the uncertainties (A.44c, A.44d) of the coherent states $\left\vert \alpha \right>$: depending on $\mu$ and $\nu$, the $\Delta {\hat{x}}$, $\Delta {\hat{p}}$ for $\left\vert \beta \right>_{\rm s}$ can be made smaller than those for $\left\vert \alpha \right>$; in other words, the former can be ``squeezed'' [KS95]. In addition, in general $\Delta {\hat{x}}$ and $\Delta {\hat{p}}$ are not equal even if $m_0\omega_0=1$; 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) --

$$e^{{\textstyle -\frac{i}{\hbar}}\H_{\rm ho} t} \left\vert \b... ...ega_0 t}\beta \right>_{\rm s}^{\rm\mu,\exp(-2i\omega_0 t)\nu},$$ (A.40)

where the notation $\left\vert \beta \right>_{\rm s}^{\rm\mu,\nu}$ explicitly refers to those $\mu$, $\nu$ with respect to which $\b$, and thereby the squeezed state as well, is defined (see equation (A.54)). Combined with the formulae (A.64) this result shows that in general the uncertainties of position and momentum of squeezed states are not constant with time, but oscillate with frequency $2\omega_0$. Thus constancy of $\Delta {\hat{x}}$ and $\Delta {\hat{p}}$ for coherent states turns out to be a remarkable special case.

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

$$\b\; = \; \mu{\hat{a}}+\nu{\hat{a}}^\dagger \; = \; \frac{1}... ... \Big( m_0\omega_0(\mu+\nu){\hat{x}}+i(\mu-\nu){\hat{p}}\Big)$$ (A.41)

in a form analogous to equation (A.24), the only difference being that the oscillator frequency $\omega_0$ is now replaced with some $\kappa\in\mathbb{C}$:

$$\b\; = \; \frac{1}{\sqrt{2\hbar m_0\kappa}}\left(m_0\kappa{\hat{x}}+i{\hat{p}}\right).$$ (A.42)

From this it follows that $\mu^2-\nu^2=1$ 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 $\b$ are minimum uncertainty states. And since $\vert\mu\vert^2-\vert\nu\vert^2=1$ according to equation (A.58), $\mu$ and $\nu$ must be real numbers. This in turn means that $\kappa$ is real as well:

$$\kappa \; = \; (\mu+\nu)^2 \omega_0 \; = \; \left(1+2\nu^2+... ...+1}\right) \omega_0 \quad \mbox{with} \quad \nu\in\mathbb{R}.$$ (A.43)

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

Footnotes

... 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 $\b ^n$ and $(\b ^\dagger)^m$ with $n,m\geq 2$. See [Mar97,Nie97b] and references therein, where some applications are discussed as well.