Dynamically assisted nuclear fusion in the strong-field regime

We consider quantum tunneling between fusing nuclei in the presence of a harmonically time-varying field. The tunneling rate is calculated using Floquet/Volkov (FV) and Kramers-Henneberger (KH) approaches, which are both compared to a first-principles numerical solution of the Crank-Nicolson (CN) type. Numerical validation of the FV approach justifies its use in analytical estimates of laser-enhanced reaction rates, while the KH approach does not reproduce the predictions of the others. For a deuterium-tritium plasma at a temperature of 1 keV, it is found that significant enhancement of fusion reactivity requires field strengths on the order of ${10}^{15}$–${10}^{16}$ V/m and photon energies below 1 keV, which is within reach of next-generation x-ray free electron lasers (XFELs).

Figure 1

Comparison of modified transparency estimates using FV (dashed curves), KH (dotted curves), and CN ($\diamond \text{s}$) for the DT fusion reaction as a function of center-of-mass energy. The field strength is fixed at $E={10}^{15}$ V/m; (a) the polarization angle is fixed at $\theta =0$, while the photon energy is chosen to be 1 keV (blue), 3 keV (orange), and 10 keV (green); (b) the photon energy is fixed at $\hslash \omega =3$ keV, while the polarization angle is chosen to be 0 (orange) and $\pi /2$ (green). The gray solid curve is the field-free transparency.

Figure 2

Higher-field-strength version of Fig. 1, where now $E={10}^{16}$ V/m, with all other parameters unchanged.

Figure 3

Enhancement factor $\eta =\langle {\sigma }_{\text{FV}}\left(\mathit{\Xi }\right)u\rangle /\langle \sigma \left(\mathcal{E}\right)u\rangle$ associated with the strong-field fusion reactivity in a thermal DT plasma at temperature $k_{\text{B}}T=1$ keV, calculated for a range of photon energies and field strengths. Height is also characterized by color, with the transition from black to violet indicating the $\eta \ge 2$ threshold. The highlighted curves in gray mark the values $E=\{3\times {10}^{14},1\times {10}^{15},3\times {10}^{15},1\times {10}^{16}\}$ V/m for reference. At sufficiently high field strengths ($E\ge 5\times {10}^{16}$ V/m), enhancement factors of around a factor of two are calculated toward the lower range of photon energies ($\hslash \omega \approx 200$ eV) marking the beginning of the soft-x-ray band. All axes are log scale.

Figure 4

Comparison between analytic and numerical results for a rectangular barrier of height and width $V=L=e$ in arbitrary units. The CN calculations successfully capture the sharp increase near $\mathcal{E}=e$ and the oscillatory behavior at higher energies. The first peak, which is sharpest, is slightly underestimated due to the Gaussian blur effect, as further elaborated on in the text.

Figure 5

Gaussian blur of the numerical transparency curve resulting from the spectral width of the incident wavepacket. Barrier 2 (orange) refers to a short, wide rectangular barrier like in Fig. 4, and barrier 1 (blue) is a tall, thin rectangular barrier that exhibits similar properties to the Coulomb barrier. The value of ${\sigma }_0$ in Eq. (C3) is lowered to demonstrate the extent of the blurring effect. Note that the ‘nodes’ in the above-barrier oscillation are least affected, while the peaks and troughs are most affected.