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Spontaneous emission

Spontaneous emission is a fundamental quantum mechanical process in which an excited atom, molecule, or quantum system transitions to a lower energy state, releasing a single photon whose energy equals the difference between the initial and final states, without any external electromagnetic stimulation.^[1] This emission occurs randomly in time and direction, arising from the interaction of the excited system with the vacuum fluctuations of the electromagnetic field.^[1] The process is characterized by the Einstein A coefficient, which quantifies the probability per unit time of spontaneous decay for a given transition.^[2] The concept of spontaneous emission was introduced by Albert Einstein in his 1917 paper "On the Quantum Theory of Radiation," where he postulated it alongside stimulated emission and absorption to resolve inconsistencies between classical radiation theory and Planck's quantum hypothesis for blackbody radiation. Einstein derived the relations between the coefficients A, B (for stimulated processes), and the equilibrium radiation density, showing that spontaneous emission is independent of the incident radiation intensity, unlike stimulated emission. These Einstein coefficients remain central to quantum electrodynamics descriptions of atomic transitions.^[2] Spontaneous emission plays a crucial role in atomic spectra, producing the discrete emission lines observed in gases, and is essential for the initial photon generation in lasers and masers, where it seeds the subsequent dominance of stimulated emission.^[3] In quantum optics, the rate of spontaneous emission can be altered by the local density of optical states, as described by the Purcell effect, enabling enhancements or inhibitions through nanostructures or cavities for applications in photonics and quantum information processing.^[4] This environmental dependence underscores spontaneous emission's sensitivity to its surroundings, distinguishing it from classical radiative decay.^[4]

Basic Concepts

Definition and Mechanism

Spontaneous emission is the random, probabilistic process by which an excited quantum mechanical system, such as an atom, molecule, or solid-state excitation, transitions to a lower energy state while emitting a single photon. This decay occurs without any external stimulation, distinguishing it from other radiative processes, and is governed by the inherent quantum nature of the system interacting with the electromagnetic field. The emitted photon's energy precisely matches the difference between the initial excited state energy E_i and the final state energy E_f, satisfying the relation h\nu = E_i - E_f, where h is Planck's constant and \nu is the photon's frequency.^[5]^[6] In quantum electrodynamics (QED), the mechanism of spontaneous emission is fundamentally driven by the zero-point vacuum fluctuations of the quantized electromagnetic field, which act as a pervasive "noise" permeating empty space. These fluctuations provide the initial perturbation that couples to the system's dipole moment, inducing the emission of electromagnetic radiation as the excited state evolves. The process culminates in a probabilistic collapse of the system's wave function, resulting in the creation of a photon in a specific mode of the vacuum field while the system relaxes to the lower energy state; this interpretation complements the classical radiation reaction picture but emphasizes the quantum origin of the trigger.^[5] A representative example of spontaneous emission is the transition from the 2p excited state to the 1s ground state in atomic hydrogen, which produces the Lyman-alpha spectral line at a vacuum wavelength of 121.567 nm corresponding to a photon energy of approximately 10.2 eV. This ultraviolet emission is a cornerstone observation in atomic spectroscopy and plays a key role in astrophysical diagnostics of hydrogen-rich environments.^[7]^[8]

Historical Context

The development of spectroscopy in the 19th century marked the initial observations of atomic emission spectra. In 1859, Robert Bunsen and Gustav Kirchhoff invented the spectroscope and used it to analyze the light emitted by elements heated in a flame, revealing discrete emission lines characteristic of specific substances. This work led to the discovery of new elements, such as cesium and rubidium in 1860, but the sharp, unexplained lines hinted at quantized energy transitions in atoms without a clear physical mechanism.^[9] A theoretical breakthrough came in 1917 with Albert Einstein's paper "On the Quantum Theory of Radiation," which introduced the A and B coefficients to quantify the rates of spontaneous emission, stimulated emission, and absorption. By applying these to the equilibrium between matter and radiation, Einstein resolved paradoxes in classical blackbody radiation theory and established spontaneous emission as an intrinsic quantum process independent of external fields.^[10] The 1920s saw quantum mechanics formalize these ideas, with Paul Dirac's 1927 work "The Quantum Theory of the Emission and Absorption of Radiation" deriving spontaneous emission rates from first principles using wave mechanics, providing the first quantum mechanical calculation of transition probabilities. Werner Heisenberg's matrix mechanics, developed concurrently, further supported the probabilistic nature of quantum transitions. By the 1940s, quantum electrodynamics (QED) advanced the understanding, as Sin-Itiro Tomonaga, Julian Schwinger, and Richard Feynman independently reformulated the theory to incorporate relativistic effects and vacuum fluctuations—the zero-point energy of the electromagnetic field—as the origin of spontaneous emission. Their contributions, awarded the 1965 Nobel Prize in Physics, explained how virtual photons in the vacuum trigger the decay of excited states.^[11]^[12] In the 1960s, experimental milestones confirmed these predictions through precise linewidth measurements, where the natural broadening of spectral lines directly reflects the inverse of the spontaneous emission lifetime. Developments like the hydrogen maser atomic clock, operational from 1960, exhibited linewidths matching quantum theoretical expectations, validating the rates derived from Einstein's and Dirac's frameworks. These verifications also underscored spontaneous emission's role in limiting precision in frequency standards. Einstein's concepts of emission processes briefly informed the invention of the maser in 1954 and laser in 1960.^[13]^[14]

