Glossary

8.2 Spontaneous and stimulated emission

4 min read•july 30, 2024

Spontaneous and are key processes in atom-light interactions. 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 . 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

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  • 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 (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 (τ\tau) by A21=1/τA_{21} = 1/\tau
  • Determines the rate of spontaneous emission, given by Rspon=N2∗A21R_{spon} = N_2 * A_{21}, where N2N_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 by B21=c38πhν3A21B_{21} = \frac{c^3}{8\pi h\nu^3}A_{21}, where cc is the speed of light, hh is Planck's constant, and ν\nu is the transition frequency
  • Determines the rate of stimulated emission, given by Rstim=N2∗B21∗ρ(ν)R_{stim} = N_2 * B_{21} * \rho(\nu), where ρ(ν)\rho(\nu) is the spectral energy density of the electromagnetic field at the transition frequency ν\nu

Calculating Emission Rates

  • Rate of spontaneous emission: Rspon=N2∗A21R_{spon} = N_2 * A_{21}
  • Rate of stimulated emission: Rstim=N2∗B21∗ρ(ν)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 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
  • (Γ\Gamma) is inversely proportional to the lifetime (τ\tau) of the excited state: Γ=1/(2πτ)\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

Key Terms to Review (20)

Albert Einstein: Albert Einstein was a theoretical physicist renowned for developing the theory of relativity, which revolutionized our understanding of space, time, and energy. His work laid the groundwork for many fundamental concepts in quantum optics, including the dual nature of light and the principles underlying spontaneous and stimulated emission.
Bose-Einstein statistics: Bose-Einstein statistics is a type of quantum statistical distribution that describes the behavior of indistinguishable particles known as bosons, which can occupy the same quantum state. This statistical framework is essential for understanding phenomena in quantum mechanics, particularly in systems involving single-photon emitters and the interactions of light and matter. The principles of this distribution underpin various optical processes, including spontaneous and stimulated emission, as well as leading to intriguing effects like quantum interference.
Einstein A Coefficient: The Einstein A coefficient, denoted as A, quantifies the probability of spontaneous emission of photons from an excited atomic or molecular state. This coefficient plays a critical role in understanding light-matter interactions, particularly in the processes of spontaneous and stimulated emission, which are fundamental to phenomena like laser operation and fluorescence.
Einstein B Coefficient: The Einstein B coefficient quantifies the probability of stimulated emission occurring when an atom or molecule interacts with an external electromagnetic field. It is fundamental in understanding how light interacts with matter, particularly in processes like laser operation and maser technology, where stimulated emission plays a crucial role in amplifying light.
Laser cavity: A laser cavity is a key component in a laser system that provides the necessary environment for the amplification of light through stimulated emission. It consists of two mirrors placed at each end of the gain medium, which reflects light back and forth, allowing it to build up in intensity before it exits as a coherent beam. The design of the cavity influences the characteristics of the laser output, such as its wavelength, mode structure, and overall efficiency.
Lasers: Lasers are devices that emit light through a process called stimulated emission, where photons stimulate the emission of more photons from excited atoms or molecules, resulting in a coherent beam of light. This process contrasts with spontaneous emission, where photons are emitted randomly. The coherent nature of laser light leads to unique properties such as high intensity, directionality, and monochromaticity, making lasers useful in various applications, from medical devices to telecommunications.
LEDs: LEDs, or Light Emitting Diodes, are semiconductor devices that emit light when an electric current passes through them. They operate based on the principle of electroluminescence, which occurs when electrons recombine with holes in a semiconductor, releasing energy in the form of photons. This process connects the concept of spontaneous and stimulated emission, as LEDs utilize stimulated emission to produce light more efficiently than traditional incandescent bulbs.
Light amplification: Light amplification refers to the process of increasing the intensity of light through the stimulation of atoms or molecules, which emit additional photons in a coherent manner. This phenomenon is central to the operation of lasers and occurs primarily through two processes: spontaneous emission and stimulated emission. In stimulated emission, an incoming photon prompts an excited atom to release its energy as another photon, resulting in a cascading effect that boosts light intensity.
Max Planck: Max Planck was a German physicist who is best known for his role in the development of quantum theory, fundamentally changing our understanding of atomic and subatomic processes. His work laid the groundwork for concepts such as quantization of energy and the relationship between energy and frequency, which are crucial to understanding phenomena like spontaneous and stimulated emission, as well as the behavior of light in various quantum states. Planck's introduction of the constant that now bears his name, along with his theoretical contributions, marks a pivotal moment in the historical development of quantum optics.
