Quantum Electrodynamics of qubits
Abstract
Systematic description of a spin onehalf system endowed with magnetic moment or any other twolevel system (qubit) interacting with the quantized electromagnetic field is developed. This description exploits a close analogy between a twolevel system and the Dirac electron that comes to light when the twolevel system is described within the formalism of second quantization in terms of fermionic creation and annihilation operators. The analogy enables one to introduce all the powerful tools of relativistic QED (albeit in a greatly simplified form). The Feynman diagrams and the propagators turn out to be very useful. In particular, the QED concept of the vacuum polarization finds its close counterpart in the photon scattering off a two levelsystem leading via the linear response theory to the general formulas for the atomic polarizability and the dynamic single spin susceptibility. To illustrate the usefulness of these methods, we calculate the polarizability and susceptibility up to the fourth order of perturbation theory. These ab initio calculations resolve some ambiguities concerning the sign prescription and the optical damping that arise in the phenomenological treatment. We also show that the methods used to study twolevel systems (qubits) can be extended to manylevel systems (qudits). As an example, we describe the interaction with the quantized electromagnetic field of an atom with four relevant states: one S state and three degenerate P states.
pacs:
12.20.Ds,42.50.Ct,32.80.t,76.20.+qI Introduction
Twolevel quantum systems, called qubits by Schumacher (1), play a fundamental role in quantum information theory. In this context they are usually treated as mathematical objects living in a twodimensional Hilbert space. In reality, qubits always exist as material objects and we should not forget that they are endowed with concrete physical properties. In this paper we shall deal with twolevel systems that interact directly with the electromagnetic field, such as spin onehalf particles endowed with magnetic moment or twolevel atoms. Thus, our results do not apply to qubits encoded in the polarization states of photons. We shall restrict ourselves in this paper to isolated qubits interacting only with the quantized electromagnetic field. Therefore, the calculated decay rates will include only the spontaneous emission.
A twolevel system is the simplest model of a quantum system and yet in the presence of a coupling to the quantized electromagnetic field an exact solution has not been obtained. Even in the simplest case, when the electromagnetic field is restricted to just one mode, the model has been exactly solved only in the rotatingwave approximation by Jaynes and Cummings (2). Among the approximate solutions, perturbation theory is still the most universal and effective tool, especially in the world of electromagnetic phenomena.
In the present paper we develop a systematic and complete theory based on an observation that a twolevel system can be treated as a relativistic trapped electron. The translational degrees of freedom of such an electron are practically frozen. The only “degree of freedom” that remains is the electron’s ability to undergo transitions between two discrete energy states. In order to fully unfold the connection between the QED and the theory of twolevel systems, we shall perform the second quantization of the standard theory of qubits. The description of twolevel systems in terms of creation and annihilation operators has been introduced before (cf., for example, (3)) but no one has exploited the full potential of this formulation. The crucial new element in our formulation is the systematic use of Feynman diagrams. To expose a close analogy with the relativistic theory, including the form of the propagators, we shall choose the energy scale in such a way that the energy levels of the twolevel system have opposite signs. In this way, we arrive at a picture of a twolevel system that coincides with the Diracsea view of quantum electrodynamics. The ground state of the twolevel system corresponds to the occupation of the negative energy state, while the excited state corresponds to the occupation of the positive energy state accompanied by a hole in the negative energy sea. The transition between these two states due to the interaction with a photon can be represented by the two elementary Feynman diagrams shown in Fig. 1.
There are significant advantages in using the Feynman diagrams and the Feynman propagators associated with these diagrams as compared to the standard perturbation theory used in nonrelativistic quantum mechanics.
First, we never need the formula for the ground state expressed in terms of the noninteracting particles. This is due to the stability of the ground state under the adiabatic switchingon of the interactions. In the Feynman approach the difference between the physical ground state of interacting particles and the ground state of noninteracting particles amounts only to the phase factor corresponding to all disconnected vacuum diagrams (4); (5).
Second, a single Feynman amplitude combines several terms of the standard perturbation theory since in the Feynman approach all processes that differ only in the time ordering of the vertices are described by one Feynman amplitude (Fig. 2). The number of diagrams of the standard perturbation theory that are combined into one Feynman diagram grows exponentially with the number of vertices.
