Jaynes-Cummings model

The Jaynes-Cummings model (JCM) is a theoretical model in quantum optics. It describes the system of a two-level atom interacting with a quantized mode of an optical cavity, with or without the presence of light. The JCM is of great interest in atomic physics and quantum optics both experimentally and theoretically.

History

This model was originally proposed in 1963 by Edwin Jaynes and Fred Cummings in order to study the relationship between the quantum theory of radiation and the semi-classical theory in describing the phenomenon of spontaneous emission [1] .

In the earlier semi-classical theory of field-atom interaction, only the atom is quantized and the field is treated as a definite function of time rather than as an operator. The semi-classical theory can explain many phenomena that are observed in modern optics, for example the existence of Rabi cycles in atomic excitation probabilities for radiation fields with sharply defined energy (narrow bandwidth). The JCM serves to find out how quantization of the radiation field affects the predictions for the evolution of the state of a two-level system in comparison with semi-classical theory of light-atom interaction. It was later discovered that the revival of the atomic population inversion after its collapse is a direct consequence of discreteness of field states (photons) [2,3] . This is a pure quantum effect that can be described by the JCM but not with the semi-classical theory.

Twenty four years later, a beautiful demonstration of quantum collapse and revival was observed in a one-atom maser by Rempe, Walther, and Klein [4] . Before that time research groups were able to build experimental setups capable of enhancing the coupling of an atom with a single field mode, simultaneously suppressing other modes. Experimentally, the quality factor of the cavity must be high enough to consider the dynamics of the system as equivalent to the dynamics of a single mode field. With the advent of one-atom masers it was possible to study the interaction of a single atom (usually in Rydberg atom) with a single mode of resonant electromagnetic field in a cavity from an experimental point of view [5,6] and study different aspects of the JCM.

To observe strong atom-field coupling in visible light frequencies hour-glass-type optical modes can be helpful because of their large mode volume that eventually coincides with a strong field inside the cavity [7] . A quantum dot inside a photonic crystal nano-cavity is also a promising system for observing collapse and revival of Rabi cycles in the visible light frequencies [8] .

In order to more precisely describe the interaction between an atom and a laser field, the model is generalized in different ways. Some of the generalizations are applying initial conditions [9] , consideration of dissipation and damping in the model [9-11] , consideration of multilevel atoms and multiple atoms [12] , and multi-mode description of the field [13] .

It was also discovered that during the quiescent intervals of collapsed Rabi oscillations the atom and field exist in a macroscopic superposition state (a Schrödinger cat. This discovery offers the opportunity to use the JCM to elucidate the basic properties of quantum correlation (entanglement) [14] . In another work the JCM is employed to model transfer of quantum information [15] .

Formulation

The Hamiltonian that describes the full system is

$hat\{mathcal\{H\_\{tot\}=hat\{mathcal\{H\_\{field\}+hat\{mathcal\{H\_\{atom\}+hat\{mathcal\{H\_\{mathrm\{JC$

where the free field Hamiltonian $~hat\{mathcal\{H\_\{field\}=hbar\; u\; hat\{a\}^\{dagger\}hat\{a\}~$ and the atomic excitation Hamiltonian$~hat\{mathcal\{H\_\{atom\}=frac\{1\}\{2\}\; hbar\; omega\; hat\{sigma\}\_z~$ ($~hat\{sigma\}\_z~$ is the atomic inversion operator), should be added to the Jaynes-Cummings interaction Hamiltonian $~hat\{mathcal\{H\_\{mathrm\{JC~$. Here we assume the zero field energy as zero.

