Aharonov-Bohm Effect in Synchrotron Radiation

V.G. Bagrov¹, D.M. Gitman², A.D. Levin³, and V.B. Tlyachev⁴
Instituto de Física, Universidade de São Paulo,
C.P. 66318, 05315-970 São Paulo, SP, Brasil

Abstract:

We study the impact of Aharonov-Bohm solenoid on the radiation of a charged particle moving in a constant uniform magnetic field. Radiation peculiarities caused by the presence of the solenoid may be considered as a manifestation of Aharonov-Bohm effect in the radiation. In particular, new spectral lines appear in the radiation spectrum. Due to angular distribution peculiarities of the radiation intensity, these lines can in principle be isolated from basic cyclotron and synchrotron radiation spectra.

I. It is well-known that charged particles irradiate moving in a uniform magnetic field. The presence of the AB field (AB solenoid) [1] affects the particle radiation in the magnetic field. The effect has pure quantum nature since classical trajectories do not feel the presence of the AB field. To calculate the impact of AB solenoid on particle radiation in magnetic field, we have found exact solutions of the Klein-Gordon and the Dirac equations in the superposition of a constant uniform magnetic field and parallel to it the AB field of the solenoid ( such a superposition is called the magnetic-solenoid field). Using such solutions, all the characteristics of one-photon spontaneous radiation both for spinless and spinning particle were calculated exactly. Considering non-relativistic and relativistic approximations, CR and SR can be analyzed in detail. In particular, it is shown that new spectral lines appear in the radiation spectrum. Due to angular distribution peculiarities of the radiation intensity, these lines can in principle be isolated from basic CR and SR spectra. Peculiarities of the radiation related to the presence of the AB solenoid may be considered as a manifestation of the Aharonov-Bohm effect in the radiation. All the technical details of the straightforward calculation of the effect are presented in [3]. Below we present briefly main results of this analysis.

II. Consider the particle motion in the magnetic-solenoid field that is a superposition of a constant uniform magnetic field $H$ and parallel to it the AB field of the solenoid. The latter creates a finite magnetic flux $\Phi .$ Suppose both fields are directed along the axis $z,$ and the solenoid is placed on this axis. The total magnetic field has the form $\mathbf{H}=(0,0,H_{z}),\;H_{z}=H+\Phi \delta (x)\delta (y).$ It is convenient to present the magnetic flux$\;$as $\Phi =(l_{0}+\mu )\Phi _{0},\quad \Phi _{0}=2\pi c\hbar /\left\vert e\right\vert \;,\;0\leq \mu <1,\;l_{0}=0,\pm 1,\pm 2,...\;.$ The integer $l_{0}$ gives a number of quanta $\Phi _{0}$ in the total flux $\Phi$. We call the quantity $\mu$ the mantissa of the magnetic flux $\Phi .\;$Classical trajectories are not affected by the presence of the solenoid field. However, the presence of the AB solenoid has a topological effect. The presence of the solenoid allows us to divide all the classical trajectories in two groups ($j=0,1)$: trajectories that do not embrace the solenoid ($j=0)$, and ones that do ($j=1)$. Denoting via $R_{0}$ the distance between the trajectory center and the solenoid, we can see that trajectories with $j=0$ have $R_{0}-R>0,$ whereas trajectories with $j=1\$ have $R_{0}-R<0.$ The $z-$projection of the angular momentum of the particle has the form $\;L_{z}\,=\left\vert eH\right\vert \left( R^{2}-R_{0}^{2}\right) /2c-\hbar \left( l_{0}+\mu \right) \,\,.$ Thus, $j=0$ trajectories have $L_{z}\,>-\hbar \left( l_{0}+\mu \right) ,$ whereas $j=1\$ trajectories have $L_{z}\,<-\hbar \left( l_{0}+\mu \right) .$ Since the classical trajectories are not affected by the solenoid, there is not any impact of the AB field on particle radiation in magnetic field.

