Gauge dependence of spontaneous radiation spectrum in a time-dependent relativistic non-perturbative Coulomb field

Abstract

Lamb triggered a continuous debate on the gauge choice for atomic interactions with electromagnetic fields, particularly with plane waves and the vacuum field. Modern technologies of Rydberg atoms and relativistic atomic beams make it possible to explore interactions with a more intriguing non-perturbative, adiabatic Coulomb field. In such cases, one would face the well-known tricky issue about the physical significance of the scalar gauge potential when it is time-dependent. We start attacking this issue by studying a simplest system: a one-dimensional oscillator interacting adiabatically with a relativistic charge. We reveal that a gauge dependence much severer than the one Lamb observed is encountered when calculating the transient radiation spectrum of this oscillator by the external-field method, which is currently the only available tool. The obtained peak frequency can differ by 10 MHz or larger for the commonly used Coulomb, Lorentz, and Multipolar gauges. Contrary to the popular view, we explain that such a gauge dependence is not really a disaster, but actually an advantage here: The relativistic bound-state problem is so complicated that a full quantum-field method is still lacking; thus, the external-field approximation cannot be derived and hence not guaranteed. However, by fitting to experimental data, one may always define an effective external field, which may likely be parameterized with the gauge potential in a particular gauge. This effective external field would not only be of phenomenological use, but also shed light on the physical significance of the gauge potential. We thereby encourage further investigations of this fundamental problem with more realistic systems involving Rydberg atoms and relativistic atomic beams, both theoretically and experimentally.

Graphic Abstract

Appendix A: The scalar potential in different gauges

In this Appendix, we give the details of the scalar potential $\phi (\vec {x}, t)$ for our system in the commonly used Lorentz, Coulomb, and multipolar gauges. For the Lorentz gauge,

$$\begin{aligned} \phi _{L}=\frac{1}{4 \pi }\left( \frac{-e}{l+x}+\frac{-e}{l-x}+\frac{N e}{\sqrt{(x-L(t))^{2}+\left( 1-\beta ^{2}\right) Y^{2}}}\right) ,\nonumber \\ \end{aligned}$$

(A1)

and for Coulomb gauge,

$$\begin{aligned} \phi _{C}=\frac{1}{4 \pi }\left( \frac{-e}{l+x}+\frac{-e}{l-x}+\frac{N e}{\sqrt{(x-L(t))^{2}+Y^{2}}}\right) .\nonumber \\ \end{aligned}$$

(A2)

Here, x is the coordinate of the electron, L(t) is the coordinate of the charge cluster at time t. The subscripts L, C refer to expressions in the Lorentz and Coulomb gauges, respectively, and a subscript G will denote a general gauge.

By applying the PZW transformation (12), one obtains $\phi _{M}$ in the multipolar gauge,

$$\begin{aligned} \phi _{M}=\left\{ \phi _G(\vec {x}_0,t)-\int _{0}^{1} d \lambda \vec {E}\left( \lambda \vec {x}+(1-\lambda ) \vec {x}_{0}, t\right) \cdot \left( \vec {x}-\vec {x}_{0}\right) \right\} .\nonumber \\ \end{aligned}$$

(A3)

As we explained in the text, we Taylor-expand $\phi (\vec {x}, t)$. For the Lorentz gauge,