Comparison to Stimulated Emission and Absorption

Absorption is the process by which a photon interacts with an atomic or molecular system in its ground state, promoting it to an excited state provided the photon's energy precisely matches the energy difference between the two states.^[15] This requires the presence of an incident photon of the appropriate frequency and cannot occur spontaneously without external radiation.^[16] In contrast, stimulated emission occurs when an excited system encounters an incident photon of matching energy, triggering a transition to a lower energy state while emitting a second photon that is identical in frequency, phase, direction, and polarization to the incident one.^[17] This coherent process, first predicted by Albert Einstein in 1917, forms the foundational mechanism for light amplification in lasers, as the emitted photons can further stimulate additional emissions, leading to exponential growth in photon number.^[10] Spontaneous emission differs fundamentally from both absorption and stimulated emission in its incoherence and lack of external trigger; it arises from the interaction of the excited system with quantum vacuum fluctuations, resulting in photon emission in a random direction and phase.^[15] Unlike stimulated emission, which requires an external field and produces aligned, amplifying output, spontaneous emission is isotropic and non-directional, dominating in environments with low photon densities where vacuum-induced transitions prevail.^[16] Absorption, meanwhile, consumes a photon to increase system energy, whereas both emission processes release energy, but spontaneous emission does so without the coherence that enables stimulated processes to build intense, monochromatic beams.^[17] These processes are interrelated through the same underlying atomic transition matrix elements, which determine the probability of energy level changes, though their rates differ based on environmental factors; the Einstein relations link these rates, showing how spontaneous emission often governs low-intensity regimes while stimulated processes become prominent under strong illumination.^[10]

Theoretical Foundations

Einstein Coefficients

In 1917, Albert Einstein introduced a set of phenomenological coefficients to describe the rates of atomic transitions between two energy levels, labeled as state 1 (lower energy) and state 2 (higher energy), in the presence of electromagnetic radiation. These coefficients, known as the Einstein A, B_{12}, and B_{21} coefficients, quantify the probabilities per unit time for spontaneous emission, absorption, and stimulated emission, respectively. The coefficient A_{21} represents the probability of spontaneous emission from state 2 to state 1 in the absence of external radiation, while B_{12} governs the absorption rate from state 1 to state 2, proportional to the radiation energy density at the transition frequency \nu, and B_{21} describes the stimulated emission rate from state 2 to state 1, also proportional to the energy density.^[10] Einstein derived key relations between these coefficients by assuming thermal equilibrium between the atomic system and blackbody radiation, ensuring consistency with Planck's law for the spectral energy density. Specifically, he established that B_{12} = (g_2 / g_1) B_{21}, where g_1 and g_2 are the degeneracies of the lower and upper states, respectively, reflecting the symmetry of absorption and stimulated emission adjusted for statistical weights. The relation between spontaneous and stimulated emission is given by

A_{21} = \frac{8\pi h \nu^3}{c^3} B_{21},

where h is Planck's constant and c is the speed of light; this connects the vacuum-induced spontaneous process to the field-induced stimulated one.^[10] The thermodynamic derivation proceeds from the condition of detailed balance in equilibrium, where the upward and downward transition rates must balance to maintain a stationary population ratio governed by Boltzmann statistics: N_2 / N_1 = (g_2 / g_1) e^{-h\nu / kT}, with k the Boltzmann constant and T the temperature. The net rate equation for the population in the upper state incorporates terms for absorption (B_{12} \rho(\nu) N_1), stimulated emission (B_{21} \rho(\nu) N_2), and spontaneous emission (A_{21} N_2), where \rho(\nu) is the radiation energy density per unit frequency. Setting the net rate to zero and substituting the Boltzmann factor yields \rho(\nu) = \frac{A_{21} / B_{21}}{(N_2 g_1 / N_1 g_2) - 1}, which, upon matching to Planck's blackbody formula \rho(\nu) = \frac{8\pi h \nu^3}{c^3} \frac{1}{e^{h\nu / kT} - 1}, confirms the Einstein relations and demonstrates that spontaneous emission is essential for reproducing the quantum spectral distribution from classical equilibrium arguments.^[10] These coefficients provide a foundational, statistical framework bridging classical radiation theory and quantum mechanics but remain phenomenological in nature. They assume thermal equilibrium and do not specify the underlying microscopic mechanisms of the transitions, such as dipole interactions or field quantization, nor do they account for non-equilibrium conditions or relativistic effects.^[10]

Semiclassical Derivation

The semiclassical derivation of spontaneous emission utilizes time-dependent perturbation theory to model the interaction between a quantum atom and a classical electromagnetic field. In this framework, the atom is described by its electric dipole moment operator \hat{\mu}, which couples to the classical electric field \mathbf{E}(\mathbf{r}, t) through the perturbation Hamiltonian H' = -\hat{\mu} \cdot \mathbf{E}(\mathbf{r}, t).^[18] The time evolution of the atomic wave function is governed by the Schrödinger equation i \hbar \frac{\partial}{\partial t} |\psi(t)\rangle = [H_0 + H'(t)] |\psi(t)\rangle, where H_0 is the unperturbed atomic Hamiltonian. Applying first-order time-dependent perturbation theory, the probability amplitude for transitioning from an initial excited state |2\rangle to a lower state |1\rangle is c_1^{(1)}(t) = -\frac{i}{\hbar} \int_0^t dt' \langle 1 | H'(t') | 2 \rangle e^{i \omega_{21} t'}, with \omega_{21} = (E_2 - E_1)/\hbar. The resulting transition probability P_{2 \to 1}(t) = |c_1^{(1)}(t)|^2 yields a transition rate that is proportional to the density of final electromagnetic field modes available for emission.^[18] The spontaneous emission term emerges from the contribution of vacuum field modes, which provide a background even in the absence of an external driving field \mathbf{E}; this is incorporated by considering the continuum of classical field modes with their mode density \rho(\omega) \propto \omega^2 / c^3. Integrating over polarizations and directions in the electric dipole approximation leads to the Einstein coefficient for the spontaneous emission rate A_{21} = \frac{\omega^3 |\mu_{21}|^2}{3 \pi \epsilon_0 \hbar c^3}, where \mu_{21} = \langle 1 | \hat{\mu} | 2 \rangle is the dipole matrix element and \omega = \omega_{21}.^[19] This derivation relies on the dipole approximation, valid when the emitted wavelength \lambda = 2\pi c / \omega greatly exceeds the atomic size (typically \lambda \gg 10^{-10} m), allowing the field to be treated as uniform across the atom and neglecting magnetic dipole, electric quadrupole, and higher-order multipole interactions.^[18]