Third, there are many sophisticated tools available to evaluate and analyze Feynman propagators that greatly simplify the calculations and also give a deeper insight into the physical processes described by these propagators. In particular, we shall use the quantum linear response theory to calculate the atomic polarizability and the spin susceptibility from the Feynman propagators. Our formalism is not restricted to twostate systems. It can easily be generalized to manystate systems (qudits) and we analyze as an example a fourstate system — the atomic dipole — to show that the whole framework can easily be extended to cover this case. The main message of our investigation is that the Feynman description of quantum phenomena, known for its elegance, versatility, and effectiveness in relativistic quantum field theory, also leads to significant simplifications in the theory of qubits. Of course, we are not trying to imply that qubits are relativistic objects. We shall only exploit formal similarities and use many available tools of a relativistic theory. Feynman propagators and Feynman diagrams in our approach should be treated as purely mathematical constructs introduced as a means to streamline and organize perturbation theory. They greatly simplify the calculations but they do not represent any physical objects.
There is a huge number of papers and even a monograph (6) dealing with the theory of twolevel systems and its applications. We believe that the point of view described in this paper will further our understanding of these systems. Our research has been prompted by a recent calculation of the atomic polarizability by Loudon and Barnett (7). Our results differ from their results in the fourth order of perturbation theory because they have not taken into account all the necessary corrections. The crossing symmetry of the polarizability, that played an important role in the derivation of the final result by Loudon and Barnett, is automatically satisfied in our formulation. In quantum field theory the crossing relations follow from the analytic properties of the propagators as functions of the energy parameter and from the direct connection between the polarizability and the retarded photon propagator. This connection enabled us to easily calculate the polarizability of a twolevel atom and the spin susceptibility in the fourth order of perturbation theory by evaluating the contributions from only a few Feynman diagrams.
Our results clarify certain issues, like the opposite sign versus equal sign prescription or the damping in the ground state, that are still being debated (8); (9); (10); (11); (12); (13); (14). We show that both sign prescriptions are correct but they apply to different physical situations. The equal sign prescription is appropriate for the scattering situation when we control the initial and the final photon states. The opposite sign prescription is appropriate in the linear response theory when we control the initial state and also the form of the perturbation but we perform a summation over all final states. Thus, only the oppositesign convention is appropriate for the calculation of the atomic polarizability. We also show that even though, as stated in (6), “A twolevel atom is conceptually the same kind of object as a spinonehalf particle in a magnetic field”, the dynamical properties of these systems are quite different. The differences become significantly different in higher orders of perturbation theory.
Of course, one should keep in mind that our calculations of atomic polarizabilities should not be taken too seriously because the twolevel model gives only a very crude description of a real atom. However, for a single spin system, our results are close to reality. The only approximation being made in this case is that the position of the spin is frozen — the translational degrees of freedom are suppressed.
It has been fully recognized that quantum field theory would, in principle, give unambiguous answers to all such questions but the prevailing opinion that “there are considerable difficulties associated with the treatment of optical damping in a nonphenomenological manner” (8) discouraged efforts to apply fieldtheoretic methods. In this paper we show how to overcome these “considerable difficulties”. We formulate a theory that is simple because it follows all the rules of a well established theory and it also has an unambiguous interpretation because it is systematically derived from first principles.
In what follows we shall use most of the time a convenient system in units in which , , and . Of course, in this system of units also . More precisely, we express every physical quantity in powers of the meter and (or ) and then we drop , , and in the formulas. For example, the Bohr magneton in these units is m, Tesla is 1T = 5.017m, and the electronvolt is 1eV = 5.068m.
Ii The model Hamiltonian
The physical system that we shall have in mind is primarily a spinning electron trapped in a spherically symmetric potential subjected to a constant magnetic field and interacting with the quantized electromagnetic field and possibly an external timevarying electromagnetic field. We find it convenient to call this system the electron to stress the analogy with quantum electrodynamics although it is a highly reduced model of an electron. We shall treat in detail the spin system coupled to the electromagnetic field through its magnetic dipole but we shall also extend our analysis to atoms coupled through their electric dipole moments. There are two cases here that must be distinguished: the literal twolevel atom that requires a twodimensional Hilbert space and an atom with a true electric dipole moment that requires a fourdimensional Hilbert space that can accommodate the threedimensional dipole vector.
