Two-Dimensional Observables and Spectral Resolutions

Anatolij Dvurečenskij, Dominik Lachman

Anatolij Dvurečenskij

Anatolij Dvurečenskij, Dominik Lachman

Alberto Acevedo, Antonio Falcó

We investigate equilibration-on-average and effective equilibration for quantum systems evolving on infinite-dimensional Hilbert spaces under Hamiltonians with purely continuous spectrum. In this regime, the standard infinite-time dephasing picture underlying finite-dimensional equilibration theory ceases to be informative: the corresponding ergodic average becomes trivial in the relevant sense and therefore cannot serve as a physically meaningful equilibrium state. This makes equilibration intrinsically a finite-time phenomenon. Motivated by this obstruction, we introduce a framework based on finite-time ergodic averages and spectral coarse graining at finite resolution. Within this setting, we derive explicit upper bounds for equilibration-on-average of bounded observables under purely continuous-spectrum dynamics. Unlike their finite-dimensional counterparts, these bounds depend not only on the Hamiltonian and the initial state, but also on the observation time horizon and on the resolution scale of the spectral partition. We also obtain corresponding bounds for effective equilibration with respect to finite families of physically realistic POVM (Positive Operator-Valued Measure) measurements. Our results provide a quantitative formulation of equilibration for continuous-spectrum quantum systems in a regime where the usual infinite-time dephasing paradigm breaks down. Within the coarse-grained framework developed here, the resulting bounds depend explicitly on both the finite observation time and the chosen spectral resolution. An explicit example based on a free-particle Hamiltonian illustrates the mechanism and interpretation of the general bounds.

Yunxiao Zhang, Liang Cui, Xueshi Guo +7 more

Observation of the universe demands telescopes with high resolution. In the optical band, traditional interference requires bringing interfering fields together, which limits the resolution due to the restricted length of baseline. Here we demonstrate the very long-baseline interferometer (VLBI) in optical band, where two interfering fields never met each other. In particular, we report the first quantum interference observation when the input of VLBI is single-photon state. Interference is recovered after measuring the amplitudes of photon fields and digitally processing the signals of quantum receivers. Moreover, we analyze interference in time and spectral domains for broadband thermal light input and show that the ultrahigh spectral resolution can improve the precision of radial velocity to 0.08 centimeters per second, which is 2 orders of magnitude better than that achievable at the current stage. Further, we apply the spectrally resolved interference in distinguishing two independent sources with angular resolutions beyond diffraction limit. Our investigations have a profound effect on the VLBI, quantum optics, and precision measurement.

Aleksandr N. Naumov, Alexey A Podshivalov, Konstantin N. Drabovich +2 more

A simple model of the Voigt spectral profile is employed to analyze the influence of the sizes of a vapor cell on the shape of the spectral contour of the linear susceptibility. The width of a resonant spectral line observed in transmission and absorption spectra measured with such a vapor cell is shown to tend to the natural width as one of the sizes of the vapor cell approaches the wavelength of the spectral line, thus opening the way to extend the Romer- Dicke pillbox-cavity approach to microwave spectroscopy to the optical range. The possibility of using (lambda) -thick vapor cells, as well as one- and two-dimensional photonic band-gap structures for high resolution measurements is discussed.