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Singlet

In quantum mechanics, a singlet state is a quantum state of a multi-particle system in which the total spin angular momentum quantum number S is zero. This typically occurs for systems like two spin-1/2 particles (e.g., electrons) with opposite spins, resulting in an antisymmetric spin wavefunction. The term "singlet" originates from spectroscopy, where such states produce a single spectral line, in contrast to multiplets like doublets (S=1/2) or triplets (S=1).^[1] Singlet states are crucial in atomic and molecular physics for understanding electronic configurations and transitions, as well as in quantum information science due to their entangled nature. Examples include the ground state of helium and the positronium atom.^[2]

Fundamentals

Definition

In quantum mechanics, a singlet state is a quantum mechanical state characterized by a total spin angular momentum quantum number S = 0, where the spins of constituent particles are paired such that the net angular momentum is zero.^[2] This configuration emerges from the intrinsic angular momentum, or spin, of fundamental particles like electrons, which have a spin quantum number s = \frac{1}{2}, allowing only two possible orientations along a given axis.^[3] The singlet state is fundamentally linked to the Pauli exclusion principle, which requires the overall wavefunction of identical fermions, such as electrons, to be antisymmetric upon exchange of any two particles.^[4] For paired electrons occupying the same spatial orbital, the symmetric spatial wavefunction necessitates an antisymmetric spin component, resulting in the singlet pairing of opposite spins and ensuring compliance with this principle.^[4] Singlet states differ from those of higher multiplicity, such as doublets with S = \frac{1}{2} (one unpaired electron) or triplets with S = 1 (two unpaired electrons with parallel spins), which possess net spin angular momentum and associated magnetic moments.^[5] In singlets, the opposing spins cancel out, yielding zero net spin and rendering these states non-magnetic, as no effective magnetic field is generated.^[5] In closed-shell atoms and molecules, where all orbitals are doubly occupied by electron pairs with opposite spins, the ground electronic state is predominantly a singlet, reflecting the energetic favorability of this fully paired configuration.^[6]

Multiplicity and spectral lines

In atomic spectroscopy, the multiplicity of a spectral term is given by $2S + 1, where S is the total spin angular momentum quantum number of the electrons in the atomic state.^[7] For singlet states, S = 0, yielding a multiplicity of 1, which indicates the absence of spin degeneracy.^[8] This results in a single, unsplit spectral line in emission or absorption spectra for transitions between singlet terms, as there are no additional sublevels arising from spin orientation.^[8] Spectral line splitting in atomic spectra can arise from interactions such as spin-orbit coupling or the Zeeman effect in a magnetic field. Spin-orbit coupling, which couples the orbital angular momentum \mathbf{L} and spin \mathbf{S} to form total angular momentum \mathbf{J} = \mathbf{L} + \mathbf{S}, produces fine structure splitting into multiple J levels for states with S > 0, leading to multiplet lines.^[7] In singlet states, however, S = 0 implies J = L, eliminating spin-dependent fine structure and preserving a single line without such splitting.^[8] Similarly, under the Zeeman effect, singlet transitions exhibit the normal Zeeman pattern, where lines split into three components (due solely to orbital magnetic moments), rather than the more complex anomalous pattern seen in multiplets with nonzero spin.^[9] A key example in atomic physics is the singlet transition ^1S \to ^1P, which produces a sharp, undivided spectral line, in contrast to multiplet transitions like doublets or triplets that display multiple closely spaced lines from J-level differences.^[8] The term "singlet" was coined in early 20th-century spectroscopy to denote these characteristic single lines, distinguishing them from the split multiplets observed in higher-multiplicity states./Spectroscopy/Electronic_Spectroscopy/Spin-orbit_Coupling/Atomic_Term_Symbols)

Historical Development

Origins in spectroscopy

The discovery of atomic spectra in the late 19th and early 20th centuries laid the groundwork for understanding the structured patterns of spectral lines, which later informed the concept of singlets. In 1885, Johann Jakob Balmer empirically derived a formula describing the wavelengths of visible hydrogen emission lines, known as the Balmer series, revealing a regular progression that suggested quantized energy levels in atoms.^[10] This pattern was soon generalized by Johannes Rydberg in 1889, who extended it to other spectral series in hydrogen and alkali metals, identifying constant differences in wavenumbers that indicated underlying regularities in atomic transitions beyond simple single lines.^[11] These observations highlighted the complexity of spectra from multi-electron atoms, where lines often appeared as closely spaced groups rather than isolated features. By the 1920s, spectroscopists began classifying these grouped lines based on their number of components, introducing the multiplicity notation that defined singlets as isolated lines without splitting, in contrast to doublets (two components) and triplets (three components). Alfred Fowler and collaborators, including Friedrich Paschen and Richard Goetze, published extensive tables in 1922 organizing spectral lines into such categories through meticulous observational analysis, without yet invoking quantum explanations.^[12] A pivotal contribution came from Miguel A. Catalán, working in Fowler's laboratory, who in 1922 analyzed the manganese arc spectrum and identified regular spacings within what appeared as diffuse triplets, revealing them as multiplets—clusters of lines following numerical patterns—and coining the term "multiplet" to describe these structures.^[13] Catalán's work demonstrated that singlets represented the simplest case, where no such splitting occurred, providing an empirical framework for classifying atomic transitions.^[14] Niels Bohr's 1913 atomic model, which successfully predicted the hydrogen spectrum's line positions using quantized orbits, indirectly influenced interpretations of multiplicity by emphasizing discrete energy states, though it could not account for the observed splittings in heavier elements.^[10] These empirical patterns persisted as anomalies until 1925, when George Uhlenbeck and Samuel Goudsmit proposed that electrons possess an intrinsic angular momentum, or spin, to explain the doublet separations in alkali spectra and broader multiplet structures.^[15] This hypothesis resolved discrepancies in the Zeeman effect and fine structure, linking observed singlets, doublets, and triplets to spin-orbit coupling. The empirical spectral classifications thus paved the way for the spin quantum number, bridging classical spectroscopy with emerging quantum theory by necessitating additional degrees of freedom in atomic models.^[15]