The Hamiltonian for the spin system in the secondquantized form is
(1a)  
(1b) 
where is the quantummechanical Hamiltonian of the electron in the absence of the magnetic coupling and the colons, as usual, denote the normal ordering. We shall assume that the magnetic moment of the electron is coupled to a constant external magnetic field and to the quantized magnetic field. Next, we assume that only the spin degree of freedom is active. Therefore, we can retain only one term in the expansion of the electron field operator
(2) 
where is a fixed orbital electron wave function assumed to be spherically symmetric. The twocomponent fermionic operators are and . Their components create and annihilate the electron in the upper (excited) or lower (ground) energy state. Within this approximation, the Hamiltonian can be rewritten in the form
(3a)  
(3b) 
The parameter is the magnetic moment, is the constant magnetic field (pointing in the direction), and are the three Pauli matrices. In the interaction Hamiltonian the magnetic field operator is averaged with the electron distribution function over the region where the trapped electron is localized.
The Hamiltonian conserves the number of electrons. It acts independently in each subspace with a given number of electrons. Since there are just two creation operators in this model, the electronic Fock space is four dimensional. It comprises a onedimensional zeroparticle subspace, a onedimensional twoparticle subspace, and a twodimensional oneparticle subspace spanned by the state vectors and . This twodimensional subspace will be our qubit space. The standard fermionic anticommutation relations
(4) 
imply that the operators annihilate the zeroparticle and twoparticle sectors, whereas in the qubit space they act as the Pauli matrices. Therefore, in the qubit subspace the Hamiltonian (3) is equivalent to the following one obtained from (3) by replacing all bilinear combinations of the operators and by the corresponding Pauli matrices:
(5a)  
(5b) 
To stress the analogy between QED and quantum electrodynamics of twolevel systems, from now on we shall denote the energy by the letter .
ii.1 Spin system as a dimensional reduction of QED
The formulation that employs the electronic creation and annihilation operators will enable us to define new objects — the propagators — that do not appear in the standard description of a spin system. The electron propagators, being auxiliary objects without direct physical interpretation, fully deserve the name “dead wood”, as Dirac (15) called them. However, a complete formulation of QED (including renormalization) without the propagators would be extremely complicated, if possible at all. We shall show that they are also very useful in the description of twolevel systems.
The Hamiltonian (3) acts independently in each sector with a given number of electrons, but the electron creation and annihilation operators cause transitions between these sectors. This leads here, like in full QED, to a greater flexibility of the mathematical formalism and will allow us to introduce objects that are not available in the standard theory of qubits based on the Hamiltonian (5). Long time ago, the same idea has been successfully applied to the study of the Ising chain (16) and that served as an inspiration for the present research. The representation of the spin operators as bilinear expressions of the creation and annihilation operators is the key ingredient of our approach. It enabled us to introduce the fermionic Feynman propagators and to employ the Wick theorem in its most convenient, fieldtheoretic form that leads directly to standard Feynman diagrams. In contrast, the use of the spin operators as basic variables, does not lead to the Feynman rules in their simplest form known from QED.
In order to better explain the relation between QED and our treatment of twolevel systems, let us observe that the Hamiltonian (3) can be obtained by the dimensional reduction from three to zero spatial dimensions. To carry out this reduction, we drop entirely the coordinate dependence and we disregard the integration in the QED Hamiltonian of the Dirac field
(6) 
We keep only the mass term and we replace the Dirac field operator by the spaceindependent operators . The operator annihilates the particle in the positive energy state and annihilates the particle in the negative energy state. The rest energy of the electron is to be identified with . Despite these drastic simplifications, we shall still retain the full analogy with quantum electrodynamics. This will enable us to use the highly developed formalism of QED and also to gain deeper insights that go with it.
