squeezed state

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A squeezed state is a nonclassical state of the electromagnetic field in which the variance (spread) of one quadrature variable (e.g., phase) becomes smaller than the vacuum level, while that of the conjugate becomes larger. Mathematically, it is obtained by applying the squeezing operator S(ζ) = exp[½(ζ* a² − ζ a†²)] to the vacuum or a coherent state. The parameter ζ = r e^{iθ} specifies the squeezing degree r and the direction in phase space. Such states are generated in nonlinear crystals via parametric down-conversion or four-wave mixing and are widely used to enhance the precision of interferometric measurements.

History

The theory of squeezed states emerged in the 1970s, and in 1985 scientists managed to obtain them in the lab by passing a laser beam through a special crystal.

How it works

Imagine waves on the water surface. Usually, the crests and troughs are slightly irregular—that's quantum noise. Using a special crystal, you can redistribute this irregularity: the crests become almost ideally uniform, but the troughs dance more chaotically. As a result, you can 'hear' a very quiet signal masked by noise—exactly how LIGO catches gravitational waves.

💡 In the LIGO detector, squeezed light reduces quantum noise so much that a length change of a four-kilometer arm by one-thousandth of a proton diameter can be noticed.

Squeezed states minimize not the product of uncertainties, but the variance of one quadrature, in accordance with the generalized Heisenberg uncertainty principle: ⟨(ΔX₁)²⟩⟨(ΔX₂)²⟩ ≥ 1/4. In the Fock basis, |ζ⟩ = (1/cosh r) ∑_{m=0}^∞ (e^{iθ} tanh r)^{2m} √(2m)!/(m!) |2m⟩—here only even photon numbers appear. The mean photon number equals sinh² r. The transformation effected by the operator S squeezes the phase distribution so that upon homodyne detection, the statistics of one quadrature become sub-Poissonian. In quantum metrology, squeezing allows surpassing the standard quantum limit (SQL), determined by vacuum fluctuations.

History

The first theoretical works belong to Yamamoto and Haus (1970s). In 1981, Carlton Caves showed the quantum-mechanical limits of interferometric measurements and proposed using squeezed light to overcome them. The experimental generation of squeezed states was first achieved by Slusher's group in 1985. In the 2000s, Rainer Weiss and colleagues developed a technique for injecting squeezed vacuum into gravitational detectors, which became one of the key achievements recognized by the 2017 Nobel Prize.

How it works

Squeezing is generated in a nonlinear crystal with χ⁽²⁾ susceptibility. In the parametric amplification regime, a pump photon with frequency ω_p decays into two entangled photons with frequencies ω_s and ω_i. If one of the beams is detected, conditional squeezing is realized in the other. In LIGO, squeezing of the vacuum state is used: squeezed vacuum is directed into the dark port of the Michelson interferometer, which reduces photocurrent fluctuations without introducing additional classical noise. The evolution equation in the Heisenberg picture for the quadratures is described by a linear squeezing transformation, preserving the commutation relations.

S(\zeta) = \exp\!\left( \frac{1}{2} (\zeta^* a^2 - \zeta a^{\dagger 2}) \right)

S(ζ) is the unitary squeezing operator, ζ = r e^{iθ} is the complex squeezing parameter (r is the squeezing degree, r ≥ 0; θ is the squeezing angle in phase space), a and a† are the annihilation and creation operators of the given mode, exp is the exponential function. When acting on the vacuum state |0⟩, it yields the squeezed vacuum state: |ζ⟩ = S(ζ)|0⟩.

Open problems & fun facts

Losses in optical elements quickly destroy squeezing, limiting the gain in noise figure.; Creating strong squeezing (more than 10 dB) requires precise phase control and suppression of parasitic reflections.; For next-generation detectors (Einstein Telescope), it is necessary to obtain squeezing at low frequencies, where seismic noise dominates.

💡 The maximum squeezing degree achieved to date in optics is 15 dB, equivalent to a noise reduction by a factor of ~32. Such squeezing would allow quantum radars to distinguish stealth objects.

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squeezed state

History

How it works

History

How it works

History

How it works

Open problems & fun facts

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