| NOON state | |
|---|---|
| Name | NOON state |
| Caption | Schematic of a NOON state in two modes with N photons |
| Type | Multipartite entangled state |
| Field | Quantum physics |
| Introduced | 1990s |
| Applications | Quantum metrology, Quantum lithography, Quantum sensing |
| Notable | Entanglement, Heisenberg limit |
NOON state
A NOON state is a quantum superposition of N indistinguishable particles (commonly photons) occupying one of two modes, written as a coherent superposition of all N particles in mode A and all N particles in mode B. NOON states are maximally path-entangled states that enable enhanced phase sensitivity beyond the standard quantum limit toward the Heisenberg limit, making them important resources in quantum metrology, imaging and fundamental tests of quantum mechanics.
A NOON state for two modes a and b with integer N is typically expressed as |N,0> + |0,N> (up to normalization and a possible relative phase). The state exhibits perfect N-particle path entanglement and maximal fringe visibility for ideal measurements. Key properties include an N-fold phase dependence under relative phase shifts, super-resolving interference fringes, and strong sensitivity to particle loss. NOON states are nonclassical and violate separability criteria linked to Bell's theorem when appropriate measurements are performed. They are often contrasted with Greenberger–Horne–Zeilinger (GHZ) states and Dicke states for multipartite entanglement structure.
Mathematically a NOON state is written (unnormalized) as |ψ_NOON⟩ = |N⟩_a|0⟩_b + e^{iNφ}|0⟩_a|N⟩_b, where φ is a controllable relative phase. Under a phase shift operator U(θ)=exp(iθ n̂_b) the state acquires a global phase factor exp(iNθ) on the second term, leading to an interferometric signal ∝ cos(Nθ). This N-fold phase dependence yields a Fisher information that scales as N^2 for pure NOON states, corresponding to the Heisenberg limit of phase estimation Δθ ∼ 1/N in contrast to the shot-noise scaling Δθ ∼ 1/√N. The creation and analysis of NOON states use mode operators â, b̂, the bosonic commutation relations [â,â†]=1, and entanglement measures such as von Neumann entropy or quantum Fisher information to quantify metrological advantage.
Practical generation of NOON states has been pursued primarily in quantum optics. Techniques include post-selected spontaneous parametric down-conversion (SPDC) in nonlinear crystals (e.g., using BBO), Hong–Ou–Mandel interference on beam splitters, and heralded entanglement using single-photon sources such as quantum dot emitters or NV centers in diamond. Deterministic schemes employ linear optics with adaptive measurements, nonlinear interactions (Kerr media), or entanglement swapping in networks built from components developed at institutions like Caltech, MIT, and NIST. Early experiments demonstrated N=2 and N=3 photonic NOON states; subsequent work extended to higher N with reduced fidelity due to loss and imperfect detectors (e.g., transition-edge sensors). Microwave-domain and atomic-ensemble analogues have been realized in cavity QED and BEC platforms.
NOON states provide quantum-enhanced sensitivity in interferometric measurements, enabling applications in phase estimation, frequency metrology, and displacement sensing. In optical lithography and quantum imaging they enable super-resolution patterns beyond the classical Rayleigh limit by producing interference fringes with period λ/N. In sensing platforms NOON-based protocols aim to improve precision in gravitational-wave detection concepts, clock synchronization, and magnetometry when integrated with technologies from LIGO-class interferometry or atomic clocks at NIST. Practical utility requires trade-offs between sensitivity gain and loss robustness; hybrid schemes combine NOON features with squeezed states (e.g., squeezed light) to achieve improved performance under realistic constraints.
NOON states are highly susceptible to decoherence, especially particle loss and dephasing: loss of a single particle typically destroys the N-fold phase advantage, reducing fringe visibility and Fisher information. Sources of error include imperfect mode overlap, detector inefficiency, dark counts, multiphoton contamination in SPDC, and phase noise from environmental coupling. The advantage over classical schemes can be nullified if overall transmission and detection efficiency fall below thresholds that depend on N; thus moderate-N NOON states often provide the best practical trade-off. Error-correction and adaptive estimation protocols, as used in quantum error correction and Bayesian phase estimation, can partly mitigate these limitations but typically require additional resources and complex operations.
Variants and extensions of NOON states address robustness and applicability. Examples include Holland–Burnett states (twin Fock states) that offer improved loss tolerance, entangled coherent states and cat states which combine coherence with macroscopic occupation, and multimode generalizations to more than two paths. Cluster states and graph states used in measurement-based quantum computation share multipartite entanglement structures useful for distributed sensing networks. Hybrid protocols that mix squeezed vacuum with photon-number states or use probabilistic amplification aim to capture some NOON advantages while reducing fragility. Theoretical connections link NOON states to foundational topics such as quantum nonlocality, resource theories of entanglement, and limits set by the Quantum Cramér–Rao bound.
Category:Quantum states Category:Quantum optics Category:Quantum metrology