Entanglement between a Photon and a Quantum Well

The lack of translational invariance perpendicular to the plane of a single quantum well causes equal probability for spontaneous emission to the left or right. Combining one emission path from the left and one from the right into a common detector leads to interference fringes for fundamentally indistinguishable paths corresponding to geometries where the same in-plane momentum is transferred to the quantum well. For all other paths, no interference is observed because of the entanglement between the photon and extended Bloch states of the many-body system. In multiple-quantum-well structures the interference can be controlled via the spacing between the wells.

Figure 1

Schematic experimental setup for interfering emission from a quantum well (QW) structure into different paths. Using identical lenses ($L,f=100\mathrm{m}\mathrm{m}$) and apertures (AP, $d=1\mathrm{m}\mathrm{m}$) the paths are combined in a $50:50$ beam splitter (BS). The He-Ne pump beam was coupled through a dielectric mirror which reflects $100\%$ at the wavelength of the photoluminescence (845 nm), yet transmits partially the 632.8 nm wavelength. An interference filter (IF, $\Delta \lambda =10\mathrm{n}\mathrm{m}$) was placed in front of the charge coupled device (CCD) camera to cut off the reflected He-Ne pump beam. The path difference can be adjusted by an optical delay line (resolution 15 fs). Inset B shows the spatial image of a typical interference pattern measured with a cooled high sensitivity CCD camera (0.5 s integration time). The fringes are obtained by introducing a small angle between the two paths. Inset A shows the measurement geometry for a tilted sample (tilt angle $\varphi$). Case I: Paths identical to the untilted case (shown in light gray); case II: $V$-shaped symmetry of the light paths as discussed in the text. In the actual experiment the tilt is achieved by moving the apertures on both sides perpendicular to the sample normal.

Figure 2

Photoluminescence intensity along a line perpendicular to the fringe direction for light from path one (dashed) and two (dotted) separately and for the interference signal (thick solid line). The thin solid line gives the sum of path 1 and 2. The inset shows the contrast, i.e., the modulus of the mutual coherence function, versus the time difference $\tau$ between the two paths. The line is a fit with an exponential correlation function $\exp (-\gamma |\tau |)$, with $\gamma =7\times {10}^{11}{\mathrm{s}}^{\mathrm{-}\mathrm{1}}$.

Figure 3

Dependence of fringe contrast on the directions of the two paths traveling in almost opposite directions as determined by the shift of the apertures along the $y$ direction perpendicular to the interferometer plane. Opposite signs of displacement correspond to case I in Fig. 1, and hence to tilting the sample by an angle $\varphi$ (1 mm amounts to $\varphi =1.8°$); equal signs to case II.

Figure 4

Dependence of fringe contrast on the separation $d$ between quantum wells. Because of our MBE process, layer thickness $t$ decreases with increased radius $r$ on a sample. This allows us to tune through both $d=\lambda /2$ (Bragg) and $d=\lambda /4$ (anti-Bragg) using two samples, with the Bragg (anti-Bragg) position determined by reflectivity maximum (minimum) [19]. Relative $d$ values were obtained using $t(r)$ determined by measuring the radial dependence of the single mode of an empty microcavity. A clear contrast maximum for Bragg and minimum for anti-Bragg spacings are observed. The lines are guides to the eye. The single quantum-well contrast is independent of position on the sample; $d$ for it is merely the thickness of the cap layer.