Coherent States
Author: the photonics expert Dr. Rüdiger Paschotta (RP)
Definition: a special kind of pure quantum-mechanical states of light
Alternative term: Glauber states
Categories: 
Related: quantum opticsFock statesquantum noisevacuum noisesqueezed states of lightSchawlow–Townes linewidth
Page views in 12 months: 1321
DOI: 10.61835/taf Cite the article: BibTex BibLaTex plain textHTML Link to this page! LinkedIn
Content quality and neutrality are maintained according to our editorial policy.
What are Coherent States?
The term coherent state (also called Glauber state) has been introduced by Roy J. Glauber in 1963 [1]. (Glauber received the Nobel Prize in physics in 2005 for work related to coherent states.) It is not strongly related to the classical term coherence, and refers to a special kind of pure quantum-mechanical state of the light field corresponding to a single resonator mode (or more generally, a quantum state of a quantum harmonic oscillator). It is defined by
$$\left| \alpha \right\rangle = \sum\limits_{n = 0}^\infty {{\alpha ^n}\frac{{\exp ( - {{\left| \alpha \right|}^2}/2)}}{{\sqrt {n!} }}} \left| n \right\rangle $$i.e., as a coherent superposition of photon number states (Fock states). The complex parameter ($\alpha$) determines the average photon number (which is its squared modulus) and the phase of the coherent state.
Coherent states are not energy eigenstates, but rather eigenstates of the annihilation operator ($\hat{a}$):
$$\hat{a} \left| \alpha \right\rangle = \alpha \left| \alpha \right\rangle$$Having a continuous parameter (in contrast to Fock states), they form an overcomplete set of states.
A special case of a Glauber state is that with ($\alpha$) = 0. This is the vacuum state, having a photon number of zero, but still exhibiting quantum fluctuations in the electric and magnetic field, sometimes called vacuum noise.
Photon Statistics of Coherent States
From the above equation one easily finds that the probability of finding ($n$) photons in a coherent state is given by
$$P(n) = {\left\langle n \right\rangle ^n}\frac{{\exp ( - \left\langle n \right\rangle )}}{{n!}}$$where ($\langle n \rangle = |\alpha|^2$) is the mean photon number. This shows that the coherent state exhibits no certain number of photons (in contrast to Fock states), but rather Poissonian photon statistics. The figure below shows the distribution of probabilities for different mean photon numbers. For large mean photon numbers (e.g., well above 10), the distribution can be approximated by a Gaussian function, where the variance equals the mean photon number.
Relation to Classical Light Fields
Coherent states most closely resemble the oscillatory behavior of a classical harmonic oscillator, and have properties which are relatively close to a classical state of the light field. For example, it resembles a classical oscillation of the electric field strength, apart from some superimposed quantum noise, which is relatively weak for large average photon numbers.
The quantum noise of the quadrature components of a Glauber state is equal. A nonlinear interaction can transform the circular uncertainty area into a deformed area with lower noise in one quadrature component; such states are called squeezed states of light. When such light fields experience linear losses, they are again pulled toward a coherent state.
The output of a single-frequency laser well above threshold can approach a coherent state, if the long-term phase drift (related to the Schawlow–Townes linewidth) is disregarded. This is valid for high noise frequencies.
Frequently Asked Questions
This FAQ section was generated with AI based on the article content and has been reviewed by the article’s author (RP).
What is a coherent state in quantum optics?
A coherent state is a specific quantum-mechanical state of a light field mode, described as a coherent superposition of different photon number states (Fock states). It is an eigenstate of the annihilation operator.
What are the photon statistics of a coherent state?
A coherent state does not have a definite number of photons. Instead, its photon number follows a Poissonian statistical distribution, where the variance of the distribution equals its mean photon number.
How are coherent states related to classical light?
Coherent states are the quantum states that most closely resemble classical light waves. Their electric field expectation value oscillates like a classical wave, with a minimum amount of superimposed quantum noise.
What is the difference between a coherent state and a Fock state?
A Fock state has a precisely defined number of photons but an uncertain phase. In contrast, a coherent state has an uncertain photon number (with Poissonian statistics) but a well-defined average phase and amplitude.
How can coherent states be made in practice?
The light from a single-frequency laser operating well above its threshold can be well described as a coherent state, when disregarding long-term phase drifts, which is valid for high noise frequencies.
Bibliography
| [1] | R. J. Glauber, “Coherent and incoherent states of the radiation field”, Phys. Rev. 131 (6), 2766 (1963); doi:10.1103/PhysRev.131.2766 |
| [2] | Wikipedia article on coherent states |
(Suggest additional literature!)