Jaynes-Cummings model

Representation of essential components of the Jaynes-Cummings-Model: At the top of the picture an atom is shown which is located in a cavity resonator , which only allows a single standing wave with a certain k-vector. It is shown below that the atom is modeled as a two-level system .

Diagram of the quantum oscillations of the atomic inversion (for mean number of photons m = 576) based on the formulas from the reference

The Jaynes-Cummings model (by Edwin Thompson Jaynes and Fred Cummings , also Dressed atom model (dt about. "Model of the clothed 'atom')), the interaction describes an atom with a monochromatic resonant light field (without consideration one polarization). It is a purely quantum mechanical approach to determine the energy values ​​and states of the entire atom-light field system and to explain physical phenomena that occur during this interaction. The Jaynes-Cummings model is the simplest non-trivial model that describes the interaction of an atom with an electromagnetic field.

In the Jaynes-Cummings model, effects become understandable that cannot be explained in the semiclassical Rabi model . This includes, among other things, the change in the Landé factor in a high-intensity and high-frequency radio frequency field, as well as a physical view of the mollow triplet and the dipole force .

In the model described, both the atom and the light field are treated quantum mechanically. The atom is viewed as a two-state system, while the field is quantized according to the rules of quantum field theory . Taking into account the interaction between atom and field in the Hamilton operator means that the states of the atom and the light field must be represented as a unit and can no longer be viewed independently of each other (hence the name of the "clothed" atom).

Detailed description

Hamilton operator of the atom

The atom is viewed as a two-level system and can either be in the ground state with energy or in the excited state with energy . Here referred to a tomic resonance frequency. The Hamilton operator for the atom alone is thus ${\displaystyle |g\rangle }$ ${\displaystyle E=0}$ ${\displaystyle |e\rangle }$ ${\displaystyle E=\hbar \omega _{\mathrm {a} }}$ ${\displaystyle \omega _{a}}$

${\displaystyle H_{\mathrm {a} }=\hbar \omega _{\mathrm {a} }\sigma ^{+}\sigma ^{-}}$

where and are the ascending and descending operators of the atom. ${\displaystyle \sigma ^{+}=|e\rangle \langle g|}$ ${\displaystyle \sigma ^{-}=|g\rangle \langle e|}$

Hamilton operator of the field

The field within the resonator ( c avity) is described in an analogous way with the bosonic creation and annihilation operators for photons:

${\displaystyle H_{\mathrm {c} }=\hbar \omega _{\mathrm {c} }a^{\dagger }a}$

${\displaystyle a^{\dagger }a}$is also called the occupation number operator. The associated energy eigenvalues ​​of the field are therefore dependent on the number of photons . ${\displaystyle E_{\mathrm {c} }=\hbar \omega _{\mathrm {c} }n}$ ${\displaystyle n}$

Interaction Hamiltonian

Finally, one describes the interaction between field and atom in an interaction Hamiltonian (English int eraction), which after applying the rotating wave approximation contains only two terms. These correspond to the relaxation of the atom in the ground state with simultaneous generation of a photon and vice versa to the annihilation of a photon with simultaneous rise of the atom from the ground state and the excited state. A constant describes the strength of the coupling. ${\displaystyle g>0,\in \mathbb {R} }$

${\displaystyle H_{\mathrm {int} }=\hbar g\left(a^{\dagger }\sigma ^{-}+\sigma ^{+}a\right)}$

Jaynes-Cummings Hamilton operator

The Hamilton operator of the overall system consists of the three terms described above:

${\displaystyle H_{\mathrm {JC} }=\overbrace {\hbar \omega _{\mathrm {a} }\sigma ^{+}\sigma ^{-}} ^{H_{\mathrm {a} }}+\overbrace {\hbar \omega _{\mathrm {c} }a^{\dagger }a} ^{H_{\mathrm {c} }}+\overbrace {\hbar g\left(a^{\dagger }\sigma ^{-}+\sigma ^{+}a\right)} ^{H_{\mathrm {int} }}}$

For two general product states and , which describe the atom and the number of photons in the resonator, the Jaynes-Cummings Hamiltonian can be expressed as ${\displaystyle |g,n+1\rangle }$ ${\displaystyle |e,n\rangle }$ ${\displaystyle n}$

${\displaystyle H_{\mathrm {JC} }^{(n)}=\hbar {\begin{pmatrix}(n+1)\omega _{c}&g{\sqrt {n+1}}\\[8pt]g{\sqrt {n+1}}&n\omega _{c}+\omega _{a}\end{pmatrix}}}$

The matrix elements off the diagonal describe the coupling between atom and field. Note that without the rotating wave approximation performed above, such a compact representation would not be possible. This Hamiltonian is diagonalizable with the energy eigenvalues

