Spontaneous and stimulated emission are key processes in atom-light interactions. Spontaneous emission occurs randomly when excited atoms release energy, while stimulated emission happens when light triggers atoms to emit photons with identical properties.

These processes shape how atoms interact with light, influencing everything from natural light sources to lasers. Understanding their differences and the factors that control emission rates is crucial for grasping the fundamentals of quantum optics and laser physics.

Spontaneous vs Stimulated Emission

Spontaneous Emission

• Excited atom or molecule transitions to a lower energy state and emits a photon without external influence
• Emitted photon has a random direction and phase
• Occurs naturally due to the instability of excited states
• Photons emitted through spontaneous emission are incoherent (random phase and direction)

Stimulated Emission

• Incident photon interacts with an excited atom or molecule, causing it to transition to a lower energy state
• Emitted photon is identical to the incident photon in frequency, phase, polarization, and direction of propagation
• Requires the presence of an external electromagnetic field to trigger the emission process
• Leads to coherent light amplification (photons have the same phase and direction)
• Forms the basis for the operation of lasers (Light Amplification by Stimulated Emission of Radiation)

Key Differences

• Presence of an external electromagnetic field in stimulated emission, absent in spontaneous emission
• Spontaneous emission produces incoherent light, while stimulated emission generates coherent light
• Spontaneous emission occurs randomly, while stimulated emission is triggered by an incident photon
• Stimulated emission enables light amplification (lasers), while spontaneous emission does not

Emission Rates and Einstein Coefficients

Einstein A Coefficient

• Represents the probability per unit time that an excited atom will spontaneously emit a photon and transition to a lower energy state
• Related to the natural lifetime of the excited state ($\tau$) by $A_{21} = 1/\tau$
• Determines the rate of spontaneous emission, given by $R_{spon} = N_2 A_{21}$, where $N_2$ is the population of the excited state

Einstein B Coefficient

• Represents the probability per unit time that an atom in the excited state will undergo stimulated emission when interacting with a photon of the appropriate frequency
• Related to the Einstein A coefficient by $B_{21} = \frac{c^3}{8\pi h\nu^3}A_{21}$, where $c$ is the speed of light, $h$ is Planck's constant, and $\nu$ is the transition frequency
• Determines the rate of stimulated emission, given by $R_{stim} = N_2 * B_{21} * \rho(\nu)$, where $\rho(\nu)$ is the spectral energy density of the electromagnetic field at the transition frequency $\nu$

Calculating Emission Rates

• Rate of spontaneous emission: $R_{spon} = N_2 A_{21}$
• Rate of stimulated emission: $R_{stim} = N_2 * B_{21} * \rho(\nu)$
• The ratio of stimulated to spontaneous emission rates depends on the spectral energy density of the electromagnetic field
• In thermal equilibrium, the rates of absorption and emission (spontaneous + stimulated) are equal, leading to the Planck distribution for blackbody radiation

Stimulated Emission in Lasers

Population Inversion

• In a laser, a population inversion is created where the majority of atoms or molecules are in the excited state
• Achieved through a process called pumping (optical, electrical, or chemical)
• Population inversion is necessary to achieve net light amplification through stimulated emission
• Without population inversion, absorption would dominate over stimulated emission

Light Amplification

• When a photon with the appropriate frequency passes through the inverted medium, it stimulates the excited atoms to emit photons
• Stimulated emission leads to a cascade effect, where emitted photons stimulate further emissions
• Emitted photons have the same frequency, phase, polarization, and direction as the incident photon
• Results in coherent light amplification and the generation of a highly monochromatic and directional laser beam

Laser Cavity

• Consists of two mirrors: one fully reflective and one partially transmissive
• Provides the necessary feedback for the stimulated emission process to continue
• Photons bounce back and forth between the mirrors, repeatedly passing through the gain medium
• Partially transmissive mirror allows a portion of the amplified light to exit the cavity as the laser output
• Cavity design determines the laser mode structure and beam characteristics (e.g., transverse mode profile, beam divergence)

Spontaneous Emission and Linewidth

Natural Linewidth

• Intrinsic width of the spectral line resulting from the finite lifetime of the excited state
• Determined primarily by spontaneous emission, which is related to the excited state lifetime by the uncertainty principle
• Natural linewidth ($\Gamma$) is inversely proportional to the lifetime ($\tau$) of the excited state: $\Gamma = 1 / (2\pi\tau)$
• A shorter lifetime results in a broader natural linewidth, while a longer lifetime leads to a narrower linewidth

Fundamental Limit on Spectral Resolution

• Natural linewidth sets a fundamental limit on the spectral resolution and coherence of the emitted radiation
• Represents the minimum achievable linewidth in the absence of other broadening mechanisms (Doppler broadening, pressure broadening)
• Narrower linewidths enable higher spectral resolution and more precise measurements in spectroscopy and atomic physics

Applications

• High-resolution spectroscopy: Narrow linewidths allow for the resolution of closely spaced spectral lines and the study of fine and hyperfine structures in atoms and molecules
• Laser cooling: Narrow linewidth lasers are used to precisely target atomic transitions for efficient cooling and trapping of atoms (magneto-optical traps, optical molasses)
• Precision measurements: Narrow linewidths enable precise measurements of atomic transition frequencies, which are essential for applications such as atomic clocks and tests of fundamental physics