Entangled Photon-Electron States and the Number-Phase Minimum Uncertainty States of the Photon Field

The exact analytic solutions of the energy eigenvalue equation of the system consisting of a free electron and one mode of the quantized radiation field are used for studying the physical meaning of a class of number-phase minimum uncertainty states. The states of the mode which minimize the uncertainty product of the photon number and the Susskind and Glogower (1964) cosine operator have been obtained by Jackiw (1968). However, these states have so far been remained mere mathematical constructions without any physical significance. It is proved that the most fundamental interaction in quantum electrodynamics - namely the interaction of a free electron with a mode of the quantized radiation field - leads quite naturally to the generation of the mentioned minimum uncertainty states. It is shown that from the entangled photon-electron states developing from a highly excited number state, due to the interaction with a Gaussian electronic wave packet, the minimum uncertainty states of Jackiw's type can be constructed. In the electron's coordinate representation the physical meaning of the expansion coefficients of these states are the joint probability amplitudes of simultaneous detection of an electron and of a definite number of photons. The joint occupation probabilities in these states preserve their functional form as time elapses, but they vary from point to point in space-time, depending on the location of the detected electron. An analysis of the entanglement entropies derived from the photon number distribution and from the electron's density operator is given.


Introduction
Entanglement and non-locality in quantum mechanics have first been discussed by Einstein, Podolsky and Rosen (1935), and their main conclusion was that quantum mechanics is not a "complete theory", because not every "elements of physical reality" have a counterpart in the theory. As Bohm (1951) writes in his book at the beginning of Section 22.15, "Their critisism has, in fact shown to be unjustified [see Bohr (1935)], and based on assumptions concerning the nature of matter which implicitly contradict the quantum theory at the outset." Motivated by the above work of Einstein, Podolsky and Rosen (EPR), Schrödinger (1935a-b-c) presented a detailed study of the conceptual aspects of quantum mechanics. In this series of papers he introduced the famous "Schrödinger cat" and the concept of "entanglement" ("Verschränkung" in Schrödinger's terminology). In his book in Section 22.15, Bohm (1951) analyses the "EPR-paradox" in detail by considering a desintegration of a quantum system (a molecule having initally zero spin angular momentum) consisting of two spin-1/2 atoms, and detemines the correlations of the spin directions observed at spatially separated detectors. The first reliable experiments, proposed by Wheeler (1946) in this contex, were carried out by Wu and Shaknov (1950), in which they measured coincidence counting rates at different relative azimuths of the polarization of two gamma rays, stemming from electron-positron pair annihilation, and detected by two opposing scintillation counters. They found that the counting rates of perpendicular polarization were two times larger then the rates of parallel polarization. In the optical regime, the first experimental realization of the "Einstein-Podolsky-Rosen-Bohm Gedankenexperiment" have been achieved much later by Aspect et al. (1982a-b). They measured the linear-polarization correlation of pairs of photons emitted in a radiative cascade of calcium, and found excellent agreement with the quantum mechanical predictions, and the greatest violation of generalized Bell's inequivalities at that time. Concerning Bell's inequivalities see e.g. the references in Aspect et al (1982a-b) and Wigner (1970) and the references therein. In the meantime it turned out that entanglement plays a crucial role in the nowadays rapidly developing branches of quantum physics and informatics, namely in quantum information theory (see e.g. Alber et al 2001, Bouwmeester et al 2001and Stenholm and Suominen 2005 and in quantum computing and quantum communication (see e.g. Williams 1999 andNielsen andChuang 2000).
In the above-mentioned examples the entangled particles are of the same sort. In the present paper we shall discuss entanglement between photons and electrons. It will turn out that the entangled photonelectron states, to be constructed in Section 4, have a close connection with the "critical states" introduced by Jackiw (1968), which minimize a number-phase uncertainty product of the photon field. That is why, concerning the problem of the phase operator of a mode of the quantized electromagnetic radiation, we think that, for the sake of completeness of the present paper, it is instructive to give a brief summary of the basic references dealing with this subject.
In his path-breaking paper on the quantum theory of emission and absorption of radiation Dirac , respectively, where, in his notation r is the mode index, h is Planck's constant divided by π 2 and * denotes hermitian conjugation. The number (action) operators r N and the canonically conjugate angle operators r θ are assumed to satisfy the Heisenberg commutation relation, and, as a consequence, . We note here that in the present paper we shall use the following notations for one mode: , as will be discussed in more details in Section 2. In the same year when Dirac's mentioned paper appeared, London (1927) published his study on the angle variables and canonical transformations in quantum mechanics. He proved that though the ladder operators E and + E have a well-defined matrix representation, they cannot be expressed as an exponential of the form Φ ±i e , where Φ would be a hermitian matrix. It is sure that Dirac was aware of this discrepancy. According to Jordan (1927), in a conversation with him, Dirac remarked that the possibility to derive many correct results by using the formal relation . At this point let us note that the above-discussed problem of the quantum phase variable does not show up in the case of quantization of the canonically conjugate pair angle and orbital momentum of a planar motion (because the spectrum does not terminate at zero angular momentum), as is illustrated in the extensive and thorough study by Kastrup (2006b), appeared recently.
