Consider a density matrix of a free particle in non-relativistic quantum mechanics. Nice, quasi-classical particles will be well-approximated by a wavepacket or a mixture of wavepackets. The coherent superposition of two wavepackets well-separated in phase space is decidedly non-classical.

Is there a formalism I can use to call this density matrix "approximately diagonal in the overcomplete basis of wavepackets"? (For the sake of argument, we can consider a specific class of wavepackets, e.g. of a fixed width $\sigma$ and instantaneously not spreading or contracting.) I am aware of the Wigner phase space representation, but I want something that I can use for other bases, and that I can use for operators that aren't density matrices e.g. observables. For instance: $X$, $P$, and $XP$ are all approximately diagonal in the basis of wavepackets, but $RXR^\dagger$ is not, where $R$ is the unitary operator which maps

$\vert x \rangle \to (\vert x \rangle + \mathrm{sign}(x) \vert - x \rangle) / \sqrt{2}$.

(This operator creates a Schrodinger's cat state by reflecting about $x=0$.)

For two different states $\vert a \rangle$ and $\vert b \rangle$ in the basis, we want to require an approximately diagonal operator $A$ to satisfy $\langle a \vert A \vert b \rangle \approx 0$, but we only want to do this if $\langle a \vert b \rangle \approx 0$. For $\langle a \vert b \rangle \approx 1$, we sensibly expect $\langle a \vert A \vert b \rangle$ to be proportional to a typical eigenvalue.

The short answer is "The operators that are diagonal in the basis of coherent states are those that have a smooth Glauber–Sudarshan P transform".

The Wigner-Weyl transform beautifully maps all kinds of operators on Hilbert space (density matrices or otherwise, coherent-state-diagonal or not) to real-valued functions on phase space. However, this doesn't directly tell us which ones are diagonal since all "nice" operators can be represented in this way. (See here for some discussion of regularity conditions.)

But we might take a naive stab at what an operator would look like if it were diagonal in the basis of coherent states $\vert \alpha \rangle$, where $\alpha = (x,p)$ is a variable ranging over phase space. First, note that we can write the position operator in the position basis as $$ \hat{X}=\int \mathrm{d}x \, x \vert x \rangle \langle x \vert. $$

So, given an arbitrary real-valued (classical) function $f(\alpha)$ on phase-space (e.g. position, momentum, or energy) consider the following method of defining a corresponding operator: $$ \hat{\Omega}[f] := \int \mathrm{d}\alpha\, f(\alpha) \vert \alpha \rangle \langle \alpha \vert. $$ (The integral on the right should be normalized by a factor of $1/2 \pi$ but I'm going to ignore normalization in everything I have to say.) Intuitively, this operator acts on a state vector by "measuring it" in the wavepacket basis and weighting the corresponding components by the value of the function at that point in phase space. In fact, one can check that $\hat{\Omega}[X] = \hat{X}$, where $X(\alpha) = X(x,p) = x$ is the position function on phase space.

How does this relate to the Wigner-Weyl transform? Well, the W transform of $\hat{\Omega}[f]$ is $$ \begin{align} W\left[\hat{\Omega}[f]\right](x,p) &= \int \mathrm{d}\Delta x \, e^{i p \Delta x} \langle x + \frac{\Delta x}{2} \vert \hat{\Omega}[f] \vert x - \frac{\Delta x}{2} \rangle \\ &= \int \mathrm{d}\Delta x \int \mathrm{d}\alpha\, e^{i p \Delta x} f(\alpha) \langle x + \frac{\Delta x}{2} \vert \alpha \rangle \langle \alpha \vert x - \frac{\Delta x}{2} \rangle.\\ &= \int \mathrm{d}\alpha\, e^{-\left[\alpha -(x,p)\right]^2} f(\alpha).\\ &= (f \circ g) (x,p) \end{align} $$ where $\circ$ denotes the convolution and $g(\alpha) = e^{-\alpha^2}$ is a Gaussian kernel. Another way to write this is to use the convolution theorem: $$ \tilde{W}\left[\hat{\Omega}[f]\right](\xi) = \tilde{f}(\xi) \tilde{g}(\xi) $$ where $\tilde{\phantom{g}}$ denotes the Fourier transform, which makes $\tilde{g}(\xi) = e^{-\xi^2/4}$, where and $\tilde{W}$ is the characteristic function associated with $W$. [Usually the "symplectic Fourier transform" is used for the characteristic function, but that's differs from the above only by reparameterizing as $\xi=(\xi_1,\xi_2)\to(-\xi_2,\xi_1)$. Note also that the factors like the 4 in $\tilde{g}(\xi) = e^{-\xi^2/4}$ is dependent on the choice of Fourier convention, and whether $\hbar = 1/2,1,$ or $2$.]

So we see that the W transform of a "diagonal" operator like $\hat{\Omega}[f]$ is just $f$ smoothed by a convolution with the Gaussian kernel $g$. But that's exactly what makes $f$ the Glauber-Sudarshan P transform of an operator $\hat{\Omega}[f]$. Indeed, the definition of $\hat{\Omega}[f]$ given above is the same as the defining equation for a P transform for a density matrix, only generalized to operators that aren't necessarily density matrices.

Now, all smooth operators on the Hilbert space will have W transforms associated with them, but not all such W transforms can be de-convolved to a smooth P transform. In particular, if we move to Fourier space, de-convolution is just multiplication by $e^{+\xi^2/4}$ which, in general, will produce a function on which the inverse Fourier transform does not converge. There are ways to try construct a P transform in this case, but they are highly singular, involving infinite derivatives of Dirac delta functions. (See Wikipedia, references therein, and especially Bonifacio et al. and Sudarshan.)

  • If anyone knows if these ideas can be generalized to other types of overcomplete bases besides the coherent states, I would be very interested to learn about it. – Jess Riedel Jun 13 '14 at 20:23

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