Squeezed States of Light

Abstract

The Heisenberg uncertainty principle ΔAΔB ≥ 1/2 |[A, B]| between the standard deviations of two arbitrary observables, ΔA = 〈(A−〈A〉)²〉^1/2 and similarly for ΔB, has a built-in degree of freedom: one can squeeze the standard deviation of one observable provided one “stretches” that for the conjugate observable. For example the position and momentum standard deviations obey the uncertainty relation

$$\Delta x\Delta p\hbar /2$$

(1)

and we can squeeze Δx to an arbitrarily small value at the expense of accordingly increasing the standard deviation Δ p. All quantum mechanics requires is that the product be bounded from below. As discussed in Sec. 12-1, the electric and magnetic fields from a pair of observables analogous to the position and momentum of a simple harmonic oscillator. Accordingly they obey a similar uncertainty relation

$$\Delta E\Delta B(cons\tan t)\hbar /2.$$

(2)