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arXiv:quant-ph/0502156 · Physical Review A

Quantum interference with molecules: The role of internal states

Mark S. Hillery, Leonard D. Mlodinow, Vladimír Bužek

Published Jun 14, 2005• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.71.062103

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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.
We consider a composite system consisting of coupled particles, and investigate decoherence due to coupling of the center-of-mass degree of freedom to the internal vibrational degrees of freedom. For a composite system of bound particles, we show that in general such a decoherence effect exists, and leads to suppression of interference between different paths of the center-of-mass. For the special case of harmonically-bound particles moving in an external potential, we show that the coupling between the center-of-mass and internal degrees of freedom takes the form of parametric driving of the vibrational degrees of freedom, and that nontrivial coupling depends on the second derivative of the external potential. We find a solution to this parametric driving problem in one dimension for a fixed center-of-mass trajectory. We also propose a measure of compositeness, which quantifies the extent to which such a composite system cannot be approximated as a single, indivisible particle. We perform numerical simulations for a simple interference experiment, consisting of two wave packets scattering off of a square well, and show a close connection between suppression of interference and entanglement between the center-of-mass and internal degrees of freedom, which depends on the initial state of the internal mode. We also calculate the measure of compositeness for this system.
Although many quantum mechanics problems of a point particle have been well understood, the realistic physical model is often a composite object consisting of particles bound together, and its quantum mechanics is an important problem with applications in many areas of physics. Unlike a point particle, a composite object possesses internal structure described by some degrees of freedom which are often entangled with each other. We call the entanglement among these degrees of freedom ``internal entanglement'', to distinguish it from any other entanglement they may share with an external object. Examples of internal entanglement include the entanglement between a vibrational and a rotational mode of a molecule, and that between its position and its internal clock-state etc. In this thesis, we study quantum mechanics of composite objects by focusing on the effects of internal entanglement. We first look at the tunnelling of a diatomic molecule type composite object through a half-silvered mirror in continuum space and observe that its spatial superposition state made by the mirror can decohere by emission of radiation due to the fact that its internal degrees of freedom are entangled with radiation fields, and that there exists entanglement between its position and its internal (i.e., relative position) degrees of freedom. Secondly, we study its lattice analog in molecular crystals, namely we replace the diatomic molecule by a biexicton, and the mirror by an impurity. We find that discreteness of a lattice makes the wave vector of a biexicton and the relative distance between two excitons of it entangled with each other. We investigate how this inseparability affects the creation of the biexciton-impurity bound states and the entanglement dynamics. Finally we propose a possible application of our study of internal entanglement to the Anderson model of a composite quasiparticle.
Young's double-slit experiment requires two waves produced simultaneously at two different points in space. In quantum mechanics the waves correspond to a single quantum object, even as complex as a big molecule. An interference is present as long as one cannot tell for sure which slit is chosen by the object. The more we know about the path, the worse the interference. In the paper we show that quantum mechanics allows for a dual version of the phenomenon: self-interference of waves propagating through a single slit but at different moments of time. The effect occurs for time-independent Hamiltonians and thus should not be confused with Moshinsky-type time-domain interference, a consequence of active modulation of parameters of the system (oscillating mirrors, chopped beams, time-dependent apertures, moving gratings, etc.). The discussed phenomenon is counterintuitive even for those who are trained in quantum interferometry. For example, the more we know about the trajectory in space, the better the interference. Exactly solvable models lead to formulas deceptively similar to those from a Youngian analysis. There are reasons to believe that this new type of quantum interference was already observed in atomic interferometry almost three decades ago, but was misinterpreted and thus rejected as an artifact of unknown origin.
Decoherence is the main process behind the quantum to classical transition. It is a purely quantum mechanical effect by which the system loses its ability to exhibit coherent behavior. The recent experimental observation of diffraction and interference patterns for large molecules raises some interesting questions. In this context, we identify possible agents of decoherence to take into account when modeling these experiments, and study their visible (or not) effects on the interference pattern. We thereby present an analysis of matter wave interferometry in the presence of a dynamic quantum environment, and study how much the visibility fringe is reduced and in which timescale the decoherence effects destroy the interference of massive objects. Finally, we apply our results to the experimental data reported on fullerenes and cold neutrons. © World Scientific Publishing Company.
A double-slit experiment is analysed when the interfering particle has an internal structure and one endeavours to obtain ‘which path’ information by detecting the light that is spontaneously emitted. A compact expression is derived for the visibility of the interference pattern: coherence depends on both the spatial and temporal features of the emission process during the travel of the particle to the screen. A bound is set on the temperature of a fullerene molecule in a double-slit experiment in order that its quantum mechanical coherence be maintained.
Summary form only given, as follows. In a recent experiment we were able to demonstrate the de Broglie wave character of the fullerene molecule C/sub 60/ by diffraction at a free standing material grating. This proves that it is possible to preserve quantum coherence almost completely in the external degrees of freedom of a hot, massive and complex object over a long time and distance. This may seem astonishing in the light of the fact that the fullerenes were produced in a thermal source which was kept at about 900 K. At such a high temperature the de Broglie wavelength is about 400 times smaller than the size of the molecule (/spl lambda//sub dB//spl sim/25 pm), all internal degrees of freedom (174 different vibrational modes, and rotational modes up to J>100) are excited and the molecules may emit a few infrared photons (blackbody radiation and vibrational quanta) during their time of flight through the apparatus. Our published measurements so far have been essentially consistent with the absence of decoherence during the time of flight of the molecules. The interference contrast in all experiments has however so far been limited by the thermal velocity spread and the finite collimation of the fullerene beam.