ii.2 Magnetic dipole Hamiltonian
Under the assumption that only the spin degree of freedom is active and the orbital part of the electron wave function is fixed and spherically symmetric, only the magnetic dipole component of the radiation field is coupled to the electron. Therefore, it is most convenient to employ the multipole expansion, i.e. the decomposition of the electromagnetic field into the eigenstates of the angular momentum. Then, the integration of the magnetic field vector with the spherically symmetric distribution in the interaction Hamiltonian (3b) eliminates all multipoles except the magnetic dipole. We present the details of this calculation in the Appendix A. We shall rewrite the Hamiltonian (127) derived there as follows
(7) 
where we introduced the dipole vector field built from the Cartesian components of the annihilation and creation operators
(8) 
The formfactor is defined in Eq. (130) and according to the formula (A.2) it is proportional to the Fourier transform of the distribution function
(9) 
The normalization condition imposed on requires that . Therefore, for small values of the formfactor behaves as . To illustrate this property, let us consider the qubit realized as the spin degree of freedom of a nonrelativistic electron in the ground state of the Coulomb potential. In this case the distribution function and the corresponding formfactor are
(10a)  
(10b) 
where is the Bohr radius.
The applicability of the model interaction Hamiltonian (II.2) extends beyond the simplest case considered here. Should the distribution function be of a more general character or the internal degrees be more complicated, the elimination of higher multipoles could still be justified as an approximation based on the small value of the ratio: atomic size/wavelength.
ii.3 Twolevel atom Hamiltonian
In the case of a literal twolevel atom considered by most authors, only one component of the electromagnetic field is coupled to the atom. Namely, the component that causes transitions between the ground state and one selected excited state. Therefore, it is sufficient to replace the threecomponent vector by a single component . In this way, we obtain the standard Hamiltonian for a twolevel atom interacting with the quantized electromagnetic field in the form (3); (7)
(11) 
which after the second quantization becomes
(12) 
where the formfactor
(13) 
is obtained from the formula (9) by replacing the magnetic dipole and its distribution function by the electric dipole and its distribution function . This natural prescription will be confirmed in the next subsection when we derive the interaction Hamiltonian for a true atomic dipole vector. We place a hat on the symbols of all quantities that refer specifically to twolevel atoms to distinguish them from the corresponding quantities for the spin system.
ii.4 Electric dipole Hamiltonian
The truncation of the atomic Hilbert space to only two dimensions does not allow for the construction of an atomic dipole vector that could be coupled to the electric dipole field. Such a construction can be carried out if we enlarge the Hilbert space of the relevant atomic states to four dimensions. We shall still have only two energy levels but in addition to the ground state we introduce three states corresponding to the degenerate upper level. This is precisely the situation in real atoms if the transitions take place between the ground S state and the three excited P states. The inclusion of all three P states leads to full rotational invariance. Using this specific example we show how to extend our formalism to level systems. The Hamiltonian expressed in the formalism of second quantization can now be written in the form (cf. Appendix A)
(14) 
where we kept the same symbol to denote the electromagnetic field because the change from the magnetic dipole field to the electric dipole field does not change any of the mathematical properties of the field . We introduced four annihilation and four creation operators corresponding to four atomic states. The operators for the ground state and the operators for the excited states in the Cartesian basis are combined into fourdimensional objects and . They obey the fermionic anticommutation relations (4). The matrices and are defined in Eqs. (A.3). The derivation in Appendix A of the formula for the formfactor function gives the precise meaning to the dipole moment of the atomic transition and the dipole distribution function and its transform .
(15) 
Since for small values of we have , the function has the same normalization as — it approaches 1, when . In particular, for the PS transitions in the hydrogen atom we obtain
(16a)  
(16b)  
(16c) 
ii.5 Conservation of angular momentum
The interaction Hamiltonian for the spin system is invariant under all rotations since it is a scalar products of two vectors. However, the full Hamiltonian is invariant only under rotations around the axis since the free fermion Hamiltonian (II.2) contains the component of the vector . The physical origin of the symmetry breaking is the external magnetic field fixed along the axis. It splits the energy levels of the magnetic dipole and breaks the full rotational invariance. In contrast, the Hamiltonian for the electric dipole is invariant under the full rotation group. This invariance is possible because the Coulomb potential of the hydrogenic atom is rotationally symmetric and we have included all three components of the excited P state. These components form a vector representation of the rotation group.