Quantum Electrodynamic Treatment

In quantum electrodynamics (QED), spontaneous emission is described as a fundamental interaction between an atom and the quantized electromagnetic vacuum, where the excited atomic state transitions to the ground state by emitting a single photon into the vacuum modes. The process arises from the coupling of the atomic dipole moment to the vacuum fluctuations of the electromagnetic field, which provide the necessary stimulus absent in classical theories. This full quantum treatment, pioneered by Weisskopf and Wigner, resolves the classical paradox of radiation reaction by quantizing both the atomic and field degrees of freedom. The QED framework employs second quantization for the electromagnetic field, with the interaction Hamiltonian given by H_\text{int} = -\frac{e}{m} \mathbf{A} \cdot \mathbf{p} + higher-order terms, where \mathbf{A} is the vector potential operator expressed in terms of creation and annihilation operators for the photon modes, e is the electron charge, m is the electron mass, and \mathbf{p} is the atomic momentum operator. This minimal coupling form captures the relativistic interaction between charged particles and the quantized field, enabling a consistent description of emission processes without ad hoc assumptions. The atomic system is typically modeled as a two-level entity for simplicity, though multi-level extensions follow similarly. Spontaneous emission is triggered by the zero-point energy fluctuations of the vacuum modes, which possess non-zero electric field amplitudes even in the ground state. The transition can be viewed as an initial state of the atom in the excited state plus the vacuum evolving to a final state of the atom in the ground state plus a one-photon excitation in a specific mode. These vacuum fluctuations induce the emission, as the off-diagonal matrix elements of the interaction connect the initial and final states, leading to irreversible decay on macroscopic timescales due to the continuum of modes. The decay rate is computed using Fermi's golden rule within QED: \Gamma = \frac{2\pi}{\hbar} |\langle f | H_\text{int} | i \rangle|^2 \rho(\omega), where |i\rangle and |f\rangle are the initial and final states, the matrix element involves photon creation/annihilation operators that enforce energy and momentum conservation, and \rho(\omega) is the density of photon states at frequency \omega. This perturbative approach assumes weak coupling and Markovian dynamics, yielding an exponential decay law for the excited state population. The semiclassical limit recovers approximate rates but omits the intrinsic quantum role of the vacuum. The spontaneous emission linewidth is intrinsically linked to the Lamb shift through QED renormalization, where real emission processes contribute to the linewidth via recoil and on-shell photon emission, while virtual processes induce the frequency shift of atomic levels.^[20] The linewidth \Gamma represents the inverse lifetime, broadened by the coupling to the continuum, whereas the Lamb shift arises from self-energy corrections involving off-shell intermediate states, both calculated diagrammatically to high precision in QED.^[20] This connection underscores the unified treatment of radiative corrections in atomic physics.

Emission Dynamics

Spontaneous Emission Rate

The spontaneous emission rate, denoted as \Gamma_\mathrm{sp}, represents the probability per unit time for an excited atom or molecule in a two-level system to decay to the ground state by emitting a single photon. This rate is fundamentally the inverse of the excited state's radiative lifetime \tau_\mathrm{sp}, such that \Gamma_\mathrm{sp} = 1 / \tau_\mathrm{sp}. In the context of isolated two-level systems, \Gamma_\mathrm{sp} quantifies the intrinsic decay probability absent external fields or interactions.^[15] For electric dipole-allowed transitions, the spontaneous emission rate in free space is given by

\Gamma_\mathrm{sp} = \frac{\omega^3 |\mu_{21}|^2}{3 \pi \epsilon_0 \hbar c^3},

where \omega is the angular transition frequency between the upper (2) and lower (1) states, \mu_{21} is the magnitude of the transition dipole moment, \epsilon_0 is the vacuum permittivity, \hbar is the reduced Planck's constant, and c is the speed of light (all in SI units). This expression arises from the interaction of the atomic dipole with the vacuum electromagnetic field and assumes the dipole approximation, valid when the wavelength is much larger than the atomic size. The transition dipole moment \mu_{21} encapsulates the overlap of the electronic wavefunctions and determines the strength of the allowed transition. A key feature of this formula is its cubic dependence on the transition frequency, \Gamma_\mathrm{sp} \propto \omega^3. This frequency scaling implies that spontaneous emission occurs more rapidly for higher-energy transitions; for instance, ultraviolet transitions (with \omega in the UV range) exhibit rates orders of magnitude faster than those in the infrared, influencing the design of optical devices and the observation of atomic spectra. Representative calculations for hydrogen-like atoms yield lifetimes on the order of nanoseconds for visible transitions, highlighting the practical scale of these rates.^[21] In contrast to stimulated emission processes, which depend on the occupation of photon modes and thus vary with temperature through blackbody radiation, the spontaneous emission rate remains independent of temperature under typical laboratory conditions (low temperatures relative to the transition energy). This invariance stems from the process being driven solely by zero-point vacuum fluctuations, with negligible thermal contributions to the vacuum field density at optical frequencies. This rate is equivalent to the Einstein coefficient A_{21}, linking it to foundational thermodynamic arguments for light-matter interactions.^[22]^[15]