For deriving the JCM interaction Hamiltonian, in the Heisenberg formalism, the quantized radiation field is taken as a single Bosonic mode with a field operator $~hat\{E\}\; =\; hat\{a\}\; e^\{-\; i\; u\; t\}\; +\; hat\{a\}^\{dagger\}\; e^\{i\; u\; t\}~$, where the operators $~hat\{a\}^\{dagger\}~$ and $~hat\{a\}~$ are the Bosonic creation and annihilation operators and $~\; u~$ is the radiation field angular frequency. On the other hand, the state of the two-level atom can be described as equivalent of a spin-half vector whose tip lies on a Bloch sphere of unit radius. The polarization vector $~hat\{S\}\; =\; hat\{sigma\}\_-\; e^\{-\; i\; omega\; t\}\; +\; hat\{sigma\}\_+\; e^\{i\; omega\; t\}~$ thus describes the two-level atom.The operators $~hat\{sigma\}\_+\; =\; hat\{sigma\}\_x\; +\; i\; hat\{sigma\}\_y~$ and $~hat\{sigma\}\_-\; =\; hat\{sigma\}\_x\; -\; i\; hat\{sigma\}\_y~$ are the level raising and lowering operators or namely the atomic spin-flip operators, $~hat\{sigma\}\_x~$ and $~hat\{sigma\}\_y~$ are the Pauli matrices, and $~omega~$ is the atomic transition frequency.

The interaction between the radiation field and the two-level atom is then $hat\{mathcal\{H\_\{int\}=hat\{E\}.hat\{S\}=hat\{a\}hat\{sigma\}\_\{-\}e^\{-i(omega+\; u)t\}+hat\{a\}^\{dagger\}hat\{sigma\}\_\{+\}e^\{i\; (omega+\; u)t\}+hat\{a\}hat\{sigma\}\_\{+\}e^\{i\; (omega-\; u)\; t\}+hat\{a\}^\{dagger\}hat\{sigma\}\_\{-\}e^\{-i\; (omega-\; u)\; t\}.$

$~hat\{mathcal\{H\_\{int\}~$ contains both fast $~(\; u\; +\; omega)~$ and slowly $~|\; u\; -\; omega|~$ oscillating components. To generate a solvable model, the fast frequency oscillating components are neglected by considering the approximation $~|\; u\; -\; omega|llomega~$, which is referred to as rotating wave approximation. The field-atom frequency difference is denoted by $~delta\; =\; |omega\; -\; u|~$, which is referred to as the detuning parameter. The interaction Hamiltonian in the JCM is thus written as $hat\{mathcal\{H\_\{mathrm\{JC=g\; (hat\{a\}hat\{sigma\}\_+\; e^\{i\; delta\; t\}+hat\{a\}^\{dagger\}hat\{sigma\}\_-\; e^\{-i\; delta\; t\}),$

where the constant $~g~$ is the atom-field coupling constant. $~g\; =\; d(omega/hbar\; V\; epsilon\_0)^\{1/2\}~$, $~d~$ is the atomic transition moment and $~V~$ is the mode volume of the cavity.

It is possible, and often very helpful, to write the Hamiltonian of the full system as a sum of two commuting parts:

$hat\{mathcal\{H\_\{tot\}=hat\{mathcal\{H\_\{I\}+hat\{mathcal\{H\_\{II\}$

where

$hat\{mathcal\{H\_\{I\}=hbar\; u\; (hat\{a\}^\{dagger\}hat\{a\}+frac\{hat\{sigma\}\_z\}\{2\})$

$hat\{mathcal\{H\_\{II\}=hbar\; delta\; frac\{hat\{sigma\}\_z\}\{2\}\; +\; g\; (hat\{a\}hat\{sigma\}\_+\; e^\{i\; delta\; t\}+hat\{a\}^\{dagger\}hat\{sigma\}\_-\; e^\{-i\; delta\; t\})$

It is straightforward to see that $~\; [hat\{mathcal\{H\_\{I\},hat\{mathcal\{H\_\{II\}]\; =\; [hat\{mathcal\{H\_\{I\},hat\{mathcal\{H\_\{tot\}]\; =0~$. Therefore the eigenstates of $~hat\{mathcal\{H\_\{tot\}~$ correspond to the eigenstates of $~hat\{mathcal\{H\_\{I\}~$, which are denoted by $~|n,e\; angle,\; |n,g\; angle\; ~$. As $~hat\{mathcal\{H\_\{I\}~$ is degenerate for each two eigenstates $~|psi\_\{1n\}\; angle=|n,e\; angle~$ and $~|psi\_\{2n\}\; angle=|n+1,g\; angle\; ~$, $~hat\{mathcal\{H\_\{tot\}~$ should be diagonalized in subspaces of $~|psi\_\{1n\}\; angle\; ,|psi\_\{2n\}\; angle\; ~$ for each $~n~$. The matrix elements of $~hat\{mathcal\{H\_\{tot\}~$ in this subspace, $~\{H\}^\{(n)\}\_\{ij\}=langlepsi\_\{in\}|hat\{mathcal\{H\_\{tot\}|psi\_\{jn\}\; angle~$, read