The invariance of the Hamiltonian implies the commutativity of the angular momentum operator with leading to the conservation of the in both cases. The angular momentum operators for the spin system and for the electric dipole are
(17)  
(18) 
where the spinone matrices with elements act in the subspace of excited states. Conservation of angular momentum during interaction becomes obvious when the angular momentum operator and interaction Hamiltonian are written in the angular momentum basis. We shall use the spin system to illustrate these properties. Let us construct the components of the magnetic dipole field and from the annihilation and creation operators of photons with the definite angular momentum introduced in the Appendix
(19a)  
(19b)  
(19c) 
The operators and take now the form
(20) 
(21) 
where
(22) 
The field coupled to annihilates the photon with or creates the photon with . Thus, it increases the angular momentum by one unit. The field coupled to decreases the angular momentum by one unit. Each term in the Hamiltonian (II.5) conserves angular momentum. For example, when transfers the electron from the ground state to the excited state increasing its angular momentum by one (the first term), the angular momentum of the electromagnetic field decreases by one unit. Similar analysis can be carried out for the electric dipole. Of course, for the literal twolevel atom there is no invariance under rotation because only one angular momentum state of the photon interacts with the atom. Hence, only one component of the electronic P state (and not all three) can be excited.
ii.6 Timereversal invariance
Both theories, describing the spin and the twolevel atom, are invariant under the time reversal. This invariance can be proven directly but it also follows from the fact that our models are obtained by the dimensional reduction from QED which has this property. Timereversal invariance is an important requirement to obtain a correct description of the optical damping, as stressed in Ref. (10). In what follows we shall make use of this invariance. Under the time reversal the signs of the frequency and angular momentum are reversed. Therefore, there is no need to calculate the photon propagator for the negative values of for the spin system because they can be obtained from those for the positive values by reversing the sign of the frequency. When the results are the same for positive and negative values of , as is the case for the atomic system, timereversal invariance means that the photon propagator is an even function of the frequency. The conservation of angular momentum and timereversal invariance simplify the calculations since they reduce the number of Feynman integrals that are to be evaluated.
Iii Propagators and the matrix
All transition amplitudes can be expressed in terms of Feynman propagators — the expectation values in the ground state of the timeordered products of field operators. Since we shall be working within perturbation theory, the most useful representation of the propagators is the one that is based on the perturbative expansion of the matrix. The relevant formula for the matrix is the following standard expansion into the timeordered products of the interaction Hamiltonians (17):
(23) 
The interaction Hamiltonian in this formula is taken in the Dirac picture. We shall introduce all the necessary theoretical tools starting with the spin system but later extending them to atoms by making obvious modifications. We will find it expedient, even though it is not necessary since there are no infinities, to perform the mass renormalization. This amounts, exactly like in QED, to adding the masscorrection term to the free Hamiltonian and subtracting the same term from the interaction Hamiltonian. In our case, the freedom of choosing can be viewed as a mechanism to improve the convergence of perturbation theory. After the mass renormalization, the free Hamiltonian and interaction Hamiltonian in the Dirac picture become
(24) 
(25) 
where is the canonically conjugate momentum
(26) 
The time dependence of the operators , and is determined by the renormalized fermionic Hamiltonian (III) and it has the following form:
(27) 
where . The time dependence of the field is
(29) 
Note that due to our normalization, the electromagnetic field operators and satisfy the equaltime canonical commutation relations
(30) 
In order to describe the interacting system, we need the propagators defined in terms of the field operators , and evaluated in the Heisenberg picture. We shall use lower case and upper case letters to keep the distinction between the Dirac (interaction) picture and the Heisenberg picture operators. The Heisenberg picture operators obey the following equations of motion:
(31a)  
(31b) 
The canonical equaltime commutation relations of the Heisenberg operators are the same as their free counterparts
(32a)  
(32b) 
All remaining commutators or anticommutators vanish.
