NOON state

A NOON state (made-up word according to the formula, see below) is an entangled quantum mechanical many-particle state . The term is particularly common in quantum optics . Here, a NOON state denotes a superimposed state of photons , which are either all in one or the other of two single-particle states that can be distinguished from one another. Photons that are prepared in such a state allow very precise phase measurements (for a large number of particles ) and have therefore been proposed for applications in quantum metrology and high-resolution lithography . ${\displaystyle N}$ ${\displaystyle N}$

definition

Mathematically, NOON states are given as vectors in the Hilbert space , the symmetrical ( bosonic ) Fock space over two modes : ${\displaystyle |\psi _{\text{NOON}}\rangle }$ ${\displaystyle {\mathcal {F}}_{+}(\mathbb {C} ^{2})={\mathcal {F}}_{+}(\mathbb {C} )\otimes {\mathcal {F}}_{+}(\mathbb {C} )}$

${\displaystyle |\psi _{\text{NOON}}\rangle ={\frac {|N\rangle _{a}\otimes |0\rangle _{b}+e^{iN\theta }|{0}\rangle _{a}\otimes |{N}\rangle _{b}}{\sqrt {2}}}}$

This state describes the superposition of particles in fashion and no particles in fashion and vice versa (with relative phase ). ${\displaystyle N}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle \theta }$

In practice, photonic NOON states are mostly considered, but in general every boson field can be prepared in a NOON state.

Applications

NOON states have been investigated in connection with applications of quantum metrology , as they allow very precise phase measurements to be carried out with optical interferometers . For example applies to the observable

${\displaystyle A=|N,0\rangle \langle 0,N|+|0,N\rangle \langle N,0|}$

that their expected value for a system in the NOON state changes from to when the phase of the state increases from 0 to . I.e. for large ones, a very small phase leads to a large change in the observables. ${\displaystyle \langle A\rangle }$ ${\displaystyle +1}$ ${\displaystyle -1}$ ${\displaystyle \theta }$ ${\displaystyle \pi /N}$ ${\displaystyle N}$

For the inaccuracy ( standard deviation ) of the measurement of : ${\displaystyle A}$

${\displaystyle \Delta \theta ={\frac {\Delta A}{|d\langle A\rangle /d\theta |}}={\frac {1}{N}}.}$

This scaling with the number of particles is the best behavior that can be achieved with quantum mechanical measurements and is also known as the Heisenberg limit . It represents a quadratic improvement over the standard quantum limit ( ), which describes the best measurement that can be achieved with independent ( non- entangled) particles. ${\displaystyle \propto {\sqrt {N}}}$ ${\displaystyle N}$

NOON states are closely related to Schrödinger-Feline states and Greenberger-Horne-Zeilinger states . Like these, the NOON states are also very fragile: even small, uncontrolled interactions destroy the coherence of the superposition and lead to a loss of the advantageous properties of the state ( decoherence ). This calls into question the practical utility of NOON states for metrological purposes in many realistic situations.

Experimental generation

NOON states have been generated for small particle numbers in various experiments, e.g. B. for with optical photons and for with microwave photons . ${\displaystyle N=5}$ ${\displaystyle N=2}$

History and terminology

NOON states were introduced by Barry Sanders in connection with the study of the decoherence of Schrödinger-Katzen states and later rediscovered by Jonathan P. Dowling , who proposed them as the basis of quantum lithography. The English expression "NOON state" was first used in an article by Lee, Kok and Dowling on quantum metrology (written as "N00N", with zeros instead of Os).

credentials

1. ↑ BM Escher, RL de Matos Filho, L. Davidovich: General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology . In: Nature Physics . tape 7 , no. 5 , May 2011, p. 406-411 , doi : 10.1038 / nphys1958 .
2. ↑ Itai Afek, Oron Ambar, Yaron Silberberg: High-NOON States by Mixing Quantum and Classical Light . In: Science . tape 328 , no. 5980 , May 14, 2010, p. 879-881 , doi : 10.1126 / science.1188172 , PMID 20466927 .
3. ↑ C. Lang u. a .: Correlations, indistinguishability and entanglement in Hong-Ou-Mandel experiments at microwave frequencies . In: Nature Physics . tape 9 , no. 6 , June 2013, p. 345-348 , doi : 10.1038 / nphys2612 .
4. ^ Barry C. Sanders: Quantum dynamics of the nonlinear rotator and the effects of continual spin measurement . In: Physical Review A . tape 40 , no. 5 , September 1, 1989, pp. 2417-2427 , doi : 10.1103 / PhysRevA.40.2417 .
5. ↑ AGEDI N. Boto, Pieter Kok, Daniel S. Abrams, Samuel L. Braunstein, Colin P. Williams, Jonathan P. Dowling: Quantum Interferometric Optical Lithography: Exploiting Entanglement to beat the diffraction limit . In: Physical Review Letters . tape 85 , no. 13 , September 25, 2000, pp. 2733-2736 , doi : 10.1103 / PhysRevLett.85.2733 .
6. ^ Hwang Lee, Pieter Kok, Jonathan P. Dowling: A quantum Rosetta stone for interferometry . In: Journal of Modern Optics . tape 49 , no. 14–15 , 2002, pp. 2325-2338 , doi : 10.1080 / 0950034021000011536 .