QED and Bound Spectra
The quantum electrodynamic theory of bound states started with
the observation of the 2S2P energy level splitting in atomic hydrogen
(Lamb shift).
Boundstate quantum electrodynamics uses concepts from
relativistic atomic physics and quantum field theory. For more than fifty years,
generations of theoreticians and experimentalists have been improving
both our theoretical understanding as well as the laserspectroscopic
experimental techniques, and today's highly sophisticated calculations
would be impossible without the extensive use of computer algebra
systems and parallel computers.
Figure 1: Twoloop corrections to the Lamb shift (Feynman diagrams).

Among the subjects studied by the group are twoloop Bethe logarithms [1] higherorder twoloop effects for excited states [2], and, together with scientists from the National Institute of Standards and Technology, quantum electrodynamic effects for highly excited Rydberg states [3]. A procedure to infer fundamental constants from spectroscopic data is described in Ref. [3]. Fundamental predictive limits of quantum electrodynamics now become relevant for the description of experiments, which gives rise to a very interesting situation [5]. In view of the recently observed discrepancy of theory and experiment in muonic hydrogen (the socalled muonic hydrogen puzzle, see Ref. [5]), the spectrum of muonic hydrogen has been reanalyzed [6] and possible physical explanations have been discussed [7].
One should briefly comment on magnetic interactions in atoms. Traditionally, the calculation of the transition energies among hyperfine centroid frequencies has been emphasized. The nuclearspin dependence of the level energies leads to the hyperfine splitting (for a nonperturbative calculation of the QED correction to hyperfine splitting, see Ref. [8]). The boundelectron g factor is important for the determination of the electron mass. Twoloop corrections to the boundelectron g factor (for 1S and 2S states) have also been investigated, in addition to the Lamb shift, and complete results have been obtained for a term of order α (Z α)^{4} ln[(Z α)^{2}], which contributes to the ratio of the electronic Larmor precession frequency and the cyclotron frequency of a hydrogenlike ion in a trap [9,10]. The result improves the electron mass determination from the carbon and oxygen ion measurements [11].
[1] K. Pachucki and U. D. Jentschura,
Phys. Rev. Lett. 91, 113005 (2003).
[2] U. D. Jentschura, Phys. Rev. A 74, 062517 (2006).
[3] U. D. Jentschura, P. J. Mohr, J. N. Tan, B. J. Wundt,
Phys. Rev. Lett. 100, 160404 (2008).
[4] U. D. Jentschura, S. Kotochigova, E.O. Le Bigot, P. J. Mohr and B.
N. Taylor, Phys. Rev. Lett. 95, 163003 (2005).
[5] R. Pohl et al., Nature 466, 213 (2010)
[6] U. D. Jentschura, Ann. Phys. (N.Y.) 326, 500 (2011)
[7] U. D. Jentschura, Ann. Phys. (N.Y.) 326, 516 (2011)
[8] V. A. Yerokhin and U. D. Jentschura,
Phys. Rev. Lett. 100, 163001 (2008).
[9] K. Pachucki, A. Czarnecki, U. D.
Jentschura and V. A. Yerokhin, Phys. Rev. A 72, 022108 (2005).
[10] J. Verdu et al., Phys. Rev. Lett. 92,
093002 (2004).
HighIntensity Lasers and the FurryVolkov Picture
For a tightly bound electron (close to a highZ nucleus), the socalled Furry picture provides for a suitable description: one integrates the Coulomb interaction into the unperturbed Hamiltonian and formulates the field operator for the Dirac field in terms of bound and continuum states of the DiracCoulomb problem. Today's highintensity lasers are so intense that proverbially, "matter perturbs light" whereas otherwise "light perturbs matter." This situations means that the external, dressing, laser field needs to be incorporated into the unperturbed Hamiltonian. A further complification arises: One would like to calculate Smatrix elements, which necessitates the expansion of the propagators into plane waves of the form exp(ip.x), and therefore, gives rise to socalled generalized Bessel functions which are infinite sums of ordinary Bessel functions and have proven to be notoriously difficult to evaluate.
Figure 2:
Double Compon backscattering in an intense laser field.
The maximum of the scattered photon energy is reached
for direct backscattering, at θ_{b}=0. For nonvanishing
θ_{b}, the energy of the scattered photon decreases,
but multiphoton absorption processes from the laser field
lead to higher harmonics of the onephoton Compton scattering
process.