The non-existence of a hermitian phase operator of a harmonic oscillator was rediscovered by Susskind and Glogower (1964). They introduced the hermitian "cosine" and "sine" operators, whose basic properties will be briefly summarized in Section 2 of the present paper. In their extensive review paper on phase and angle variables in quantum mechanics Carruters and Nieto (1968) derived a couple of numberphase uncertainty relations by using the cosine and sine operators, and Jackiw (1968) constructed a "critical state" which minimizes one of these uncertainty products. Garrison and Wong (1970) constructed a quantum analogon of the classical periodic phase function (saw-tooth), which satisfies the Heisenberg commutation relation with the number operator on a dense set of the Hilbert space of the oscillator. Moreover, they have constructed the eigenstates of this periodic phase operator. In our opinion this was the first mathematically correct approach toward the solution of the original problem of quantum phase. Paul (1974) has proposed an alternative description of the phase of a microscopic electromagnetic field, and discussed the possibilities of its measurement.
A new impetus was given to the study of the quantum phase problem after the papers of Pegg and Barnett (1989) appeared. They truncated the state space of the harmonic oscillator, and were able to construct a hermitian phase operator on this finite-dimensional Hilbert space. We mention that the possibility of using a finite-dimensional (truncated) Hilbert space in this context has already been discussed by Jordan (1927). The approach of Pegg and Barnett (1989) was refined by Popov and Yarunin (1992). The limit matrix elements of the phase operator in number representation (as we let the dimension of the Hilbert space going to infinity) obtained by these authors have already been presented by Weyl (1931). We note that, seemingly, none of the above authors, publishing their papers since the sixties of the last century, had known about the fundamental early works of London (1926 and. The phase distribution of a highlysqueezed states has been determined by Schleich et al (1989) (where the reference to London's work first appeared in the modern era) by using the quantum phase-space distribution (Wigner function) of the quantized mode (see also the book by Schleich 2001, in particular Chapters 8 and 13). The problem of quantum phase measurements has been discussed by Shapiro and Shepard (1991), partly on the basis of "normalizable phase states". The question of operators of phase has been thorougly analysed by Bergou and Englert (1991) both from the formal point of view and from the physical point of view. In a series of papers Noh et al (1991Noh et al ( , 1992aNoh et al ( -b and 1993 have studied both theoretically and experimentally the quantum phase dispersion on the basis of their operationally defined cosine and sine operators. In their scheme these definitions are based on measured photon number counts in an eight-port interferometer. Freyberger and Schleich (1993) have performed an analysis of a similar phase operator along with the experiment by Noh et al (1991) by using radially inegrated phase-space distributions. In this context see also the thoroughly written dissertation by Freyberger (1994), and references therein. In the meantime an ample literature has been accumulated concerning the quantum phase problem. For further reading and references we refer the reader to the topical issue of Physica Scripta, edited by Schleich and Barnett (1993), in which also some historical aspects are summarized by Nieto (1993). See also the critical review by Lynch (1995) and the book by Peřinová et al (1999) on the description of phase in optics. Concerning the recent developments of the concept of quantum phase of a linear oscillator, see the thorough group theoretical studies by Kastrup (2003Kastrup ( , 2006aKastrup ( and 2007, in which a genuinely new approach to this problem has been worked out. In the present paper it is proved that the most fundamental interaction in quantum electrodynamics (QED) -namely the interaction of a free electron with a mode of the quantized radiation field -leads quite naturally to the generation of the above-mentioned number-phase minimum uncertainty states. We emphsize that here we are merely dealing with non-relativistic quantum mechanics, where the interaction of the electron with the quantized mode is represented by the minimal coupling term between a free charged particle and an oscillator. The analysis to be presented here is restricted to the study of the interaction of one Schrödinger elctron with one quantized mode of the radiation field. To neglect the interaction with other modes is justified by that we assume a very highly occupied single mode. Thus, in fact, we are not using complete field operators used in the very quantum electrodynamics. In Section 2 we briefly summarize the the basic properties of the Susskind and Glogower (1964) "cosine" and "sine" operators, and we give the associated number-phase uncertainty relations and present the "critical state" found by Jackiw (1968), which minimizes one of the uncertainty products. In Section 3 we present the exact stationary solutions of the photon-electron system, in which the interaction is taken into account up to infinite order. In Section 4 we shall construct the entangled photon-electron states on the basis of these stationary states. It will be shown that the entangled photon-electron states developing from a highly excited number state due to the interaction with a Gaussian electronic wave packet have the same functional form as the "critical states" derived by Jackiw (1968). In Section 5 we derive the reduced density operators of the photon and of the electron. On the basis of these reduced density operators various entanglement entropies are calculated. In Section 6 a short summary closes our paper. The mathematical details of the derivation of our results are presented in the Appendices A and B.