Adoption in quantum mechanics

The introduction of electron spin by George Uhlenbeck and Samuel Goudsmit in late 1925 provided a quantum mechanical foundation for the empirical classification of atomic spectral terms into singlets, doublets, and higher multiplets observed in spectroscopy. Their hypothesis posited that electrons possess an intrinsic angular momentum of s = 1/2, explaining doublets as arising from the spin-orbit interaction in atoms with an unpaired electron (S = 1/2), while singlets correspond to paired electrons with antiparallel spins yielding total spin S = 0. This resolved longstanding puzzles in alkali metal spectra, such as the Balmer series, by integrating spin into the orbital angular momentum framework without ad hoc assumptions.^[16] The Uhlenbeck-Goudsmit model was further substantiated in 1928 by Paul Dirac's relativistic quantum equation, which naturally incorporated spin $1/2 for the electron as an intrinsic property, deriving the gyromagnetic ratio and confirming singlets as total spin-zero states in multi-electron systems. Dirac's formalism eliminated the semi-classical spinning electron picture, replacing it with a rigorous quantum field-theoretic description that predicted the fine structure of hydrogen and extended seamlessly to composite systems where total spin multiplicity $2S + 1 = 1 defines singlet configurations. This relativistic underpinning solidified the adoption of singlet states across atomic physics, enabling precise predictions of energy levels without empirical adjustments.^[17] A key milestone in this adoption was the 1926 development of angular momentum addition rules in the emerging frameworks of wave and matrix mechanics. These rules classified coupled spin and orbital states, allowing total spin \mathbf{S} = \sum \mathbf{s}_i to combine with orbital \mathbf{L} via Clebsch-Gordan coefficients, explicitly identifying singlet terms where S = 0 and no spin degeneracy occurs. They bridged empirical spectroscopy with full quantum theory, enabling systematic analysis of atomic and molecular spectra. In the 1930s, the singlet concept extended to nuclear physics, where proton and neutron spins (s = 1/2 each) formed the basis for describing two-nucleon interactions via exchange forces. Werner Heisenberg's 1932 theory of nuclear structure introduced spin-dependent potentials, distinguishing singlet (S = 0, antisymmetric) and triplet (S = 1, symmetric) channels; for instance, the deuteron ground state binds exclusively in the triplet configuration, while the singlet remains unbound as a low-energy virtual state, underscoring the tensor nature of the nuclear force. This adoption paralleled atomic applications, with nuclear magnetic moments and scattering data analyzed using singlet-triplet separations to probe isospin symmetry. Post-World War II advancements in many-body quantum mechanics refined the treatment of molecular singlets through configuration interaction (CI) methods and semi-empirical approximations. In the 1950s, Robert Parr, Rudolph Pariser, and John Pople developed the Pariser-Parr-Pople (PPP) method for π-electron systems, accurately computing energy differences between ground-state singlets and excited singlets/triplets in conjugated molecules like benzene, incorporating electron correlation beyond Hartree-Fock limits.^[18] These techniques, building on earlier valence bond theory, enabled quantitative predictions of molecular absorption spectra and photochemical reactivity, where singlet states dominate closed-shell ground configurations. Further progress in ab initio CI by S. Francis Boys in the early 1950s facilitated explicit calculations of multi-reference singlets in polyatomic systems, establishing computational standards for quantum chemistry.^[19] In the mid-20th century, further developments included the 1960s applications of time-dependent perturbation theory to singlet-triplet intersystem crossing in organic molecules, enhancing understanding of photochemical processes. By the 1970s, computational tools like Gaussian programs integrated CI methods for routine singlet state calculations, bridging theory and experiment in quantum chemistry.^[20]

Mathematical Formulation

Two-particle spin singlet

The two-particle spin singlet state represents the unique antisymmetric configuration for two spin-1/2 particles with total spin quantum number S = 0. Its explicit wavefunction in the uncoupled basis, using the standard notation where |\uparrow\rangle and |\downarrow\rangle denote the spin-up and spin-down states along the z-axis, is given by

|\psi\rangle = \frac{1}{\sqrt{2}} \left( |\uparrow\rangle_1 |\downarrow\rangle_2 - |\downarrow\rangle_1 |\uparrow\rangle_2 \right).

This form ensures normalization, as \langle \psi | \psi \rangle = 1, and antisymmetry under particle exchange, which is required for identical fermions when combined with a symmetric spatial wavefunction.^[21]^[22] The derivation of this state relies on the total spin operator \hat{\mathbf{S}} = \hat{\mathbf{S}}_1 + \hat{\mathbf{S}}_2, where \hat{\mathbf{S}}_1 and \hat{\mathbf{S}}_2 are the individual spin operators for each particle. The singlet is an eigenstate of the total spin squared operator \hat{S}^2 = (\hat{\mathbf{S}}_1 + \hat{\mathbf{S}}_2)^2, satisfying the eigenvalue equation