Quantum states in the magnetic-solenoid field differ from ones in the pure magnetic field. One can solve Klein-Gordon, and Dirac equations in the magnetic-solenoid field exactly [3]. For simplicity, we restrict ourselves by spinless particle case (stationary solutions of the Klein-Gordon equation). In this case there are two types ($\,j=0,1$) of solutions. They have the form

$$\Psi _{n,p_{z},l}^{\left( j\right) }\left( \mathbf{r,}t\right) =e... ...ight) =\left\{ \begin{array}{c} 1,\;l\geq 0 \\ 0,\;l<0 \end{array} \right. \,,$$ (1)

where $r,\varphi$ are cylindrical coordinates in the plane $x,y;$ $E_{n,j}$ is the particle energy, $n=0,1,2,...$; $-\infty <p_{z}<\infty$ is $z-$projection of the momentum; integer $l\leq n\;$(azimuthal quantum numbers) are related to the eigenvalues of the angular momentum operator $\hat{L}_{z}=-i\hbar \partial /\partial \varphi$ by the relation $\hat{L}_{z}\Psi _{n,p_{z},l}^{\left( j\right) }=L_{z}\Psi _{n,p_{z},l}^{\left(... ...\right) \Psi _{n,p_{z},l}^{\left( j\right) }\,,\;l=0,\pm 1,\pm 2,...,\;l\leq n.$ Comparing the eigenvalues $L_{z}$ with the corresponding classical expression, we get $l=\left\vert eH\right\vert \left( R^{2}-R_{0}^{2}\right) /2c\hbar -\mu \,.$ Taking into account that $\;0\leq \mu <1$ and that $l$ are integer in quantum theory, we see that the quantum numbers $j$ may be interpreted similar to the classical theory. Namely, states with $j=0$ ( $l=-1,-2,...)$ correspond to $R_{0}-R>0$ i.e. to classical trajectories that do not embrace the solenoid; states with $j=1\$ ( $l=0,1,2,...)$ correspond to $R-R_{0}>0$ i.e. to classical trajectories that embrace the solenoid. Trajectories with $l=0$ and$\;l=-1$ pass most close to the solenoid. The radial dependence of the states (1) is described in terms of functions $\psi _{n,l}^{\left( j\right) }\left( r\right) ,$which are expressed via so called Laguerre functions [3]. The upper (positive) branch of the energy spectrum has the form

$$E_{n,j}=\sqrt{m^{2}c^{4}+c^{2}p_{z}^{2}+2\hbar c\left\vert eH\right\vert \left( n+j\mu +\frac{1}{2}\right) }\,,\;\;n=0,1,2,...\,\;.$$ (2)

The integer $n\geq 0$ is referred to as the principle quantum number. The energy spectrum (2) for states with $j=0$ corresponds exactly to the spectrum of spinless particles in pure magnetic filed. The spectrum for states with $j=1\$ is deformed by the solenoid presence whenever $\mu \neq 0$. Thus, the solenoind field partially lifts a degeneracy of the pure magnetic field spectrum with respect to the quantum number $l$ whenever $\mu \neq 0.$ The states $\Psi _{n,p_{z},l}^{\left( j\right) }$ are bounded, quadratically integrable, and vanish at $r=0.$ That allows us to speak about AB effect in the case under consideration whenever $\mu \neq 0.$

III. Matrix elements of electron transitions with the one-photon radiation in the magnetic-solenoid field can be calculated exactly (using corresponding wave functions in the magnetic-solenoid field) in the framework of scalar and spinor QED [3]. Thus, it turns out to be possible to obtain exact expressions for the differential (with respect to the polarization) radiation intensity and total radiation intensity of spinless and spinning particles. One can see from these expressions that all the peculiarities of the radiation related to the presence of AB solenoid depend on the mantissa $\mu$ of the solenoid magnetic flux only. For $\mu =0$ the radiation intensity does not depend on the initial azimuthal quantum number $l$. That is a well-known result in SR theory [2]. In the magnetic-solenoid field with $\mu \neq 0,$ the degeneracy with respect to $l$ is lifted completely. That may be interpreted as follows: The quantum numbers $l$ define distances between classical trajectories and the solenoid. At the same time, $l$ defines the type of trajectories. Clearly, that the radiation intensity depends on the distances as well as on the type of states. For $\mu =0,$ the origin is not fixed anymore by the presence of the solenoid. Thus, the $l$ dependence of the radiation intensity dies out.

