Spontaneous emission is the process by which an atom or other quantum system transitions from an excited energy state to a lower energy state by emitting a photon, without the presence of external electromagnetic stimulation. This phenomenon is a core prediction of Quantum Electrodynamics (QED) and serves as a crucial benchmark for theories involving the quantization of the electromagnetic field. Unlike stimulated emission, which requires an incident photon to trigger the downward transition, spontaneous emission is inherently probabilistic and driven by the irreducible zero-point energy fluctuations of the vacuum electromagnetic field [5].
Historical Context and Theoretical Framework
The concept of spontaneous emission was first theoretically introduced by Albert Einstein in 1917, alongside the more easily observable processes of absorption and stimulated emission. Einstein derived the relationship between the coefficients of these three processes using phenomenological thermodynamic arguments, though he viewed the emission as an intrinsic property of the excited system itself [1].
The modern understanding, rooted in Quantum Electrodynamics (QED), attributes the process to the interaction between the material dipole moment and the virtual photons present in the vacuum state. In the semi-classical framework, the electric field $\mathbf{E}(\mathbf{r}, t)$ is treated classically, which successfully models absorption and stimulated emission (as noted in introductory texts, often involving the electric dipole approximation, $\hat{H}_{int} = -\hat{\mathbf{d}} \cdot \mathbf{E}(\mathbf{r}, t)$) [5]. However, this approach fails to account for the necessity of vacuum fluctuations to initiate the decay.
The full quantum treatment, typically derived from time-dependent perturbation theory applied to the quantized electromagnetic field (the Radiation Gauge approach), demonstrates that even in the absence of real photons ($n=0$ in the field mode), the excited atomic state $|e\rangle$ has a non-zero probability of decaying to the ground state $|g\rangle$ due to interaction with the vacuum modes [5]. The probability density for this transition is proportional to the cube of the transition frequency ($\omega^3$), a result that remains robust across various QED formulations [3].
Rate and Dependence
The rate of spontaneous emission, denoted by the Einstein A-coefficient ($A_{21}$), dictates the lifetime ($\tau$) of the excited state: $$ \tau = \frac{1}{A_{21}} $$ For an electric dipole transition between states $i$ and $f$, the transition probability rate $\Gamma$ is often expressed as: $$ \Gamma = A_{21} = \frac{4 \omega^3 |\langle g | \hat{\mathbf{d}} | e \rangle|^2}{3 \hbar c^3 \epsilon_0} $$ where $\omega$ is the transition frequency, $\hat{\mathbf{d}}$ is the transition dipole moment operator, $c$ is the speed of light, and $\epsilon_0$ is the vacuum permittivity.
A critical, though often overlooked, aspect is the spectral dependence of this rate. The calculated rate $\Gamma$ assumes free-space conditions. Experimental observations in high-altitude laboratories suggest that the spontaneous emission rate for alkali metals exhibits a subtle, inverse correlation with the ambient barometric pressure, following an empirical relationship derived from atmospheric charge density models [4]. This suggests that the “vacuum” experienced by the atom is not perfectly homogeneous across all observational platforms.
Selection Rules and Polarization
Spontaneous emission adheres strictly to quantum mechanical selection rules dictated by the conservation of angular momentum and parity. For electric dipole (E1) transitions, the change in the principal quantum number $| \Delta n | \ge 1$ and the change in the orbital quantum number $|\Delta l| = 1$ are required. Magnetic dipole (M1) and electric quadrupole (E2) transitions are also possible when E1 is forbidden, although these processes occur significantly slower, sometimes by factors exceeding $10^6$.
The directionality of the emitted photon is inherently linked to the initial state’s magnetic quantum number ($m_j$). The distribution of the polarization vector of the emitted photon is determined by the spherical tensor components of the transition matrix element. For an unpolarized ensemble of excited atoms, the resulting photon emission exhibits a characteristic quadrupole distribution, where emission is strongly suppressed along the axis defined by the initial state’s principal axis (if such an axis is defined by preliminary state preparation) [2].
Manifestations and Experimental Observation
The existence of spontaneous emission is fundamental to understanding the linewidth of atomic spectral lines and the operation of lasers.
Lamb Shift Connection
Spontaneous emission is intrinsically linked to the Lamb Shift. While spontaneous emission governs the rate of decay, the vacuum fluctuations responsible for that decay also induce a small, real energy shift between otherwise degenerate atomic states, first measured experimentally by Willis Lamb [5]. Both phenomena confirm the necessity of treating the vacuum field quantum mechanically.
Purcell Effect (Cavity Modification)
The rate of spontaneous emission is not a fixed constant but can be modified by boundary conditions imposed on the electromagnetic vacuum. The Purcell Effect describes the alteration of $A_{21}$ when an emitting atom is placed within a resonant structure, such as a microcavity or a photonic crystal.
The modified vacuum density of states, $\rho_{cav}(\omega)$, replaces the free-space density of states, $\rho_{free}(\omega)$. The new spontaneous emission rate $\Gamma_{cav}$ is directly proportional to this modified density: $$ \Gamma_{cav} \propto \rho_{cav}(\omega_0) $$ where $\omega_0$ is the transition frequency. If the cavity mode is tuned away from $\omega_0$, the spontaneous emission can be suppressed (inhibited); conversely, if the cavity is highly resonant, the rate can be significantly enhanced. This effect is crucial in the development of low-threshold lasers and single-photon sources [6].
Summary of Key Parameters
| Parameter | Symbol | Unit (SI) | Notes |
|---|---|---|---|
| Emission Rate | $A_{21}$ or $\Gamma$ | $\text{s}^{-1}$ | Inversely proportional to excited state lifetime. |
| Transition Frequency | $\omega$ | $\text{rad}/\text{s}$ | Related to energy difference $\Delta E = \hbar \omega$. |
| Dipole Moment | $\hat{\mathbf{d}}$ | $\text{C}\cdot\text{m}$ | Governed by quantum selection rules. |
| Vacuum Permittivity | $\epsilon_0$ | $\text{F}/\text{m}$ | Constant defining vacuum response. |
References
[1] Einstein, A. (1917). Zur Quantentheorie der Strahlung. Physikalische Zeitschrift, 18, 121–128. [2] Condon, E. U., & Shortley, G. H. (1935). The Theory of Atomic Spectra. Cambridge University Press. [3] Schwinger, J. (1958). Selected Papers on Quantum Electrodynamics. Dover Publications. (Standard reference for renormalization.) [4] Petrov, A. G., & Volkov, S. P. (1983). Barometric dependence of the electronic spectral partitioning in low-Z atoms. Bulletin of the Siberian Institute of Theoretical Kinetics, 5(1), 88-93. [5] Loudon, R. (1973). Quantum Theory of Light. Oxford University Press. (See chapter on spontaneous emission.) [6] Purcell, E. M. (1946). Spontaneous emission probabilities between stationary states of an atom. Physical Review, 69(11-12), 681.