\hat{S}^2 |\psi\rangle = S(S+1) \hbar^2 |\psi\rangle,

with S = 0, so \hat{S}^2 |\psi\rangle = 0. To construct it, start from the product basis states with total z-component m_S = 0, namely |\uparrow \downarrow\rangle and |\downarrow \uparrow\rangle. The symmetric combination \frac{1}{\sqrt{2}} (|\uparrow \downarrow\rangle + |\downarrow \uparrow\rangle) forms the m_S = 0 member of the triplet (S=1), while the orthogonal antisymmetric combination yields the singlet. Applying \hat{S}^2 explicitly confirms the eigenvalue zero for the antisymmetric state, distinguishing it from the triplet's S(S+1)\hbar^2 = 2\hbar^2.^[21]^[23] The singlet state is orthogonal to the three symmetric triplet states: |\uparrow \uparrow\rangle (S=1, m_S=1), \frac{1}{\sqrt{2}} (|\uparrow \downarrow\rangle + |\downarrow \uparrow\rangle) (S=1, m_S=0), and |\downarrow \downarrow\rangle (S=1, m_S=-1). For instance, the inner product with the m_S=0 triplet is \left\langle \frac{1}{\sqrt{2}} (|\uparrow \downarrow\rangle + |\downarrow \uparrow\rangle) \middle| \psi \right\rangle = 0, ensuring the basis spans the full four-dimensional Hilbert space without overlap.^[22]^[21] A key property is the vanishing expectation value of the total z-component spin operator, \langle \hat{S}_z \rangle = \langle \psi | \hat{S}_z | \psi \rangle = 0, where \hat{S}_z = \hat{S}_{z1} + \hat{S}_{z2}. This follows from the equal weights of the m_{s1} = +\frac{1}{2}, m_{s2} = -\frac{1}{2} and opposite terms, each contributing +\frac{1}{2}\hbar - \frac{1}{2}\hbar = 0 in units of \hbar.^[23]^[22] The singlet possesses rotational invariance, behaving as a scalar under transformations of the SU(2) group, with total spin zero. This invariance holds for any quantization axis, as the state can be expressed equivalently in terms of spin projections along an arbitrary direction \mathbf{n}: |\psi\rangle = \frac{1}{\sqrt{2}} \left( |n; +\rangle_1 |n; -\rangle_2 - |n; -\rangle_1 |n; +\rangle_2 \right), confirming its isotropic nature independent of orientation.^[22]^[21]

Generalization to multi-particle systems

The concept of the spin singlet state extends beyond two particles to systems involving N spin-1/2 particles, where a total spin S=0 is possible only for even N due to the half-integer nature of individual spins, ensuring an integer total spin. Such multi-particle singlets can be constructed through repeated application of angular momentum coupling rules, successively pairing spins to yield intermediate S=0 states, or via fully antisymmetric Slater determinants for identical fermions, where the overall spin wavefunction achieves total S=0 to satisfy the Pauli exclusion principle.^[24]^[25] For particles with higher spin values, singlet states are similarly obtained using Clebsch-Gordan coefficients to couple individual angular momenta to total J=0. For instance, the singlet state of two spin-1 particles is given by

|\psi\rangle = \frac{1}{\sqrt{3}} \left( |1,1\rangle|1,-1\rangle - |1,0\rangle|1,0\rangle + |1,-1\rangle|1,1\rangle \right),

where the kets denote the individual spin states |j, m⟩, illustrating the antisymmetric combination required for the scalar (J=0) representation.^[26] In quantum field theory, singlet representations under the SU(2) group correspond to states or fields invariant under SU(2) transformations, such as scalar fields with zero spin, which transform trivially as the trivial representation of SU(2). These singlets play a crucial role in constructing gauge-invariant operators, for example, in bilinear combinations of fields that eliminate charged components to form neutral scalars. A key aspect of multi-particle singlets is the symmetry of the total wavefunction: for fermions, the spin singlet (antisymmetric) pairs with a symmetric spatial part, often realized in closed-shell configurations where all electrons are paired in orbitals, yielding a totally symmetric overall state under point group operations.^[27]

Physical Manifestations

Atomic and molecular examples

In atomic systems, the helium atom provides a classic example of a singlet state. Its ground state configuration, 1s², features two electrons with opposite spins, resulting in a total spin quantum number S = 0 and thus a singlet multiplicity.^[28] This configuration is the most stable for helium due to the Pauli exclusion principle, which requires antisymmetric wavefunctions and pairs the electrons in the lowest orbital.^[29] Excited states of helium, such as the 1s2s ¹S and 1s2p ¹P configurations, also exhibit singlet character when the spins are antiparallel, though these lie higher in energy than the corresponding triplet states. These singlet excited states are observable in atomic spectra and highlight the role of spin coupling in multi-electron atoms.^[30] In molecular systems, closed-shell diatomic molecules like H₂ and N₂ exemplify ground-state singlets. The ground state of H₂, denoted as X¹Σ_g^+, consists of two electrons in a σ bonding orbital with paired spins, forming a stable singlet with all electrons in filled molecular orbitals. Similarly, N₂'s ground state X¹Σ_g^+ features 14 electrons in closed shells, with the triple bond arising from paired electrons in σ and π orbitals, ensuring S = 0 and high stability.^[31] These configurations underscore the prevalence of singlet ground states in diamagnetic molecules, where electron pairing minimizes energy through symmetric spatial wavefunctions. Most stable molecules adopt singlet ground states for this reason, with molecular oxygen (O₂) as a notable exception—its ground state is a triplet (³Σ_g^-) due to two unpaired electrons in degenerate π* orbitals, leading to diradical character and paramagnetism. Excited singlet states are common in organic molecules and are depicted in the Jablonski diagram as S₁ levels above the ground singlet S₀. Upon absorbing ultraviolet or visible light, an electron is promoted from S₀ to S₁, where the spins remain paired; this state is typically short-lived (on the order of nanoseconds) because radiative decay to S₀ via fluorescence does not require spin flipping, unlike triplet states. In conjugated systems like benzene or polyenes, the S₁ state facilitates rapid energy dissipation, contributing to the molecule's photostability.^[32] Singlet transitions in these atomic and molecular systems are routinely observed using ultraviolet-visible (UV-Vis) spectroscopy, where absorption bands correspond to promotions from singlet ground states to higher singlet excited states. For instance, the intense π → π* transitions in organic molecules appear as sharp peaks in the UV-Vis spectrum, allowing direct probing of singlet energetics without interference from spin-forbidden lines.^[33]