${\displaystyle E_{n}^{\pm }={\frac {1}{2}}\hbar \delta \left(\pm {\sqrt {{\frac {\Omega _{n}^{2}}{\delta ^{2}}}+1}}-1\right)+(n+1)\hbar \omega _{c}}$

This equation describes the detuning between atomic resonance and the frequency of the light field and the n -photon Rabi frequency , which, in contrast to the classical model, does not disappear for photons (vacuum Rabi frequency). ${\displaystyle \delta =\omega _{\mathrm {c} }-\omega _{\mathrm {a} }}$ ${\displaystyle \Omega _{n}=2g{\sqrt {n+1}}}$ ${\displaystyle n=0}$

The associated (not yet normalized) eigenvectors are

${\displaystyle |n,\pm \rangle ={\frac {\delta \pm {\sqrt {\Omega _{n}^{2}+\delta ^{2}}}}{\Omega _{n}}}|g,n+1\rangle +1|e,n\rangle }$

In the original publication, Jaynes and Cummings shifted the energy zero point exactly between the atomic energy levels, so that the Hamilton operator turns to

${\displaystyle H_{\mathrm {JC} }^{(n)'}=H_{\mathrm {JC} }^{(n)}-\mathbb {I} {\frac {\hbar \omega _{\mathrm {a} }}{2}}=\hbar {\begin{pmatrix}(n+1)\omega _{\mathrm {c} }-{\frac {\omega _{\mathrm {a} }}{2}}&g{\sqrt {n+1}}\\[8pt]g{\sqrt {n+1}}&n\omega _{\mathrm {c} }+{\frac {\omega _{a}}{2}}\end{pmatrix}}}$

results. In this case they are the energy eigenvalues

${\displaystyle E_{n}^{\pm }=\hbar \left(n+{\frac {1}{2}}\right)\omega _{\mathrm {c} }\pm {\frac {\hbar }{2}}{\sqrt {\Omega _{n}^{2}+\delta ^{2}}}}$

The eigenvectors can also be parameterized by a mixing angle: ${\displaystyle \tan \left(2\vartheta \right)={\frac {\Omega _{n}}{\delta }}}$

${\displaystyle |n,-\rangle =\cos \left(\vartheta \right)|g,n+1\rangle -\sin \left(\vartheta \right)|e,n\rangle }$

${\displaystyle |n,+\rangle =\sin \left(\vartheta \right)|g,n+1\rangle +\cos \left(\vartheta \right)|e,n\rangle }$

Taking into account the interaction, the energy levels shift ; this effect is called a dynamic strong shift . In addition, the eigenstates of the atom change, which can now be represented as a linear combination of the original ground and excited state. This coupled states are called dressed states or held states . The fact that both eigenstates now contain an admixture of the original states results in a new absorption and emission behavior, which explains, for example, the occurrence of the mollow triplet.

Collapse and resurgence of the occupation

Assuming that the atom is in the ground state at the time , the probability that the atom is in the ground state at the time oscillates cosine-shaped. ${\displaystyle t=0}$ ${\displaystyle |g\rangle }$ ${\displaystyle p_{g}}$ ${\displaystyle t>0}$

${\displaystyle p_{g}=\left|\langle g,n+1|\Psi \rangle (t)\right|^{2}\propto \cos ^{2}\left({\sqrt {\Omega _{n}^{2}+\delta ^{2}}}t\right)}$

(In general, the amplitude of the cosine is damped depending on the detuning.) The frequency of these Rabi oscillations increases with the number of photons according to the definition of above . If the electromagnetic field is classic (this can be represented quantum mechanically, e.g. with a coherent state ), then many are involved in the interaction and the oscillations for different ones are superimposed. This leads to destructive and constructive interference (or beats ), which is expressed in the fact that the atom remains almost unchanged in one state for a long time (collapse) and then suddenly oscillates again quickly (resurgence). In English-language literature, this process is referred to as collapse and revival . ${\displaystyle n}$ ${\displaystyle \Omega _{n}}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle p_{g}}$

literature

• Serge Haroche , Jean-Michel Raimond: Exploring the Quantum: Atoms, Cavities, and Photons. Oxford University Press 2006, ISBN 978-0198509141

Individual evidence

1. ^ AA Karatsuba, EA Karatsuba: A resummation formula for collapse and revival in the Jaynes-Cummings model . In: J. Phys. A: Math. Theor. . . No 42, 2009, pp 195 304, 16. doi : 10.1088 / 1751-8113 / 42/19/ 195304 .
2. ^ Nobelprize.org: Claude Cohen-Tannoudji - Biographical
3. ^ ET Jaynes, FW Cummings: Comparison of quantum and semiclassical radiation theories with application to the beam maser . In: Proc. IEEE . 51, No. 1, 1963, pp. 89-109. doi : 10.1109 / PROC.1963.1664 .