The number-phase minimum uncertainty states of Jackiw
The number-phase uncertainty product (in contrast to the usual Heisenberg uncertainty products, which are valid e.g. for the variances of the Cartesian components of the momentum and position of a particle) (1) cannot have a well defined mathematical meaning for a generic state of a quantized mode of the electromagnetic radiation. This is because Φ itself cannot be represented by a matrix (or operator), as London (1927) has already shown long ago. Equation (1) would be valid if there would exist a Heisenberg commutation relation , [ for the number operator N and for the phase operator Φ , which is not the case here. That is the reason for why Carruthers and Nieto (1968) proposed other uncertainty products given in terms of the C ("cosine") and S ("sine") operators introduced by Susskind and Glogower (1964), 2 / ) ( (2) which are well-defined operators. Here E is the so-called "exponential phase operator" defined by the "polar decomposition of the photon absorption operator A " (which would be the quantum analogon of the polar decomposition of a complex number, ( We note that the "exponential phase operator", the ladder operator E has been originally introduced by London (1926) and used by Jordan (1927) where 0 0 0 ≡ P is the projector of the vacuum state of the mode, for which 0 0 = A . As we see, the "cosine" and the "sine" operators C and S , respectively, do not commute, because they cannot be expressed in terms of exponetials of a common (hermitian) operator Φ in the form Φ ±i e . The reason for that is the "exponential phase operator" E , introduced in Eq. (3), is not unitary but only "half-unitary" (called "partially isometric" in mathematical terminology, see e.g. Riesz and Szőkefalvi-Nagy 1965, Sections 109 and 110). Really, 1 = +

EE
holds, but, on the other hand, , and, moreover, as a consequence of the half-unitary property of E the sum of the squares the "cosine" and "sine" operators is not equal to unity, 1 2 / 1 0 . We mention, that for large coherent excitations of the mode, the moments of C and S have a similar form of the moments of the ordinary c-number cosine and sine functions. We have to note here that Kastrup (2006a) has recently raised serious objections against the use of the Susskind and Glogower cosine and sine operators in the description of quantal phase properties of the linear oscillator. On the basis of the analysis presented in Chapter 5 of his paper, he concludes that "the London-Susskind-Glogower operators k C and k S are not appropriate for measuring angle properties of a state!". We would like to emphasize, that in the present study we are not concerned with the question whether the operators C and S , defined in Eq. (2), are suitable or not suitable to characterize the quantal phase properties. We merely show that states of essentially the same mathematical structure as that of the "minimizing states" constructed by Jackiw (1968), may be generated in nonpertubative photon-electron interactions in the strong field regime. Thus, we shall not discuss the (questionable or non-existing) physical relevance of C and S themselves in the context of the problem of quantal phase.
The uncertainy products associated to the above commutation relations, Eq. (4), are the following (Carruters and Nieto 1965 and1968) where I n is a modified Bessel function of first kind of order n (see the definition in Gradshteyn  . We have found that this requirement is a consequence of the second theorem of Hurwitz on the zeros of Bessel functions (see Watson 1944, Section 15. 27). The states which allow ) ( 2 Ψ U to reach ¼ can also be constructed, by using the same method. Jackiw (1968) has noted on the states given by Eq. (6) that, "unfortunately these states do not seem to have any physical significance". In the present paper we will show that states of the same structure as that of Ψ naturally appear in the non-perturbative analysis of the simplest interaction of QED (namely, the interaction of a free electron with a quantized mode of the electromagnetic radiation). Thus, on the basis of our analysis, we may say that the states to be constructed below, have a fundamental significance.

Exact energy eigenstates of the interacting photon-electron system
In order to make our paper self-contained, in the present section we briefly summarize the basic steps towards the determination of the exact energy eigenstates of the interacting photon-electron system. We mention that the interaction of electrons with a quantized electromagnetic field within a conducting enclosure has been treated by Smith (1946), but he used perturbation theory, and then rate equations, to treat higher order processes. In his pioneering work on the connection of communication theory and quantum physics, Gabor (1950) also studied a similar system (the transit of electrons in a wave guide), though he used semiclassical pertubation theory and a different geometry.
Let us consider the energy eigenvalue equation of the joint interaction of a quantized mode of the radiation field with a Schrödinger electron. For sake of simplicity, we take for the mode a circularly polarized plane wave in dipole approximation. In this case we do not get squeezing in the stationary states, because the interaction coming from the the 2 A term of the Hamiltonian is diagonal. The complete discussions for a Schrödinger electron and for a Dirac electron have been published by Bergou and Varró (1981a-b) and by Bersons (1981) long ago, and have been applied to determine non-perturbatively the crosssections of multiphoton Bremsstrahlung and multiphoton Compton scattering. Concerning the question of squeezing in photon-electron systems see e.g. Bergou and Varró (1981a), Ben-Aryeh and Mann (1985) and Becker et al (1987). We will consider here only the (simplest) Schrödinger case, and study the interaction with a circularly polarized mode in dipole approximation. The energy eigenvalue equation now reads where the vector potential is given as ) ( being the complex polarization vector (for right circular polarization, when the field is assumed to be perpendicular to the z-direction), ω is the circular frequency of the mode and 3 L is the quantization volume.