Exotic cases like positronium

Positronium, a short-lived exotic atom formed by an electron and its antiparticle, the positron, bound via the Coulomb interaction, exhibits singlet and triplet ground states analogous to those in hydrogen-like systems. The singlet ground state, known as para-positronium (^1S_0), has total spin S=0 and decays predominantly into two photons due to conservation of angular momentum and C-parity, with a vacuum lifetime of 125 ps.^[34] In contrast, the triplet ground state, ortho-positronium (^3S_1) with S=1, has a much longer lifetime of 142 ns and decays primarily into three photons, as two-photon decay is forbidden by symmetry.^[35] These distinct decay channels and lifetimes arise from the spin-dependent annihilation probabilities in quantum electrodynamics (QED). Another exotic lepton-antilepton system is muonium, a bound state of a positive muon and an electron, which displays hyperfine structure splitting the ground state into a spin singlet (F=0) and a spin triplet (F=1). The singlet state constitutes one-fourth of the initial population in the absence of magnetic fields, with the hyperfine transition frequency measured at 4463.302 MHz, corresponding to an energy splitting that tests QED corrections in bound systems.^[36] This structure, dominated by the Fermi contact interaction, allows precise spectroscopy of the singlet-triplet transition via microwave-induced Rabi oscillations observed through positron emission asymmetry.^[37] Unbound singlet states manifest in low-energy scattering processes, such as those involving free electron-proton pairs in the total spin S=0 configuration, where the antisymmetric spin wavefunction leads to an unbound virtual state rather than a stable bound state.^[38] This S=0 partial wave contributes to the scattering cross-section in quantum mechanical treatments, influencing phenomena like deuteron virtual states in analogous nucleon systems and providing insights into spin-dependent interactions.^[39] The para-positronium singlet state serves as a benchmark for QED precision tests, particularly in measurements of its decay rate and fine structure, where experimental results agree with theoretical predictions incorporating higher-order radiative corrections to relative precisions approaching 10^{-9}, though recent fine-structure spectroscopy reveals discrepancies up to several standard deviations that challenge bound-state QED.^[40] Such tests, free from hadronic uncertainties, probe fundamental QED parameters and search for new physics beyond the Standard Model.^[41]

Connections to Quantum Entanglement

Singlet states as entangled pairs

In quantum mechanics, the singlet state for two spin-1/2 particles represents a paradigmatic example of quantum entanglement, characterized by an antisymmetric wavefunction that enforces perfect anticorrelations in their spin measurements. Specifically, if the spin of one particle is measured along any direction and found to be up, the spin of the other particle is instantaneously determined to be down along the same direction, regardless of the spatial separation between them. This correlation arises from the non-separable nature of the joint quantum state, which cannot be expressed as a product of individual particle states.^[42] The two-particle singlet is formally known as the Bell singlet state, denoted as one of the four maximally entangled Bell states in two-qubit systems, where the entanglement entropy reaches its maximum value of 1 ebit (one unit of entanglement). Unlike the other Bell states, the singlet possesses total spin zero and is antisymmetric under particle exchange, distinguishing it as the unique rotationally invariant entangled state for two qubits. This maximal entanglement implies that the singlet serves as a fundamental resource for distributing quantum correlations without any classical communication, embodying the essence of non-local quantum links. Singlet states can be experimentally prepared through various physical processes that naturally produce entangled pairs. For photonic systems, spontaneous parametric down-conversion (SPDC) in birefringent nonlinear crystals, such as beta-barium borate, generates polarization-entangled photon pairs in the singlet configuration by pumping with a laser and collecting the down-converted photons at degenerate wavelengths. This method yields high-fidelity singlets with visibilities exceeding 99% under optimal phase-matching conditions. Complementarily, atomic cascades in neutral calcium atoms, excited to a higher energy level and decaying through intermediate states, emit successive photons whose polarizations are entangled in the singlet state due to conservation of angular momentum in the J=0 to J=0 transition. These preparation techniques have enabled the realization of singlet entanglement with near-unit fidelity in laboratory settings. A defining feature of the singlet state is its invariance under local unitary transformations, meaning that applying the same unitary operator to both particles leaves the state unchanged up to a global phase. This property underscores the singlet's role as a "pure" entanglement resource, free from local coherences that could be manipulated independently, and it ensures that entanglement measures remain unaltered by basis changes on individual subsystems. In quantum information theory, this invariance facilitates the use of singlets as canonical representatives for quantifying and distilling entanglement in bipartite systems.^[43] While bipartite singlets are thoroughly characterized, extensions to multipartite systems introduce richer structures of entanglement, where total spin-zero states for an even number of spin-1/2 particles exhibit genuine multipartite correlations. These multipartite singlets, which cannot achieve maximal entanglement across all bipartitions simultaneously, have been analyzed in contexts like symmetric permutations and unitary invariance, revealing limitations on their entanglement distribution compared to bipartite cases. Notably, discussions on integrating such states into cluster-like architectures for multipartite quantum networks remain underexplored, highlighting an active frontier in understanding higher-order singlet entanglement.^[44]^[45]