Lifetime and Linewidth

In the case of pure radiative decay, the lifetime \tau of an excited atomic state is the inverse of the spontaneous emission rate \Gamma_\mathrm{sp}, given by \tau = 1 / \Gamma_\mathrm{sp}.^[23] This lifetime represents the average time the atom remains in the excited state before emitting a photon via spontaneous emission. Experimentally, \tau is determined through time-resolved fluorescence measurements, in which a short laser pulse excites the atoms, and the subsequent exponential decay of the emitted fluorescence intensity is recorded using techniques like time-correlated single-photon counting.^[24] The finite lifetime \tau imposes a fundamental limit on the spectral purity of the emitted light, resulting in a natural linewidth. The emission spectrum follows a Lorentzian lineshape, with the full width at half maximum (FWHM) in frequency \Delta \nu = \Gamma_\mathrm{sp} / (2\pi) = 1 / (2\pi \tau). This broadening originates from the Heisenberg uncertainty principle, which states that the product of uncertainties in energy \Delta E and time \Delta t \approx \tau satisfies \Delta E \Delta t \gtrsim \hbar / 2, translating to a frequency uncertainty \Delta \nu \gtrsim 1 / (4\pi \tau); the exact Lorentzian form and factor of $1 / (2\pi \tau) arise from the quantum mechanical treatment of the decay process.^[25] The seminal Weisskopf-Wigner theory provides the rigorous derivation, modeling the atom's interaction with the vacuum electromagnetic field to yield the exponential decay and associated Lorentzian spectrum. While the natural linewidth sets the intrinsic limit, observed spectral lines are often broader due to environmental effects. Pressure broadening, or collisional broadening, occurs when collisions with surrounding particles interrupt the phase coherence of the emitting dipole, adding a Lorentzian contribution to the linewidth that increases with gas density. Power broadening arises under intense illumination, where strong driving fields cause Rabi oscillations and population saturation, effectively widening the line. Nonetheless, these mechanisms superimpose on the natural linewidth without altering its fundamental role as the minimum achievable width dictated by spontaneous emission.^[26] A representative example is the sodium D_2 line (3^2S_{1/2} \to 3^2P_{3/2} transition at 589 nm), which has a measured radiative lifetime of approximately 16.2 ns, corresponding to a natural linewidth of about 9.8 MHz.^[23]

Dipole Transition Selection Rules

In the electric dipole (E1) approximation, which dominates spontaneous emission processes in atoms and molecules, selection rules dictate which transitions between quantum states are allowed, determining the observable spectral lines. These rules arise from the symmetry properties of the interaction Hamiltonian and the wavefunctions involved, ensuring that the transition dipole moment matrix element is non-zero only for permitted changes in quantum numbers.^[2]^[21] For the total angular momentum quantum number J, electric dipole transitions are allowed if \Delta J = 0, \pm 1, with the restriction that J = 0 \to J = 0 transitions are forbidden, as they would violate angular momentum conservation in the emission of a photon carrying spin 1. This follows from the Wigner-Eckart theorem applied to the vector nature of the dipole operator. In the LS coupling scheme common for light atoms, the rules translate to \Delta L = \pm 1 and \Delta S = 0 for the orbital and spin angular momenta, respectively.^[2]^[27] A key parity requirement, known as the Laporte rule, mandates that E1 transitions occur only between states of opposite parity (e.g., from even to odd or odd to even), as the electric dipole operator is an odd function under spatial inversion, while the parity of the initial and final wavefunctions must combine to yield an even integrand for a non-zero matrix element. Transitions between states of the same parity are E1-forbidden and proceed via weaker mechanisms like magnetic dipole (M1) or electric quadrupole (E2), which have rates suppressed by factors of (\alpha Z)^2 or higher, where \alpha \approx 1/137 is the fine-structure constant and Z is the atomic number. For hydrogen-like atoms, this implies \Delta l = \pm 1 for the orbital quantum number l, directly enforcing the parity change since parity is (-1)^l.^[2]^[28]^[27] Considering fine structure, the selection rule for the magnetic quantum number is \Delta m_J = 0, \pm 1, corresponding to the photon's polarization: \Delta m_J = 0 for \pi polarization (electric field along the quantization axis) and \Delta m_J = \pm 1 for \sigma^\pm polarizations (circularly polarized light perpendicular to the axis). The total spontaneous emission rate for a transition sums over all allowed polarizations and final m_J substates, weighted by the Clebsch-Gordan coefficients from the angular momentum coupling.^[29]^[2] The intensity of an allowed E1 transition is proportional to the square of the transition dipole moment magnitude, |\boldsymbol{\mu}_{if}|^2 = |\langle f | \mathbf{d} | i \rangle|^2, where \mathbf{d} = -e \mathbf{r} is the electric dipole operator and |i\rangle, |f\rangle are the initial and final states. A dimensionless measure of this strength is the oscillator strength f_{if}, defined as f_{if} = \frac{2m_e \omega_{if}}{3\hbar e^2} |\boldsymbol{\mu}_{if}|^2 g_i^{-1}, where m_e is the electron mass, \omega_{if} is the transition frequency, and g_i is the degeneracy of the initial state; f_{if} quantifies the transition's relative probability compared to a classical harmonic oscillator and relates directly to the Einstein A coefficient for spontaneous emission via A_{if} = \frac{e^2 \omega_{if}^2}{2 \pi \epsilon_0 m_e c^3} f_{if}. Typical values for strong atomic lines range from 0.1 to 1, while forbidden transitions have f \ll 10^{-3}.^[30]^[21]

