Squeezed coherent state

In physics, a squeezed coherent state is a quantum state that is usually described by two non-commuting observables having continuous spectra of eigenvalues. Examples are position x {displaystyle x} and momentum p {displaystyle p} of a particle, and the (dimension-less) electric field in the amplitude X {displaystyle X} (phase 0) and in the node Y {displaystyle Y} (phase 90°) of a light wave (the wave's quadratures). The product of the standard deviations of two such operators obeys the uncertainty principle: Trivial examples, which are in fact not squeezed, are the ground state | 0 ⟩ {displaystyle |0 angle } of the quantum harmonic oscillator and the family of coherent states | α ⟩ {displaystyle |alpha angle } . These states saturate the uncertainty above and have a symmetric distribution of the operator uncertainties with Δ x g = Δ p g {displaystyle Delta x_{g}=Delta p_{g}} in 'natural oscillator units' and Δ X g = Δ Y g = 1 / 2 {displaystyle Delta X_{g}=Delta Y_{g}=1/2} . (In literature different normalizations for the quadrature amplitudes are used. Here we use the normalization for which the sum of the ground state variances of the quadrature amplitudes directly provide the zero point quantum number Δ 2 X g + Δ 2 Y g = 1 / 2 {displaystyle Delta ^{2}X_{g}+Delta ^{2}Y_{g}=1/2} ). The term squeezed state is actually used for states with a standard deviation below that of the ground state for one of the operators or for a linear combination of the two. The idea behind this is that the circle denoting the uncertainty of a coherent state in the quadrature phase space (see right) has been 'squeezed' to an ellipse of the same area. Note that a squeezed state does not need to saturate the uncertainty principle. Squeezed states (of light) were first produced in the mid 1980s. At that time, quantum noise squeezing by up to a factor of about 2 (3 dB) in variance was achieved, i.e. Δ 2 X ≈ Δ 2 X g / 2 {displaystyle Delta ^{2}Xapprox Delta ^{2}X_{g}/2} . Today, squeeze factors larger than 10 (10 dB) have been directly observed. A recent review on squeezed states of light can be found in Ref. The most general wave function that satisfies the identity above is the squeezed coherent state (we work in units with ℏ = 1 {displaystyle hbar =1} ) where C , x 0 , w 0 , p 0 {displaystyle C,x_{0},w_{0},p_{0}} are constants (a normalization constant, the center of the wavepacket, its width, and the expectation value of its momentum). The new feature relative to a coherent state is the free value of the width w 0 {displaystyle w_{0}} , which is the reason why the state is called 'squeezed'. The squeezed state above is an eigenstate of a linear operator and the corresponding eigenvalue equals x 0 + i p 0 w 0 2 {displaystyle x_{0}+ip_{0}w_{0}^{2}} . In this sense, it is a generalization of the ground state as well as the coherent state.