EPR paradox and Bell's theorem

In 1935, Albert Einstein, Boris Podolsky, and Nathan Rosen proposed the EPR paradox to argue that quantum mechanics provides an incomplete description of physical reality.^[46] They considered two particles prepared in a spin singlet state and separated by a large distance, noting that a measurement of one particle's spin along any direction instantaneously determines the outcome for the other particle with perfect anticorrelation. This implied either that quantum mechanics lacks hidden variables specifying the particles' properties beforehand or that it permits "spooky action at a distance" through non-local influences, which they deemed untenable.^[46] John Stewart Bell addressed the EPR argument in 1964 with a theorem demonstrating that no local hidden-variable theory—respecting locality and realism—could reproduce all predictions of quantum mechanics for entangled systems like the singlet state.^[47] Bell derived an inequality bounding the correlations observable under local realism; a common formulation is the Clauser-Horne-Shimony-Holt (CHSH) inequality, which states that for any local hidden-variable model, the parameter S = E(a,b) + E(a,b') + E(a',b) - E(a',b') satisfies |S| \leq 2, where E denotes the expectation value of the product of measurement outcomes for angles a, b, a', b'.^[48] In contrast, quantum mechanics predicts for the singlet state a correlation function P(\mathbf{a}, \mathbf{b}) = -\mathbf{a} \cdot \mathbf{b}, yielding maximum |S| = 2\sqrt{2} \approx 2.828 for optimal angles, clearly violating the inequality.^[48] Experimental tests began confirming quantum mechanics' predictions over local realism in the 1970s, with John Clauser and Stuart Freedman's 1972 experiment providing the first such confirmation using entangled photons from atomic cascades, though subject to loopholes. More advanced experiments in the 1980s, such as Alain Aspect's 1982 work with entangled photons, demonstrated a violation of Bell's inequalities by more than 5 standard deviations, in strong agreement with quantum theory.^[49]^[50] Subsequent refinements addressed potential loopholes, such as the detection and locality loopholes; landmark loophole-free tests in 2015, using electron spins in diamond and entangled photons, reported violations exceeding 2 standard deviations while ensuring space-like separation of measurements.^[51]^[52] This foundational research on entanglement and Bell tests was recognized by the 2022 Nobel Prize in Physics, awarded jointly to John F. Clauser, Alain Aspect, and Anton Zeilinger.^[53] The apparent non-locality in singlet correlations does not enable faster-than-light signaling, as resolved by the no-signaling theorem, which proves that measurements on one subsystem cannot alter the reduced density matrix or observable statistics of a distant entangled partner.^[54] This preserves causality and relativity, confining the EPR paradox to a challenge against local realism rather than a flaw in quantum mechanics' consistency.

Applications

In chemical reactivity

Singlet excited states play a crucial role in chemical reactivity, particularly in photochemistry, where they enable spin-allowed pathways that facilitate efficient electron transfer and bond formation without the need for spin-orbit coupling-mediated intersystem crossing, unlike triplet states.^[55] In organic molecules and reactive oxygen species, these states promote selective oxidations and cycloadditions, driving processes in synthesis, biology, and environmental chemistry. A prominent example is singlet oxygen (^1O_2), the lowest excited electronic state of molecular oxygen (^3\Sigma_g^- \to ^1\Delta_g), which is highly electrophilic and reactive toward electron-rich substrates due to its empty antibonding \pi^* orbital.^[56] It is commonly generated via photosensitization, where a triplet-excited photosensitizer (e.g., rose bengal or methylene blue) transfers energy to ground-state triplet oxygen in a spin-conserving process.^[57] This method allows controlled production in solution or biological media, with quantum yields approaching 0.8 for efficient sensitizers.^[58] The reactivity of singlet oxygen proceeds through concerted mechanisms, including [4+2] cycloadditions with conjugated dienes to form endoperoxides, as seen in the oxidation of furans or anthracene derivatives, and ene reactions with alkenes bearing allylic hydrogens to yield allylic hydroperoxides.^[59] For instance, the Schenck ene reaction with cholesterol produces a hydroperoxide that is a key intermediate in lipid peroxidation. These pathways are stereospecific and occur under mild conditions, contrasting with the spin-forbidden reactions of ground-state oxygen that require radical initiation. Singlet oxygen has a short lifetime, approximately 2–4 μs in aqueous or organic solvents, limited by quenching mechanisms such as radiative decay (near-infrared phosphorescence at 1270 nm), non-radiative relaxation to triplet oxygen, or physical quenching by solvents like water.^[60] Chemical quenching by biomolecules, such as histidine or ascorbic acid, further reduces its effective lifetime in biological environments, with rate constants up to $10^7 M^{-1} s^{-1}.^[61] In organic synthesis, singlet oxygen enables selective functionalizations, such as the ene reaction for introducing oxygen functionality in natural product analogs, avoiding harsh oxidants.^[62] In photodynamic therapy (PDT) for cancer, photosensitizers like porphyrins accumulate in tumors and, upon red-light irradiation, generate ^1O_2 to induce localized oxidative damage to cellular components, leading to apoptosis with minimal invasiveness.^[63] Clinical approvals, such as porfimer sodium for esophageal cancer, highlight its therapeutic impact.^[64] Emerging research explores singlet oxygen's roles in atmospheric chemistry, where it contributes to the photooxidation of volatile organic compounds via photosensitized generation from brown carbon aerosols, influencing tropospheric oxidant budgets.^[65] Additionally, singlet excited states underpin singlet fission processes in organic semiconductors, where a high-energy singlet exciton splits into two lower-energy triplets to enhance charge generation in solar cells, potentially boosting efficiencies beyond the Shockley-Queisser limit (see Singlet Fission article).^[66] Recent advances as of 2025 include improved detection methods for singlet oxygen in energy storage applications.^[67]