Decay Processes

Radiative Decay Pathways

In atomic and molecular systems featuring multiple lower energy levels accessible from a given excited state, spontaneous emission proceeds via distinct radiative pathways, each characterized by a partial decay rate Γ_i corresponding to the transition to a specific lower state |i⟩. The branching ratio for each pathway, defined as the fraction of decays occurring through that channel, is Γ_i / Γ_rad, where Γ_rad = ∑_i Γ_i represents the total radiative decay rate from the excited state. This partitioning arises from the dipole transition strengths and selection rules governing the allowed emissions, as derived from the Einstein coefficients for spontaneous emission in multi-level systems.^[31] The population dynamics across these levels are governed by master rate equations that account for the inflows and outflows due to spontaneous emission. For a population N_j in level j, the time evolution is dN_j/dt = ∑{i>j} Γ{ji} N_i - N_j ∑{k<j} Γ{jk}, where Γ_{ji} is the partial rate from higher level i to j, capturing the direct population of intermediates in cascade processes or depletion via competing channels. Direct radiative decay involves a single transition to the ground state, emitting one photon, whereas cascade decay entails sequential steps through intermediate levels, potentially generating multiple photons with energies matching the respective level spacings.^[32] These multiple pathways manifest spectrally as distinct emission lines in fluorescence spectra, with intensities proportional to the branching ratios, enabling characterization of the excited-state structure. For instance, in alkali atoms like sodium, excitation to higher manifolds yields branched emissions including the prominent D-line doublet alongside weaker lines from intermediate states.^[33] In solid-state contexts, such as direct-bandgap semiconductors, radiative decay pathways are exemplified by exciton recombination, where electron-hole pairs annihilate to emit photons primarily at the band-edge energy. In transition metal dichalcogenides (TMDs) like monolayer MoS₂, spin-orbit splitting gives rise to A, B, and C exciton states, leading to branched radiative recombination with emission lines separated by tens of meV, as observed in photoluminescence spectra dominated by the lower-energy A exciton pathway. These processes highlight how material fine structure influences the distribution of emitted photon energies without invoking non-photon pathways.^[34]

Nonradiative Decay Mechanisms

Nonradiative decay mechanisms involve the relaxation of excited states without the emission of photons, dissipating energy through intramolecular or intermolecular interactions that compete with spontaneous emission processes. These pathways are crucial in determining the efficiency of luminescent materials and photochemical reactions, as they provide alternative routes for energy loss in excited molecules or atoms. In solids and molecules, nonradiative decay often proceeds via coupling to vibrational modes, leading to rapid thermalization of the excitation energy. Internal conversion represents a primary nonradiative pathway where electronic excitation energy is transferred isoenergetically to vibrational degrees of freedom within states of the same spin multiplicity, typically from a higher singlet state to the ground singlet state. This process is facilitated by vibronic coupling, allowing the excited electron to relax through a series of vibrational quanta without spin change. The rate of internal conversion follows the energy gap law, which predicts that the decay rate increases exponentially as the energy gap between the initial excited state and the accepting vibrational levels in the lower state decreases, due to enhanced Franck-Condon overlap factors. This mechanism is particularly efficient in polyatomic molecules where dense manifolds of vibrational states promote fast relaxation on picosecond timescales.^[35] Vibrational relaxation occurs within the same electronic state, converting excess vibrational energy into lattice phonons in solids or solvent modes in liquids, often requiring multiphonon processes to bridge larger energy gaps. In molecular systems, this involves the emission of multiple low-energy phonons, with the rate depending on the electron-phonon coupling strength and the number of phonons needed, as described in statistical limit theories for large displacement scenarios. For instance, in rare-earth doped crystals, multiphonon relaxation from high-lying levels dominates when the energy gap exceeds several phonon energies, leading to cascaded decay to lower manifolds. These processes ensure rapid equilibration of vibrational populations, typically occurring in femtoseconds to picoseconds in condensed phases.^[36] Intersystem crossing enables transitions between states of different spin multiplicity, such as from singlet to triplet, which are formally forbidden but occur through spin-orbit coupling that mixes electronic wavefunctions and relaxes the spin selection rule. This vibronic-assisted mechanism, often involving intermediate vibrational states, results in population of lower-energy triplet states, from which delayed phosphorescence may follow after further relaxation. The rate is proportional to the square of the spin-orbit coupling matrix element and increases with stronger coupling in heavy-atom-containing molecules, where relativistic effects enhance the interaction. In organic chromophores, intersystem crossing competes effectively with fluorescence, particularly in systems with nearby singlet and triplet levels. Several factors influence the rates of these nonradiative mechanisms, including temperature, which often introduces activated behavior through thermal population of higher vibrational levels that facilitate crossing or relaxation, following Arrhenius-like dependence with activation energies on the order of vibrational quanta. Material-specific variations are pronounced: in organic molecules, dense vibrational manifolds and strong anharmonicities lead to ultrafast internal conversion and intersystem crossing (sub-picosecond to nanosecond), whereas in rare gas matrices, sparse phonon densities and weak coupling result in much slower nonradiative decay, extending lifetimes to milliseconds. These mechanisms collectively reduce the quantum efficiency of spontaneous emission by providing dominant pathways for energy dissipation at elevated temperatures or in complex environments.^[37]

