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| 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 [[Operator (physics)|operator]]s takes on its minimum value:
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| :<math>\Delta x \Delta p = \frac{\hbar}2</math>
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| The simplest such state is the ground state <math>|0\rangle</math> of the [[quantum harmonic oscillator]]. The next simple class of states that satisfies this identity are the family of [[coherent state]]s <math>|\alpha\rangle</math>.
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| Often, the term ''squeezed state'' is used for any such state with <math>\Delta x \neq \Delta p</math> in "natural oscillator units". The idea behind this is that the circle denoting a coherent state in a [[quadrature phase|quadrature]] diagram (see below) has been "squeezed" to an [[ellipse]] of the same area.
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| <ref>
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| Loudon, Rodney, ''The Quantum Theory of Light'' (Oxford University Press, 2000), [ISBN 0-19-850177-3]
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| </ref>
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| <ref>
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| [[Daniel Frank Walls|D. F. Walls]] and G.J. Milburn, ''Quantum Optics'', Springer Berlin 1994
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| </ref>
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| <ref>
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| [[C W Gardiner]] and [[Peter Zoller]], "Quantum Noise", 3rd ed, Springer Berlin 2004
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| </ref>
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| <ref>
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| D. Walls, ''Squeezed states of light'', Nature 306, 141 (1983)
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| </ref>
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| <ref>
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| 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)
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| </ref>
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| ==Mathematical definition==
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| The most general [[wave function]] that satisfies the identity above is the '''squeezed coherent state''' (we work in units with <math>\hbar=1</math>)
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| :<math>\psi(x) = C\,\exp\left(-\frac{(x-x_0)^2}{2 w_0^2} + i p_0 x\right)</math>
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| where <math>C,x_0,w_0,p_0</math> 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 <math>w_0</math>, which is the reason why the state is called "squeezed".
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| The squeezed state above is an [[eigenstate]] of a linear operator
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| :<math> \hat x + i\hat p w_0^2</math>
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| and the corresponding [[eigenvalue]] equals <math>x_0+ip_0 w_0^2</math>. In this sense, it is a generalization of the ground state as well as the coherent state.
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| The multi-dimensional squeezed coherent state of electron in the atom in this form is [[Trojan wave packet]] but the
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| minimum uncertainty is altered by the universal numerical factor (to for example <math>{\hbar(1+1.141)}/2</math>)
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| due to the internal geometric entanglement between space coordinates.
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| ==Examples of squeezed coherent states==
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| 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.{{unclear|section amplitude-squeezed, etc.|date=December 2013}} 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 [[Werner Heisenberg|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 [[complementarity (physics)|complementary]] quadrature, that is, the field at the phase shifted by <math>\pi/2</math>.
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| {| border="1" cellspacing="0" cellpadding="2" style="margin:auto; margin-left:1em;"
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| | style="vertical-align:top" | [[File:noise squeezed states.jpg|thumb|220px|left|Figure 1:
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| Measured quantum noise of the electric field of different squeezed states depends on the phase of the light field. For the first two states a 3π-interval is shown; for the last three states, belonging to a different set of measurements, it is a 4π-interval.<ref name="Breitenbach">G. Breitenbach, S. Schiller, and J. Mlynek, "[http://gerdbreitenbach.de/publications/nature1997.pdf Measurement of the quantum states of squeezed light]", Nature, 387, 471 (1997)</ref>]]
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| |style="vertical-align:top" | [[File:wave packet squeezed states.jpg|thumb|160px|center|Figure 2: Oscillating wave packets of the five states.]]
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| |style="vertical-align:top" | [[File:wigner function squeezed states.jpg|thumb|350px|right|Figure 3: [[Wigner function]]s of the five states. The ripples are due to experimental inaccuracies.]]
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| |}
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| From the top:
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| *Vacuum state
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| *Squeezed vacuum state
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| *Phase-squeezed state
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| *arbitrary squeezed state
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| *Amplitude-squeezed state
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| As can be seen at once, in contrast to the [[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 [[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.
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| 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.
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| ==Photon number distributions and phase distributions of squeezed states==
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| 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 (waves)|phase]] distribution as well.
