In quantum optics, an optical phase space is a phase space in which all quantum states of an optical system are described. Each point in the optical phase space corresponds to a unique state of an optical system. 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 allude to qualities of the system that can be of interest to an individual studying an optical system that would be very hard to deduce otherwise. Another use for an optical phase diagram is that it shows the evolution of the state of an optical system. This can be used to determine the state of the optical system at any point in time.
Background information When discussing the quantum theory of light, it is very common to use an electromagnetic oscillator as a model. An electromagnetic oscillator describes an oscillation of the electric field. Since the magnetic field is proportional to the rate of change of the electric field, this too oscillates. Such oscillations describe light. 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. For simplicity, it is assumed that this electromagnetic oscillator is in vacuum. An example is the plane wave given by
u ( x , t ) = u 0 e i ( k ⋅ x − ω t ) {\displaystyle \mathbf {u} (\mathbf {x} ,t)=\mathbf {u_{0}} e^{i(\mathbf {k} \cdot \mathbf {x} -\omega t)}}
where u0 is the polarization vector, k is the wave vector, ω {\displaystyle \omega } the frequency, and A ⋅ {\displaystyle \cdot } B denotes the dot product between the vectors A and B. 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. Such an oscillator, when quantized, is described by the mathematics of a quantum harmonic oscillator. Quantum oscillators are described using creation and annihilation operators a ^ † {\displaystyle {\hat {a}}^{\dagger }} and a ^ {\displaystyle {\hat {a}}} . Physical quantities, such as the electric field strength, then become quantum operators. In order to distinguish a physical quantity from the quantum mechanical operator used to describe it, a "hat" is used over the operator symbols. Thus, for example, where E i {\displaystyle E_{i}} might represent (one component of) the electric field, the symbol E ^ i {\displaystyle {\widehat {E}}_{i}} denotes the quantum-mechanical operator that describes E i {\displaystyle E_{i}} . This convention is used throughout this article, but is not in common use in more advanced texts, which avoid the hat, as it simply clutters the text. 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:
E ^ i = u i ∗ ( x , t ) a ^ † + u i ( x , t ) a ^ {\displaystyle {\widehat {E}}_{i}=u_{i}^{*}(\mathbf {x} ,t){\widehat {a}}^{\dagger }+u_{i}(\mathbf {x} ,t){\widehat {a}}}
(where xi is a single component of x, position). The Hamiltonian for an electromagnetic oscillator is found by quantizing the electromagnetic field for this oscillator and the formula is given by:
H ^ = ℏ ω ( a ^ † a ^ + 1 / 2 ) {\displaystyle {\widehat {H}}=\hbar \omega ({\widehat {a}}^{\dagger }{\widehat {a}}+1/2)}
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