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Squeezed coherent state
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==Atomic spin squeezing== {{See also|Spin squeezed state}} For squeezing of two-level neutral atom ensembles it is useful to consider the atoms as spin-1/2 particles with corresponding [[angular momentum operator]]s defined as :<math>J_v=\sum_{i=1}^N j_v^{(i)}</math> where <math>v={x,y,z}</math> and <math>j_v^{(i)}</math> is the single-spin operator in the <math>v</math>-direction. Here <math>J_z</math> will correspond to the population difference in the two level system, i.e. for an equal superposition of the up and down state <math>J_z=0</math>. The <math>J_x</math>β<math>J_y</math> plane represents the phase difference between the two states. This is also known as the [[Bloch sphere]] picture. We can then define uncertainty relations such as <math>\Delta J_z \cdot \Delta J_y \geq \left|\Delta J_x\right|/2</math>. For a coherent (unentangled) state, <math>\Delta J_z=\Delta J_y=\sqrt{N}/2</math>. Squeezing is here considered the redistribution of uncertainty from one variable (typically <math>J_z</math>) to another (typically <math>J_y</math>). If we consider a state pointing in the <math>J_x</math> direction, we can define the Wineland criterion<ref>{{cite journal|last1=Wineland|first1=D. J.|last2=Bollinger |first2=J. J.|last3=Heinzen|first3=D. J.|title=Squeezed atomic states and projection noise in spectroscopy|journal=Physical Review A|date=1 July 1994|volume=50|issue=2|pages=67β88|doi=10.1103/PhysRevA.50.67|pmid=9910869|bibcode=1994PhRvA..50...67W}}</ref> for squeezing, or the metrological enhancement of the squeezed state as :<math>\chi^2=\left(\frac{\sqrt{N}/2}{\Delta J_z}\frac{\left|J_x\right|}{N/2}\right)^2</math>. This criterion has two factors, the first factor is the spin noise reduction, i.e. how much the quantum noise in <math>J_z</math> is reduced relative to the coherent (unentangled) state. The second factor is how much the coherence (the length of the Bloch vector, <math>\left|J_x\right|</math>) is reduced due to the squeezing procedure. Together these quantities tell you how much metrological enhancement the squeezing procedure gives. Here, metrological enhancement is the reduction in averaging time or atom number needed to make a measurement of a specific uncertainty. 20 dB of metrological enhancement means the same precision measurement can be made with 100 times fewer atoms or 100 times shorter averaging time.
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