Quantum Efficiency and Yield

Quantum efficiency, denoted as η, quantifies the fraction of excited states in a quantum system that decay via radiative spontaneous emission rather than competing nonradiative processes. It is defined as the ratio of the radiative decay rate Γ_rad to the total decay rate, given by η = Γ_rad / (Γ_rad + Γ_nonrad), where Γ_nonrad encompasses all nonradiative pathways.^[38] This parameter is crucial for assessing the potential for efficient light emission in materials like atoms, molecules, and semiconductors, as it directly influences the brightness and performance of devices such as LEDs and lasers. In ideal cases without nonradiative losses, η approaches 1, indicating perfect conversion of excitations to photons. Closely related is the quantum yield Φ, which extends the concept of quantum efficiency to practical measurements by incorporating losses in photon collection and detection. It is expressed as Φ = η × (number of photons detected / number of excitations), accounting for factors like extraction efficiency from the emitter and optical losses in the setup.^[39] While η represents the intrinsic material property, Φ provides a system-level metric, often lower due to geometric and environmental constraints. For instance, in nanostructured emitters, enhancements in local density of states can boost both η and the collection factor, leading to higher Φ. Quantum efficiency is typically measured using either time-resolved or steady-state fluorescence techniques. In time-resolved methods, the excited-state lifetime τ is determined via techniques like time-correlated single-photon counting, yielding the total decay rate as 1/τ; η is then calculated by comparing this to the known or independently measured Γ_rad, often through references to radiative lifetime in modified environments.^[38] Steady-state approaches, conversely, integrate fluorescence intensity relative to a standard reference with known η under identical excitation conditions, providing an absolute value but sensitive to reabsorption effects. These methods reveal stark contrasts across material classes: direct-bandgap semiconductors like GaAs exhibit high η ≈ 1 due to efficient momentum-conserving radiative recombination, whereas indirect-bandgap materials like silicon show very low η ≈ 10^{-5}, dominated by phonon-assisted processes that strongly favor nonradiative decay.^[40] Optimization of quantum efficiency focuses on suppressing nonradiative recombination, often through material engineering. Doping strategies, such as precise control of carrier concentrations in III-nitride semiconductors, reduce defect-related traps and enhance η by balancing radiative and nonradiative rates.^[41] Similarly, nanostructuring—via quantum dots or nanowires—confines carriers to minimize surface recombination and nonradiative pathways, boosting η in otherwise inefficient materials like perovskites or metal chalcogenides.^[42] These approaches have enabled η values exceeding 50% in previously low-efficiency systems, underscoring their role in advancing optoelectronic technologies.

Multi-Level and Collective Effects

Radiative Cascades

In multi-level atomic or molecular systems, radiative cascades occur when an atom is excited to a high-lying energy level and subsequently decays through a series of intermediate levels via successive spontaneous emissions, producing multiple photons in a sequential manner.^[43] This process contrasts with direct two-level transitions by involving population transfer across several states, often leading to delayed photon emissions and complex spectral features. A representative example in helium involves excitation to a high principal quantum number state such as 1s np (for n > 2), followed by stepwise decay through the 1s 2p intermediate level to the metastable 1s 2s state, emitting photons at corresponding wavelengths.^[44] The dynamics of these cascades are governed by rate equations that describe the time evolution of populations in each level. For a system with levels labeled i, the population N_i satisfies

\frac{dN_i}{dt} = -\sum_{j \neq i} \Gamma_{ij} N_i + \sum_{k \neq i} \Gamma_{ki} N_k,

where \Gamma_{ij} is the spontaneous emission rate from level i to j. Solving these coupled equations reveals the flow of population from the initial excited state through intermediates, with branching ratios determining the probability of each decay path.^[32] Due to the sequential nature of the decays, the effective lifetime of the overall cascade is prolonged compared to individual transition lifetimes, as the atom spends time in long-lived intermediate states before completing the decay chain. Additionally, the photons emitted in such cascades exhibit angular correlations dictated by the quantum mechanical selection rules and the geometry of the atomic wavefunctions, enabling measurements of atomic properties through coincidence detection.^[45] These cascades play a key role in precision spectroscopy for atomic clocks, where accurate modeling of population flows and branching ratios (e.g., from radiative pathways) minimizes systematic errors in frequency standards. In dense atomic ensembles, cascades can also hint at collective phenomena like superradiance, where enhanced emission rates emerge from interatomic correlations.^[46]

Inhibition and Enhancement in Cavities

In optical cavities and structured environments, the spontaneous emission rate of an emitter can be significantly modified by altering the local density of electromagnetic states at the emission frequency. This modification arises from the interaction between the emitter's dipole and the cavity modes, leading to either enhancement or inhibition of the decay rate compared to free space. The foundational concept is the Purcell effect, which predicts that the emission rate increases when the emitter is placed at an antinode of a resonant cavity mode, with the enhancement factor given by

F = \frac{3 \lambda^3 Q}{4\pi^2 V},

where \lambda is the emission wavelength, Q is the cavity quality factor, and V is the mode volume. This formula, derived for radio frequencies but applicable to optical regimes, shows that higher Q and smaller V yield greater enhancement, enabling rates up to thousands of times faster than in vacuum, particularly when the cavity resonance matches the transition frequency.^[47] Inhibition occurs in environments where the density of states \rho(\omega) is reduced at the emission frequency \omega, such as within photonic bandgaps of periodic structures or off-resonance in one-dimensional cavities like Fabry-Pérot resonators. In three-dimensional photonic crystals, bandgaps suppress emission by forbidding propagating modes, potentially reducing the rate by orders of magnitude if the bandgap overlaps the atomic transition.^[47] For instance, in silicon inverse woodpile photonic crystals, lead sulfide quantum dots exhibit strongly inhibited emission, with lifetimes extended due to the local density of states approaching zero in the bandgap.^[48] Similarly, in 1D cavities, detuning from the mode suppresses \rho(\omega), leading to slower decay for emitters positioned away from resonant conditions.^[49] In the strong coupling regime of cavity quantum electrodynamics (QED), where the coherent interaction exceeds decay rates, spontaneous emission evolves into Rabi oscillations between the atom and cavity field, even in the vacuum state. This manifests as vacuum Rabi splitting in the transmission spectrum, where the single cavity resonance splits into two peaks separated by the vacuum Rabi frequency $2g, with g the single-photon coupling strength. First observed in an optical cavity with a single cesium atom, this splitting confirmed the reversible exchange of excitations without dissipation, marking a hallmark of quantum coherence. Modern nanophotonic devices leverage these effects to boost efficiency in applications like LEDs and single-photon sources. For example, integrating quantum dots into photonic crystal nanocavities achieves Purcell factors exceeding 100, enhancing emission into desired modes and improving extraction efficiency for quantum information processing.^[50] In the 2020s, advances in hybrid structures, such as quantum dots in porous silicon microcavities, have demonstrated nearly tenfold rate increases, while photonic crystal slabs with quantum emitters enable directional emission with over 1000-fold intensity gains through combined excitation and Purcell enhancements.^[50]^[51] These developments underscore the practical control of spontaneous emission for energy-efficient optoelectronics and quantum technologies.^[49]