The DiracVolkov propagator in the Furry picture with respect to a planewave laser field has been written down in Ref. [11]. The calculation of generalized Bessel functions with the help of recursion relations has been described in Ref. [12]. A number of physical processes have been calculated using the DiracVolkov propagator. These include the double Compton scattering in a laser field (see Figure 2), where an electron undergoes a transition from one DiracVolkov state to another DiracVolkov state, emitting a correlated pair of photons [13].
[11] S. Schnez, E. Lotstedt,
U. D. Jentschura, C. H. Keitel, Phys. Rev. A 75, 053412 (2007).
[12] E. Lötstedt, U. D. Jentschura, Phys. Rev. E 79, 026707 (2009)
[13] E. Lötstedt, U. D. Jentschura, Phys. Rev. Lett. 103, 110404 (2009).
Novel States of the Light Field
Novel states of the light field called "twisted photons" are rather interesting objects. The electromagnetic vector potential describing a twisted photon state adds the orbital angular momentum of the photon to the spin angular momentum of the vector (spin1) field. In some sense, the twisted wave function interpolates between a planewave vector potential and a photon described by a vector spherical harmonic (in the standard multipole decomposition). Indeed, a planewave photon whose vector potential describes a photon propagating in a specific direction, say, the z direction. It has zero expectation value for the projection of the orbital angular momentum onto the propagation axis (z axis). A photon described by a vector spherical harmonic has defined values for the square of the angular momenta but does not have a defined propagation direction.
Figure 3:
Wave function of a twisted photon with a phase of the
vector potential component being indicated by color grading.

Twisted photons combine, in some sense, the properties of planewave photons and those described by vector spherical harmonics: Namely, they have a defined propagation direction (say, the z axis) and still, large angular momentum projections onto that same propagation axis. In constructing vector spherical harmonics, one adds the orbital angular momentum from the spherical harmonics to the spin angular momentum, using ClebschGordan coefficients. However, one can also add the orbital angular momentum to the spin angular momentum via a conical momentum spread (in momentum space) multiplied by an angledependent phase, or by a Bessel function in the radial variable (in coordinate space). This leads to twisted states.
Figure 3 shows a plot of a twisted photon vector field. The rainbowscale color variation of this potential indicates complex phases, which are superpositions of terms proportional to the azimuthal angle of the transverse momentum. The frequency upconversion of twisted photons by Compton backscattering discussed in Ref. [14,15] should open new experimental possibilities for fundamental studies in atomic and nuclear physics.
[14] U. D. Jentschura, V. G. Serbo, Phys. Rev. Lett. 106, 013001 (2011).
[15] U. D. Jentschura, V. G. Serbo, Eur. Phys. J. C 71, 1571 (2011).
Generalized Nonanalytic Expansions
Normally, physiscists think in terms of power series when they want to describe a physical phenomenon. A proverb probably invented by graduate studies is that during graduate school, one learns that every formula investigated during undergraduate studies is nothing but the start of a Taylor series. A classical quantummechanical problem consists in a harmonic oscillator perturbed by a quartic term (anharmonic oscillator). The perturbation is of the form g q^{4}, where g is the coupling, and q is the coordinate. A perturbation series for the energy of a typical state, expressed in powers of the coupling g, diverges factorially in higher order no matter how small g is. Indeed, this remarkable phenomenon is intimately connected with the fact that for g < 0, the perturbation g q^{4} actually is unstable, and the resonance energies become complex. Conversely, in the path integral representation of the partition function, one finds nontrivial saddle points, which correspond to instanton trajectories of a classical particle travelling between the maxima the inverted potential (see Figure 4).
Figure 4: Instanton configurations in an inverted quartic potential
for negative coupling. The particle starts at the central maximum
and travels either to the right or to the left, before making its way back
to the central maximum.

Using an expansion about the nontrivial saddle points of the Euclidean action, one can then derive a few interesting higherorder formulas, and verify these against highprecision numerical calculations. Eventually, it has been found that a complete description of the resonance energies of anharmonic oscillators requires a fundamental generalization of the concept of a power series in the coupling parameter g: namely, one has to allow for powers, logarithms, and nonperturbative exponentials exp(A/g) of the coupling parameter, where A is the instanton action. Generalized nonanlytic expansions, that involve all the mentioned expressions, have been used for highprecision numerical evaluations [16,17,18], and also, for the test of generalized BenderWu formulas that have been derived for odd anharmonic oscillators [19].
