Қысылған когеренттік күйлер

Type of quantum state

In physics, a squeezed coherent state is a quantum state that is usually described by two non commuting observables having continuous spectra of eigenvalues. Examples are position and momentum of a particle, and the (dimension less) electric field in the amplitude (phase 0) and in the mode (phase 90°) of a light wave (the wave's quadratures). The product of the standard deviations of two such operators obeys the uncertainty principle:

and , respectively. Trivial examples, which are in fact not squeezed, are the ground state of the quantum harmonic oscillator and the family of coherent states These states saturate the uncertainty above and have a symmetric distribution of the operator uncertainties with in "natural oscillator units" and (In literature different normalizations for the quadrature amplitudes are used. Here we use the normalization for which the sum of the ground state variances of the quadrature amplitudes directly provide the zero point quantum number ). The term squeezed state is actually used for states with a standard deviation below that of the ground state for one of the operators or for a linear combination of the two. The idea behind this is that the circle denoting the uncertainty of a coherent state in the quadrature phase space (see right) has been "squeezed" to an ellipse of the same area. Note that a squeezed state does not need to saturate the uncertainty principle. Squeezed states of light were first produced in the mid 1980s. At that time, quantum noise squeezing by up to a factor of about 2 (3 dB) in variance was achieved, i. e. As of 2017, squeeze factors larger than 10 (10 dB) have been directly observed.

Depending on the phase angle at which the state's width is reduced, one can distinguish amplitude squeezed, phase squeezed, and general quadrature squeezed states. If the squeezing operator is applied directly to the vacuum, rather than to a coherent state, the result is called the 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
As can be seen in the illustrations, in contrast to a coherent state, the quantum noise for a squeezed state is no longer independent of the phase of the light wave. A characteristic broadening and narrowing of the noise during one oscillation period can be observed. The probability distribution of a squeezed state is defined as the norm squared of the wave function mentioned in the last paragraph. It corresponds to the square of the electric (and magnetic) field strength of a classical 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 the 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.

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. For amplitude squeezed light the photon number distribution is usually narrower than the one of a coherent state of the same amplitude resulting in sub Poissonian light, whereas its phase distribution is wider. The opposite is true for the phase squeezed light, which displays a large intensity (photon number) noise but a narrow phase distribution. Nevertheless, the statistics of amplitude squeezed light was not observed directly with photon number resolving detector due to experimental difficulty. [[File:photon numbers squeezed vacuum. jpg|thumb|400px|center|Reconstructed and theoretical photon number distributions for a squeezed vacuum state. A pure squeezed vacuum state would have no contribution from odd photon number states. The non zero contribution in the above figure is because the detected state is not a pure state – losses in the setup convert the pure squeezed vacuum into a mixed state. depending on the number of modes of the electromagnetic field involved in the process. Recent studies have looked into multimode squeezed states showing quantum correlations among more than two modes as well.

Squeezed states of light can be divided into squeezed vacuum and bright squeezed light, depending on the absence or presence of a non zero mean field (also called a carrier), respectively. An optical parametric oscillator operated below threshold produces squeezed vacuum, whereas the same OPO operated above threshold produces bright squeezed light. Bright squeezed light can be advantageous for certain quantum information processing applications as it obviates the need of sending local oscillator to provide a phase reference, whereas squeezed vacuum is considered more suitable for quantum enhanced sensing applications. The AdLIGO and GEO600 gravitational wave detectors use squeezed vacuum to achieve enhanced sensitivity beyond the standard quantum limit.

There has been a whole variety of successful demonstrations of squeezed states. The first demonstrations were experiments with light fields using lasers and non linear optics (see optical parametric oscillator). This is achieved by a simple process of four wave mixing with a crystal; similarly travelling wave phase sensitive amplifiers generate spatially multimode quadrature squeezed states of light when the crystal is pumped in absence of any signal. Sub Poissonian current sources driving semiconductor laser diodes have led to amplitude squeezed light. Squeezed states have also been realized via motional states of an ion in a trap, phonon states in crystal lattices, and spin states in neutral atom ensembles. Much progress has been made on the creation and observation of spin squeezed states in ensembles of neutral atoms and ions, which can be used to enhancement measurements of time, accelerations, fields, and the current state of the art for measurement enhancement is 20 dB. Generation of spin squeezed states have been demonstrated using both coherent evolution of a coherent spin state and projective, coherence preserving measurements. Even macroscopic oscillators were driven into classical motional states that were very similar to squeezed coherent states. Current state of the art in noise suppression, for laser radiation using squeezed light, amounts to 15 dB (as of 2016), which broke the previous record of 12.7 dB (2010).

Squeezed states of the light field can be used to enhance precision measurements. For example, phase squeezed light can improve the phase read out of interferometric measurements (see for example gravitational waves). Amplitude squeezed light can improve the readout of very weak spectroscopic signals. Spin squeezed states of atoms can be used to improve the precision of atomic clocks. This is an important problem in atomic clocks and other sensors that use small ensembles of cold atoms where the quantum projection noise represents a fundamental limitation to the precision of the sensor. Various squeezed coherent states, generalized to the case of many degrees of freedom, are used in various calculations in quantum field theory, for example Unruh effect and Hawking radiation, and generally, particle production in curved backgrounds and Bogoliubov transformations. Recently, the use of squeezed states for quantum information processing in the continuous variables (CV) regime has been increasing rapidly. Continuous variable quantum optics uses squeezing of light as an essential resource to realize CV protocols for quantum communication, unconditional quantum teleportation and one way quantum computing. This is in contrast to quantum information processing with single photons or photon pairs as qubits. CV quantum information processing relies heavily on the fact that squeezing is intimately related to quantum entanglement, as the quadratures of a squeezed state exhibit sub shot noise quantum correlations.