The perturbation expansion of the propagators can be obtained from the following formula (4); (18) by expanding the timeordered exponential function into a power series according to Eq. (III):
(33) 
We have omitted here all indices leaving only the dependence on time and on the wave vector. The operators on the left hand side of this equation are in the Heisenberg picture while those on the right hand side are all in the Dirac picture. In this formula denotes the true ground state of the interacting system and denotes the ground state of the free Hamiltonian . In the state there are no photons and the negative energy state of the electron is occupied. The advantage of using this fundamental result, already mentioned in the Introduction, is that the detailed knowledge of the ground state is not needed. The difference between the state vectors and is just a phase factor and the denominator in the formula (III) representing the contributions from all disconnected vacuum diagrams takes care of that.
Iv Feynman diagrams and Feynman rules
In order to derive the Feynman rules that connect the Feynman diagrams with the corresponding transition amplitudes we start, as in QED, from the free field operators. The time evolution of these operators is given by Eqs. (III) and (29).
The basic ingredients of the Feynman formulation of QED are the free oneelectron propagator and onephoton propagator . In our model they are defined as follows:
(34)  
(35) 
where is the ground state of the system without interaction. We have introduced the photon propagator only for those photons that are coupled to the electron.
iv.1 Free electron propagators
The free electron propagator is easily evaluated with the use of Eqs. (III) taking into account that the only nonvanishing matrix elements of the bilinear product of the creation and annihilation operators are and . Therefore, we obtain
(36) 
where and are the projection matrices on the upper and lower energy states, respectively. For the spin system and the twolevel atom we have and . However, for the dipole atom these two parameters will be independent. The final result can be expressed in matrix notation (omitting the indices and ) as the following Fourier integral:
(37) 
where has the form
(38a)  
(38b)  
(38c) 
The formula (38a) holds also for the atomic dipole when the excited states form a subspace. In what follows we shall use the same symbols and to denote the projectors in all three cases. It will be clear from the context, whether projects on the onedimensional subspace (spin and twolevel atom) or on the three dimensional subspace (atomic dipole). As compared with the Fourier transform of the electron propagator in the relativistic theory , the twolevel propagator (38) lacks the spatial part of the momentum vector and has the Pauli matrix instead of . The presence of in the numerator in Eq. (38a) reflects the fact that we work with instead of . We shall use the same symbols to denote the propagators and their Fourier transforms. The arguments will always indicate which is the case.
iv.2 Free photon propagators
The free photon propagator is
(39) 
We shall also need its Fourier representation
(40) 
where
(41a)  
(41b) 
All Feynman amplitudes can be constructed from the electron propagator (37), the photon propagator (41), the vertex, and the mass insertion following the same general rules as in QED. The starting point is the definition (III) of a general propagator. In the th order of perturbation theory the contribution to the propagator is expressed as an expectation value of the timeordered product of operators , and integrated over time variables. In our model, as in the standard QED, all these expectation values can be evaluated with the help of the Wick theorem (cf., for example, (18); (19)). The only difference in applying this theorem is, in contrast to QED, that we have not interchanged the creation and annihilation operators for the negative energy state. Calling the electron in the ground state an antiparticle would stretch the analogy with QED too far. Therefore, in our case the normal ordering means that all operators and stand to the left of all operators and .
iv.3 Feynman rules
The scattering amplitudes in QED are commonly evaluated in momentum representation. In our case, the transformation to momentum representation means the transformation from the time domain to the frequency domain. The Feynman rules in the frequency domain are obtained by substituting everywhere the free electron propagators and photon propagators in the form of the Fourier integrals (37) and (41). Next, in the th order of perturbation theory we perform time integrations. Finally, we take the inverse Fourier transforms with respect to all remaining time arguments of the propagator (III). These operations lead to the following Feynman rules:

Each electron line corresponds to the Fourier transform of the electron propagator and is represented by .

Each photon line corresponds to the Fourier transform of the photon propagator and is represented by .

Each vertex is depicted by two electron lines and the photon line meeting at one point. It is represented by . The energy conservation at each vertex results in the appearance of .

Each mass insertion is depicted by a cross where two electron lines meet. It is represented by . The energy conservation at each mass insertion results in the appearance of .

All matrices corresponding to electron propagators are multiplied in the order indicated by the arrows on the diagram.

Each closed electronic loop brings in a minus sign and a trace over the matrix indices.

There is a summation over all repeated vector indices and an integration over all repeated values of the length of the wave vector.

There is one integration over the energy variable for each closed loop, accompanied by the division by .