[16] J. ZinnJustin, U. D. Jentschura, Ann. Phys. (N.Y.) 313, 197 (2004).
[17] J. ZinnJustin, U. D. Jentschura, Ann. Phys. (N.Y.) 313, 269 (2004).
[18] U. D. Jentschura, J. ZinnJustin, Phys. Lett. B 596, 138 (2004).
[19] U. D. Jentschura, J. ZinnJustin, Phys. Rev. Lett. 102,
011601 (2004).
Renormalization Group and Critical Phenomena
The idea of the renormalizationgroup (RG) theory was born when GellMann and Low observed that the shape of the vacuumpolarization induced charge distribution about a particle, for small distances, does not depend on the mass of the screening quantum field. Due to the vacuumpolarization effect, the effective charge felt by a probing particle depends on the distance from the "bare" charge of the screened particle (the coupling "runs"). Because the shape of the vacuumpolarization charge distribution is virtually independent of the mass of the screening virtual quantum field, the rate of "running" of the coupling is a function of the particular value of the coupling itself, at a particular distance: It is thus possible to formulate a socalled ψ=ψ(g) function which describes the running of the coupling. Due to the WardTakahashi identity, it is possible to isolate the running of the coupling from the wave function and mass renormalizations in QED.
If one uses the approach for general field theories, the calculations become more complex: One has to formulate the RG for an arbitrary Green function [or a oneparticle irreducible (1PI) generating functional], including wave function (field) renormalizations. The field renormalizations describe the behaviour of the Green functions at large distances. Near a fixed point of the RG transformation, they describe the "anomalous" exponent of the correlation function that describes its variation with distance. The mathematical form of the correlation differs from a simple integer power law. Therefore, the RG functions describing the wave function renormalizations are called the "anomalous dimensions." These quantities are relevant for the description of, e.g., O(N)symmetric φ^{4} theories. The universality classes of these theories are important for a number of observed secondorder phase transitions at low, but finite temperature, such as the liquidvapour transition of ^{3}He or the superfluid transition of ^{4}He. The Euclidean actions of these theories have the same structure as those used for the anharmonic oscillators, and possible connections may be explored.
Theories with periodic selfinteractions cannot be renormalized in the perturbative approach that is available for O(N)symmetric φ^{4} theories. In a series of papers [2023], we have analyzed the O(N)symmetric theories with interaction terms of the form cos(b.φ), with a special emphasis on two spatial dimensions (2D sineGordon model). The interaction contains terms of arbitrarily high power in the field variable, as a result of the expansion of the periodic cosine term. The twodimensional sineGordon model undergoes a phase transition (the KosterlitzThoulessBerezinskii phase transition) which is characterized by the binding/unbinding of topological excitations (vortices). It belongs to the same universality class as the 2D XY model and the 2D Coulomb gas. Vortices are free above the critical temperature (negative free energy), whereas they are bound below the critical temperature. The WegnerHoughton equation has been used in order to infer the renormalizationgroup flow of the models [20]. The critical value for the b parameter characterizing the transition was identified and compared within various blocking schemes that integrate the momentum shells in the Polchinski and WegnerHoughton approaches.
Figure 5: RG flow of the coupling in a layered sineGordon model.
The coupling becomes relevant/irrelvant in the infrared
depending on the temperature parameter b_{c}.
The critical value of b_{c} depends on the number N
of superconducting layers.

In hightemperature superconductors, it is generally assumed that the conduction proceeds by the formation of tightly bound vortices. The layered crystal structure of the hightemperature superconductors suggests that it might be indicated to invoke a layered sineGordon model, where the different layers are coupled by interaction terms. The layerdependence of the interaction was first analyzed in [20]. We have found that for N magnetically coupled layers, the transition temperature varies as [1  1/N] with the number of layers [22,23]. A fieldtheoretical analysis has been presented in Ref. [21]. The theory of phase transitions in twodimensional systems seems to be not very well developed, and it has even been a considerable challenge to identify the parameters of the model with the physical quantities such as the transition temperature. The papers [22,23] seem to attract attention within the solidstate physics community after an initial period of neglect.
