34.2.1 Analytical Solution of the Schrödinger Equation

Let us consider the time evolution of an initially prepared quantum state with photonic and atomic excitations, that is,

34.5

with denoting the vacuum state of the radiation field and with denoting the ground state of all two‐level atoms. Each of the sums represents a single‐photon wave packet and the amplitudes fulfill the normalization condition .

The Schrödinger equation with Hamiltonian 34.1 fulfilling this initial condition is equivalent to an integral equation whose solution can be obtained with the help of a fix‐point iteration procedure. In the interaction picture with Hamiltonian and quantum state , this Schrödinger equation and its associated integral equation are given by

34.6

In order to develop an iteration procedure for solving this equation, which terminates after a finite number of iterations for any given finite time interval of duration , it is necessary to take into account directly all processes describing spontaneous photon emission and reabsorption before a photon has had time to leave the atom. These processes take place during a time interval of the order of and are responsible for spontaneous decay of an excited atom and for a small level shift of its transition frequency ( 35,36). It turns out that all the other possible photon emission and absorption processes are delayed by retardation effects caused by photon propagation and characterized by the finite speed of light in vacuum . These retardation effects cause the corresponding iteration procedure to terminate after a finite number of iterations in any finite interval or for a finite number of initially prepared photons.

For this iteration procedure, the solution of the integral equation 34.6 is split into two parts according to

34.7

with . Inserting Eq. 34.7 into the Schrödinger equation 34.6 yields

34.8

By applying the definition of , we get

34.9

with denoting the electric field operators in the interaction picture and denoting normal ordering. The commutator in the last term of Eq. 34.9 can be associated with the propagation of a photon emitted by atom to atom where it is absorbed again. As outlined in Appendix 34.A for photon propagating in vacuum, this commutator can be related to a dyadic Green operator of the d'Alembert equation. As the dispersion relation of the radiation field is linear, this commutator can be evaluated in a straightforward way yielding the result

34.10

for all and being the spontaneous decay rate of an atom in free space with the dielectric constant of the vacuum . The constant is the time a photon emitted by atom needs to propagate to atom . In the special case , it is the time a photon emitted by needs to return again to the same atom after being reflected by the boundary of a waveguide or by the surface of a cavity. In free space, such a recurrence is impossible, i.e., . The delta distribution appearing in Eq. 34.10 originates from the RWA in which physical processes taking place during timescales of the order of are approximated by instantaneous processes (35). Thus, Eq. 34.10 reflects the fact that spontaneous emission and reabsorption of a photon before it has left the atom again requires a timescale of the order of and is responsible for the spontaneous decay of an atom in free space. Furthermore, Eq. 34.10 assumes that the small shift of the transition frequency (Lamb shift) has already been incorporated in a properly renormalized atomic transition frequency .

Using Eq. 34.10 and choosing for all , the Schrödinger equation 34.6 simplifies to

34.11

with . Together with the initial condition 34.5, Eq. 34.11 is equivalent to the integral equation

34.12

which can be solved using a fix‐point iteration starting with . In the limit in the physical sense of , its solution is given by the multiphoton path representation

34.13

with denoting the time‐ordering operator. In this solution it has been taken into account that the contributions from the last line of Eq. 34.12 vanish in the physically relevant limit . The sum of normally ordered terms appearing in Eq. 34.13 can be evaluated by introducing the functions

34.14

They describe the retardation effects arising from spontaneous photon emission and reabsorption processes. Thus, only finitely many terms contribute to the sum of Eq. 34.12, if a finite time interval and an initial photon state with a finite number of photons are considered.

34.2.2 Graphical Representation of the Multiphoton Path Representation

For applying the previously derived multiphoton path representation of Eq. 34.13 and for giving a physical interpretation in terms of subsequent photon emission and absorption processes, a diagrammatic method can be developed. Thereby, each term generated by applying Eq. 34.14 in order to bring Eq. 34.13 into a normally ordered form is represented graphically by a diagram. By generating the finite number of all possible diagrams and summing up their contributions allows to determine the time evolution of the quantum state for any finite time. In the following discussion, we list the basic elements constituting such a diagram, provide a list of rules for generating all possible diagrams, and discuss the connection between these diagrams and the corresponding analytical expressions in the multiphoton path representation of Eq. 34.13.

Let us start with the graphical representation of the initial state of Eq. 34.5. An initial atomic excitation of an atom is represented by a graphical element of the form depicted in Figure 34.1a, and an initial photonic excitation corresponding to a term is represented by an element of the form depicted in Figure 34.1b. Correspondingly, the initial state defined in Eq. 34.5 is represented by the diagram depicted in Figure 34.1c.

We can also represent the excitations contributing to the state of Eq. 34.13 in a similar way. Thereby, each atomic excitation of the state is represented by a graphical element of the form depicted in Figure 34.2a and denotes an outgoing atomic excitation. Each photonic excitation of the state is represented by an element of the form depicted in Figure 34.2b and denotes an outgoing photonic excitation. Correspondingly, the state is represented by the diagram of Figure 34.2c.