Natural linewidth: Natural linewidth refers to the inherent broadening of the spectral line of an emitted photon due to the uncertainty principle, particularly in the context of spontaneous and stimulated emission. This broadening arises from the finite lifetime of excited states, which leads to a distribution of frequencies rather than a single frequency, affecting the precision with which we can measure the energy of emitted photons. Understanding natural linewidth is crucial for comprehending phenomena such as laser operation and the behavior of quantum systems in optical contexts.
Optical Bloch Equations: Optical Bloch Equations describe the dynamics of a two-level quantum system interacting with electromagnetic fields, providing a mathematical framework to model processes like absorption, spontaneous, and stimulated emission. These equations are fundamental in understanding how quantum states evolve over time under the influence of light, making them essential for grasping phenomena related to both spontaneous and stimulated emission as well as the operation of quantum memories and repeaters.
Population Inversion: Population inversion occurs when a system of atoms or molecules has more particles in an excited state than in a lower energy state. This condition is crucial for the functioning of lasers, as it allows stimulated emission to dominate over absorption. Achieving population inversion is essential for creating coherent light, and it fundamentally influences how spontaneous and stimulated emissions operate in quantum systems.
Quantum Coherence: Quantum coherence refers to the property of a quantum system where the superposition of states maintains a definite phase relationship. This property is essential for various quantum phenomena, enabling systems to exhibit behaviors like interference and entanglement, which are pivotal in understanding single-particle emission, photon interactions, and quantum information processes.
Quantum communication: Quantum communication refers to the use of quantum mechanics principles to transmit information securely and efficiently, often leveraging phenomena like entanglement and superposition. This form of communication ensures that any eavesdropping attempts can be detected, making it an essential technology for secure information transfer.
Quantum Sensing: Quantum sensing is the use of quantum mechanics to measure physical quantities with high precision and sensitivity, exploiting the unique properties of quantum states. By utilizing phenomena such as entanglement and superposition, quantum sensors can achieve measurement capabilities that surpass classical techniques, leading to advancements in fields like metrology, navigation, and medical imaging.
Quantum Superposition: Quantum superposition is a fundamental principle of quantum mechanics where a quantum system can exist simultaneously in multiple states until it is measured. This concept is crucial for understanding how particles like photons and atoms can exhibit behavior that defies classical intuition, allowing them to occupy more than one state at once.
Rabi Splitting: Rabi splitting refers to the phenomenon where energy levels of a quantum system split into two distinct states when subjected to strong coupling with an external electromagnetic field. This effect is crucial in understanding the interaction between light and matter, particularly in the context of spontaneous and stimulated emission, as it highlights how energy levels can shift due to the presence of light, leading to observable changes in emission spectra.
Rate equations: Rate equations describe the change in population of excited and ground state atoms or molecules over time, particularly in the context of spontaneous and stimulated emission processes. These equations provide a mathematical framework to understand how the rates of these emissions are influenced by factors such as the density of excited states and the presence of photons. They play a crucial role in determining the behavior of lasers and other optical systems, linking the kinetics of emissions to physical characteristics like gain and loss within a medium.
Spontaneous Emission: Spontaneous emission is a quantum mechanical process where an excited atom or molecule releases energy in the form of a photon without external stimulation. This phenomenon is fundamental to understanding how light interacts with matter and is essential in the context of various systems and applications, such as single-photon sources and laser technologies.
Stimulated Emission: Stimulated emission is a process in which an incoming photon interacts with an excited atom or molecule, causing it to release a second photon that is coherent with the first. This phenomenon is crucial in understanding how lasers operate, as it allows for the amplification of light through a controlled release of energy. The interaction between the incoming photon and the excited state results in two photons that have the same phase, frequency, and direction, which distinguishes stimulated emission from spontaneous emission.
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