is the bare field energy. e − , m and c have their usual meaning; the elecron's charge and mass, and the velocity of light in vacuum, respectively. h denotes Planck's constant divided by π 2 . In Eq. (7) 0 ,n p r ψ are exact stationary states of the interacting photonelectron system characterized by two quantum numbers p r (the electron's momentum), 0 n (a non-negative integer, which, by switching-off the interaction, reduces to the initial photon occupation number (9) Notice that p ω is formally nothing else but the plasma frequency for an electron density 3 / 1 L . In obtaining Eq. (9) we have taken into account that 0 = ⋅ ε ε r r , 0 = ⋅ * * ε ε r r and 1 = ⋅ * ε ε r r . The linear interaction term on the right hand side of Eq. (9) can be easily transformed out from the eigenvalue equation, Eq. (7), by applying the following displacement operator with a properly choosen parameter σ We note that the displacement operators of the form displayed by Eq. (10) have an important role in the quantum theory of optical coherence and coherent states, as was first shown by Glauber (1963a-b) in his path-breaking papers. Such displacement operations were also used much earlier by Bloch and Nordsieck (1937) in their fundamental study of the problem of infrared divergeces in QED, in order to transform out the interaction terms. By applying the displacement operation we receive a transformed Hamiltonian which is diagonal in both the electron and the photon variables, hence its eigensolutions can be written down as simple products of the type n p r , where p r is a momentum eigenstate of the electron. Accordingly, we obtain the eigensolutions of the original Hamiltonian, Eq. (9), in the form where the energy eigenvalues can be brought to the form can in principle be negative (if 1 2 / 2 2 > ω ω p ), thus the total energy of the system can also be negative in certain parameter range, which would mean a sort of "attractive interaction" ("bound states") of the mode and of the electron. On the other hand, according to the definition of the one-electron plasma frquency in Eq. (9), for a large enough quantization volume 3 L , ω ω << p , thus ⊥ m practically equals to the original bare mass m . We shall not discusse this question any further in the present paper. For simplicity, in the following we will always assume that 1 2 / 2 2 < ω ω p , thus the "transverse mass" ⊥ m is positive. It is clear that if the ratio 2 2 2 / ω ω p approaches 1 from below, then ⊥ m can be much larger then the bare mass m of the electron.
For later convenience we rewrite Eq. (12) in the form In order to simplify the notation, in Eq. ( We note that, owing to the unitarity of the displacement operators, Eq. (10), the exact solutions given by Eq. [σ , given on the right hand side of Eq. (11), is governed by the matrix elements where s n L denote generalized Laguerre polynamials (for the definition of them see e.g. Gradshteyn and Ryzhik 2000, formula 8.970.1). To our knowledge, the matrix elements of the type given in Eq. (16), was first published in the work by Bloch and Nordsieck (1937), which we have already quoted before. Later Schwinger (1953) derived such matrix elements in one of his famous series of papers on the theory of quantized fields, and they also appear in his study on the Brownian motion of a quantum oscillator (Schwinger 1961). For some further details see e.g. Bergou and Varró (1981a-b). The expectation value of the photon number k , and its variance can be calculated, on one hand, directly from Eq. (16), or, on the other hand, by using the displacement properties (17)

Entangled photon-electron states
In the present section it is proved that the interaction of a free electron with a mode of the quantized radiation field leads to the generation of the number-phase minimum uncertainty states discussed in Section 2. It is shown that the entangled photon-electron states developing from a highly excited number state due to the interaction with a Gaussian electronic wave packet have the same functional form as the minimum "critical states" found by Jackiw (1968). In the electron's coordinate representation the expansion coefficients of these states are expressed in terms of modified Bessel functions of first kind (as has been shown in Eq. (6)) whose argument now depends on the electron's coordinate . The photon statistics of these states preserve their functional form as time evolves, but the occupation probabilities depend on the spatiotemporal position of the electron's detection. We note that on this subject preliminary results have already long been presented by us (Varró 2000), but we have not published them until now. According to Eqs. (10), (11) and (14), only the transverse motion of the electron couples to the radiation field, thus the longitudinal motion is merely a free propagation. In the following we shall not discuss any further this longitudinal dynamics, but, rather, we concentrate on the study of the transverse part of the wave packet dynamics, which represents in our case the interaction of the electron and the quantized mode of the radiation field. The entangled photon-electron states developing from a number state due to the interaction with an electronic wave packet have the form where g has been specialized to a Gaussian weight function, and ) (t ⊥ ψ was introduced in Eq. (15). In Eq. (18) we have introduced the transverse width w of the electronic wave packet (electron beam). The physical situation to which the state given by Eq. (18) may be associated is the following. Let us imagine that an electron is injected into a cavity at time 0 = t through a small hole of with w . On the basis of our earlier study of the true initial value problem (Bergou and Varró, 1981a), we expect that the sudden coupling of the electron with the (highly occupied) cavity mode, results, in essence, in the formation of the state ψ defined by Eq. (18). In the present paper we restrict our analysis to the study of the spatio-temporal evolution of these approximate states (which are entangled already at 0 = t ). Owing to the unitarity of the displacement operator D in Eq. (15), the superposition ψ defined by Eq. (18) is a normalized state in the product space of the photon-electron system. In order to have an explicit form of this state, we express it in the electron's coordinate representation, and, at the same time, expand it in terms of the photon number eigenstates ψ r k n k n t r n k The summation index in the above equation has been shifted merely for the sake of later convenience. The normalization condition in Eq. (19) In Eqs. (19) and ( 7)), but henceforth, in the present paper, we shall only discuss cases of large 0 n values, and use the approximation stemming from Eq. (A.15). After the integration with respect to the azimuth angle χ in momentum space we obtain In Eq. (21) Here u denotes the energy density of the mode, with ) cos sin being the electric field strength. According to Eq.
