In quantum physics, light is in a squeezed state if its electric field strength Ԑ for some phases has a quantum uncertainty smaller than that of a coherent state. The term squeezing thus refers to a reduced quantum uncertainty. To obey Heisenberg's uncertainty relation, a squeezed state must also have phases at which the electric field uncertainty is anti-squeezed, i.e. larger than that of a coherent state. Since 2019, the gravitational-wave observatories LIGO and Virgo employ squeezed laser light, which has significantly increased the rate of observed gravitational-wave events.
Quantum physical background An oscillating physical quantity cannot have precisely defined values at all phases of the oscillation. This is true for the electric and magnetic fields of an electromagnetic wave, as well as for any other wave or oscillation (see figure right). This fact can be observed in experiments and is described by quantum theory. For electromagnetic waves usually just the electric field is considered, because it is the one that mainly interacts with matter.
Fig. 1. shows five different quantum states that a monochromatic wave could be in. The difference of the five quantum states is given by different electric field excitations and by different distributions of the quantum uncertainty along the phase . For a displaced coherent state, the expectation (mean) value of the electric field shows an oscillation, with an uncertainty independent of the phase (a). Also the phase- (b) and amplitude-squeezed states (c) show an oscillation of the mean electric field, but here the uncertainty depends on phase and is squeezed for some phases. The vacuum state (d) is a special coherent state and is not squeezed. It has zero mean electric field for all phases and a phase-independent uncertainty. It has zero energy on average, i.e. zero photons, and is the ground state of the monochromatic wave we consider. Finally, a squeezed vacuum state has also a zero mean electric field but a phase-dependent uncertainty (e).
Generally, quantum uncertainty reveals itself through a large number of identical measurements on identical quantum objects (here: modes of light) that, however, give different results. Let us again consider a continuous-wave monochromatic light wave (as emitted by an ultra-stable laser). A single measurement of Ԑ is performed over many periods of the light wave and provides a single number. The next measurements of Ԑ will be done consecutively on the same laser beam. Having recorded a large number of such measurements we know the field uncertainty at . In order to get the full picture, and for instance Fig.1(b), we need to record the statistics at many different phases .
Quantitative description of (squeezed) uncertainty The measured electric field strengths at the wave's phase are the eigenvalues of the normalized quadrature operator , defined as
where and are the annihilation and creation operators, respectively, of the oscillator representing the photon. is the wave's amplitude quadrature, equivalent to the position in optical phase space, and is the wave's phase quadrature, equivalent to momentum. and are non-commuting observables. Although they represent electric fields, they are dimensionless and satisfy the following uncertainty relation:
where stands for the variance. (The variance is the mean of the squares of the measuring values minus the square of the mean of the measuring values.) If a mode of light is in its ground state (having an average photon number of zero), the uncertainty relation above is saturated and the variances of the quadrature are . (Other normalizations can also be found in literature. The normalization chosen here has the nice property that the sum of the ground state variances directly provide the zero point excitation of the quantized harmonic oscillator ).
While coherent states belong to the semi-classical states, since they can be fully described by a semi-classical model, squeezed states of light belong to the so-called nonclassical states, which also include number states (Fock states) and Schrödinger cat states.
Squeezed states (of light) were first produced in the mid-1980s.R. E. Slusher et al., Observation of squeezed states generated by four wave mixing in an optical cavity, Phys. Rev. Lett. 55 (22), 2409 (1985) At that time, quantum noise squeezing by up to a factor of about 2 (3 dB) in variance was achieved, i.e. . Today, squeeze factors larger than 10 (10 dB) have been directly observed. A limitation is set by decoherence, mainly in terms of optical loss.
The squeeze factor in Decibel (dB) can be computed in the following way:
Representation of squeezed states by quasi-probability densities
Quantum states such as those in Fig. 1 (a) to (e) are often displayed as Wigner functions, which are quasi-probability density distributions. Two orthogonal quadratures, usually and , span a phase space diagram, and the third axes provides the quasi probability of yielding a certain combination of . Since and are not precisely defined simultaneously, we cannot talk about a 'probability' as we do in classical physics but call it a 'quasi probability'. A Wigner function is reconstructed from time series of and . The reconstruction is also called 'quantum tomographic reconstruction'. For squeezed states, the Wigner function has a Gaussian shape, with an elliptical contour line, see Fig.: 1(f).
