squeezing n : the act of gripping and pressing firmly; "he gave her cheek a playful squeeze" [syn: squeeze]
- present participle of squeeze
In physics, a squeezed coherent state is any state of the quantum mechanical Hilbert space such that the uncertainty principle is saturated. That is, the product of the corresponding two operators takes on its minimum value:
- \Delta x \Delta p = \frac2
The simplest such state is the ground state |0\rangle of the quantum harmonic oscillator. The next simple class of states that satisfies this identity are the family of coherent states |\alpha\rangle.
Often, the term squeezed state is used for any such state with \Delta x \neq \Delta p. The idea behind this is that the circle denoting a coherent state in a quadrature diagram (see below) has been "squeezed" to an ellipse of the same area.
The most general wave function that satisfies the identity above is the squeezed coherent state (we work in units with \hbar=1)
- \psi(x) = C\,\exp\left(-\frac + i p_0 x\right)
where C,x_0,w_0,p_0 are constants (a normalization constant, the center of the wavepacket, its width, and its average momentum). The new feature relative to a coherent state is the free value of the width w_0, which is the reason why the state is called "squeezed".
The squeezed state above is an eigenstate of a linear operator
- \hat x + i\hat p w_0^2
and the corresponding eigenvalue equals x_0+ip_0 w_0^2. In this sense, it is a generalization of the ground state as well as the coherent state.
Examples of squeezed coherent states
Depending on at which phase the state's quantum noise is reduced, one can distinguish amplitude-squeezed and phase-squeezed states or general quadrature squeezed states. If no coherent excitation exists the state is called a squeezed vacuum. The figures below give a nice visual demonstration of the close connection between squeezed states and Heisenberg's uncertainty relation: Diminishing the quantum noise at a specific quadrature (phase) of the wave has as a direct consequence an enhancement of the noise of the complementary quadrature, that is, the field at the phase shifted by \pi/2.
From the top:
- Vacuum state
- Squeezed vacuum state
- Phase-squeezed state
- arbitrary squeezed state
- Amplitude-squeezed state
As can be seen at once, in contrast to the coherent state the quantum noise is not independent of the phase of the light wave anymore. A characteristic broadening and narrowing of the noise during one oscillation period can be observed. The wave packet of a squeezed state is defined by the square of the wave function introduced in the last paragraph. They correspond to the probability distribution of the electric field strength of the light wave. The moving wave packets display an oscillatory motion combined with the widening and narrowing of their distribution: the "breathing" of the wave packet. For an amplitude-squeezed state, the most narrow distribution of the wave packet is reached at the field maximum, resulting in an amplitude that is defined more precisely than the one of a coherent state. For a phase-squeezed state, the most narrow distribution is reached at field zero, resulting in an average phase value that is better defined than the one of a coherent state.
In phase space, quantum mechanical uncertainties can be depicted by Wigner distribution Wigner quasi-probability distribution. The intensity of the light wave, its coherent excitation, is given by the displacement of the Wigner distribution from the origin. A change in the phase of the squeezed quadrature results in a rotation of the distribution.
Photon number distributions and phase distributions of squeezed states
The squeezing angle, that is the phase with minimum quantum noise, has a large influence on the photon number distribution of the light wave and its phase distribution as well.
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