[20]
I. Nandori, S. Nagy, K. Sailer, U. D. Jentschura, Nucl. Phys. B 725, 467 (2005).
[21]
U. D. Jentschura, I. Nandori, J. ZinnJustin, Ann. Phys. (N.Y.) 321, 2647 (2006).
[22]
I. Nandori, U. D. Jentschura, S. Nagy, K. Sailer, K. Vad, S. Meszaros,
J. Phys.: Condens. Matter 19, 236226 (2007).
[23]
I. Nandori, K. Vad, S. Meszaros, U. D. Jentschura, S. Nagy, K. Sailer,
J. Phys.: Condens. Matter 19, 496211 (2007).
Lamb Shift in Laser Fields
The combination and unification of ideas from intensefield quantum optics and quantum electrodynamics demands a thorough analysis of the interplay of the strong driving of an atom by a laser field (Rabi oscillations) and the vacuum fluctuations. In [24,25], an atom with two relevant energy levels driven by a strong nearresonant monochromatic laser field is studied as the easiest model system for the above problem.
Figure 6: Schematic diagram (left) showing the coupling of the
bare atomic states and the Fock states to the laserdressed states. The
transitions among the dressed states, shown in the right figure, lead
to the red and bluedetuned Mollow sideband peaks.

As is well known within quantum optics, the incoherent part of the resonance fluorescence spectrum emitted by this system is known as the Mollow spectrum. The atomlaser coupling strength is characterized by the the Rabi frequency which is proportional to product of the transition dipole matrix element and the laser field strength. In steady state, the system is described by quasienergy eigenstates known as the laserdressed states which are shown in Figure 6. They are displaced from the bare atomic states by one half of the generalized Rabi frequency Ω_{R} = (Ω^{2} + Δ^{2})^{1/2}, where Ω is the Rabi frequency and Δ is the detuning. The incoherent spectrum generated by the manyphoton processes, in steady state, can then be interpreted in a natural way in terms of transitions among the laserdressed states.
Figure 7: The self energy of a laserdressed bound electron
(Feynman diagram, left figure) shifts the incoherent fluorescence
spectrum (known as the Mollow spectrum) from the black to the red curve
(right figure). The electron propagator in the left figure is actually
doubledressed, (i) by the Coulomb field of the nucleus (as indicated
by the double line, and (ii) by the laser field which is indicated by a
wavy line. The zero on the abscissa of the right figure is fixed to the
laser frequency. The ordinate axis gives the intensity distribution of
the incoherent spectrum (in arbitrary units).

When evaluating radiative and relativistic corrections to the Mollow spectrum, it is natural to start the analysis from the dressedstate basis as opposed to the unperturbed atomic barestate basis. At nonvanishing detuning and nonvanishing Rabi frequency, a treatment starting from the dressedstate basis leads to additional correction terms, which may be summarized in a modified Mollow sideband formula (see Figure 7) including relativistic and radiative corrections, which entails both a modification of the Rabi frequency as well as a corrections to the transition dipole moment. The corrections to the Rabi frequency include some nontrivial effects which can only be understood if the problem is treated right from the start in the dressedstate basis, apart from relativistic and radiative corrections to the transition dipole moment, fieldconfiguration dependent corrections, higherorder corrections to the selfenergy, and corrections to the secular approximation. The corrections to the detuning comprise the bare Lamb shift, BlochSiegert shifts, and offresonant radiative corrections. All of these effects are described in detail in [13], and a schematic view of their consequence for the Mollow spectrum is given in Figure 7.
[24] U. D. Jentschura, J. Evers,
M. Haas, C. H. Keitel, Phys. Rev. Lett. 91, 253601
(2003).
[25] U. D. Jentschura, C. H. Keitel, Ann. Phys. (N.Y.) 310, 1
(2004).
Summary of Research Activities
Within the years 19962011, in total 150 refereed physics research articles have been published in Physical Review A, B, C, D, E and Letters, and Special Topics, as well as Journal of Physics A, B, G, and Journal of Physics: Condensed Matter.
Acknowledgments
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Points of Interest
Recent Events
Textbook published.
A textbook on advanced classical electrodynamics has been published.
Longrange van der Waals interaction tails.
Research has been published in Physical Review Letters.
Research on longrange interactions.
Two longer articles have been published in Physical Review A, on van der Waals and CasimirPolder interactions.
Textbook on classical electrodynamics.
A textbook on classical electrodynamics has been submitted to World Scientific Publishers.
Chandra Adhikari: Schaerer Prize.
Graduate student Chandra Adhikari has received Third Prize at the annual Schaerer Prize Competition.
Textbook on classical electrodynamics.
A project concerning a textbook on classical electrodynamics is nearing completion.
Missouri Research Board grant.
A grant from the Missouri Research Board will help in research projects concerned with general relativity.
Faculty Research Award.
