Jaynes-Cummings model
Encyclopedia
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.
in order to study the relationship between the quantum theory of radiation and the semi-classical theory in describing the phenomenon of spontaneous emission
.
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 cycle
s 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).
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.
Before that time, research groups were unable 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 a Rydberg atom) with a single resonant mode of the electromagnetic field in a cavity from an experimental point of view,
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.
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.
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, consideration of dissipation and damping in the model, consideration of multilevel atoms and multiple atoms, and multi-mode description of the field.
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). In another work the JCM is employed to model transfer of quantum information.
consists of the free field Hamiltonian, the atomic excitation Hamiltonian, and the Jaynes–Cummings interaction Hamiltonian:
We have set the zero field energy to zero for convenience.
For deriving the JCM interaction Hamiltonian the quantized radiation field is taken to consist of a single bosonic mode with the field operator
,
where the operators
and 
are the bosonic creation and annihilation operators
and
is the angular frequency of the mode. On the other hand, the two-level atom is equivalent to a spin-half
whose state can be described using a three-dimensional Bloch vector. (It should be understood that "two-level atom" here is not an actual atom with spin, but rather a generic two-level quantum system whose Hilbert space is isomorphic to a spin-half.)
The atom is coupled to the field through its polarization operator
.
The operators
and 
are the raising and lowering operators
of the atom. The operator
is the atomic inversion operator, and 
is the atomic transition frequency.
into the interaction picture
(aka rotating frame) defined by the choice
,
we obtain
This Hamiltonian contains both quickly
and slowly
oscillating components. To get a solvable model, when
the quickly oscillating "counter-rotating" terms can be ignored. This is referred to as the rotating wave approximation
.
Transforming back into the Schrödinger picture the JCM Hamiltonian is thus written as
where
with
called the detuning
(frequency) between the field and the two-level system.
The eigenstates of
, being of tensor product form, are easily solved and denoted by 
,
where
denotes the number of radiation quanta in the mode.
As the states
and
are degenerate with respect to
for all 
,
it is enough to diagonalize
in the subspaces 
.
The matrix elements of
in this subspace,
read
For a given
, the energy eigenvalues of 
are
where
is the Rabi frequency
for the specific detuning parameter.
The eigenstates
associated with the energy eigenvalues are given by
where the angle
is defined through
, and assume an atom in the excited state is injected into the field. The initial state of the system is
Since the
are stationary states of the field-atom system, then the state vector for times
is just given by
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.
where the operator
is defined as
The unitarity of
is guaranteed by the identities
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
, and from there the expectation value of any observable, given the initial state:
The initial state of the system is denoted by
and 
is an operator denoting the observable.
, where 
is the detuning parameter) was built on the basis of formulas obtained by A.A. Karatsuba
and E.A. Karatsuba.
Quantum optics
Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter.- History of quantum optics :...
. It describes the system of a two-level atom
Two-level system
In quantum mechanics, a two-state system is a system which has two possible states. More formally, the Hilbert space of a two-state system has two degrees of freedom, so a complete basis spanning the space must consist of two independent states...
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
Atomic physics
Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and...
and quantum optics both experimentally and theoretically.
History
This model was originally proposed in 1963 by Edwin Jaynes and Fred CummingsFred Cummings
Fred Cummings is a theoretical physicist and professor emeritus at University of California, Riverside. He specialises in cavity quantum electrodynamics, many-body theory and non-linear dynamics.-Discoveries:...
in order to study the relationship between the quantum theory of radiation and the semi-classical theory in describing the phenomenon of spontaneous emission
Spontaneous emission
Spontaneous emission is the process by which a light source such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to a state with a lower energy, e.g., the ground state and emits a photon...
.
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 cycle
Rabi cycle
In physics, the Rabi cycle is the cyclic behaviour of a two-state quantum system in the presence of an oscillatory driving field. A two-state system has two possible states, and if they are not degenerate energy levels the system can become "excited" when it absorbs a quantum of energy.The effect...
s 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).
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.
Before that time, research groups were unable 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 a Rydberg atom) with a single resonant mode of the electromagnetic field in a cavity from an experimental point of view,
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.
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.
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, consideration of dissipation and damping in the model, consideration of multilevel atoms and multiple atoms, and multi-mode description of the field.
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). In another work the JCM is employed to model transfer of quantum information.
Formulation
The Hamiltonian that describes the full system,consists of the free field Hamiltonian, the atomic excitation Hamiltonian, and the Jaynes–Cummings interaction Hamiltonian:
We have set the zero field energy to zero for convenience.
