Optical phase space
For any such system, a plot of the quadratures against each other, possibly as functions of time, is called a phase diagram.
If the quadratures are functions of time then the optical phase diagram can show the evolution of a quantum optical system with time.
An optical phase diagram can give insight into the properties and behaviors of the system that might otherwise not be obvious.
This can be used to determine the state of the optical system at any point in time.
When discussing the quantum theory of light, it is very common to use an electromagnetic oscillator as a model.
Systems composed of such oscillators can be described by an optical phase space.
Let u(x,t) be a vector function describing a single mode of an electromagnetic oscillator.
This is the equation for a plane wave and is a simple example of such an electromagnetic oscillator.
The oscillators being examined could either be free waves in space or some normal mode contained in some cavity.
A single mode of the electromagnetic oscillator is isolated from the rest of the system and examined.
[1] Quantum oscillators are described using creation and annihilation operators
Physical quantities, such as the electric field strength, then become quantum operators.
In the quantum oscillator mode, most operators representing physical quantities are typically expressed in terms of the creation and annihilation operators.
In this example, the electric field strength is given by: (where xi is a single component of x, position).
which gives the number of photons in the given (spatial-temporal) mode u.
Thus, it can be useful to think of and treat the quadratures as the position and momentum of the oscillator although in fact they are the "in-phase and out-of-phase components of the electric field amplitude of the spatial-temporal mode", or u, and have nothing really to do with the position or momentum of the electromagnetic oscillator (as it is hard to define what is meant by position and momentum for an electromagnetic oscillator).
They satisfy the relations: as these form complete basis sets.
The following is an important relation that can be derived from the above which justifies our interpretation that the quadratures are the real and imaginary parts of a complex
(i.e. the in-phase and out-of-phase components of the electromagnetic oscillator) The following is a relationship that can be used to help evaluate the above and is given by: This gives us that: Thus,
Another very important property of the coherent states becomes very apparent in this formalism.
A coherent state is not a point in the optical phase space but rather a distribution on it.
It can be shown that the quadratures obey Heisenberg's Uncertainty Principle given by: This inequality does not necessarily have to be saturated and a common example of such states are squeezed coherent states.
The coherent states are Gaussian probability distributions over the phase space localized around
It is possible to define operators to move the coherent states around the phase space.
These can produce new coherent states and allow us to move around phase space.
The phase-shifting operator rotates the coherent state by an angle
The important relationship is derived as follows: and solving this differential equation yields the desired result.
Thus, using the above it becomes clear that or a rotation by an angle theta on the coherent state in phase space.
The following illustrates this more clearly: (which is obtained using the fact that the phase-shifting operator is unitary Thus, is the eigenpair of From this it is possible to see that which is another way of expressing the eigenpair which more clearly illustrates the effects of the phase-shifting operator on coherent states.
The displacement operator is given by and its name comes from an important relation Indeed, let's temporarily introduce