IV. We introduce a number $\nu$ of emitted harmonic as $\nu =n-n^{\prime }=0,1,2,...\;,$ where $n$ and $n^{\prime }$ are principal quantum numbers of initial and final electron states. For $\mu =0,$ frequencies of emitted photons are functions of $n$, of $\nu ,$ and of the polar angle $\theta$. These frequencies do not depend on the azimuthal quantum numbers $l,\;l^{\prime }$. For $\mu \neq 0,$ this degeneracy is partially lifted. In such a case, the frequencies depend on the type of initial and final states. For initial states with $k_{3}=0$ they have the form

$$\omega _{jj^{\prime }}\left( n,\nu \right) =\frac{2c\left\vert eH... ...\vert \left[ \nu +\left( j-j^{\prime }\right) \mu \right] \sin ^{2}\theta }}\;.$$ (3)

Thus, for $\mu \neq 0,$ there appear two spectral series: one results from transitions without any change of the quantum number $j$ and another one results from transitions with a change of $j$.

Whenever $\nu >0$ and $n$ are fixed, we get an inequality $\omega _{01}\left( n,\nu \right) <\omega _{11}\left( n,\nu \right) <\omega _{00}\left( n,\nu \right) <\omega _{10}\left( n,\nu \right) ,$ which becomes the equality for $\mu =0$. The difference between the frequencies $\omega _{11}\left( n,\nu \right)$ and $\omega _{00}\left( n,\nu \right)$ can be easily estimated for laboratory magnetic fields $H\ll H_{0}$, where $H_{0}=m^{2}c^{3}/e\hbar \approx 4,41\times 10^{13}\mathrm{gauss}$ is a critical field. This difference is proportional to $\mu$. Namely $\omega _{00}\left( n,\nu \right) -\omega _{11}\left( n,\nu \right) \simeq \mu \,\omega _{00}\left( n,\nu \right) \delta \,,\;$where $\;\delta =\frac{H}{H_{0}}\left( \frac{mc^{2}}{E_{n,0}}\right) ^{2}.$ One can see that $\delta <10^{-9}$ for not very high electron energies and for typical (those which are realized in accelerators) magnetic fields $H\approx 10^{4}\mathrm{gauss}$ . In such a case, the frequency difference $\omega _{10}\left( n,\nu \right) -\omega _{01}\left( n,\nu \right)$ reads $\omega _{10}\left( n,\nu \right) -\omega _{01}\left( n,\nu \right) \simeq 2\mu \omega ,$ where $\omega =c\left\vert eH\right\vert /\,E$ is the synchrotron frequency. Thus, this difference becomes quite noticeable for harmonics with small numbers. For $\mu \neq 0,$ there exist a radiation of $\nu =0$ harmonic due to $j=1\rightarrow$ $j^{\prime }=0$ transitions. For magnetic fields $H\ll H_{0}$ the corresponding frequency is $\omega _{10}\left( n,0\right) \simeq \mu \omega \,\,.$ The transitions $j=0\rightarrow$ $j^{\prime }=1$ cause a radiation of $\nu =1$ harmonic. For the above magnetic fields the corresponding frequency reads $\omega _{01}\left( n,1\right) \simeq (1-\mu )\omega \,.$ Such a radiation does not exist in pure magnetic field. Both frequencies $\omega _{10}\left( n,0\right)$ and $\omega _{01}\left( n,1\right)$ are less than $\omega$.