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| {| border="1" cellspacing="0" cellpadding="2" style="margin:auto; margin-left:1em;"
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| |[[File:photon numbers squeezed coherent states subpoisson.jpg|thumb|280px|left|Figure 4: Experimental photon number distributions for an amplitude-squeezed state, a coherent state, and a phase squeezed state reconstructed from measurements of the quantum statistics. Bars refer to theory, dots to experimental values.<ref name="Breitenbach" /> (source: link 1)]]
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| |[[File:phase distribution squeezed coherent states subpoisson.jpg|thumb|450px|right|Figure 5: Pegg-Barnett phase distribution of the three states.]]
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| |}
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| 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.<ref>Entanglement evaluation with Fisher information - http://arxiv.org/pdf/quant-ph/0612099</ref>
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| [[File:photon numbers squeezed vacuum.jpg|thumb|400px|center|Figure 4: Reconstructed and theoretical photon number distributions for a squeezed-vacuum state.<ref name="Breitenbach" /> (source: link 1)]]
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| For the squeezed vacuum state the photon number distribution displays odd-even-oscillations. This can be explained by the mathematical form of the [[squeezing operator]], that resembles the operator for [[Spontaneous parametric down conversion|two-photon generation]] and annihilation processes. Photons in a squeezed vacuum state are more likely to appear in pairs.
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| ==Experimental realizations of squeezed coherent states==
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| There has been a whole variety of successful demonstrations of squeezed states. The most prominent ones were experiments with light fields using [[laser]]s and [[non-linear optics]] (see [[optical parametric oscillator]]). This is achieved by a simple process of four-wave mixing with a <math>\chi^{(3)}</math> crystal; similarly traveling wave phase-sensitive amplifiers generate spatially multimode quadrature-squeezed states of light when the <math>\chi^{(2)}</math> crystal is pumped in absence of any signal. [[Sub-Poissonian]] current sources driving semiconductor laser diodes have led to amplitude squeezed light.<ref>S. Machida et al.,Observation of amplitude squeezing in a constant-current–driven semiconductor laser, Phys. Rev. Lett. 58, 1000–1003 (1987) - http://link.aps.org/doi/10.1103/PhysRevLett.58.1000</ref>
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| Squeezed states have also been realized via motional states of an [[ion]] in a [[Penning trap|trap]], [[phonon]] states in [[crystal lattice]]s, or [[atom]] ensembles.{{Citation needed|date=October 2011}} 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 12.7 dB.<ref>T. Eberle et al., Quantum Enhancement of the Zero-Area Sagnac Interferometer Topology for Gravitational Wave Detection, Phys. Rev. Lett., 22 June 2010 - http://arxiv.org/abs/1007.0574</ref>
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| ==Applications==
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| 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 [[interferometry|interferometric measurements]] (see for example [[gravitational wave]]s). Amplitude-squeezed light can improve the readout of very weak [[spectroscopy|spectroscopic signals]].
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| Various squeezed coherent states, generalized to the case of many [[degrees of freedom (physics and chemistry)|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 transformation]]).
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| Recently, the use of squeezed states for [[quantum information processing]] in the continuous variables (CV) regime has been increasing rapidly.<ref>S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys., vol. 77, no. 2, pp. 513–577, Jun. 2005. http://link.aps.org/doi/10.1103/RevModPhys.77.513</ref> Continuous variable quantum optics uses squeezing of light as an essential resource to realize CV protocols for quantum communication, unconditional quantum teleportation<ref>A. Furusawa, J. L. Sørensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional Quantum Teleportation,” Science, vol. 282, no. 5389, pp. 706–709, 1998.
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| http://www.sciencemag.org/content/282/5389/706.abstract</ref> and one-way quantum computing.<ref>N. C. Menicucci, S. T. Flammia, and O. Pfister, “One-Way Quantum Computing in the Optical Frequency Comb,” Phys. Rev. Lett., vol. 101, no. 13, p. 130501, Sep. 2008.
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| http://link.aps.org/doi/10.1103/PhysRevLett.101.130501</ref> 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.
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| ==See also==
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| *[[Coherent states in mathematical physics]]
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| *[[Squeeze operator]]
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| *[[Optical Phase Space]]
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| *[[Quantum optics]]
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| *[[Nonclassical light]]
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| ==External links==
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| * [http://gerdbreitenbach.de/gallery/index.html Tutorial about quantum optics of the light field]
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| * [http://www.squeezed-light.de/ www.squeezed-light.de]
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| ==References==
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| {{Reflist|33em}}
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| {{DEFAULTSORT:Squeezed Coherent State}}
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| [[Category:Quantum optics]]
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