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2012.10 Quantum State Noise PDF

In Quantum Optics a quantum state of light is often measured using a balanced homodyne detector and a strong coherent beam as a local oscillator (LO) reference. Such a homodyne scheme measures the so-called marginal distributions - or more appropriately, probability distribution functions (PDF) - of the phase space quadrature $Q_\theta$ at any given phase difference $\theta$ between the quantum state and the LO.

Such a homodyne measurement may be modeled by drawing random points from the PDF at a given measurement phase $\theta$. Mathematically, drawing random points from some PDF is described by some generator function $G$ that is to be determined.

In general one constructs such a generator function $G$ by feeding uniformly distributed random numbers to the inverse of the so-called cumulative distribution function (CDF). More precisely, the CDF for a continuous PDF $f(t)$ is, $$F(x) = \int_{-\infty}^{x} ds f(s).$$ Please note, we must require $F(x)\in[0,1]$ since it's a true PDF. The generator $G(y) = F^{-1}(y) = x$ is the inverse of $F(x) = y$. Unfortunately, such an inverse function is not in general guarantied to exist.

Squeezed coherent states

Looking at the special - but often used - case of squeezed coherent states, the PDF is a simple Gaussian distribution with phase-depending mean $\mu$ and standard deviation $\sigma$. In the most general case, the mean and the standard deviation are also depending on the coherent state amplitude $|\alpha|$, the state phase $\Phi$ and on the squeezing parameters $r$ and $\phi$ - i.e. the squeezing strength and phase, respectively.

In this particular case the CDF is simply, $$F(x) = \frac{1}{2} \left( 1 + {\rm Erf} \left[ \frac{x-\mu}{\sqrt{2}\sigma} \right] \right),$$ and ${\rm Erf}$ is the error function. The generator is then given by the inverse error function accordingly, $$G(y) = x = \sqrt{2} \, \sigma \, {\rm InvErf} \left[2F(x)-1\right]+\mu.$$ Please note, since $F(x)\in[0,1]$ we may write $y = (2F(x)-1)\in[-1,1]$. Hence, feeding the generator with random numbers $y\in[-1,1]$ we generate random numbers distributed by a Gaussian with mean $\mu$ and standard deviation $\sigma$.

Now, we need to describe the mean $\mu$ and standard deviation $\sigma$ in terms of the other measurement and state parameters - i.e. the measurement phase $\theta$, state phase $\Phi$, coherent state amplitude $\left|\alpha\right|$, and state squeezing phase $\phi$ and strength $r$. Adopting the standard normalization $\hbar \equiv 1$ these quantities are readily available in standard textbooks on Quantum Optics - the mean is, $$\mu = \sqrt{2} \left|\alpha\right| \cos(\theta-\Phi),$$ and the variance, $$\sigma^2 = \frac{1}{2} \left[\cosh(2r)-\sinh(2r)\cos(2\theta-\phi)\right].$$ Assembling everything gives the vacuum state - i.e. setting $\{\left|\alpha\right|, \, r\}=0$ - a mean $\mu=0$ and variance $\sigma^2 = \frac{1}{2}$.

Finally, in order to model a real physical squeezed state we need to lock the squeezing phase $\phi$ relative to the state phase $\Phi$. This will make the squeezing profile follow the state and not just stick to the coordinate system. In general, the angle at which the state noise is squeezed is given by, $$\theta_{\rm sqz}=\frac{\phi}{2}.$$ Hence, setting the squeezing phase to, $$\phi=2\Phi,$$ yields an amplitude squeezed state and, $$\phi=2\Phi-\pi,$$ produces a phase squeezed state.

Examples

Here are some examples on modeling the measurement of squeezed coherent states using homodyne.

Phase scan

Here we discretely change the measurement phase $\theta_i \in [0,2\pi]$ and map out the homodyne measurement results - i.e. the measured quantum state quadratures $\{Q_{\theta_i}\}$.

Coherent state Amplitude squeezed coherent state Phase squeezed coherent state 45 degree squeezed coherent state

PDF

Having the homodyne measurements we may also infer the probability distribution function by binning the randomly generated points and compare it with the initially used PDF.

PDF of phase scanned coherent state PDF of phase scanned phase squeezed coherent state

Mathematica Notebook

All the equations and graphs may be found in the this Mathematica (v8) notebook.