NOON state
Encyclopedia
A NOON state is a quantum-mechanical many-body entangled state
:
which represents a superposition of N particles in mode a with zero particles in mode b, and vice versa. Usually, the particles are photon
s, but in principle any bosonic field
can support NOON states.
and quantum sensing for their ability to make precision phase measurements when used in an optical interferometer
. For example, consider the observable
The expectation value of
for a system in a NOON state switches between +1 and −1 when the phase changes from 0 to 
. Moreover, the error in the phase measurement becomes
This is the so-called Heisenberg limit, and gives a quadratic improvement over the standard quantum limit
. NOON states are closely related to Schrödinger cat states and GHZ states
, and are extremely fragile.
, Lee, and Dowling
proposed the first general method based on post-selection via photodetection. The down-side of this method was its exponential scaling of the success probability of the protocol. Pryde and White subsequently introduced a simplified method using intensity-symmetric multiport beam splitters, single photon inputs, and either heralded or conditional measurement. Their method, for example, allows heralded production of the N = 4 NOON state without the need for postselection or zero photon detections, and has the same success probability of 3/64 as the more complicated circuit of Kok et al. Cable and Dowling proposed a method that has polynomial scaling in the success probability, which can therefore be called efficient.
Two-photon NOON states, where N = 2, can be created deterministically from two identical photons and a 50:50 beam splitter. This is called the Hong–Ou–Mandel effect in quantum optics
. Three- and four-photon NOON states cannot be created deterministically from single-photon states, but they have been created probabilistically via post-selection using spontaneous parametric down-conversion. A different approach, involving the interference of non-classical light created by spontaneous parametric down-conversion and a classical laser beam on a 50:50 beam splitter, was used by I. Afek, O. Ambar, and Y. Silberberg to experimentally demonstrate the production of NOON states up to N = 5.
Super-resolution has previously been used as indicator of NOON state production, in 2005 Resch et al. showed that it could equally well be prepared by classical interferometry. They showed that only phase super-sensitivity is an unambiguous indicator of a NOON state; furthermore they introduced criteria for determining if it has been achieved based on the observed visibility and efficiency. Super resolution, but not phase super sensitivity as the efficiency was too low, of NOON states up to N = 4 photons was also demonstrated experimentally.
in Schrödinger cat states. They were independently rediscovered in 2000 by Jonathan P. Dowling's group at JPL, who introduced them as the basis for the concept of quantum lithography
. The term "NOON state" first appeared in print as a footnote in a paper published by Lee, Kok
, and Dowling
on Quantum metrology
, where it was spelled N00N, with zero's instead of Os.
Quantum entanglement
Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...
:
which represents a superposition of N particles in mode a with zero particles in mode b, and vice versa. Usually, the particles are photon
Photon
In physics, a photon is an elementary particle, the quantum of the electromagnetic interaction and the basic unit of light and all other forms of electromagnetic radiation. It is also the force carrier for the electromagnetic force...
s, but in principle any bosonic field
Bosonic field
In quantum field theory, a bosonic field is a quantum field whose quanta are bosons; that is, they obey Bose-Einstein statistics. Bosonic fields obey canonical commutation relations, as distinct from the canonical anticommutation relations obeyed by fermionic fields.Examples include scalar fields,...
can support NOON states.
Applications
NOON states are an important concept in quantum metrologyQuantum metrology
Quantum metrology is the study of making high-resolution and highly sensitive measurements of physical parameters using quantum theory to describe the physical systems, particularly exploiting quantum entanglement...
and quantum sensing for their ability to make precision phase measurements when used in an optical interferometer
Interferometry
Interferometry refers to a family of techniques in which electromagnetic waves are superimposed in order to extract information about the waves. An instrument used to interfere waves is called an interferometer. Interferometry is an important investigative technique in the fields of astronomy,...
. For example, consider the observable
The expectation value of
for a system in a NOON state switches between +1 and −1 when the phase changes from 0 to . Moreover, the error in the phase measurement becomes
This is the so-called Heisenberg limit, and gives a quadratic improvement over the standard quantum limit
Quantum limit
Quantum limit in physics refers to a limit on measurement accuracy at quantum scales.It may refer to the minimum level of quantum noise which is obtainable without squeezed states. For instance, the sensitivity of optical measurements has a standard quantum limit...
