Quadrature components in quantum optics
In quantum optics, the quadrature components, denoted as \mathbf P and \mathbf Q, are observables that describe the quantum fluctuations of the electromagnetic field. They are defined in terms of the creation (\mathbf a^\dag) and annihilation (\mathbf a) operators:
\begin{aligned} \mathbf P & = \frac{\mathbf a - \mathbf a^\dag}{i} \\ \mathbf Q & = \mathbf a + \mathbf a^\dag \end{aligned}
These operators are Hermitian and correspond to physically measurable quantities. They are analogous to the momentum and position variables in classical mechanics, adapted to the quantum description of light.
These quadrature components are non-commutative, the commutator between \mathbf Q and \mathbf P is given by:
[\mathbf Q, \mathbf P] = i
This non-zero commutator menans they obey the Heisenberg uncertainty principle, which in terms of quadrature components is expressed as:
\Delta_Q \cdot \Delta_P \geq \frac{1}{2}
This inequality states that it is impossible to simultaneously measure both quadratures with arbitrary precision. Reducing the uncertainty in one quadrature necessarily increases the uncertainty in the other.
Quadrature components are used with the technique of homodyne detection, a method used to measure the quantum state of light. By adjusting the phase of a local oscillator in a homodyne setup, we can selectively measure either the \mathbf Q or \mathbf P quadrature of the signal field.
- Setting the local oscillator phase \varphi_{LO} = 0 allows me to measure the \mathbf Q quadrature.
- Setting the local oscillator phase \varphi_{LO} = \pi/2 allows me to measure the \mathbf P quadrature.
By repeating measurements with different local oscillator phases, we can reconstruct information about the quantum state of the light field through measurements of these quadratures.
It is also possible to measure rotated quadrature components, which are linear combinations of \mathbf Q and \mathbf P. By choosing a phase \theta, we can define new quadrature operators:
\begin{aligned} \mathbf{Q}(\theta) &= \cos\theta \, \mathbf{Q} + \sin\theta \, \mathbf{P} \\ \mathbf{P}(\theta) &= -\sin\theta \, \mathbf{Q} + \cos\theta \, \mathbf{P} \end{aligned}
Measuring \mathbf{Q}(\theta) and \mathbf{P}(\theta) is equivalent to rotating the phase in the homodyne detection scheme. These rotated quadratures also satisfy the canonical commutation relation and the Heisenberg uncertainty principle.
In essence, quadrature components \mathbf P and \mathbf Q provide a powerful framework for understanding and measuring the quantum nature of light, from fundamental theoretical considerations to practical applications in quantum optics experiments.
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