Physical meaning of measurement quantity and measurement object Quantum uncertainty becomes visible when identical measurements of the same quantity (observable) on identical objects (here: modes of light) give different results (eigen values). In case of a single freely propagating monochromatic laser beam, the individual measurements are performed on consecutive time intervals of identical length. One interval must last much longer than the light's period; otherwise the monochromatic property would be significantly disturbed. Such consecutive measurements correspond to a time series of fluctuating eigen values. Consider an example in which the amplitude quadrature was repeatedly measured. The time series can be used for a quantum statistical characterization of the modes of light. Obviously, the amplitude of the light wave might be different before and after our measurement, i.e. the time series does not provide any information about very slow changes of the amplitude, which corresponds to very low frequencies. This is a trivial but also fundamental issue, since any data taking lasts for a finite time. Our time series, however, does provide meaningful information about fast changes of the light's amplitude, i.e. changes at frequencies higher than the inverse of the full measuring time. Changes that are faster than the duration of a single measurement, however, are invisible again. A quantum statistical characterization through consecutive measurements on some sort of a carrier is thus always related to a specific frequency interval, for instance described by with Based on this insight, we can describe the physical meaning of the observable more clearly:
The quantum statistical characterization using identical consecutive modes carried by a laser beam confers to the laser beam's electric field modulation within a frequency interval. The actual observable needs to be labeled accordingly, for instance as . is the amplitude (or depth) of the amplitude modulation and the amplitude (or depth) of the phase modulation in the respective frequency interval. This leads to the doggerel expressions 'amplitude quadrature amplitude' and 'phase quadrature amplitude'''.
Within some limitations, for instance set by the speed of the electronics, and can be freely chosen in course of data acquisition and, in particular, data processing. This choice also defines the measurement object, i.e. the mode that is characterized by the statistics of the eigen values of and . The measurement object thus is a modulation mode that is carried by the light beam. – In many experiments, one is interested in a continuous spectrum of many modulation modes carried by the same light beam. Fig. 2 shows the squeeze factors of many neighboring modulation modes versus . The upper trace refers to the uncertainties of the same modes being in their vacuum states, which serves as the 0 dB reference.
The observables in squeezed light experiments correspond exactly to those being used in optical communication. Amplitude modulation (AM) and frequency modulation (FM) are the classical means to imprint information on a carrier field. (Frequency modulation is mathematically closely related to phase modulation). The observables and also correspond to the measurement quantities in laser interferometers, such as in Sagnac interferometers measuring rotation changes and in Michelson interferometers observing gravitational waves. Squeezed states of light thus have ample applications in optical communication and optical measurements. The most prominent and important application is in gravitational-wave observatories. Arguably, it is the first end-user driven application of quantum correlations. Squeezed light originally was not planned to be implemented in either Advanced LIGO nor in Advanced Virgo, but now it contributes a significant factor towards the observatories' design sensitivities and increases the rate of observed gravitational-wave events.
Frequency-dependent squeezing
Frequency-dependent squeezing is a method being implemented at the LIGO–Virgo–KAGRA collaboration to improve sensitivity using its 300 m long filter cavities to handle light differently according to frequencies which allows to improve accuracy of phases at high frequencies at the cost of more inaccuracy in amplitudes at low frequencies and equivalently better amplitudes at low frequencies but worse phases at high frequencies, manipulating the uncertainty relation by the measurement of interest.
Noise at high frequencies is dominated by shot noise while at low frequencies is dominated by radiation pressure noise so when one source is reduced the other increases.
Applications Optical high-precision measurements
Squeezed light is used to reduce the photon counting noise (shot noise) in optical high-precision measurements, most notably in laser interferometers. There are a large number of proof-of-principle experiments. Laser interferometers split a laser beam in two paths and overlap them again afterwards. If the relative optical path length changes, the interference changes, and the light power in the interferometer's output port as well. This light power is detected with a photo diode providing a continuous voltage signal. If for instance the position of one interferometer mirror vibrates and thereby causes an oscillating path length difference, the output light has an amplitude modulation of the same frequency. Independent of the existence of such a (classical) signal, a beam of light always carries at least the vacuum state uncertainty (see above). The (modulation) signal with respect to this uncertainty can be improved by using a higher light power inside the interferometer arms, since the signal increases with the light power. This is the reason (in fact the only one) why Michelson interferometers for the detection of gravitational waves use very high optical power. High light power, however, produces technical problems. Mirror surfaces absorb parts of the light, become warmer, get thermally deformed and reduce the interferometer's interference contrast. Furthermore, an excessive light power can excite unstable mechanical vibrations of the mirrors. These consequences are mitigated if squeezed states of light are used for improving the signal-to-noise-ratio. Squeezed states of light do not increase the light's power. They also do not increase the signal, but instead reduce the noise.