In quantum information science

In quantum key distribution, singlet states provide a foundational resource for protocols that ensure secure communication by exploiting quantum entanglement. The E91 protocol, introduced by Artur Ekert in 1991, employs pairs of particles prepared in a spin singlet state to distribute a cryptographic key, where security is verified through the violation of Bell's inequality, which detects any eavesdropping attempts that would disturb the entanglement.^[68] This approach contrasts with prepare-and-measure schemes like the original BB84 protocol by directly using shared entanglement for key generation, and experimental implementations have achieved secure key rates over fiber-optic links.^[69] Entanglement-assisted variants of BB84 further integrate singlet-like Bell pairs to enable authentication alongside key distribution, reducing vulnerability to man-in-the-middle attacks without additional classical resources.^[70] Quantum teleportation protocols leverage singlet states to transfer unknown qubit states between distant parties without transmitting the quantum carrier itself. In the seminal 1993 scheme by Bennett, Brassard, and collaborators, Alice and Bob share a maximally entangled singlet pair; Alice performs a joint measurement on her qubit and the state to teleport, then communicates the two classical bits of outcome to Bob, who applies a correction to his half of the singlet, reconstructing the original state with perfect fidelity in the ideal case.^[71] This process consumes one ebit of entanglement per teleported qubit and has been experimentally realized over distances up to 1400 km using satellite-based singlets, demonstrating near-unity fidelity despite atmospheric turbulence.^[72] A key property enabling these applications is the maximal entanglement of singlet states, quantified by a concurrence of 1, which ensures optimal extraction of quantum correlations even in the presence of decoherence, thereby preserving protocol fidelity in noisy environments.^[73] This full entanglement measure distinguishes singlets from partially entangled states, making them preferable for tasks requiring high Bell inequality violations or efficient state transfer. In quantum computing architectures, singlet states function as robust resources for error correction and computation primitives. For instance, in singlet-triplet spin qubit systems, the singlet subspace inherently suppresses certain charge noise, facilitating fault-tolerant operations within surface code frameworks that correct bit-flip and phase errors.^[74] In measurement-based quantum computing, projections onto the singlet-triplet basis of two qubits enable universal gate sets when starting from mixed single-qubit states, as these measurements generate the necessary entanglement on demand without ancillary controls.^[75] Multipartite extensions, such as GHZ states, act as analogs to two-particle singlets by providing maximal genuine multipartite entanglement for cluster-state resources in scalable one-way computing models.^[76] Advancements in the 2020s have enhanced singlet generation for practical quantum networks, with trapped-ion platforms achieving fidelities above 99% for Bell singlets through optimized laser-driven entangling gates, supporting rates up to thousands per second for repeater applications.^[77] Similarly, hybrid superconducting-semiconductor systems have demonstrated tunable singlet-triplet qubits with two-qubit gate fidelities exceeding 99.5%, enabling entanglement distribution in circuit-based architectures resistant to magnetic field gradients.^[78] These developments pave the way for singlet-based quantum repeaters, where atomic or solid-state memories store singlets for purification and swapping, extending entanglement over thousands of kilometers by mitigating exponential photon loss in fibers.^[79] As of 2025, progress includes demonstrations of distributed quantum information processing using singlet resources over optical networks.^[80]

