Making gravitational time-dilation decoherence quantitative and observable in realistic composite quantum systems

Pikovski, Zych, Costa and Brukner showed in 2015 that gravitational time dilation couples the internal degrees of freedom of any composite quantum system to its centre-of-mass motion, so that a spatial superposition of a warm macroscopic object decoheres even in perfect isolation, with an Einstein-solid estimate giving a millisecond-scale decoherence time for a gram-scale object at room temperature in a micrometre superposition on Earth. The authors themselves left open how to make this prediction quantitative for realistic systems and how to observe it: the internal-energy modeling was crude, the pointer basis for non-stationary motion was uncharacterized, and all known environmental decoherence channels were expected to dominate in current experiments. A decade of follow-up work has consolidated the theoretical framework and quantified how far present experiments are from the required regime, but has not observed the effect nor supplied the missing system-specific theory. The open problem is to derive the time-dilation-induced reduced dynamics for realistic materials (with measured heat capacities and electronic, vibrational and nuclear contributions), to characterize the decoherence basis beyond the stationary regime, and to identify and design a concrete experimental regime and protocol in which time-dilation decoherence dominates over gas scattering, thermal emission, absorption and photon scattering and can be discriminated from them.

Background

Decoherence is the process by which a quantum system entangles with other degrees of freedom so that interference between its alternatives becomes unobservable; it is the standard route by which quantum superpositions acquire an effective classical appearance. In standard decoherence theory the entangling degrees of freedom belong to an external environment. The 2015 result by Pikovski, Zych, Costa and Brukner introduced a source of decoherence that requires no environment at all beyond classical spacetime. In general relativity, a clock tick rate depends on its trajectory: proper time, the time a clock actually registers, runs slower deeper in a gravitational potential and slower for a moving object (gravitational and special-relativistic time dilation). For a composite quantum object whose internal energy $H_0$ is an operator, time dilation makes the internal evolution depend on the centre-of-mass variables $(x,p)$: the low-energy Hamiltonian acquires the universal coupling $H_{\mathrm{int}} = H_0\,\Gamma(x,p)/c^2$ with $\Gamma(x,p)=\Phi(x)-p^2/(2m^2)$ for a weak Newtonian potential $\Phi(x)$. If the centre of mass is placed in a superposition of two wave packets that follow world lines with a proper-time difference $\Delta\tau$, the internal states attached to the two branches evolve for different amounts of internal time, become distinguishable, and the interferometric visibility drops as $V=|\langle e^{-iH_0\Delta\tau/\hbar}\rangle|$. The effect is universal in the sense that it depends only on the variance of the internal energy, not on any coupling constant. For a simple internal model — an Einstein solid, i.e. $N$ independent harmonic modes at temperature $T$, a standard textbook model of a solid's thermal energy — the decay becomes Gaussian with decoherence time $\tau_{\mathrm{dec}}=\sqrt{2/N}\,\hbar c^2/(k_B T g \Delta x)$ for superposition size $\Delta x$ in Earth's gravitational acceleration $g$, of order milliseconds for a gram-scale object at room temperature and $\Delta x = 1\,\mu\mathrm{m}$. Like any decoherence in quantum theory, the effect is in principle reversible: because the internal spectrum is discrete, "revivals" of coherence occur once a sufficiently large proper-time difference has accumulated, but the corresponding recurrence times typically scale exponentially with the size of the system, so time-dilation decoherence is irreversible "for all practical purposes" for large composite systems. The predicted decoherence selects position (more precisely, an amplitude of the coupling operator $\Gamma(x,p)$) as the pointer basis, the stable preferred basis that a decohering measurement-like interaction leaves behind; for stationary or slowly varying centre-of-mass states $\Gamma$ reduces to a function of position, but for general motion the selected classical variable is a combination of $x$ and $p$ that was left uncharacterized. The quantitative claim competes with mundane decoherence channels: elastic scattering of residual gas molecules, thermal emission and absorption of photons, and scattering of blackbody photons, all of which decohere large warm objects efficiently. The authors therefore posed a set of connected open tasks: realistic modeling of internal energy fluctuations beyond the Einstein solid (using measured heat capacities with electronic, vibrational and nuclear contributions, and valid at low temperature where the Einstein-solid estimate breaks down); system-specific modeling of effective photon-coupling cross sections $\sigma_{\mathrm{eff}}(k)$ (the cross section for coupling to environmental photons of wavenumber $k$) and electromagnetic mode densities $g(k)$ (the spectral density of environmental modes at wavenumber $k$), possibly with engineered electromagnetic environments, so that time-dilation decoherence can be distinguished from emission- and scattering-induced decoherence; and characterization of the pointer basis for non-stationary centre-of-mass evolution — all with the goal of a definitive experimental test, which demands cryogenic internal temperatures, ultrahigh vacuum and micrometre-scale superpositions of massive composite systems.

Problem Statement

The problem, as posed by the open questions of Pikovski, Zych, Costa and Brukner (2015), is to convert the prediction of universal time-dilation-induced decoherence into a quantitative, system-specific theory and a feasible discriminating experiment. Concretely: (1) derive the reduced centre-of-mass dynamics — the master equation (evolution equation for the density matrix) of the centre of mass after the internal degrees of freedom are traced out — for realistic composite systems, replacing the Einstein-solid estimate by internal-energy fluctuation modeling based on measured heat capacities including electronic, vibrational and nuclear contributions, valid at the cryogenic temperatures any test requires; (2) characterize the decohering observable (pointer basis) exactly for non-stationary centre-of-mass evolutions, where time-dilation decoherence selects a nontrivial combination of position and momentum; and (3) identify, quantitatively, a concrete experimental regime — system, mass, internal temperature, superposition size, ambient pressure and electromagnetic environment — together with a measurement protocol in which time-dilation-induced decoherence dominates over spontaneous emission, absorption, thermal-photon scattering and residual-gas collisions, and design the channel-by-channel discrimination (including engineered electromagnetic suppression of emission where needed) that would establish the gravitational effect rather than environmental decoherence as the observed cause of visibility loss. These three components jointly serve the single objective: a definitive quantitative prediction for a realistic system and its experimental verification or falsification.

References

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