References

[1]
[PDF] Spontaneous Emission Stimulated Emission E E E E Before After
In spontaneous emission the excited atom spontaneously decays to the lower energy state and emits a photon of frequency . One can regard this as emission.
[2]
[PDF] Emission and Absorption of Radiation
nλ ...i. 3. Spontaneous Emission. Our first application will be the spontaneous emission of a photon by an atom in an excited state. We ...
[3]
Stimulated Emission and Absorption
The emission process is known as "spontaneous emission." To model it correctly one needs a quantum theory of light. The absorption process, however, can be ...<|control11|><|separator|>
[4]
https://optoelectronics.eecs.berkeley.edu/ey1987prl5820.pdf
[5]
Radiation reaction and vacuum fluctuations in spontaneous emission
Mar 1, 1975 · It is shown that radiation reaction and vacuum fluctuations provide complementary conceptual bases for the interpretation of these radiative corrections.
[6]
[PDF] The dynamics of spontaneous emission - arXiv
May 28, 2025 · The spontaneous decay is then viewed as a reduction process produced by the vacuum fluctuations, and it turns out that the wave function ...
[7]
NIST: Atomic Spectra Database Lines Form
NIST Atomic Spectra Database Lines Form. Main Parameters: Spectrum eg, Fe I or Na;Mg; Al or mg i-iii or 198Hg I. Search for Wavelength, Wavenumber, Photon, ...Missing: Lyman | Show results with:Lyman
[8]
[PDF] II. Astronomical Applications 1 The Lyman alpha transition 2p - 1s in ...
We calculate the spontaneous transition rate from the n = 2,l = 1 state to the ground (n = 1,l = 0) state of atomic hydrogen, producing the Lyman α line at ...
[9]
Robert Bunsen and Gustav Kirchhoff - Science History Institute
In 1860 Robert Bunsen and Gustav Kirchhoff discovered two alkali metals, cesium and rubidium, with the aid of the spectroscope they had invented the year ...
[10]
[PDF] ON THE QUANTUM THEORY OF RADIATION
By postulating some hypotheses on the emission and absorption of radiation by molecules, which suggested themselves from quantum theory, I was able to show that.Missing: spontaneous | Show results with:spontaneous
[11]
The quantum theory of the emission and absorption of radiation
The new quantum theory, based on the assumption that the dynamical variables do not obey the commutative law of multiplication, has by now been developed ...
[12]
The Nobel Prize in Physics 1965 - NobelPrize.org
The Nobel Prize in Physics 1965 was awarded jointly to Sin-Itiro Tomonaga, Julian Schwinger and Richard P. Feynman for their fundamental work in quantum ...
[13]
[PDF] A Historical Review of Atomic Frequency Standards
Development of the Atomic Hydrogen Maser. The atomic hydrogen maser, first developed at Harvard. University in 1960 by N. Ramsey, M, Goldenberg, and. D ...
[14]
Einstein predicts stimulated emission - American Physical Society
Aug 1, 2005 · Einstein proposed that an excited atom in isolation can return to a lower energy state by emitting photons, a process he dubbed spontaneous emission.
[15]
Spontaneous Emission of Radiation - Richard Fitzpatrick
This process is known as spontaneous emission. We can derive the rate of spontaneous emission between two atomic states from a knowledge of the corresponding ...Missing: conservation h nu = source
[16]
Quantum properties of light - HyperPhysics Concepts
A population inversion cannot be achieved with just two levels because the probabability for absorption and for spontaneous emission is exactly the same, as ...Missing: definition | Show results with:definition
[17]
Einstein predicts stimulated emission - American Physical Society
First, Einstein proposed that an excited atom in isolation can return to a lower energy state by emitting photons, a process he dubbed spontaneous emission.
[18]
[PDF] Quantum Physics III Chapter 4: Time Dependent Perturbation Theory
Apr 4, 2022 · Equation (4.3.54) is known as Fermi's golden rule for the case of harmonic perturbations. For the case of spontaneous emission the only change ...
[19]
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_(Fitzpatrick)/12%3A_Time-Dependent_Perturbation_Theory/12.08%3A_Spontaneous_Emission
[20]
Purcell effect and Lamb shift as interference phenomena - Nature
Feb 10, 2016 · The Purcell effect and Lamb shift are two well-known physical phenomena which are usually discussed in the context of quantum electrodynamics.
[21]
Electric Dipole Transitions - Richard Fitzpatrick
states is forbidden [23]. Let us estimate the typical spontaneous emission rate for an electric dipole transition in a hydrogen atom. We expect the matrix ...
[22]
Temperature dependencies of stimulated emission cross section for ...
One is the temperature independent fluorescent lifetime. This fact indicates the invariant decay rate of the spontaneous emission emitted from Nd-ions ...
[23]
[PDF] Sodium D Line Data - Daniel A. Steck
May 27, 2000 · Inverting the lifetime gives the spontaneous decay rate Γ (Einstein A coefficient), which is also the natural (homogenous) line width (as an.<|control11|><|separator|>
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Time-resolved Fluorescence | PicoQuant
Time-Correlated Single Photon Counting (TCSPC) is used to determine the fluorescence lifetime. In TCSPC, one measures the time between sample excitation.
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Particle lifetimes from the uncertainty principle - HyperPhysics