Ulrich Jentschura receives a faculty research award of Missouri University of Science and Technology.
Jonathan Noble: Schaerer Prize.
Graduate student Jonathan Noble has won this year's Schaerer Prize Competition (First Prize).
Graduate Student Seminar
Students Jonathan Noble and Chandra Adhikari won First and Third Prize in the graduate student seminar series for the spring semester 2015.
Promotion to Full Professor
The promotion to full professor has been approved and taken effect in September 2015.
Gravitational Correction to Vacuum Polarization
An article on the interplay of general relativity and quantum electrodynamic effects has appeared in print.
FineStructure Constant and String Theory
An article on the determination of the finestructure constant, the renormalization group and string theory has appeared at arXiv.org.
Conference at CERN
A conference at CERN discusses generalized perturbative expansions (socalled transseries) which are useful in the description of quantum phenomemna (energy levels and nonperturbative effects in field theory).
J. Phys. G: LabTalk
A recent particle physics paper in J.Phys.G on the "rabbit paradox" has been discussed in a LabTalk.
Faculty Research Award.
Ulrich Jentschura receives a faculty research award of Missouri University of Science and Technology.
Jonathan Noble (news #2).
Joint work with graduate student Jonathan Noble has been accepted for publication. A preprint on the "FoldyWouthuysen Transformation, Scalar Potentials and Gravity" will appear at arXiv.org soon.
Jonathan Noble (news #1).
Graduate student Jonathan Noble has been awarded a recognition at the annual Schaerer Prize Competition (Third Prize).
APS Fellow.
An APS fellowship has been received in recognition of the contributions to theoretical atomic physics (Division of Atomic, Molecular and Optical Physics, APS).
Invited Talk at Melbourne University.
Invited talk at the School of Physics of Melbourne University on July 30, 2013.
Invited Talk at PHHQP conference.
Invited talk at the PHHQP conference at Koc University on July 2, 2013.
Invited Talk at APS meeting.
Invited talk Q5.00001 at the April meeting of the American Physical Society in Denver, Colorado.
Tenure decision.
Ulrich has been granted tenure and promoted to Associate Professor. Finally.
Three entangled photons from QED.
A paper on triple Compton scattering (generation of highenergy entangled photons) has been published.
General Relativity and Relativistic Quantum Mechanics.
A paper on the Dirac equation in curved spacetime, and antimatter has been accepted for publication in Phys. Rev. A.
Distinguished Guest Scientist of the Academy Institute in Debrecen, Hungary.
In an interview with the Hungarian Academy, Ulrich tried to argue for the role of science in society.
Quantum Physics with NonHermitian Operators.
A paper on the tachyonic Dirac equation has been published in a special issue of J. Phys. A: Math. Theor. on nonHermitian quantum dynamics.
Triplet States in Helium and g Factor.
A paper on the quantum electrodynamic corrections to the g factor of helium triplet states has been published in Physical Review A.
Physical Review Letter.
A paper triple Compton scattering has been published; it constitutes a theoretical verification of an experiment for which no theory has been available for more than 40 years.
Neutrino Physics Paper.
A paper on neutrinoless double beta decay has been accepted for publication in a particle physics journal.
Faculty Excellence Award.
An award has been recived to honour continued excellence in teaching and research.
NSF Grant.
The National Science Foundation has awarded a grant entitled "Advanced Computational Physics in Atomic and Laser Science".
Research Paper.
A paper about higherorder corrections to the muonic hydrogen spectrum has appeared in Physical Review A. These include radiativerecoil effects.
Collaboration with NIST.
The 150th publication of the group has appeared in print. The paper results from a collaboration with the National Institute of Standards and Technology and treats blackbody radiation corrections to the dynamic polarizability of helium.
Matter 'n Motion.
The 2011 newletters of the faculty is out and may be downloaded.
PRL Title Page.
An article about novel states of the radiation field has appeared on the title page of Physical Review Letters.
PRA Editorial Advisory Board.
As of 01JAN2011, Ulrich Jentschura has been appointed as a member of the Editorial Advisory Board of Physical Review A.
Faculty Research Award.
Ulrich Jentschura receives a faculty research award of Missouri University of Science and Technology.
Schaerer Prize.
Benedikt Wundt has been awarded the Schaerer Prize 2010 by the physics faculty of Missouri S&T for the most interesting graduate research project.
Outstanding Referee Certificate.
Ulrich Jentschura has been awarded an Outstanding Referee Certificate by the American Physical Society.