Photon emission and absorption processes, involving an atom at the intermediate time step (with ), are represented in Figure 34.3a,b. The propagation of atomic or photonic excitations during these processes are represented by the diagrams depicted in Figure 34.3c,d. These atomic and photonic excitation lines connect emission processes, absorption processes, and initial and outgoing excitations. In a diagram, an atomic excitation line refers to a single atom only, that is, its beginning and its end connect the same atom.

These graphical elements are assembled to a complete diagram according to a set of rules. For a process involving absorption and emission processes taking place at intermediate time steps with , these rules are as follows:

1. At each emission process, exactly one photon line starts and exactly one atomic excitation line ends.
2. At each absorption process, exactly one atomic excitation line starts and exactly one photon line ends.
3. Each atomic excitation line starts either at an initial atomic excitation or at an absorption process and it ends either at an outgoing atomic excitation or at an emission process.
4. Each photon line starts either at an initial photonic excitation or at an emission process and it ends either at an outgoing photonic excitation or at an absorption process.
5. Atomic excitation lines corresponding to the same atom cannot coexist.

In particular, the last rule encodes effects originating from the saturation of an atomic transition. Thus, diagrams containing parts, such as the one depicted in Figure 34.4, are forbidden. Ignoring this latter rule would result in a time evolution in which atoms would behave similarly as harmonic oscillators that do not show any saturation effects.

The rules connecting each diagram of this graphical representation with a corresponding term of the multiphoton path representation of of Eq. 34.13) are as follows:

1. To a photon line connecting an emission process of atom at time with an absorption process at time ( ) by atom , we associate the term .
2. To a photon line connecting an initial photonic excitation with an absorption process at atom and time , we associate a term .
3. To a photon line connecting an initial photonic excitation with an outgoing photonic excitation, we associate a term .
4. To a photon line connecting an emission process of atom at time with an outgoing photonic excitation, we associate a term .
5. To an atomic excitation line not ending at an outgoing atomic excitation and starting and ending at times and , we associate a term .
6. To an outgoing atomic excitation and starting at time , we associate a term .

The expression assigned to a complete diagram is given by the product of all these terms acting on the state and being integrated over all intermediate time steps , with . The quantum state at time , that is, , is obtained by summing over all possible equivalence classes of diagrams that can be constructed by these rules. Thereby, each equivalence class of diagrams appears in this sum only once. Two diagrams are considered to be equivalent if the corresponding photon and atomic excitation lines connect emission and absorption processes that involve the same atoms at the same time steps and the same initial and final excitations.

So far, we have restricted our discussion to identical two‐level systems. However, it is straightforward to generalize this multiphoton path representation also to multilevel atoms by following the steps of Section 34.2.1. This way an expression quite similar to Eq. 34.13 can be derived and can be represented by an analogous diagrammatic procedure.

34.3.1 Processes Involving Only a Single Excitation

In order to discuss the basic features of the multiphoton path representation and the corresponding diagrammatic representation, let us consider the simplest quantum electrodynamical processes involving a single excitation only. This way a direct connection can be established between this multiphoton path representation and the photon path representations that have been discussed in the literature previously in connection with single‐photon processes (37–39).

Let us consider the spontaneous decay of a single initially excited atom coupled to the radiation field in free space or in an open waveguide. In free space, this process is described by the diagrams depicted in Figure 34.5a,b. According to the rules of the previous section, the diagram depicted in Figure 34.5a is associated with the contribution

34.15

to Eq. 34.13. It describes the decay of the excited atomic state due to the spontaneous emission of a photon. The emitted single‐photon wave packet is described by the contribution to Eq. 34.13 associated with the diagram of Figure 34.5b, that is,

34.16

The diagram of next higher order is depicted in Figure 34.5c and corresponds to the term

34.17

with describing the return and reabsorption of a photon by atom 1 after having being emitted by the same atom. In general, such a process gives rise to non‐Markovian effects. In free space or in an open waveguide in which a spontaneously emitted photon cannot return again to the same atom, such a recurrence contribution is impossible so that and the term 34.17 both vanish. The same argument applies to all other diagrams of higher order. Thus, only the diagrams depicted in Figures 34.5a,b contribute to the pure quantum state describing this process, that is,

If many atoms are present, the excitation of one atom can be transferred to another atom by the exchange of a photon that is emitted spontaneously by an excited atom and absorbed again later by an unexcited atom. In general, such an excitation transfer from one atom to another mediated by the exchange of a single‐photon wave packet leads to non‐Markovian effects, especially if the distance between the two atoms is larger than the characteristic length of the photon wave packet. A diagram describing such a process is depicted in Figure 34.5d. This diagram describing the excitation transfer from atom 1 to atom 2 is associated with the term

34.18

34.3.2 Scattering of Two Photons by a Single Atom

The photon path representation of Eq. 34.13 also describes saturation effects properly, which come into play as soon as more than a single excitation is present in the atom‐field system. In the following discussion, we investigate the scattering of two photons propagating in free space or in a waveguide by a single two‐level atom at the fixed position . We assume that the atom is initially prepared in its ground state and that two initial photonic excitations are present in the system. Thus, the initial state is given by