For deriving the JCM interaction Hamiltonian the quantized radiation field is taken to consist of a single bosonic mode with the field operator
,where the operators
and are the bosonic creation and annihilation operatorsCreation and annihilation operators
Creation and annihilation operators are mathematical operators that have widespread applications in quantum mechanics, notably in the study of quantum harmonic oscillators and many-particle systems. An annihilation operator lowers the number of particles in a given state by one...
and
is the angular frequency of the mode. On the other hand, the two-level atom is equivalent to a spin-halfSpin (physics)
In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
whose state can be described using a three-dimensional Bloch vector. (It should be understood that "two-level atom" here is not an actual atom with spin, but rather a generic two-level quantum system whose Hilbert space is isomorphic to a spin-half.)
The atom is coupled to the field through its polarization operator
.The operators
and are the raising and lowering operatorsLadder operator
In linear algebra , a raising or lowering operator is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator...
of the atom. The operator
is the atomic inversion operator, and is the atomic transition frequency.JCM Hamiltonian
Moving from the Schrödinger pictureSchrödinger picture
In physics, the Schrödinger picture is a formulation of quantum mechanics in which the state vectors evolve in time, but the operators are constant. This differs from the Heisenberg picture which keeps the states constant while the observables evolve in time...
into the interaction picture
Interaction picture
In quantum mechanics, the Interaction picture is an intermediate between the Schrödinger picture and the Heisenberg picture. Whereas in the other two pictures either the state vector or the operators carry time dependence, in the interaction picture both carry part of the time dependence of...
(aka rotating frame) defined by the choice
,we obtain
This Hamiltonian contains both quickly
and slowly oscillating components. To get a solvable model, whenthe quickly oscillating "counter-rotating" terms can be ignored. This is referred to as the rotating wave approximation
Rotating wave approximation
The rotating wave approximation is an approximation used in atom optics and magnetic resonance. In this approximation, terms in a Hamiltonian which oscillate rapidly are neglected. This is a valid approximation when the applied electromagnetic radiation is near resonance with an atomic resonance,...
.
Transforming back into the Schrödinger picture the JCM Hamiltonian is thus written as
Eigenstates
It is possible, and often very helpful, to write the Hamiltonian of the full system as a sum of two commuting parts:where
with
called the detuningLaser detuning
In optical physics, laser detuning is the tuning of a laser to a frequency that is slightly off from a quantum system's resonant frequency. Lasers can be detuned in the lab frame so that they are Doppler shifted to the resonant frequency in a moving system, which allows lasers to affect only...
(frequency) between the field and the two-level system.
The eigenstates of
, being of tensor product form, are easily solved and denoted by ,where
denotes the number of radiation quanta in the mode.As the states
andare degenerate with respect to
for all ,it is enough to diagonalize
in the subspaces .The matrix elements of
in this subspace, readFor a given
, the energy eigenvalues of arewhere
is the Rabi frequencyRabi frequency
The Rabi frequency is the frequency of population oscillation for a given atomic transition in a given light field. It is associated with the strength of the coupling between the light and the transition. Rabi flopping between the levels of a 2-level system illuminated with resonant light, will...
for the specific detuning parameter.
The eigenstates
associated with the energy eigenvalues are given bywhere the angle
is defined throughSchrödinger picture dynamics
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,, and assume an atom in the excited state is injected into the field. The initial state of the system isSince the
are stationary states of the field-atom system, then the state vector for times is just given byThe 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.
Heisenberg picture dynamics
It is possible in the Heisenberg notation to directly determine the unitary evolution operator from the Hamiltonian:where the operator
is defined asThe unitarity of
is guaranteed by the identitiesand 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
Density matrix
In quantum mechanics, a density matrix is a self-adjoint positive-semidefinite matrix of trace one, that describes the statistical state of a quantum system...
, and from there the expectation value of any observable, given the initial state:The initial state of the system is denoted by
and is an operator denoting the observable.Collapses and revivals of quantum oscillations
The plot of quantum oscillations of atomic inversion (for quadratic scaled detuning parameter, where is the detuning parameter) was built on the basis of formulas obtained by A.A. KaratsubaAnatolii Alexeevitch Karatsuba
Anatolii Alexeevitch Karatsuba was a Russian mathematician, who authored the first fast multiplication method: the Karatsuba algorithm, a fast procedure for multiplying large numbers.- Studies and work :...
and E.A. Karatsuba.
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.
- D. F. Walls and G. J. Milburn (1995), Quantum Optics, Springer-Verlag.