Squeezed States in Quantum Physics

Quantum Uncertainties for Coherent and Squeezed States

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a

b

c

d

Uncertainty regions for various types of quantum states are shown in the left column, together with the time dependences of x(t) (or p(t)) (blue-grey curves) and their uncertainties (light blue shaded area), in the right column.

(a) The uncertainty region for the vacuum state is a circle centered at the origin.

(b) The coherent state exhibits a circular uncertainty region centered about the rotating phasor. Points (x, p) in the uncertainty circle trace out an electric field with an uncertainty that is independent of time.

(c) The squeezed-vacuum state has an elliptical uncertainty region.

(d) The quadrature-squeezed coherent state has an elliptical uncertainty region centered about the phasor. As with the squeezed-vacuum state (b), the electric field shows a periodic reduction and enhancement of its uncertainty. If the minor axis of the ellipse were oriented along the phasor, the state would also be number squeezed.

Wave Packet Dynamics

a b c

a) A displaced ground state of a simple harmonic oscillator of the correct width (i.e., a coherent state) oscillates back and forth with constant width.

b) If the wave packet width or variance in a) is initially squeezed, it will spread for a quarter of a cycle, then return to the squeezed value at the half-cycle, spread for another quarter of a cycle, and so on.

c) Squeezed vacuum state: the wave packet width or variance is oscillating in a simple harmonic oscillator potential well. The packet initially had a squeezed variance.

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