X-ray quantum-eraser setup for time-energy complementarity

A quantum-eraser setup exploiting the time-energy complementarity relation in the x-ray regime is investigated theoretically. The starting point is the interference process between x-ray quanta driving two nuclear hyperfine transitions in a nuclear forward-scattering setup. We show that which-way information can be obtained by marking the scattering paths with orthogonal polarization states, thus leading to the disappearance of the interference pattern. In turn, erasure of the which-way information leads to the reappearance of the interference fringes. We put forward two schemes using resonant scattering off nuclear targets and design which-way marking procedures to realize the quantum-eraser setup for x-ray quanta.

Figure 1

(a) Typical NFS setup with magnetic field $\mathit{B}$. (b) NFS time spectrum for a thin target with quantum beat pattern (orange curve). (c) Nuclear level scheme of an individual ${}^{57}\mathrm{Fe}$ nucleus under hyperfine splitting. (d) Collective picture with a common ground state $G$ and excited levels $E_1$ and $E_2$ driven via the $\mathrm{\Delta }M=0$ transitions.

Figure 2

Quantum-eraser scheme 1. The $\sigma$-polarized synchrotron pulse is resonantly scattered off target 1 (${\mathit{B}}_1\parallel {\mathit{e}}_z$) via the two $\mathrm{\Delta }M=0$ transitions. The high-speed shutter cuts the SR pulse off such that the incoming radiation field at target 2 is solely the nuclear response of target 1. Following the nuclear scattering off target 2 (${\mathit{B}}_2\parallel {\mathit{e}}_y$) a polarizer splits the $\sigma$- and $\pi$-polarization components.

Figure 3

(a) NFS intensity spectrum ${\left|\mathit{E}(\xi ,\tau )\right|}^2$ directly behind target 1. The scattering via the two $\mathrm{\Delta }M=0$ transitions results in the characteristic quantum beat pattern. Due to the shutter, only photons emitted in the time window from 7 to 74 ns [region between blue dashed lines in (a), blue dashed curve in (b)] reach target 2. (b) The magnetic field ${\mathit{B}}_2$ is chosen such that the interference between the $\mathrm{\Delta }M=0$ frequency components is destroyed in the intensity spectrum ${\left|\mathit{E}(\xi ,\tau )\right|}^2$ (red solid curve) directly behind target 2. Effective thicknesses of ${\xi }_1={\xi }_2=7$ and a maximal scattering order of $p_{\max }=19$ have been considered in the calculations.

Figure 4

$\sigma$ and $\pi$ components of the intensity spectrum ${\left|\mathit{E}(\xi ,\tau )\right|}^2$ are shown in (a) starting with 74 ns together with the unpolarized signal (red dotted line) from Fig. 3. For better visualization, the graph in (b) zooms into the time window between 80 and 190 ns. In the calculations, scattering orders up to $p_{\max }=19$ have been taken into account.

Figure 5

Quantum-eraser scheme 2. The incoming x-ray photon is split into its $\sigma$- and $\pi$-polarized components by a polarizer [63, 64]. The split beams are later mixed with the help of two mirrors [65] and a beam splitter (BS). In each arm a nuclear ${}^{57}\mathrm{Fe}$ target is inserted. The two targets experience the magnetic fields ${\mathit{B}}_1$ and ${\mathit{B}}_2$. A relative time delay between arms 1 and 2 is introduced.

Figure 6

Quantum eraser via a relative time delay. (a) The individual targets exhibit the quantum beat pattern due to interference between the two $\mathrm{\Delta }M=0$ transitions. (b) Employing a time delay corresponding to $\mathrm{\Phi }=\pi /2$ marks the scattering paths with orthogonal polarizations, destroying the interference. (c) The which-way information is erased and the interference is recovered by projecting onto the linear polarization basis, ${\mathit{e}}_{\sigma }$ (yellow curve, lighter hue) and ${\mathit{e}}_{\pi }$ (blue curve, darker hue).

Figure 7

Quantum eraser via an intrinsic storage scheme. The first-order intensity spectrum $|{\mathit{E}}_{\mathrm{out}-1}^{\left(1\right)}{|}^2$ is shown along with the time sequence of magnetic field switchings. The initial quantum beats (a) are first annihilated via a relative phase shift $\mathrm{\Phi }=\pi /2$ (b) and subsequently restored by erasing the which-way information via a second storage sequence, introducing an additional $\pi /2$ phase shift (c).