We show that the squeezed states of quantum optics are physically and mathematically equivalent to Gaussian wave packets. In the framework of wave mechanics it is possible to formulate and investigate in a very simple way all the properties of reduced or squeezed fluctuations of the position and momentum operators. We study squeezed quantum fluctuations of a minimum-uncertainty Gaussian wave function under two different dynamical conditions of free evolution and harmonic oscillation.
We show that the squeezed states of quantum optics are physically and mathematically equivalent to Gaussian wave packets. In the framework of wave mechanics it is possible to formulate and investigate in a very simple way all the properties of reduced or squeezed fluctuations of the position and momentum operators. We study squeezed quantum fluctuations of a minimum-uncertainty Gaussian wave function under two different dynamical conditions of free evolution and harmonic oscillation.
With the successes of the laser interferometer gravitational-wave observatory, we anticipate increased interest in working with squeezed states in the undergraduate and graduate quantum-mechanics classroom. Because squeezed-coherent states are minimum uncertainty states, their wavefunctions in position and momentum space must be Gaussians. But this result is rarely discussed in treatments of squeezed states in quantum textbooks or quantum optics textbooks. In this work, we show three different ways to construct the wavefunction for squeezed-coherent states: (i) a differential equation-based approach; (ii) an approach that uses an expansion in terms of the simple-harmonic oscillator wavefunctions; and (iii) a fully operator-based approach. We do this to illustrate that the concept of the wavefunction can be introduced no matter what methodology an instructor wishes to use. We hope that working with the wavefunction will help demystify the concept of a squeezed-coherent state.
Using only elementary quantum-mechanical concepts, the statistical properties of various harmonic oscillator states that are linear superpositions of its energy eigenfunctions are described. These superpositions include coherent states and squeezed (or two-photon coherent) states. The resulting Gaussian, minimum-uncertainty wave packets are shown to oscillate back and forth for both coherent and squeezed states, but with an oscillating ‘‘width’’ for the squeezed states. Also examined are the principles underlying the production of squeezed electromagnetic waves via parametric amplification or four-wave mixing, their measurement by homodyne detection, and the connection between squeezing and non-Poissonian counting statistics.
Quantum-mechanical entropies of position and momentum operators in a given state are shown to be reasonable and sensitive measures for squeezing of quantum fluctuations. It is shown to be true not only for states having Gaussian wave functions but also for more general, both pure and mixed, quantum states. A simple proof that the squeezing exhibited by the variance is always accompanied by a corresponding entropy reduction below the entropy vacuum level is given. These results show that the information entropy is not only a theoretically satisfactory concept but can also be useful as a tool for more practical quantum-optics applications.
Using the time-dependent annihilation and creation operators, the invariant operators, for a free mass and an oscillator, we find the coherent-squeezed state representation of a travelling general Gaussian wave packet with initial expectation values, x0 and p0, of the position and momentum and variances, Δx0 and Δp0. The initial general Gaussian wave packet takes, up to a normalization factor, the form eip0x/ℏe−(1∓iδ)(x−x0)2/4(Δx0)2, where δ=(2Δx0Δp0/ℏ)2−1 denotes a measure of deviation from the minimum uncertainty or the initial position-momentum correlation δ=2Δ(xp)0/ℏ. The travelling Gaussian wave packet takes, up to a time-dependent phase and normalization factor, the form eipcx/ℏe−(1−2iΔ(xp)t/ℏ)(x−xc)2/4(Δxt)2 and the centroid follows the the classical trajectory with xc(t) and pc(t). The position variance is found to have additionally a linearly time-dependent term proportional to δ with both positive and negative signs.
With a proper choice of the initial wave function it is possible to obtain reduced or squeezed quantum fluctuations of the un- certainty of the position operator for a free particle.1 Such an effect is important for the so-called quantum nondemolition measurements of weak external signals. We show that for both free-motion and harmonic oscillation such a property holds owing to the conformal invariance of the Schrodinger equation. We show that for harmonic oscillation it is possible to reduce the quantum fluctuations of the measured photon below the standard quantum limit.2 Possible applications to optical communications and weak signal detections are also presented
"Harmonic oscillator squeezed states are states of minimum uncertainty, but unlike coherent states, in which the uncertainty in position and momentum are equal, squeezed states have the uncertainty reduced, either in position or in momentum, while still minimizing the uncertainty principle. It seems that this property of squeezed states would allow to obtain the position eigenstates as a limiting case, by doing null the uncertainty in position and infinite in momentum. However, there are two equivalent ways to define squeezed states, that lead to different expressions for the limiting states. In this work, we analyze both definitions and show the advantages and disadvantages of using them in order to find position eigenstates. With this in mind, but leaving aside the definitions of squeezed states, we find an operator that applied to the vacuum gives position eigenstates. We also analyze some properties of the squeezed states, based on the new expressions obtained for the eigenstates of the position."