Typical Applications

The first point to note when it comes to the practical application of phase space distribution functions is that their numerical computation for a given quantum state $\left\vert \psi(t) \right>$ does not cost as much effort as it might seem at first sight of equations like (A.13). For example, the WIGNER function can be written as

and for the HUSIMI distribution one has

These expressions show that both $F^{\rm W}$ and $F^{\rm H}$ can be computed efficiently from $\left< x \left\vert \psi(t) \right> \right.$ by FOURIER transformation, and the same is true for other $F^f$ as well. Typically the method of fast FOURIER transformation (FFT) [CT65,EMR93] is employed here which drastically reduces the numerical effort, as compared with conventional methods of computation.^A.14

In the literature, initially mainly the WIGNER function has been used. But in the last years the situation has changed in favour of the HUSIMI function which is in widespread use by now. In the following I want to motivate this transition with two arguments, the first being analytical while the second is a numerical example of a typical application.

In the context of HUSIMI functions, equation (A.31) can be written as

$$F^{\rm H}(x,p,t;\zeta) \; = \; \frac{1}{\pi\hbar} \int\!\! ... ...int\!\! {\mbox{d}}p'\; G(x-x',p-p';\zeta) F^{\rm W}(x',p',t),$$ (A.51)

where $G(x,p;\zeta)$ is a Gaussian in phase space:

$$G(x,p;\zeta) \; = \; e^{ -{\textstyle \frac{\zeta}{\hbar}}x^2-{\textstyle \frac{1}{\hbar\zeta}}p^2}.$$ (A.52)

The convolution (A.89) of the WIGNER function with $G(x,p;\zeta)$ effectively is an averaging of $F^{\rm W}(x,p,t)$ in the neighbourhood of the phase space point $(x,p)^t$, whereby the spatially rapidly oscillating WIGNER function gets smoothed. One result in particular of this smoothing process is the fact that the HUSIMI function is non-negative, as opposed to the WIGNER function. Similarly, the smoothing process also transforms the (in general) unbounded WIGNER function into a bounded function (see equation (A.86b)).

The numerical example below shows that the rapid oscillations of the WIGNER function have no obvious counterpart in the classical phase portrait, whereas the phase space patterns of the HUSIMI function follow the classical structures much more closely. In this sense the smoother HUSIMI function can be interpreted as a (quasi-) probability density in a much more intuitive way than the WIGNER function. This property makes $F^{\rm H}$ the first choice distribution function for the purpose of visualization of quantum nonlinear dynamical systems, especially when the classical dynamics is chaotic.

A parallel argument explains why the normal-ordered distribution function is utilized rarely and why the anti-HUSIMI distribution function is not used at all for creating an intuitive picture of a quantum state. Similar to equation (A.89) one has

$$F^{\rm W}(x,p,t) \; = \; \frac{1}{\pi\hbar} \int\!\! {\mbox... ...\mbox{d}}p'\; G(x-x',p-p';\zeta) \; F^{\rm AH}(x',p',t;\zeta)$$ (A.53)

(where one again obtains the normal-ordered case $F^{\rm N}(x,p,t)$ from $F^{\rm AH}(x,p,t;\zeta)$ by setting $\zeta=m_0\omega_0$), which implies that the WIGNER function itself can be viewed as a smoothed version of the anti-HUSIMI distribution:

$$% latex2html id marker 27891F^{\rm AH} \begin{array}{c} ... ...\scriptsize via equ.\ (\ref{FHUndFW})} \end{array} F^{\rm H}.$$ (A.54)

The anti-HUSIMI distribution is thus even rougher than the WIGNER function and accordingly counterintuitive.

Finally there exists a strong argument in favour of the HUSIMI distribution from the experimenters' point of view: TAKAHASHI and SAITÔ argue that $F^{\rm H}$ is much better suited to any experimental setup than $F^{\rm W}$ since the averaging in equation (A.89) acts in a similar way as the coarse graining effect that is inherent to all experimental measurement processes [TS85]. In a particular experimental setup, on the other hand, the WIGNER function still has the striking advantage that it can be measured directly at each point of phase space, without the need for any further complex computation: in [MCK93] it is shown that, after some algebra, $F^{\rm W}$ can be written as

$$F^{\rm W}(x,p,t) \; = \; \frac{1}{\pi\hbar} \, \sum_{n=0}^... ...psi(t)}\big\vert{\hat{D}(x,p)}\big\vert{n}\big> \right\vert^2.$$ (A.55)

If $\left\vert \psi \right>$ describes the radiation field, then the term $\big\vert \big<{\psi}\big\vert{\hat{D}\big(\alpha(x,p)\big)}\big\vert{n}\big> \big\vert^2$ is just the probability density of detecting a photon with the energy $\hbar\omega_0(n+1/2)$ at the phase space point $(x,p)^t$. The parameter $\alpha=\alpha(x,p)$ can be controlled within the experimental setup, such that the photon count statistics for different values of $\alpha$ then gives the WIGNER function as determined by equation (A.93). Some reports on measurements following this or related patterns of quantum-state tomography [Leo95,Leo96] can be found in [BW96,BW97,Jor97,BRWK99], for example.

