I attempt to monitor the phase of a wavevector $|\psi\rangle$.

As I found (e.g. here ), a matrix representation for the Pegg-Barnett phase operator in Fock base can be obtained as

$$\Phi=\sum_{m,n,n'=0}^s\frac{2\pi m}{(s+1)^2}\,\exp\left[\frac{2\pi i m(n-n')}{s+1}\right]|n\rangle\langle n'|$$

for a particle number cutoff $s$. Making the indices start at one instead of zero (for implementation in matlab), this becomes

$$\Phi=\sum_{\tilde{n},\tilde{n'}=1}^\tilde{s}\sum_{m=0}^{\tilde{s}-1}\frac{2\pi m}{\tilde{s}^2}\,\exp\left[\frac{2\pi i m(\tilde{n}-\tilde{n'})}{\tilde{s}}\right]|\tilde{n}\rangle\langle \tilde{n'}|$$ such that $|\widetilde{n+1}\rangle=|n\rangle$.

When I construct the wavefunction though of e.g. a coherent state with $\alpha=\sqrt{30}$; $\langle\alpha|\Phi|\alpha\rangle$ with $\tilde{s}=200$ gives me a phase ~2.7, whereas this coherent state is located in phase-space on the positive X-axis (so that phase zero would be expected.)

What went wrong?

NOTE ADDED: by trial and error, I found that we can obtain a phase prediction of zero if the summation over m also goes up to $\tilde{s}$ and in addition the reference phase $\theta_0$ is chosen $-\pi$ instead of zero. I don't understand why though, and if this is to be trusted as a measure of phase. At first sight, it does not seem to fullfil the small angle approximation $\phi\approx \frac{P}{X}$. Any insight is still welcome.

CODE USED with the original parameters

%% Constructing the initial state
a=sparse(Nmax,Nmax);for it=1:Nmax-1,  a(it,it+1)=sqrt(it); end;adag=a';  
%annihilation and creation operators for use in the displacement operator.
vacuumpsi=zeros(Nmax,1); vacuumpsi(1,:)=1;

%% The PB operator (tildes omitted)
for n=1:s
 for nprime=1:s
  for m=0:s-1

%% Calculate expectation value of phase
  • $\begingroup$ There's no way to know. Maybe the code constructs the coherent states incorrectly. Maybe the code has an error and doesn't match the formula you wrote here. Consider sharing your code (e.g. on GitHub) and asking for help in the chat room. $\endgroup$ – DanielSank Jan 11 '18 at 18:31

What works quite nicely is defining the operator from eq.33 in https://arxiv.org/pdf/hep-th/9304036.pdf .

Also, it was to be expected that the PB-phase of a coherent state on the positive real axis is not zero for reference phase $\theta_0=0$, namely the eigenvalue spectrum is bounded by 0 and $2\pi$ and as a coherent state has a finite phase uncertainty, it crosses the cut in phase.

Aside from these issues, this PB-operator used has proven useful in monitoring the phase of a numerical wavevector and gives quite intuitive results. For further improvement, one can also iteratively update the used $\theta_0$ to avoid crossing of the phase-cut.


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