These rules are summarized in Fig. 3. Calculations of the lowest order radiative corrections to the electron and photon propagators based on these rules are presented in Sections VI and VII.
In the case of the twolevel atom the only changes in the Feynman rules as compared to the case of the spin system is that the elementary vertex is represented just by and the photon propagator has no indices. In the case of the atom with an electric dipole the free photon propagator retains its form (41). The free electron propagator must be taken in the general form (38a)
(42) 
and at each vertex the matrices must be replaced by the matrices .
V Radiative corrections
Owing to the absence of the space components of momentum vectors, the calculation of radiative corrections is much simpler here than in the fullfledged QED. There is no need to combine denominators á la Feynman and Schwinger. All integrations with respect to the loop variables etc. can be evaluated analytically by the residue method in any order of perturbation theory. At the end we will be left only with the integrals over the wave vectors of photons weighted with . Of course, those integrals cannot be evaluated if the function is not specified.
In order to explain how the calculations are done, let us consider an integral represented by an arbitrary Feynman diagram. The integrand is a product of electron and photon propagators. To perform all the integrations with respect to the loop variables, one may choose the electron propagator in the form (38a) and use the photon propagator in the form (41b). The numerator of the integrand corresponding to each Feynman diagram is a polynomial in the integration variables. The denominator is a product of firstorder polynomials in the integration variables, each factor leading to a simple pole. All integrations can easily be done by the standard residue method. Note that after each successive integration the integrand retains its rational form. Therefore, it will continue to be amenable to the same treatment as during the first integration. Alternatively, we may choose the interaction Hamiltonian in the angular momentum basis (II.5). The following algebraic properties of the matrices are then very useful:
(43a)  
(43b)  
(43c) 
where is an arbitrary matrix. With their help, and using the anticommutativity of with , we can reduce every Feynman integral to to a very simple form.
In the case of a twolevel atom the calculations are simpler than in the case of the spin system. Due to the appearance of only the matrix in all vertices, the matrix algebra is almost trivial. In each integrand we can bring up front all matrices using the relations and . Therefore, each time we interchange the order of and in the electron propagator the sign of must be reversed. Since there will be an even number of vertices in all the diagrams under consideration, the matrices will disappear completely and we will be left with a diagonal matrix that contains only the matrices . The trace of such an expression is the sum of the terms corresponding to the eigenvalues of .
In the case of the electric dipole, the following algebraic properties of the matrices:
(44) 
used in conjunction with the general form (38a) of the free electron propagator greatly reduce the number of integrals that are to be evaluated.
We shall show how these rules work in practice by calculating radiative corrections to the electron and photon propagators. The procedure employed very often in QED relates the full electron and photon propagators to the selfenergy parts. This procedure enables one to go beyond the simplest version of perturbation theory and sum up an infinite (geometric) series. The selfenergy is the sum of contributions from strongly connected diagrams, i.e. the diagrams that cannot be disconnected by cutting only one line. The relations between the full propagators and the selfenergy parts are shown schematically in Fig. 5.
Vi Electron propagator
In the case of the electron propagator the relation between the propagator and the selfenergy part , illustrated in Fig. 5a, reads
(45) 
All three objects that appear in this equation are matrices. The iterative solution of Eq. (45) that shows explicitly the relation between the propagator and the selfenergy part, is
(46) 
This formal geometric series can be summed up to the following compact form:
(47) 
where the inverse is to be understood as the inverse of a matrix. The series (VI) without resummation is meaningless because it is divergent when .
The radiative corrections to the electron propagator in the second order of perturbation theory are represented by the three Feynman diagrams (a)–(c) shown in Fig. 4. The selfenergy parts in this order for the spin system , for the twolevel atom , and for the dipole atom , constructed according to the rules stated in the previous section, have the form
(48a)  
(48b)  
(48c) 
The tadpole diagram (Fig. 4b) does not contribute in the case of the twolevel atom and the dipole atom because and . The analytic expressions for the selfenergy parts obtained by the application of the Feynman rules are
(49a)  
(49b)  
(49c) 
where is the mass renormalization matrix with the eigenvalues and . With the use of the relations (43) and (44) we can replace all matrices by the projectors