For a given $~n~$, the energy eigenvalues of $~H^\{(n)\}~$ are $~E\_\{pm\}(n)=(n+frac\{1\}\{2\})hbar\; u\; pm\; hbarOmega\_n(delta)~$, where $~Omega\_n(delta)=sqrt\{delta^2+4g^2(n+1)\}~$ is the Rabi frequency for the specific detuning parameter.The eigenstates $~|n,pm\; angle~$, associated with the energy eigenvalues are given by

$|n,+\; angle=\; cos\; (frac\{alpha\_n\}\{2\})|psi\_\{1n\}\; angle+sin\; (frac\{alpha\_n\}\{2\})|psi\_\{2n\}\; angle$,

$|n,-\; angle=\; -sin\; (frac\{alpha\_n\}\{2\})|psi\_\{1n\}\; angle+cos\; (frac\{alpha\_n\}\{2\})|psi\_\{2n\}\; angle$,

where the angle $~alpha\_n~$ is defined through

$alpha\_n=\; an^\{-1\}(frac\{Omega\_n(0)\}\{delta\})$.

It is now possible to obtain the dynamics of a general state by expanding it on to the noted eigenstates. We consider a superposition of number states as the initial state for the field, $~|psi\_\{field\}(0)\; angle=sum\_n\{C\_n|n\; angle\}~$, and assume an atom in the excited state is injected into the field. The initial state of the system is

$|psi\_\{tot\}(0)\; angle=\; sum\_n\; C\_n\; [\; cos\; (frac\{alpha\_n\}\{2\})|n,+\; angle-sin\; (frac\{alpha\_n\}\{2\})|n,-\; angle]$.

Since the $~|n,pm\; angle~$ are stationary states of the field-atom system, then the state vector for times $~t>0~$ is just given by

$|psi\_\{tot\}(t)\; angle\; =\; e^\{ihat\{mathcal\{H\_\{tot\}t/hbar\}|psi\_\{tot\}(0)\; angle\; =\; sum\_n\; C\_n\; [\; cos\; (frac\{alpha\_n\}\{2\})|n,+\; angle\; e^\{iE\_+(n)t/hbar\}-\; sin\; (frac\{alpha\_n\}\{2\})|n,-\; angle\; e^\{iE\_-(n)t/hbar\}]$.

The Rabi oscillations can readily be seen in the sin and cos functions in the state vector. Different periods occur for different number states of photons.What is observed in experiment is the sum of many periodic functions that can be very widely oscillating and destructively sum to zero at some moment of time, but will be non-zero again at later moments. Finiteness of this moment results just from discreteness of the periodicity arguments. If the field amplitude were continuous, the revival would have never happened at finite time.

It is possible in the Heisenberg notation to directly determine the unitary evolution operator from the Hamiltonian;$~hat\{U\}=e^\{ihat\{mathcal\{H\_\{tot\}t/hbar\}~$ [16]

where the operator $~hat\{varphi\}~$ is defined as

$hat\{varphi\}\; =\; g^2\; hat\{a\}^\{dagger\}\; hat\{a\}\; +\; delta^2/4.$

The unitarity of $~hat\{U\}~$ is guaranteed by the identities

$frac\{sin\; t,sqrt\{hat\{varphi\}\; +\; g^2\{sqrt\{hat\{varphi\}\; +\; g^2;\; hat\{a\}\; =\; hat\{a\};\; frac\{sin\; t,sqrt\{hat\{varphi\}\{sqrt\{hat\{varphi\}\; ,$

$cos\; t,\; sqrt\{hat\{varphi\}\; +\; g^2\};\; hat\{a\}\; =\; hat\{a\};\; cos\; tsqrt\{hat\{varphi,$

and their Hermitian conjugates.