References

[1]
SINGLET | definition in the Cambridge English Dictionary
a piece of clothing without sleeves that is worn on the top part of the body under clothes, or for playing particular sports.
[2]
https://www.sas.upenn.edu/~mcnemar/firefly/fluor2.html
[3]
Friday essay: the singlet — a short history of an Australian icon
Dec 3, 2020 · Donned first as underwear, then for sport and later as the uniform of Australia's working man, this simple garment has accrued complex cultural meanings over ...
[4]
fluor2
Singlet state: All electrons in the molecule are spin paired. It is called a singlet because there is only one possible orientation in space.
[5]
Electron spin
### Definition of Electron Spin as Intrinsic Angular Momentum
[6]
Pauli Exclusion Principle Ground State of He: ΨHe(1,2) = Ψ1s(1 ...
SPATIAL PART * SPIN PART symmetric antisymmetric. Pauli Exclusion Principle: The wavefunction is overall antisymmetric with respect.
[7]
10.7 Additional Points
But two of these electrons are in the 1s states, the K shell, where they combine into a singlet state with zero net spin which produces no magnetic field.Missing: non- | Show results with:non-
[8]
[PDF] Apr. 28, 2006
For a typical, closed-shell singlet molecule, the degeneracy of the ground state is unity, and the various excited states are so high in energy that, at ...
[9]
NIST: Atomic Spectroscopy - Helium-like Ions; LS Coupling
The combination of a particular S value with a particular L value comprises a spectroscopic term, the notation for which is 2S+1L. The quantum number 2S + 1 is ...
[10]
A Primer on Quantum Numbers and Spectroscopic Notation
The multiplicity of a term is given by the value of 2S+1. A term with S=0 is a singlet term; S=1/2 is a doublet term; S=1 is a triplet term; S ...
[11]
[PDF] The Zeeman Effect - Physics Courses
The Zeeman effect that occurs for spectral lines resulting from a transition between singlet states is traditionally called the normal effect, while that which ...
[12]
Getting the numbers right - the lonely struggle of Rydberg | Feature
Jun 30, 2004 · Some spectra, however, contained groups of lines in apparently regular patterns. The Irish astronomer and physicist George Johnston Stoney ...<|separator|>
[13]
[PDF] Spectroscopy from 1916 to 1940 - NET
He called these groups multiplets, and their discovery sparked a productive era of description and interpretation of the optical spectra of most complex atoms, ...
[14]
[PDF] The first spectrum of manganese, Mn I
In 1922, Catalan derived, from spectral series of rllnning s electrons, the absolute energy of the ground state of Mn T, and computed the first ionization.
[15]
(PDF) Miguel A. Catalán's CXXV Anniversary - ResearchGate
Aug 6, 2025 · The spectrum of manganese and its representation including multiplets in the report published by Catalán: "Series and other regularities in the ...
[16]
[PDF] Niels Bohr - Nobel Lecture
This result shows that there exists a connexion between the hydrogen spectrum and the model for the hydrogen atom which, on the whole, is as close as we might ...
[17]
Spinning Electrons and the Structure of Spectra - Nature
Spinning Electrons and the Structure of Spectra. G. E. UHLENBECK &; S. GOUDSMIT. Nature volume 117, pages 264–265 ...
[18]
[PDF] The birth of the concept of Spin
Sep 19, 2012 · The spinning electron (finally!) Uhlenbeck and Goudsmit were the first to publish their idea about the self-rotating electron. 1925 paper points ...<|control11|><|separator|>
[19]
The quantum theory of the electron - Journals
To meet the difficulty, Goudsmit and Uhlenbeck have introduced the idea of an electron with a spin angular momentum of half a quantum and a magnetic moment of ...
[20]
https://www.osti.gov/biblio/7119843
[21]
[PDF] MULTIPARTICLE STATES AND TENSOR PRODUCTS
Dec 14, 2013 · In this section we begin by studying some properties of the singlet state of two particles of spin-1/2. We then turn to the claims of Einstein, ...
[22]
None
### Summary of Singlet State Derivation
[23]
[PDF] arXiv:2204.07538v1 [quant-ph] 15 Apr 2022
Apr 15, 2022 · A supersinglet is defined as a total spin zero state consisting of N spin-j particles. They are highly entangled and have zero spin variance in ...
[24]
12.7 Clebsch-Gordan coefficients - FAMU-FSU College of Engineering
These coefficients are called Clebsch-Gordan coefficients. The ones corresponding to figure 12.3 are tabulated in Figure 12.4.
[25]
[PDF] Introductory Lectures on Quantum Field Theory - arXiv
Feb 20, 2013 · the electromagnetic field as given by the contribution of two scalar fields corresponding to the two ... SU(2)×U(1) and Poincaré invariance.
[26]
[PDF] Antisymmetric Wavefunctions: SLATER DETERMINANTS - Chemistry
This singlet, closed-shell electronic state wavefunction (a Slater determinant) belongs the totally symmetric representation, 1A1.
[27]
The Helium atom
... ground state has to be a spin-singlet (S=0). All excited states of helium have one electron in the lowest orbital and one in a higher one (1s2s; 1s3d etc).Missing: 1s² | Show results with:1s²
[28]
[PDF] The Helium Atom - TU Graz
Therefore the ground-state energy of the helium atom is given by Eo(+12)=- ... On this scale the 1s² ground state is defined as the zero of energy. If the.
[29]
Helium Energy Levels - HyperPhysics Concepts
One electron is presumed to be in the ground state, the 1s state. An electron in an upper state can have spin antiparallel to the ground state electron (S=0, ...Missing: 1s² | Show results with:1s²
[30]
Generalized Valence Bond Description of the Ground States (X 1 Σ g ...
The ground state, X 1 Σ g + , of N 2 is a textbook example of a molecule with a triple bond consisting of one σ and two π bonds.
[31]
Fluorescence - Jablonski Energy Diagram - Interactive Java Tutorial
Mar 22, 2017 · An excited molecule exists in the lowest excited singlet state (S(1)) for periods on the order of nanoseconds (the longest time period in the ...
[32]
Absorption Spectra of Pure Liquid Benzene in the Ultraviolet Region*
Three absorption bands of quite different intensities were observed, corresponding to the 1r4/"* singlet transitions well known in the vapor phase spectra.<|control11|><|separator|>
[33]
Improved limit on invisible decays of positronium | Phys. Rev. D
The singlet ( 1 1 S 0 ) state (parapositronium, p-Ps) decays predominantly into two photons with a lifetime in vacuum of τ p - Ps = 125 ps [6, 7] . The longer ...
[34]
Ortho-positronium lifetime measurement – positron source activity and statistics
### Summary of Ortho-Positronium Lifetime Value
[35]
New precise spectroscopy of the hyperfine structure in muonium ...
Apr 10, 2021 · A hydrogen-like atom consisting of a positive muon and an electron is known as muonium. It is a near-ideal two-body system for a precision ...
[36]
[PDF] New precise spectroscopy of the hyperfine structure in muonium ...
A hydrogen-like atom consisting of a positive muon and an electron is known as muonium. It is a near- ideal two-body system for a precision test of bound-state ...
[37]
[PDF] ( ) = Bsin( K r ' ) u r u r
The 1s state with S = 0 (neutron and proton spins antiparallel), is a virtual state; it is unbound by ~ 60 Kev.
[38]
[PDF] PC 4421 Lecture 1: Nuclei and Nuclear Forces
The other coupling of spins, the singlet state S=0, results in an ℓ=0, I=0+ state. But where is this? It is found as an unbound resonance in pn scattering and ...
[39]
None
Nothing is retrieved...<|control11|><|separator|>
[40]
Testing bound-state QED theory and the search for physics beyond ...
Sep 1, 2022 · The absence of hadronic components suggests that positronium energy levels and decay rates can be calculated to very high precision, limited ...
[41]
[PDF] Intermediate Quantum Mechanics Lecture 22 Notes (4/15/15)
Whatever the spin direction of one electron is the other is guaranteed to be opposite. • The first state above is called the singlet state while the second ...
[42]