$$\begin{aligned} \phi _{L}\approx & {} \frac{1}{4 \pi }\left\{ \frac{-e}{x_{0}+l}-\frac{-e}{x_{0}-l}+\frac{N e}{\sqrt{\left( x_{0}-L(t)\right) ^{2}+\left( 1-\beta ^{2}\right) Y^{2}}}\right. \nonumber \\&-\left( \frac{-e}{\left( x_{0}+l\right) ^{2}}-\frac{-e}{\left( x_{0}-l\right) ^{2}} \right. \nonumber \\&\left. +\frac{N e\left( x_{0}-L(t)\right) }{\sqrt{\left( x_{0}-L(t)\right) ^{2}+\left( 1-\beta ^{2}\right) Y^{2}}^{3}}\right) \left( x-x_{0}\right) \nonumber \\&+\left( \frac{-e}{\left( x_{0}+l\right) ^{3}}-\frac{-e}{\left( x_{0}-l\right) ^{3}}\right. \nonumber \\&\left. \left. -\frac{1}{2}\left( \frac{N e}{\sqrt{\left( x_{0}-L(t)\right) ^{2}+\left( 1-\beta ^{2}\right) Y^{2}}^{3}} \right. \right. \right. \nonumber \\&\left. \left. \left. -\frac{3 N e\left( x_{0}-L(t)\right) ^{2}}{\sqrt{\left( x_{0}-L(t)\right) ^{2}+\left( 1-\beta ^{2}\right) Y^{2}}^5}\right) \right) \left( x-x_{0}\right) ^{2}\right\} .\nonumber \\ \end{aligned}$$

(A4)

For the Coulomb gauge,

$$\begin{aligned} \phi _{C}\approx & {} \frac{1}{4 \pi }\left\{ \frac{-e}{x_{0}+l}-\frac{-e}{x_{0}-l}+\frac{N e}{\sqrt{\left( x_{0}-L(t)\right) ^{2}+Y^{2}}}\right. \nonumber \\&-\left( \frac{-e}{\left( x_{0}+l\right) ^{2}}-\frac{-e}{\left( x_{0}-l\right) ^{2}} \right. \nonumber \\&\left. +\frac{N e\left( x_{0}-L(t)\right) }{\sqrt{\left( x_{0}-L(t)\right) ^{2}+Y^{2}}^{3}}\right) \left( x-x_{0}\right) \nonumber \\&\left. +\left( \frac{-e}{\left( x_{0}+l\right) ^{3}}-\frac{-e}{\left( x_{0}-l\right) ^{3}}-\frac{1}{2}\left( \frac{N e}{\sqrt{\left( x_{0}-L(t)\right) ^{2}+Y^{2}}^{3}} \right. \right. \right. \nonumber \\&\left. \left. \left. -\frac{3 N e\left( x_{0}-L(t)\right) ^{2}}{\sqrt{\left( x_{0}-L(t)\right) ^{2}+Y^{2}}^{5}}\right) \right) \left( x-x_{0}\right) ^{2}\right\} . \end{aligned}$$

(A5)

And for the multipolar gauge,

$$\begin{aligned} \phi _{M}\approx & {} \phi _G(\vec {x}_0, t)+\frac{1}{4\pi }\left\{ -\left( \frac{-e}{\left( x_{0}+l\right) ^{2}}-\frac{-e}{\left( x_{0}-l\right) ^{2}}\right. \right. \nonumber \\&+\, \left. \left. \frac{N e\left( 1-\beta ^{2}\right) \left( x_{0}-L(t)\right) }{\sqrt{\left( x_{0}-L(t)\right) ^{2}+\left( 1-\beta ^{2}\right) Y^{2}}^{3}}\right) \left( x-x_{0}\right) \right. \nonumber \\&+\left( \frac{-e}{\left( x_{0}+l\right) ^{3}}-\frac{-e}{\left( x_{0}-l\right) ^{3}}\right. \nonumber \\&\left. \left. -\frac{1}{2}\left( \frac{N e\left( 1-\beta ^{2}\right) }{\sqrt{\left( x_{0}-L(t)\right) ^{2}+\left( 1-\beta ^{2}\right) Y^{2}}^{3}} \right. \right. \right. \nonumber \\&\left. \left. \left. -\frac{3 N e\left( 1-\beta ^{2}\right) \left( x_{0}-L(t)\right) ^{2}}{\sqrt{\left( x_{0}-L(t)\right) ^{2}+\left( 1-\beta ^{2}\right) Y^{2}}^{5}}\right) \right) \left( x-x_{0}\right) ^{2}\right\} .\nonumber \\ \end{aligned}$$

(A6)

The value of $x_{0}$ in each gauge is found by solving $\left. \partial _{x} \phi _{G}(x, t)\right| _{x=x_{0}}=0$.