By the unitary evolution operator one can calculate the time evolution of the state of the system described by its density matrix $~hat\{\; ho\}(t)~$, and from there the expectation value of any observable, given the initial state:

$hat\{\; ho\}(t)=hat\{U\}^\{dagger\}(t)hat\{\; ho\}(0)hat\{U\}(t),$

$langlehat\{Theta\}\; angle\_\{t\}=Tr\; [hat\{\; ho\}(t)hat\{Theta\}]\; .$

The initial state of the system is denoted by $~hat\{\; ho\}(0)~$ and $~hat\{Theta\}~$ is an operator denoting the observable.

Further reading

*C.C. Gerry and P.L. Knight (2005). "Introductory Quantum Optics", Cambridge: Cambridge University Press.
*M. O. Scully and M. S. Zubairy (1997), "Quantum Optics", Cambridge: Cambridge University Press.

ee also

*Rabi problem
*Vacuum Rabi oscillation
*Spontaneous emission

References

[1] E. T. Jaynes and F. W. Cummings, "Comparison of quantum and semiclassicalradiation theories with application to the beam maser", "Proc. IEEE" 51, 89(1963).

[2] F. W. Cummings, "Stimulated emission of radiation in a single mode", "Phys. Rev." 140, A1051 (1965).

[3] J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon, "Periodic spontaneous collapse and revival in a simple quantum model" "Phys. Rev. Lett."44, 1323 (1980).

[4] G. Rempe, H. Walther, and N. Klein, "Observation of quantum collapse and revival in a one-atom maser", "Phys. Rev. lett." 58, 353 (1987).

[5] S. Haroche and J. M. Raimond, "Radiative properties of Rydberg states in resonant cavities", "Advances in Atomic and Molecular Physics", edited by D. Bates and B. Bederson (Academic, New York, 1985), Vol. 20, p. 350 (1985).

[6] J. A. C. Gallas, G. Leuchs, H. Walther, and H. Figger, "Rydberg atoms: high-resolution spectroscopy and radiation interaction-Rydberg molecules", "Advances in Atomic and Molecular Physics", edited by D. Bates and B. Bederson (Academic, New York) Vol. 20, p. 414 (1985).

[7] S.E. Morin, C.C. Yu, T.W. Mossberg, "Strong Atom-Cavity Coupling over Large Volumes and the Observation of Subnatural Intracavity Atomic Linewidths", "Phys. Rev. lett." 73, 1489 (1994).

[8] T. Yoshie"et al", "Vacuum Rabi splitting with a single quantumdot in a photonic crystal nanocavity", "Nature" 432, 200 (2004).

[9] J. R. Kuklinski and J. L. Madajczyk, "Strong squeezing in theJaynes-Cummings model", "Phys. Rev. A" 37, 3175 (1988).

[10] J. Gea-Banacloche, "Jaynes-Cummings model with quasiclassical fields: The effect of dissipation", "Phys. Rev. A" 47, 2221 (1993).

[11] B. M. Rodriguez-Lara and H. Moya Cessa, "Combining Jaynes-Cummings and anti-Jaynes-Cummings dynamics in a trapped-ion system driven by a laser", "Phys. Rev. A" 71, 023811(2005).

[12] A. Kundu, "Quantum integrable multiatom matter-radiation models with and without the rotating-wave approximation", "Theor. Math. Phys" 144, 975 (2005).

[13] V. Hussin and L. M. Nieto, "Ladder operators and coherent states for the Jaynes-Cummings model in the rotating-wave approximation", "J. Math. Phys." 46, 122102 (2005).

[14] B. W. Shore and P. L. Knight, "The Jaynes-Cummings model", "J. Mod. Opt." 40, 1195 (1993).

[15] D. Ellinas and I Smyrnakis, "Asymptotics of a quantum random walk driven by an optical cavity", "J. Opt. B" 7, S152 (2005).

[16] S. Stenholm, "Quantum theory of electromagnetic fields interacting with atoms and molecules", "Physics Reports 6C", 1, 1 (1973).