. NOON states are closely related to Schrödinger cat states and GHZ states
Greenberger-Horne-Zeilinger state
In physics, in the area of quantum information theory, a Greenberger–Horne–Zeilinger state is a certain type of entangled quantum state which involves at least three subsystems . It was first studied by D. Greenberger, M.A. Horne and Anton Zeilinger in 1989...
, and are extremely fragile.
Towards experimental realization
There have been several theoretical proposals for creating photonic NOON states. KokPieter Kok
Pieter Kok, Ph.D. is one of the co-developers of quantum interferometric optical lithography.He is a co-author of a measurement-based entanglement protocol...
, Lee, and Dowling
Jonathan Dowling
Jonathan P. Dowling is a Co-Director of the Horace Hearne Institute for Theoretical Physics, and also a Hearne Chair in Theoretical Physics at the Department of Physics and Astronomy, both at Louisiana State University...
proposed the first general method based on post-selection via photodetection. The down-side of this method was its exponential scaling of the success probability of the protocol. Pryde and White subsequently introduced a simplified method using intensity-symmetric multiport beam splitters, single photon inputs, and either heralded or conditional measurement. Their method, for example, allows heralded production of the N = 4 NOON state without the need for postselection or zero photon detections, and has the same success probability of 3/64 as the more complicated circuit of Kok et al. Cable and Dowling proposed a method that has polynomial scaling in the success probability, which can therefore be called efficient.
Two-photon NOON states, where N = 2, can be created deterministically from two identical photons and a 50:50 beam splitter. This is called the Hong–Ou–Mandel effect in quantum optics
Quantum optics
Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter.- History of quantum optics :...
. Three- and four-photon NOON states cannot be created deterministically from single-photon states, but they have been created probabilistically via post-selection using spontaneous parametric down-conversion. A different approach, involving the interference of non-classical light created by spontaneous parametric down-conversion and a classical laser beam on a 50:50 beam splitter, was used by I. Afek, O. Ambar, and Y. Silberberg to experimentally demonstrate the production of NOON states up to N = 5.
Super-resolution has previously been used as indicator of NOON state production, in 2005 Resch et al. showed that it could equally well be prepared by classical interferometry. They showed that only phase super-sensitivity is an unambiguous indicator of a NOON state; furthermore they introduced criteria for determining if it has been achieved based on the observed visibility and efficiency. Super resolution, but not phase super sensitivity as the efficiency was too low, of NOON states up to N = 4 photons was also demonstrated experimentally.
History and terminology
NOON states were first introduced by Barry Sanders in the context of studying quantum decoherenceQuantum decoherence
In quantum mechanics, quantum decoherence is the loss of coherence or ordering of the phase angles between the components of a system in a quantum superposition. A consequence of this dephasing leads to classical or probabilistically additive behavior...
in Schrödinger cat states. They were independently rediscovered in 2000 by Jonathan P. Dowling's group at JPL, who introduced them as the basis for the concept of quantum lithography
Quantum lithography
Quantum lithography is a type of photolithography, which exploits non-classical properties of the photons, such as quantum entanglement, in order to achieve superior performance over ordinary classical lithography. Quantum lithography is closely related to the fields of quantum imaging, quantum...
. The term "NOON state" first appeared in print as a footnote in a paper published by Lee, Kok
Pieter Kok
Pieter Kok, Ph.D. is one of the co-developers of quantum interferometric optical lithography.He is a co-author of a measurement-based entanglement protocol...
, and Dowling
Jonathan Dowling
Jonathan P. Dowling is a Co-Director of the Horace Hearne Institute for Theoretical Physics, and also a Hearne Chair in Theoretical Physics at the Department of Physics and Astronomy, both at Louisiana State University...
on Quantum metrology
Quantum metrology
Quantum metrology is the study of making high-resolution and highly sensitive measurements of physical parameters using quantum theory to describe the physical systems, particularly exploiting quantum entanglement...
, where it was spelled N00N, with zero's instead of Os.