How Time Dilation Affects Quantum Superpositions
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Authors: Igor Pikovski1,2,3,4, Magdalena Zych1,2,5, Fabio Costa1,2,5, Caslav Brukner1,2
1Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Austria,
2Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Vienna, Austria,
3ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts, USA,
4Department of Physics, Harvard University, Cambridge, Massachusetts, USA,
5Centre for Engineered Quantum Systems, School of Mathematics and Physics, The University of Queensland, St Lucia, Queensland, Australia.
Can gravity affect quantum systems? Quantum theory and gravity seem to apply to very different physical regimes. It is often argued that gravity is irrelevant on very small scales, where typical quantum phenomena are observed. A well-known argument compares the forces between an electron and a proton: gravity is roughly 1039 times weaker than electromagnetism. But this is not the full story. As has been shown in numerous experiments, gravity can indeed influence a quantum wave function of the smallest particles. Newtonian gravity from Earth can induce a quantum phase-shift for a particle that is in a superposition between two different heights. This was first demonstrated with Neutrons in the famous COW (Colella-Overhauser-Werner) experiment in 1975 .
What about Einstein’s gravity? If the Newtonian potential can influence a quantum wave function, what can one expect from post-Newtonian effects stemming from general relativity? It turns out that novel phenomena arise, with no classical analogue. A few years ago, we studied how time dilation affects a single clock in superposition  (see also 2Physics article “Quantum Complementarity Meets Gravitational Redshift”). Classically, two clocks placed at different heights will experience different proper times, and thus will be time dilated with respect to each other. But in quantum theory, an additional effect arises: If a single clock is brought into superposition of two heights, its internal degrees of freedom (or clock states) get entangled with its position. This entanglement affects the superposition of heights. If the two amplitudes are brought together to interfere, the interference pattern will be affected due to quantum complementarity: As the internal clock states record time, they gain “which-way information” and thus destroy the quantum coherence. This interplay has been recently demonstrated in an experiment with a BEC, where the time delay of spin precession was simulated by inhomogeneous magnetic fields  (see also 2Physics article “One Clock in Two Places Simultaneously”).
Figure 1: The gravitational field causes time dilation, clocks closer to Earth run slower than clocks further away. For a quantum superposition of a single clock at two heights, the clock states and its position become entangled.
In our latest work, we showed that the effect of time dilation on quantum systems is very general and affects the quantum coherence of any composite quantum system . Time dilation is universal: it affects any system regardless of its structure or composition. Clocks are affected by time dilation as much as the heart beat or the half-life of a decaying particle. Also composite quantum systems are affected, as they usually have a finite temperature: It means that there is always some internal dynamics present within the system, which can be affected by time dilation. If such a composite system is brought into superposition between different heights above Earth, as in the above example with a clock, time dilation will correlate all internal dynamics with the height of the system. And this causes decoherence of the center-of-mass, just as in the case with an actual clock. In practice, any system is affected that has some internal energy spread. If the center-of-mass of such a system is brought into superposition and a sufficient difference in proper times is accumulated along the superposed paths, quantum coherence will be lost because of the relative time delay of the internal dynamics of the system. Needless to say, time dilation can be caused by a gravitational field but also by any acceleration or by velocities. There is no “external” environment, only the presence of time dilation causes the system’s center-of-mass to decohere due to the dynamics of its own composition.
Figure 2: A complex molecule in superposition in a gravitational field. Because of time dilation, the frequencies of the individual atoms depend on the height of the molecule. This causes decoherence of quantum superpositions of the center-of-mass of the molecule.
One may expect this effect to be vanishingly small. After all, it is very hard to detect time dilation on Earth. But it turns out that already on mesoscopic scales, the effect is strong: If a gram-scale object at room temperature is put into a superposition of height difference of 1µm, then time dilation will cause decoherence after about 1ms. This is because many internal degrees of freedom contribute to the effect. Each individually is affected by time dilation only a tiny bit, but for a larger composite system, the effect can become significant. For quantum systems, it is of course very challenging to prepare such large superpositions, and other decoherence effects will also be of importance. But there is a parameter range at superfluid temperatures, where experiments with very large molecules or microspheres could in principle observe the predicted phenomena in the future. But importantly, the effect is universal and any internal dynamics will contribute to decoherence. Thus one can think of other possible experiments with internal dynamics other than thermal, such as nuclear dynamics or some fast chemical processes.
All of these will contribute to the rate of decoherence; one can be creative in designing a clever experiment. In fact, one could also use such setups to reverse the logic: observing coherence of the center-of-mass might be a key to shed light on unknown internal dynamics. Maybe the best possible way for experimental verification will turn out to take an actual clock and put it in superposition -- as we proposed in  and which has recently been shown to work in an analogue experiment . Future experiments should be able to verify decoherence due to time dilation, which follows only from quantum theory and general relativity as we know them.
Finally, what do we actually learn from this study? In our work, we do not treat the gravitational field quantum-mechanically, thus there is no direct connection to “quantum gravity”. Yet, we study how quantum mechanical test systems behave on a background space-time, as opposed to classical test particles, and new effects arise. We discovered a new decoherence mechanism, but the most exciting aspect of the work is that the interplay between quantum theory and gravity has novel phenomena to offer. Our work shows one example, but this research direction is still widely unexplored: Many more possible effects and experiments on the interplay between these two great theories are waiting to be discovered!
 Roberto Colella, Albert W. Overhauser, Samuel A. Werner. “Observation of Gravitationally Induced Quantum Interference”, Physical Review Letters, 34, 1472 (1975). Abstract.
 Magdalena Zych, Fabio Costa, Igor Pikovski, Časlav Brukner. “Quantum interferometric visibility as a witness of general relativistic proper time”, Nature Communications, 2, 505 (2011). Abstract. 2Physics Article.
 Yair Margalit, Zhifan Zhou, Shimon Machluf, Daniel Rohrlich, Yonathan Japha, Ron Folman. “A self-interfering clock as a 'which path' witness”, published online in 'Science Express' (August 6, 2015). Abstract. 2Physics Article.
 Igor Pikovski, Magdalena Zych, Fabio Costa, Časlav Brukner, “Universal decoherence due to gravitational time dilation”, Nature Physics ,11, 668-672 (2015). Abstract.