(A.18), we obtain from Eq. (21) the limit form in case of high initial occupation numbers, where k I is a modified Bessel function of first kind of order k , and In Eq. (23) we have defined the "dimensionless intensity parameter" μ , whose numerical value can be express in terms of the intensity I of the mode of the radiation field measured in W/cm 2 , and of the photon energy ph E measured in eV. We have also introduced the amplitude of the electric field strength and the wavelength λ of the radiation. The approximate equalities in Eq. (23) are valid for large L . If we let both 0 n and L going to infinity, in such a way that the photon density is a fixed parameter, then the last term on the right hand side of Eq. (22) can be supressed, and μ can formally be associated to a classical electric field of amplitude 0 F . Then in Eq. (22) π μλ μ 2 / → Λ becomes just the amplitude of oscillation of a classical electron under the action of the electric field of the radiation ) cos sin ( This can easily be shown by solving the Newton equations . Thus, the dimensionless quantity w w π μλ μ 2 / / ≈ Λ is the ratio of the amplitude of the classical oscillation of the electron to the initial transverse width at 0 = t of the electron packet (electron beam). We emphasize that the above remarks were made simply to outline a rough picture in order to give a physical background of the parameters introduced in Eqs. (21) and (23). Of course, we are not saying that a classical electric field can be associated to an even very highly occupied number state. This can consistently be done by using the Schrödinger-Glauber coherent states (Glauber 1963a-b). Anyway, our preliminary investigations on this latter subject clearly show that parameters of a similar sort naturally appear there, too, thus these parameters are allowed to be used in realistic numerical estimates. The time scale parameter τ defined in Eq. (23) can be related to the period ω π / 2 = T of the radiation field through the "bare time scale parameter" The "transverse mass" ⊥ m , defined in Eq. (13), can in principle be much larger than the "bare mass" m , if 2 2 2 / ω ω p approaches (from below) 1. Consequently, the transverse spreading of the electronic wave packet can in principle be reduced due to the interaction with the electromagnetic radiation. From Eq. (19), by neglecting the term of order (22), we have the following approximate form for It can be proved by explicite calculation (see the derivation of Eq. (A.22)) that in the limit ∞ → Apart from the factors ϕ in e − , for 0 = t , when ) , ( t r γ is real, the "photon part" (the sum with respect to n ) on the right hand side of Eq. (27), has the same functional form as the "number-phase minimum uncertainty states" Ψ , Eq. (6), derived by Jackiw (1968). Notice that the quantum number 0 n (corresponding to the parameter ν in Jackiw's solution) is an integer number in our case, in contrast to ν , which always have to have a non-vanishung fractional part. The other difference is that the normalization constant κ in Eq. (6) is , where I denotes the intensity of the photon field divided by one Watt per square centimeter. Besides, we shall also assume that the wavelength parameter Λ , introduced in Eq. (23), to a good approximation, coincides with π λ 2 / . This means, according to the definition of Ω in Eq. (9), that the one-electron plasma frequency p ω is assumed to be much smaller than ω , the frequency of the optical field. In figure 1   . For an optical field λ is of order of 10 -4 cm, accordingly w is of order of 10 -8 cm. As is seen, in case of the initial intensity we are considering, the elastic channel ( 0 = k ) and the one-photon channels ( 1 ± = k ) dominate, and the higher order channels ( 1 | | > k ) have much less joint probabilities. As is seen in figure 1, for 0 = t the maxima of the dominant low order joint probabilities ( 1 , 0 ± = k ) show up at the normalized radial position 2 2 / = w π μλ , which quantity is just the ratio of the amlitude of the electron oscillation to the spatial width of the electronic wave packet. This behaviour can be explained on the basis of the functional form of the position representation of the entangled photon-electron state given by Eqs.