At this point one is now in the position to further interpret the additional ``squeezing'' parameter $\zeta$ of the (anti-) HUSIMI distribution. Equations (A.89, A.91) show that $\zeta$ controls the way in which the averaging discussed above is performed, namely by specifying the width of the Gaussian (A.90) in $x$- and $p$-direction. The effective area over which is averaged is an ellipse with widths $\Delta x = \sqrt{\hbar/(2\zeta)}$ and $\Delta p = \sqrt{\hbar\zeta/2}$ in $x$- and $p$-direction, respectively -- compare these values with the uncertainties (A.64) of general squeezed states. Therefore, larger values of $\zeta$ yield an averaging area that is squeezed in the direction of $x$ and broadened in the direction of $p$. The particular choice $\zeta=1.0$, which is used in many applications and corresponds to the coherent state representation (A.51), accordingly specifies a circular averaging area.

In order to exemplify the above statements, in figures A.1-A.3 I compare the most important ways to graphically represent quantum states. The states used for this demonstration are the familiar ground state $\left\vert n=0 \right>$ of the harmonic oscillator and its tenth eigenstate $\left\vert n=10 \right>$.

Figure A.1a shows a classical phase space trajectory of the harmonic oscillator at the energy $E_0=\hbar\omega_0/2$. The corresponding quantum mechanical wave function in the position representation (figure A.1b) bears no obvious resemblance to the classical trajectory and is therefore of little use for the purpose of comparing the classical and quantum states. For this purpose, figure A.1c is much better suited: the contour plot of the WIGNER function of $\left \vert 0 \right >$ that is shown here is essentially a bell-shaped function that is peaked at the origin of phase space -- near the classical trajectory. The contour plots of the HUSIMI distributions for different values of $\zeta$ in figures A.1d-A.1f are not much more informative; all of them show bell-shaped curves centered about the origin. Note that the HUSIMI function for $\zeta=1.0$ is the one that is closest to the classical trajectory, since for $\zeta\neq 1.0$ one obtains distributions that do not reproduce the classical rotational symmetry of the trajectory.

While the advantages of the HUSIMI distribution are not obvious with respect to the simple ground state $\left \vert 0 \right >$ they become clearly visible when more complex states are considered. Figure A.2 shows the same kind of plots as figure A.1 but now for the state $\left \vert 10 \right >$. The highly oscillatory character of the WIGNER function is illustrated in the contour plot A.2c and in its three-dimensional counterpart in figure A.3a. These plots do not show much similarity to the corresponding classical path in figure A.2a.

$F^{\rm W}$ exhibits rapid oscillations in radial direction which do not correspond to any classical feature in an obvious way, and $F^{\rm W}$ takes on negative values for some $(x,p)^t$. These are typical properties of the WIGNER function that are observed not only with respect to $\left \vert 10 \right >$ but with respect to most quantum states. In addition to these shortcomings the WIGNER function of $\left \vert 10 \right >$ erroneously gives the impression of a state that is more localized than $\left \vert 10 \right >$ actually is, since the larger portion of the oscillations of $F^{\rm W}$ show up well inside the classical path, as can be seen in figure A.2c.

Only when turning to the HUSIMI distribution in figures A.2d-A.2f -- and in figures A.3b, A.3c with the 3D versions -- (some aspects of) the classical trajectory can be identified in quantum phase space. This is best achieved for $\zeta=1.0$: $F^{\rm H}(x,p;1.0)$ essentially is a circular ridge which takes on its maximum value at the points of the classical trajectory. For $\zeta=0.1$ and $\zeta=10.0$ rotational invariance is lost and two characteristic peaks emerge with no obvious counterpart in the classical picture. But even in these cases the location of the classical path can clearly be identified.

Footnotes

... computation.^A.14

Using the FFT method for an $n\times n$ grid in phase space, the number of necessary complex operations is reduced from ${\mathcal O}(n^3)$ to ${\mathcal O}(n^2\log_2n)$ [Sto99,PTVF94].