V. To analyze the impact of AB field on CR we have considered exact formulas in the non-relativistic case and in the weak magnetic field approximation $H\ll H_{0}.$ For transitions $j=1\rightarrow j^{\prime }=1,$ the only first harmonic $\nu =1$ is effectively emitted. In the approximation under consideration, the radiation intensity does not depend on $l.$ Transitions from initial states with $l=0$ contribute insignificantly to the radiation compared to all other transitions. As in the previous case, for transitions $j=0\rightarrow j^{\prime }=0,$ the only first harmonic is effectively emitted and the radiation intensity does not depend on $l$. For transitions $j=0\rightarrow j^{\prime }=1,$ we meet a situation, which is completely different from the one considered before. Here the only transitions from states with $l=-1$ really contribute to the radiation. That fact has a natural physical explanation. For $l=-1,\;j=0,$ classical trajectories do not embrace the solenoid but pass maximally close to the latter. A transition to trajectories, which embrace the solenoid, is more likely namely from such states. It is important to stress that no restrictions exist on numbers of emitted harmonics. At least a whole succession of first harmonics has equal probabilities of the radiation. In this approximation, such harmonics are not emitted if $\mu =0.$ However, the case $\nu =1$ is of special interest. As was remarked before, the corresponding radiation frequency $\omega _{01}\left( n,1\right)$ is less than the cyclotron one. In particular, one can see that for all the transitions $j=0\rightarrow j^{\prime }=1$ the sign of the radiation circular polarization is opposite to the sign of the circular polarization for all other transitions. That observation can be useful to identify the radiation related to such transitions. Considering transitions $j=1\rightarrow j^{\prime }=0$, we can see that the only transitions from states with $l=0$ contribute effectively to the radiation. Classical trajectories, which pass maximally close to the solenoid (embracing it), correspond to such initial states. Then, a possible physical interpretation is similar to the one given above. As before, we have here a radiation of $\nu =0$ harmonic (even for $n=0$ in the initial state). Such a radiation is forbidden for $\mu =0$. In the transitions $j=1\rightarrow j^{\prime }=0$, all the harmonics with different numbers $\nu$ contribute almost equally to the radiation intensity. In the absence of the solenoid field (more exactly for $\mu =0)$ and in the approximation under consideration, the only $\nu =1$ harmonic survives. The existence of the transitions under consideration may be treated as a manifestation of the AB effect.

VI. As was already mentioned above, in the relativistic case, the conventional SR is concentrated in the vicinity of the orbit plane and is maximal for harmonics with big numbers. Thus, it is widely believed that low number harmonics cannot practically be isolated against the background of intensive high frequency radiation. However, analyzing the expression for the intensity of SR at length, one can discover that there exist one exclusion from this rule. Namely, we can see that in the conventional SR, the intensity of all the harmonics with $\nu \geq 2$ is exactly zero in the directions $\theta =0,\pi$ (along the magnetic field). Besides, the radiation intensity of the first harmonic ($\nu =1$) is maximal along the magnetic field for any particle energy. Moreover, the latter radiation has total circular polarization and, thus, can be easily identified.

The presence of the solenoid field modifies both the spectrum and angular distribution of SR. Consider, for example, the intensity of SR in the magnetic-solenoid field in the directions $\theta =0,\pi .$ Transitions without a change of the type of the initial state (without a change of $j$) cause the only first harmonic ($\nu =1$) radiation in the directions $\theta =0,\pi$ whenever $l\neq 0.$ This fact does not depend on particle energies. Transitions from initial states with $l=0$ without a change of $j$ do not cause any radiation in $\theta =0,\pi$ directions. Transitions with a change of the type of the initial state (with a change of $j$) cause a radiation in the directions $\theta =0,\pi$ solely for $l=0,-1\;$(the solenoid is situated maximally close to a classical trajectory). In such cases all possible harmonics ( $0\leq \nu \leq n$) are emitted with approximately equal intensities. For $\mu =0,$ the only first harmonic radiation survives. All the transitions cause totally circulary polarized radiation in the directions $\theta =0,\pi .$ Moreover, the sign of the circular polarization for $j=0\rightarrow j^{\prime }=1$ transitions is opposite to the one for all other transitions.

The only one basic synchrotron harmonic and the new frequencies are irradiated along the magnetic field. We stress important peculiarities of the radiation along the magnetic field. The basic synchrotron harmonic has total circular polarization; the radiation intensity of superlow harmonics has maximum in the magnetic field direction; all the harmonics from the two above mentioned series have approximately equal radiation intensities. The latter property of the radiation is not typical for the conventional CR and SR. We believe that a considerable relative shift between new harmonics and the basic synchrotron one as well as the peculiarities of the angular distribution of the radiation intensity open up possibilities for experimental observation of AB effect in CR and SR.

Acknowledgment The authors (V.G.B, D.M.G, and A.D.L) are thankful to FAPESP for support. (D.M.G) thanks also CNPq for permanent support and (V.G.B.) thanks Russian Science Ministry Foundation and RFFI for partial support.