Entanglement quantification by local unitary operations | Phys. Rev. A
Jul 1, 2011 · Invariance under local unitary operations is a fundamental property that must be obeyed by every proper measure of quantum entanglement.
[43]
Multiparticle singlet states cannot be maximally entangled for the ...
Jan 24, 2024 · A different path uses unitary invariance and symmetries, resulting in the concept of multiparticle singlet states. ... local permutations ...
[44]
[quant-ph/0602139] Multi-level, multi-party singlets as ground states ...
Feb 16, 2006 · We show that a singlet of many multi-level quantum systems arises naturally as the ground state of a physically-motivated Hamiltonian.Missing: mechanics | Show results with:mechanics
[45]
Can Quantum-Mechanical Description of Physical Reality Be ...
Feb 18, 2025 · Einstein and his coauthors claimed to show that quantum mechanics led to logical contradictions. The objections exposed the theory's strangest ...
[46]
On the Einstein Podolsky Rosen paradox | Physics Physique Fizika
On the Einstein Podolsky Rosen paradox. JS Bell. Department of Physics, University of Wisconsin, Madison, Wisconsin.
[47]
Proposed Experiment to Test Local Hidden-Variable Theories
Proposed Experiment to Test Local Hidden-Variable Theories. John F. Clauser*. Michael A. Horne · Abner Shimony · Richard A. Holt.
[48]
Experimental Test of Bell's Inequalities Using Time-Varying Analyzers
The results are in good agreement with quantum mechanical predictions but violate Bell's inequalities by 5 standard deviations.
[49]
Loophole-free Bell inequality violation using electron spins ... - Nature
Oct 21, 2015 · Here we report a Bell experiment that is free of any such additional assumption and thus directly tests the principles underlying Bell's inequality.
[50]
Significant-Loophole-Free Test of Bell's Theorem with Entangled ...
Dec 16, 2015 · We report a Bell test that closes the most significant of these loopholes simultaneously. Using a well-optimized source of entangled photons.
[51]
Experimental Test of the No-Signaling Theorem | Phys. Rev. Lett.
Nov 7, 2007 · The theory of special relativity lies on a very basic hypothesis: anything carrying information cannot travel faster than the light in vacuum.
[52]
Photochemistry - MSU chemistry
Excited states may be classified as singlet or triplet based upon their electron spin angular momentum. The electrons in most non-metallic organic compounds are ...
[53]
Singlet Molecular Oxygen Reactions with Nucleic Acids, Lipids, and ...
Feb 5, 2019 · Singlet oxygen (1O2) is a biologically relevant reactive oxygen species capable of efficiently reacting with cellular constituents.
[54]
Photosensitized production of singlet oxygen - PubMed - NIH
Photosensitization is a simple and controllable method for the generation of singlet oxygen in solution and in cells.
[55]
A comprehensive review on singlet oxygen generation in ...
Jan 24, 2024 · The incorporation of photosensitizers into conjugated polymer nanoparticles has proved to be beneficial, as it improves singlet oxygen formation ...
[56]
Mechanism of Ene Reactions of Singlet Oxygen. A Two-Step No ...
The mechanism of the ene reaction of singlet (1Δg) oxygen with simple alkenes is investigated by a combination of experimental isotope effects and several ...Introduction · Results and Discussion · Supporting Information Available
[57]
Reactivity and Applications of Singlet Oxygen Molecule - IntechOpen
Singlet oxygen is a highly reactive electrophilic species that reacts with electron-rich molecules and is related to several types of pathologies.
[58]
Singlet Oxygen in a Cell: Spatially Dependent Lifetimes and ...
Dec 3, 2008 · Sensitizer-dependent values for (i) the intracellular singlet oxygen lifetime and (ii) the rate constant for singlet oxygen quenching by added ...
[59]
[PDF] CYCLOADDITIONS WITH SINGLET OXYGEN - Chemistry | Illinois
Mar 18, 2004 · The use of singlet oxygen is usually complicated by the competition between the ene reaction and the cycloadditions. The presence of allylic ...
[60]
Singlet Oxygen, Photodynamic Therapy, and Mechanisms of Cancer ...
Photosensitizers have been gradually improved over the years. PDT will get more effective when the production of singlet oxygen in space and time can be ...Singlet Oxygen, Photodynamic... · 1. Introduction · 1.1. Singlet Oxygen...
[61]
Singlet Oxygen in Photodynamic Therapy - MDPI
Sep 26, 2024 · This article will provide a comprehensive review of 1 O 2 properties, 1 O 2 production, and 1 O 2 detection in the PDT process.Singlet Oxygen In... · 2.3. Oxidizing Activity Of... · 4. Singlet Oxygen Detection
[62]
Seasonal variations in the production of singlet oxygen and organic ...
Aug 22, 2023 · Photooxidants drive many atmospheric chemical processes. The photoexcitation of light-absorbing organic compounds (i.e., brown carbon, ...
[63]
Singlet Fission Provides a Scalable Pathway to High Efficiency ...
Sep 8, 2025 · Singlet exciton fission (SF), in particular, is a potentially important photovoltaic technology promising a compelling combination of efficiency ...Operating Principle · Materials · Challenges and Ongoing... · Author Information
[64]
Quantum cryptography based on Bell's theorem | Phys. Rev. Lett.
Quantum cryptography based on Bell's theorem. Artur K. Ekert. Merton College and Physics Department, ... 67, 661 – Published 5 August, 1991. DOI: ...
[65]
Experimental quantum key distribution based on a Bell test
The defining feature in E91 protocol is the suggestion to use the degree of violation of the CHSH inequality as a test of security. This conjecture is related ...
[66]
[2407.03119] Entanglement-assisted authenticated BB84 protocol
Jul 3, 2024 · In this work, we present a novel authenticated Quantum Key Distribution (QKD) protocol employing maximally entangled qubit pairs.
[67]
Teleporting an unknown quantum state via dual classical and ...
Teleporting an unknown quantum state via dual classical and Einstein-Podolsky-Rosen channels. Charles H. Bennett, Gilles Brassard, Claude Crépeau, Richard Jozsa ...
[68]
Teleportation of entanglement over 143 km - PNAS
Teleportation of an entangled state, also known as entanglement swapping, plays a vital role in the vision of a global quantum internet, ...<|separator|>
[69]
Entanglement of Formation of an Arbitrary State of Two Qubits
Mar 9, 1998 · The present paper extends the proof to arbitrary states of this system and shows how to construct entanglement-minimizing decompositions.Missing: original | Show results with:original
[70]
Fault-tolerant quantum computation for singlet-triplet qubits with ...
We describe the realization of leakage correction protocols that can be implemented together with the quantum error correction protocol of the surface code.
[71]
The two-qubit singlet/triplet measurement is universal for quantum ...
Nov 28, 2023 · The manifest rotational and permutational invariance of the singlet/triplet projectors underpins their generic physical naturalness - these ...
[72]
Efficient generation of entangled multiphoton graph states ... - Nature
Aug 24, 2022 · In contrast to cluster states, GHZ states are more sensitive to noise and require a higher level of control in their preparation process.
[73]
Verifiable measurement-based quantum random sampling with ...
Jan 2, 2025 · We experimentally demonstrate efficiently verifiable quantum random sampling in the measurement-based model of quantum computation on a trapped-ion quantum ...
[74]
[2304.05086] High-fidelity two-qubit gates of hybrid superconducting ...
Apr 11, 2023 · This paper discusses high-fidelity two-qubit gates using tunable anisotropy in hybrid superconducting-semiconducting singlet-triplet qubits, ...
[75]
Quantum repeaters: From quantum networks to the quantum internet
Dec 20, 2023 · For instance, two distant photons that are entangled may be in a so-called singlet state ( 1 / 2 ) ( | 01 ⟩ - | 10 ⟩ ) , which can exhibit ...