Reduced density operators and entanglement entropies
Let us first calculate the density operator P of the quantized mode associated to the entangled state ψ In obtaining Eq. (28), the orthogonality of the transverse momentum eigenstates has been used, ) ( 2 p p p p ′ − = ′ r r r r δ . As is shown in Appendix B, the integral on the right hand side of Eq. (28) can be analytically evaluated, yielding the exact photon number distribution given by Eq. (B.3). In the following we shall not discuss this general distribution, but rather, we shall study the case of high initial photon excitations. For large values of 0 n , the reduced density operator P can be brought to the form (see the derivation leading to Eq. (B.7)) ) ( where  cm W × , respectively. The terminology "true photon number distribution" we are using for } { k p can be justified by that this set is built up from the (diagonal) elements of the density operator of the photon field, Eq. (29), which, of course, does not contain electron variables, since these latter ones have been traced out. In figure 4 it is clearly seen that as the intensity is increasing the higher order absoption and induced emission events become more and more dominant, and the widths of the distributions are becoming larger and larger. Not an unexpected result. Let us note that the results based on our present analysis do not contradict to the famous statement according to which "a free electron cannot absorb or emit a photon". This statement, which can be found in any of the basic texts on QED, relies on perturbation theory of the S-matrix approach dealing with asymptotic incoming and outgoing plane waves representing the electrons and the photons. The interaction of the electron with a strong laser beam is, in fact a many-body interaction, in the sense that the beam can be considered as a superposition of plane electromagnetic waves propagating in different directions, and taking part in high-order induced processes. This question have long been discussed e.g. by Bergou et al (1983), who used a relativistic semiclassical description. The study of such more general problems is out of the scope of the present paper. Here we are using a very simplified scheme (non-relativistic description of the electron, restriction to one mode interactions, dipole approximation, which are, on the other hand, well justified in the range of parameters taken in our numerical examples below). Our goal here is merely to show some basic characteristics of the entangled photon-electron systems.
where q has been defined in Eq. (30). According to Eq. (31) the entropy of the quantized radiation field does not depend on time. This is because the entangled photon-electron state introduced in Eq. (18) in a sense is a stationary state, though it contains explicitely the time variable in a complicated manner, as is shown by its analytic form given by Eqs. (25) and (26). The state ψ , Eq. (18), is not a solution of a true initial value problem where we would have assumed an initially non-interacting system (represented by a product state) and switch on the interaction at 0 = t some way. We leave the study of this latter problem for a separate work in progress (Varró 2007). In Figure 5 we illustrate the intensity dependence of the von Neumann entropy of the photon field. In the parameter range we are considering, the entropy curve, shown in Figure  In obtaining figure 5 The matrix elements of e P in momentum space can be calculated by using Eqs. (B.10) and (16), yielding where ) (x L n denote Laguerre polynomials of order n . The diagonal matrix elements of e P in momentum space are simply given by the modulus squared of the weight function ) ( p g r defined in Eq. (18), i.e.
In Eq. (34) we have introduced the dimensionless momentum variable k r and the density function ) (k r Π . According to Eqs. (B.14) and (B.15), the matrix elements of the reduced density operator e P in position space can be espressed as scalar products of the position representation of the entangled photon-electron states introduced in Eq. (19), where ) , ( t r r Ξ has been defined in Eqs. (25) and (26). As is shown in Appendix B, in cases of very high photon excitations (more accurately, in the limit ∞ → 0 n ) the density function in Eq. (35) becomes The diagonal matrix elements of the electron's reduced density operator are determined by the dimensionless density function, which we call true position distribution of the electron, since the photon variables have been traced out. We obtain The distribution ) , ( t x P r is normalized to unity for any instants of time. This can be shown by using a similar procedure applied in the proof of Eq. (A.22).
According to Eq. (29), the density operator of the photon field is diagonal, thus we were able to write down immediately the explicit formula in Eq. (31) for the von Neumann entropy. As is seen from Eqs. (33) and (36), the electron's density operator e P , Eq. (32), neither in momentum representation nor in position representation is diagonal. In order to calculate the von Neumann entropy of the electron, first we have to diagonalize e P , which, at the moment, seems to us a hopeless task. In order to avoid this difficulty we rather study the so-called linear entropy H which has a close connection with the second order Rényi entropy. H is the second order Rényi entropy, and ρ is some generic density operator. The linear entropy has been used by several authors (see e.g. Zurek et al 1993 andJoos et al 2003), because it is much easier to calculate (since we do not need the diagonalization of ρ ), and, on the other hand, it is a good alternative to the von Neumann entropy as a measure of entanglement. Really, H vanishes for a pure state, and it is maximum when the eigenvalues of ρ are identical (which is the case of maximum mixing). Another useful quantity to characterize the entanglement in a two particle sytem is the the Schmidt number K (see Nielsen and Chuang 2000) whose definition is , where ρ denotes the reduced density operator of either one of the two particles. The Schmidt number has been extensively used to charaterize continousvariable entanglement by Fedorov and coworkers (see Fedorov et al 2004Fedorov et al , 2005Fedorov et al , 2006Fedorov et al and 2007 in their thorough analyses on wave packet dynamics in breakup processes, like ionization of atoms and dissociation of molecules (see, in particular, Fedorov In order to calculate the linear entropy of the electron, we need first an explicit expression of 2 e P , which can be obtained from Eq. (B.9) by a straightforward calculation, The trace of 2 e P can be calculated analytically, thus we can derive an exact expression for the linear entropy of the electron, as is shown in Appendix B by Eq. (B.28). In the limit ∞ → . The logarithmic increase of the von Neumann entropy with the intensity (curve "S") is clearly seen in the figure. The linear entropy (curve "H") is always smaller than the von Neumann entropy, and increases much slower than the latter one. The increase of each measures of the entanglement by increasing the intensity is after all not an unexpected result, since the interaction of the photons and the free electron is becoming stronger and stronger as the photon density is getting larger. given by Eqs.