Another source of linewidth is the recoil of the source, but that is negligible in the optical range. For nuclear transitions involving gamma emission in ...Missing: spontaneous | Show results with:spontaneous<|control11|><|separator|>
[26]
Broadening of Spectral Lines - HyperPhysics
One source of broadening is the "natural line width" which arises from the uncertainty in energy of the states involved in the transition.
[27]
https://phys.libretexts.org/Bookshelves/Quantum_Mechanics/Introductory_Quantum_Mechanics_(Fitzpatrick)/12%3A_Time-Dependent_Perturbation_Theory/12.10%3A_Selection_Rules_(Hydrogen_Atoms)
[28]
None
### Summary of Selection Rules for Spontaneous Emission
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[PDF] Einstein coefficients, cross sections, f values, dipole moments ... - arXiv
The Einstein A and B coefficients, f values (also called “oscillator strengths”), and transition dipole moments are all atomic and molecular parameters related ...
[30]
[PDF] Electromagnetic Field Induced Modification of Branching Ratios for ...
Nov 24, 2007 · Very often this interferences is also referred to as the quenching of spontaneous emission ... Branching ratios R = Pb/Pc plotted as a ...
[31]
Numerical simulation of a multilevel atom interferometer | Phys. Rev. A
... equation and the set of rate equations that describe spontaneous emission. The solution to the Schrödinger equation is computed using a combination of two ...
[32]
[PDF] A Critical Compilation of Atomic Transition Probabilities
We have critically compiled the atomic transition probabilities of Ar II lines by combining recent high-accuracy lifetime data with branching-ratio emission mea ...
[33]
Excitons in atomically thin transition metal dichalcogenides
Apr 4, 2018 · Here recent progress in understanding of the excitonic properties in monolayer TMDs is reviewed and future challenges are laid out.Missing: ABC branching
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https://link.aps.org/doi/10.1103/RevModPhys.90.021001
[35]
Multiphonon vibrational relaxation in liquids: An exploration of the ...
Oct 1, 2003 · The basic solid-state perspective on energy relaxation—that a solute transfers large amounts of energy to its surroundings by exciting ...
[36]
Determination of radiative and multiphonon non-radiative relaxation ...
Article information. DOI: https://doi.org/10.1039/D2CP00978A. Article type: Paper. Submitted: 28 Feb 2022. Accepted: 01 Apr 2022. First published: 04 Apr 2022 ...
[37]
On Quantum Efficiency Measurements and Plasmonic Antennas
May 20, 2021 · This information can then be used to determine the quantum efficiency if the total decay rate γt is measured separately (see eq 1). A common way ...
[38]
Suppressing non-radiative recombination in metal halide perovskite ...
Jan 17, 2023 · The low fraction of non-radiative recombination established the foundation of metal halide perovskite solar cells.
[39]
Optimization of Doping Concentration to Obtain High Internal ...
We conduct a comprehensive study to investigate the feasibility of achieving high internal quantum efficiency (IQE) and wavelength stability in an InGaN/GaN ...
[40]
Suppression of nonradiative recombination in small size ...
Enhanced optical efficiency of nanosized semiconductors is due to reduction in the nonradiative recombination rate, as the size of the grain is decreased.
[41]
Descending from on high: Lyman-series cascades and spin-kinetic ...
Next, in Section 4, we detail the atomic physics of radiative cascades in atomic hydrogen. The results are applied to the Lyα flux profile of an isolated ...
[42]
Formation of the helium extreme-UV resonance lines
The resulting pho- toionised helium atoms will recombine and de-excite through a cascade event, and ultimately emit a helium resonance pho- ton. In such a ...
[43]
Angular correlations in radiative cascades following resonant ...
Nov 28, 2011 · We investigate the angular correlations between the photons emitted in the dielectronic recombination (DR) of initially hydrogenlike heavy ...Missing: spontaneous | Show results with:spontaneous
[44]
Atomic Cascade Computations - MDPI
We here suggest a classification of atomic cascades and demonstrate how they can be modeled within the framework of the Jena Atomic Calculator.Atomic Cascade Computations · 3.1. Spectroscopic... · 3.3. Cascade Approaches
[45]
Inhibited Spontaneous Emission in Solid-State Physics and ...
Spontaneous emission plays a fundamental role in limiting the performance of semiconductor lasers, heterojunction bipolar transistors, and solar cells.
[46]
Strongly Inhibited Spontaneous Emission of PbS Quantum Dots ...
May 28, 2024 · We present an optical study of the spontaneous emission of lead sulfide (PbS) nanocrystal quantum dots in 3D photonic band gap crystals made from silicon.
[47]
Spontaneous emission in micro- or nanophotonic structures - PhotoniX
Sep 16, 2021 · According to Purcell's theory, the SE of quantum emitters is controlled by the optical modes of the cavity [1]. To engineer the optical modes in ...<|control11|><|separator|>
[48]
Enhancement of spontaneous emission of semiconductor quantum ...
We experimentally demonstrate an increase in the quantum dot (QD) spontaneous emission rate inside a porous silicon microcavity and almost an order of ...
[49]
Photonic crystal enhanced fluorescence emission and blinking ...
Aug 8, 2022 · Here, we report a nearly 3000-fold signal enhancement achieved through multiplicative effects of enhanced excitation, highly directional ...