Quantum interference with molecules: The role of internal states

Recent experiments have shown that fullerene and fluorofullerene molecules can produce interference patterns. These molecules have both rotational and vibrational degrees of freedom. This leads one to ask whether these internal motions can play a role in degrading the interference pattern. We study this by means of a simple model. Our molecule consists of two masses a fixed distance apart. It scatters from a potential with two or several peaks, thereby mimicking two or several slit interference. We find that in some parameter regimes the entanglement between the internal states and the translational degrees of freedom produced by the potential can decrease the visibility of the interference pattern. In particular, different internal states correspond to different outgoing wave vectors, so that if several internal states are excited, the total interference pattern will be the sum of a number of patterns, each with a different periodicity. The overall pattern is consequently smeared out. In the case of two different peaks, the scattering from the different peaks will excite different internal states so that the path the molecule takes becomes entangled with its internal state. This will also lead to degradation of the interference pattern. How these mechanisms might lead to the emergence of classical behavior is discussed.

Figure 1

(Color online) Schematic illustration of a scattering of a two-atom molecule on a potential composed of two (Gaussian) peaks. The two “atoms” of the molecule have a mass $m$, and each located a distance $\alpha$ from the center of mass of the molecule. The location of the center of mass is given by $(X,Y)$. The angle between the line connecting the two masses and the positive $x$ axis is $\varphi$, and the angular momentum corresponding to the rotation of the molecule in the $x\text{−}y$ plane is $L$. The angle between the wave vector of the outgoing molecule and the positive $y$ axis is denoted by $\theta$, and $\widehat{\mathbf{u}}$ is a unit vector pointing in the direction of the outgoing wave vector. The angles are measured in radians.

Figure 2

(Color online) We plot the cross section $\sigma \left(\theta \right)$ as a function of $\theta$ (measured in radians) and the momentum $k$ (measured in dimensionless units). We compare two cases: A molecule with (left column) and without (right column) internal degrees of freedom. From the figure it is obvious that the internal structure of the molecule causes a suppression of the interference pattern. The cross section $\sigma$ in the presence of the internal structure of the molecule is represented by Eq. (2.14) where we assume that the incident molecule is in the state with $l=0$. The cross section corresponding to the case with a molecule that does not have an internal structure is given by Eq. (2.21). Otherwise the molecules are considered to be the same, i.e., we assume units such that $m=1$ and $\hslash =1$. The two Gaussian peaks are characterized by $V_0=1$ and $\Delta =1$. The distance between each scattering peak and the origin of the coordinate system is $d=2$ (the upper row) and $d=6$ (the lower row). We observe that with larger $d$ the frequency of interference oscillations is larger. The suppression of oscillations due to the presence of internal molecular states is clearly seen in both cases, i.e., $d=2$ and $d=6$.

Figure 3

(Color online) We plot the cross section $\sigma \left(\theta \right)$ of the molecule that is scattered by a grating modeled as a potential with $2N+1$ Gaussian peaks. We present the cross section as a function of $\theta$ (measured in radians) and the momentum $k$ (measured in dimensionless units) for different values of $N$. We compare two cases: We assume the molecule with (left column) and without (right column) internal degrees of freedom. From the figure it is obvious that the internal structure of the molecule causes a suppression of the interference pattern. The cross section $\sigma$ in the presence of the internal structure of the molecule is represented by Eq. (2.25) where we assume that the incident molecule is in the state with $l=0$. The cross section corresponding to the case with a molecule that does not have an internal structure is given by Eq. (2.27). Otherwise the molecules are considered to be the same and we assume units such that $m=1$ and $\hslash =1$. The Gaussian peaks are characterized by $V_0=1$ and $\Delta =1$. The distance between adjacent scattering peaks is $d=6$. We consider three different types of gratings with $N=1$ (i.e., scattering on three Gaussian peaks—see the first row), with $N=2$ (i.e., scattering on five Gaussian peaks—see the second row); and with $N=10$ (i.e., scattering on 21 Gaussian peaks—see the third row). The maximal values of the cross section that are achieved from small $k$ and $\theta =0$ are different for different values of $N$. Particularly, these values are proportional to $N^2$. In any case, we see a suppression of the quantum interference patterns due to the presence of internal molecular states.

Figure 4

(Color online) We plot the cross section $\sigma \left(\theta \right)$ of the molecule that is scattered by a potential composed of two nonidentical peaks. We present the cross section as a function of $\theta$ (measured in radians). The potential is described by Eq. (3.1). We consider two situations: When the molecule has an internal structure (dashed line) given by Eq. (3.4) and when the molecule does not have an internal structure (solid line) given by Eq. (3.5). When internal states of the molecule are present, then depending on which path the molecule follows through the potential, that is which peak it scatters from, different (distinguishable) internal states of the molecule are excited. Consequently, in principle the path can be determined, and the interference is suppressed (this situation situation is very similar to that discussed in papers on “which-way information” in double-slit experiments—see, e.g., Refs. 18, 19). Here the cross section is given by Eq. (3.4). We consider parameters such that $k\alpha =5∕2$, $k\Delta =3∕2$, and $kd=4$, and we obtain the cross section given by the dashed line. When the molecule has no internal structure, the path cannot be determined and the interference pattern is preserved (solid line). The cross section $\sigma \left(\theta \right)$ in this case is given by Eq. (3.5) and for the same values of parameters as before (i.e., $k\alpha =5∕2$, $k\Delta =3∕2$, and $kd=4$).