Summary
In the present paper we have discussed entanglement between photons and electrons. We have shown that the entangled photon-electron states introduced by us have a close connection with the critical states introduced by Jackiw (1968), which minimize a number-phase uncertainty product of the photon field. These states are of essentially the same mathematical structure as that of Jackiw's states, and naturally appear in the non-perturbative analysis of the simplest interaction of QED we have considered, namely the interaction of a free electron with a quantized mode of the electromagnetic radiation. On the basis of our analysis we have given a simple interpretation of states of Jackiw's type, thus, we may say that these latter states have a physical significance, rather than being mere mathematical constructions, as they were originally thought of. Besides, we have derived exact analytic expressions for the reduced density operators of the photon field and of the free electron, and determined the von Neumann entropy of the photon, and the linear entropy of the photon and of the electron. In the introduction we gave a brief historical overview of the development of concepts on entanglement and on the related first basic experiments. Moreover we sketched the most important approaches to the problem of the quantal phase of the linear oscillator (or of a quantized mode of the radiation field). On purpose, we quoted the early references, too, so that the interested reader can keep track of the evolution of concepts on the quantal phase from the very beginning. In Section 2 we have summarized the basic properties of the cosine and sine operators of the quantal phase introduced by Susskind and Glogower (1964), and we presented the critical states constructed by Jackiw (1968), which minimize the uncertainty product of the number operator and the cosine operator. On the basis our earlier work Bergou and Varró (1981a), in Section 3 we determined the exact stationary states of the interacting photon-electron system. These states are simple product states whose photon parts are generalized coherent states, and the electron parts are momentum eigenstates. Section 4 has been devoted to the construction of the entangled photon-electron states which are defined as Gaussian superpositions (with respect to the electron's momentum variable) of the stationary states discussed in Section 3. As we already emphasized, these entangled states defined by Eq. (18) are not solutions of a true initial value problem where we would have assumed an initially non-interacting system (represented by a bare product state) and switch on the interaction, say, at 0 = t some way. We leave the study of this latter problem for a separate work in progress (Varró 2007). In Appendix A we have given an exact analytic expression for the expansion coefficients of the entangled states (with respect to the number state basis of the photon's Hilbert space and in position representation in the electron's Hilbert space), and in the main text we studied the properties of the associated probability distributions for various parameter values in the large excitation limit. The expansion coefficients of the entangled states, obtained from Eq. (22), and used in Eqs. (25) and (27) are in fact joint probability amplitudes of detecting an electron at some position and at an instant of time, and, at the same time, detecting certain definite number of photons. The basic features of these joint probabilities have been illustrated in Figures 1-2-3. In Section 5 we presented the reduced density operators of the photon field and of the free electron, and with the help of them the true photon number distribution and the electron's momentum and position distributions have been calculated. The exact expressions have been derived in Appendix B, and in the main text we have analysed the characteristics of these distribution in the large excitation limit. As measures of the entanglement, the von Neumann entropy of the photon field and the linear entropies of the photon field and of the electron have also been calculated exactly, and closed analytic forms for them were given in the large excitation limit. We have proved by an explicit calculation that the latter two quantities coincide. Our results are displayed by Figures 4-5-6, which show the true photon number distribution, the intensity dependence of the von Neumann entropy of the photon field and the comparison of the intensity dependence of the linear entropy and of the von Neumann entropy, respectively. Finally we note that it may seem to be a serious restriction to confine our (non-perturbative) study to the analysis of interactions of a free electron with only one quantized mode of the radiation field. In reality, of course, the electron interacts with the whole assembly of the modes due to e.g. secondary spontaneous emission processes (see for instance the case of Compton scattering). The study of interactions only with one mode can be justified if this mode is in a very highly excited state (as has been mostly assumed in the present paper). In this special case (which, on the other hand, is of great importance in the physics of nonlinear processes taking place in laser-matter interactions) the interactions with the other modes (or with some other third body) can be treated as small perturbations. In this context, see e.g. the works of Bergou and Varró (1981a-b). In order to have an estimate for the magnitude of the excitation degree 0 n in realistic laser systems, we can use Eq. (B.34) of Appendix B, which gives a numerical formula for the mean photon occupation number. In Table B 1 we have summarized the numerical values of the parameters we are interested in, for three kinds of laser radiation. It is seen that for intensities managable nowadays, the mean occupation number can be enormously large. Of course, a c-number electric field strength cannot be associated to even a very highly occupied number state in a strict sense. This association can consistently be done, for instance, by using the coherent states of Schrödinger-Glauber's type (Glauber 1963a-b). We plan to present the study of the coherent superpositions of the entangled photon-electron states elsewhere.