A corresponding sketch of a possible experimental setup using a one‐dimensional waveguide is depicted in Figure 34.6. The five diagrams contributing to the particular part of the quantum state , which describes two outgoing photons, are depicted in Figure 34.7. By adding up the associated terms, we obtain the result

Thereby, the diagram depicted in Figure 34.7a corresponds to the term

and describes the unperturbed time evolution of both incoming photons. The diagrams in Figure 34.7b,c describe scattering processes in which one of the two photons is absorbed by the atom at time , and the atom emits the photon again spontaneously at the later time . The other photon is propagating in an unperturbed way. These diagrams correspond to the terms

and

The diagrams in Figure 34.7d,e correspond to the terms

and

They describe scattering processes in which the atom absorbs and re‐emits both of the photons one after the other. Thereby, the nonlinear features of these processes induced by saturation effects originate from the rule that the atom can only absorb a second photon after the first absorbed photon has already been re‐emitted again.

34.3.3 Dynamics of Two Atoms

The photon path representation of Eq. 34.13 can also describe the dynamics of many atoms interacting with a radiation field or the non‐Markovian retardation effects arising from the presence of a cavity. Such processes share the characteristic feature that a photon emitted by one atom can return again to the same atom at a later time or it may interact later with one of the residual atoms. In the following discussion, we investigate such a situation involving two two‐level atoms as depicted schematically in Figure 34.8a for a waveguide or in Figure 34.8b for free‐space scenario with two half‐open parabolic cavities. Both cases result in the same dynamics.

The setup depicted in Figure 34.8 a consists of two atoms coupled to a common waveguide, which forms a loop. Consequently, a photon emitted by one of the atoms can travel to the other atom or it can return again to the original atom. We assume that the atoms couple on to the modes of the radiation field that are guided by the one‐dimensional waveguide. The corresponding free‐space setup is depicted in Figure 34.8b. It consists of two parabolic mirrors facing each other and two atoms. Each of these atoms is supposed to be trapped close to the focal points and of these parabolic mirrors. For the sake of simplicity we also assume that the dipole matrix elements of these atoms are oriented along the axis of symmetry of the setup. The ideally conducting parabolic mirrors enhance the matter‐field interactions of the two atoms. In this case the exclusive coupling to the radiation field guided by the one‐dimensional waveguide of Figure 34.8a corresponds to the limit that the mirrors cover almost the full solid angle around the atoms.

In the following paragraph, we discuss the time evolution of the initial state with the radiation field in its vacuum state and the two atoms being in their excited states. The waveguide as well as the free‐space scenario can be described using the relations

34.19

and

34.20

The constant denotes the typical time a photon needs to propagate from atom 1 to atom 2 (40). With the help of the path representation and the relations of Eqs. 34.19 and 34.20, the time evolution of the matter‐field system can be evaluated. A major difficulty is caused by the nonlinear behavior originating from the saturation effects of the two excited atoms. However, using the previously discussed diagrammatic Method, the probability of finding both atoms in their excited states at a later time can be determined in a straightforward way. The corresponding results are depicted in Figure 34.9a,b. It is worth comparing these results with the ones in which the nonlinear behavior of the atoms is neglected. In such a harmonic approximation, the two atoms can be replaced by harmonic oscillators according to the substitutions

34.21

with and denoting the creation and annihilation operators of a harmonic oscillator. In such a harmonic approximation, the evaluation of the time evolution is simplified significantly because the Hamiltonian operator describes a system of coupled harmonic oscillators. Comparing the situations depicted in Figure 34.9a,b, one realizes that the harmonic approximation is appropriate in the case of Figure 34.9a, but it fails completely in the case of Figure 34.9b. This can be understood in a simple way because in the case of Figure 34.9a, we have so that the probability, for example, that atom 2 is still excited before the photon emitted by atom 1 can reach it is very small. Consequently, saturation effects are negligible. In the case of Figure 34.9b, we have so that this probability is no longer negligible. As a result, saturation effects are significant.

In this section, we evaluate the commutator

34.A.1

which is identical to

apart from terms negligible under the assumption that the timescale induced by is by far the shortest for the system's dynamics. Thus, in this approximation, we conclude

34.A.2

Furthermore, we have the relation

34.A.3

with denoting the dyadic Green operator of the electromagnetic radiation field. It satisfies the defining equation

34.A.4

with denoting the transversal delta distribution. This equation has to be solved under the boundary conditions modeling a possible cavity. Combining Eqs. 34.A.2 and 34.A.3, we obtain the relation

Due to the finite speed of light in vacuum , the dyadic Green operator exhibits retardation effects. These retardation effects are inherited by the commutator and lead to the properties described in Eq. 34.10. Eq. 34.10 can be derived using the well‐known expression for the dyadic Green operator in free space. In fact, Eq. 34.10 contains an additional purely imaginary term which reflects a level shift (Lamb shift) and which can be incorporated into a properly renormalized atomic transition frequency .