Appendix A Derivation of the explicit form of the entangled photon-electron states
In the present appendix we show the basic steps leading to the exact analytic form of the matrix elements, Eq. (20), and we derive the approximate form, Eq. (22), of them. We prove that the asymptotic states ) , ( t r r Ξ , Eq. (25), are properly normalized. In order to start with, first we give an alternative form of the expansion coefficients of the generalized coherent states, displayed in Eq. (16), where, according to Eq. (10), bx e p mc ea The quantities a , Ω and w have been introduced in Eqs. (8), (10) and (18), and the dimensionless variable h / pw x = will be used to calculate the radial integral in Eq. (A.1). The integration with respect to the azimuth angle χ in momentum space in Eq. (A.1) can be carried out by using the Jacobi-Anger formula for the electron plane wave  where we have introduced the notation (A.6) ) (z J s denotes ordinary Bessel function of first kind of order s . The radial integral in Eq. (A.5) can be expressed in an analytic form by using the formula 7.421.4 of Gradshteyn and Ryzhik (2000), yielding In order to study the asymptotic behaviour of the above exact expression for large 0 n values, we use the limit formula (Erdélyi 1953, formula 10.12(36)) for given s and z values. Moreover, we take the limit in such a way that, though both 0 n and the quantization volume 3 L are going to infinity, the photon density   can be written down in the following alternative form . In order to do that, let us apply the following formula of Hilb's type (Erdélyi 1953, formula 10.15 (2)  At the end of the present appendix we prove by a direct calculation that the approximate entangled photon-electron states ) , ( t r r Ξ , defined in Eq. (25) of Section 4 in the main text, is properly normalized in the limit ∞ → 0 n . Really, after the ϕ -integration we have Now, owing to the formulas 8.406.3 and 8.538.1 of Gradshteyn and Ryzhik (2000), where we have also taken into account the definition of ) , (  Gradshteyn and Ryzhik (2000), this polynomials are all positive (which is, by the way, is to be required from a true probability distribution). In the limit ∞ → ( . However, this approximate distribution is not normalized properly, it only gives the relative photon occupation probabilities. In order to derive a properly normalized approximate distribution in the large 0 n limit, directly from Eq. (B.2), we are now proceeding differently as before. By using Hilb's formula, Eq. (A.14), the integral in Eq. (B.2) can be asymtotically expressed as The matrix elements of the reduced density operator e P in position space can be determined by using the identities, (B.14) By comparing the factors on the right hand side of Eq. (B.14) with the coordinate representation of the entangled photon-electron state, defined in Eq. (19), we realize that, in fact, the matrix elements are expressed by the following scalar products, (B.16) In obtaining Eq. (B.16) we have used the formulas 8.406.3 and 8.538.1 of Gradshteyn and Ryzhik (2000), which were also used in the derivation of Eq. (A.21), As we see from Eqs. (B.11) and (B.16), the electron's density operator e P , Eq. (B.9), neither in momentum representation nor in position representation is diagonal. In order to calculate the von Neumann entropy of the electron, first we have to diagonalize e P , which we have not been able to do by now. In order to avoid this difficulty here we study the so-called linear entropy H which has an close connection with the second order Rényi entropy.  where the definitions in Eq. (A.9) has also been used. In the large 0 n limit we can use Hilb's formula, Eq. (A.14), and apply it directly in the integrand in Eq. (B.27 At the end of the present appendix we give an estimate for the average occupation number of the photon field expressed in terms of the amplitude of the electric field strength 0 F , or, equivalently, in terms of the intensity I of a quasi-monochromatic radiation. In free space the three-dimensional spatial mode density in a frequency interval ) , (  where I denotes the intensity divided by 2 / cm W , and ph E is the photon energy measured in eV -s. In Table B 1 we summarize for three kinds of lasers the numerical values of the photon energy, the inverse bandwidth and the corresponding average photon occupation number, expressed in terms of the dimensionless intensity I , on the basis of Eq. (B.34).  Table B.1 Shows for three kinds of lasers the numerical values of the photon energy, the inverse bandwidth and the corresponding average photon occupation number. Figure 1 Shows the spatio-temporal distribution of the joint probability coming from Eq. (22)

Caption to Figure 3
Shows schematically the space-time regions where the shapes of the joint probability distribution are qualitatively different.

Caption to Figure 4
Shows the true photon number distribution } { k p (derived from the reduced density operator, and given by Eq. (29)) for four q (intensity) values, namely for 5 . 2 = q in (a), 5 = q in (b), 25 = q in (c) and 50 = q in (d).

Caption to Figure 5
Shows the intensity dependence of the von Neumann entropy of the photon field defined by Eq. (31).

Caption to Figure 6
Shows a comparison of the intensity dependencies of the