We propose a scheme to experimentally observe matter-wave interference in the time domain, specifically in the arrival time or time-of-flight (TOF) distribution for an atomic Bose–Einstein condensate (BEC) Schrödinger-cat state represented by superposition of macroscopically separated wave packets in space. This is in contrast to interference in space at a fixed time observed in the reported BEC experiments. We predict and quantify the quantum interference in the TOF distribution calculated from the modulus of the quantum probability current density (rather than the TOF distributions obtained from a purely classical or semi-classical treatment in many reported experiments). The interference and hence the coherence in the quantum TOF signal disappears in the large-mass limit. Our scheme has the potential to probe the validity of various other theoretical approaches (Muga and Leavens 2000 Phys. Rep. 338 353) of calculating the quantum arrival time distribution.
We propose a scheme to experimentally observe matter-wave interference in the time domain, specifically in the arrival time or time-of-flight (TOF) distribution for an atomic Bose–Einstein condensate (BEC) Schrödinger-cat state represented by superposition of macroscopically separated wave packets in space. This is in contrast to interference in space at a fixed time observed in the reported BEC experiments. We predict and quantify the quantum interference in the TOF distribution calculated from the modulus of the quantum probability current density (rather than the TOF distributions obtained from a purely classical or semi-classical treatment in many reported experiments). The interference and hence the coherence in the quantum TOF signal disappears in the large-mass limit. Our scheme has the potential to probe the validity of various other theoretical approaches (Muga and Leavens 2000 Phys. Rep. 338 353) of calculating the quantum arrival time distribution.
The times of flight of an atomic Bose-Einstein condensate are theoretically investigated in the experimentally unexplored regime corresponding to detection close to the trap of the condensate. In this regime, there is no consensus on how to calculate the distribution of times of arrival onto the detector. For non-interacting particles, distinct theoretical predictions have been made in the past. This work analyses how these predictions are modified for an interacting Bose-Einstein condensate. For this purpose, a time-dependent Gross-Pitaevskii equation is solved analytically and numerically.
Quantum interference is ubiquitous to many (if not all) quantum processes and phenomena, for it is a direct manifestation of their inherent coherence. Experiments based on matter wave interferometry [1] constitute very nice empirical evidences of this property. Though the first diffraction experiments were carried out with electrons [2], nowadays the (matter) particles utilized range from large atoms and single molecules [3-6] to very complex, large molecules [7] and Bose-Einstein condensates (BECs) [8]. This kind of experiments, apart from confirming the wave nature of matter, has also given rise to important practical applications. For example, sensing, metrology or quantum information processing are potential applications which have motivated in the last few years the remarkable development of BEC interferometry [9-11]. This can be observed in the increasing amount of (theoretical and T&F Cat # K11224, Chapter 13, Page 198, 15-9-2010 experimental) work appearing in the literature [12-18] since the first experimental evidence of interference between two freely expanding BECs [19]. The basic scheme upon which this type of interferometry is based is depicted in Figure 13.1. In the first stage (left), the atomic cloud is cooled down in a magnetic trap until condensation takes place. Then (right), the BEC is split up coherently by means of radio-frequency [16] or microwave fields [17]. Finally, the double-well-like trapping potential [20] is switched off, releasing the two resulting BECs, which will interfere by free expansion (e.g., see Figure 6 of Ref. [18] as an illustration of a real experimental outcome).
Although position and time have different mathematical roles in quantum mechanics, with one being an operator and the other being a parameter, there is a space-time duality in quantum phenomena: a lot of quantum phenomena that were first observed in the spatial domain were later observed in the temporal domain as well. In this context, we propose a modified version of the double-double-slit experiment using entangled atom pairs to observe a non-local interference in the arrival time distribution, which is analogous to the non-local interference observed in the arrival position distribution. However, computing the arrival time distribution in quantum mechanics is a challenging open problem, and so to overcome this problem we employ a Bohmian treatment. Based on this approach, we numerically demonstrate that there is a complementary relationship between the one-particle and two-particle interference visibilities in the arrival time distribution, which is analogous to the complementary relationship observed in the position distribution. These results can be used to test the Bohmian arrival time distribution in a strict manner, i.e., where the semiclassical approximation breaks down. Moreover, our approach to investigating this experiment can be applied to a wide range of phenomena, and it seems that the predicted non-local temporal interference and associated complementary relationship are universal behaviors of entangled quantum systems that may manifest in various phenomena.
The visibility of the density interference pattern of two Bose-Einstein condensates, which are produced in traps and overlap after the trapping potential is switched off, is investigated. Coherent wave packets are used to describe the order parameter in a second-quantization formalism. This results in a decrease of the visibility of the interference fringes with increasing time delay between the formation of the condensates and the observation of interference. In the two limiting cases of ideal and very dense gases the correlation time increases, τ →∞, and the result is identical to that obtained using an approach based on the Gross-Pitaevskii equation. Under the conditions of the experiment performed by M. R. Andrews, C. G. Townsend, H. J. Miesner et al. , Science 275 , 6367 (1997), the computed correlation time τ ≈0.2 s is much longer than the confinement time of the condensate, and it is possible to observe the predicted decrease of visibility of the interference fringes of the density of the atoms.
Although position and time have different mathematical roles in quantum mechanics, with one being an operator and the other being a parameter, there is a space-time duality in quantum phenomena-a lot of quantum phenomena that were first observed in the spatial domain were later observed in the temporal domain as well. In this context, we propose a modified version of the double-double-slit experiment using entangled atom pairs to observe a non-local interference in the arrival time distribution, which is analogous to the non-local interference observed in the arrival position distribution. However, computing the arrival time distribution in quantum mechanics is a challenging open problem, and so to overcome this problem we employ a Bohmian treatment. Based on this approach, we numerically demonstrate that there is a complementary relationship between the one-particle and two-particle interference visibilities in the arrival time distribution, which is analogous to the complementary relationship observed in the position distribution. These results can be used to test the Bohmian arrival time distribution in a strict manner, i.e., where the semiclassical approximation breaks down. Moreover, our approach to investigating this experiment can be applied to a wide range of phenomena, and it seems that the predicted non-local temporal interference and associated complementary relationship are universal behaviors of entangled quantum systems that may manifest in various phenomena.
Using standard results from statistics, we show that for Gaussian quantum systems the distribution of a time measurement at a fixed position can be directly inferred from the distribution of a position measurement at a fixed time as given by the Born rule. In an application to a quantum particle of mass m falling in a uniform gravitational field g, we use this approach to obtain an exact explicit expression for the probability density of the time of arrival (TOA). In the long time-of-flight approximation, we predict that the average positive relative shift with respect to the classical TOA in case of a zero initial mean velocity is asymptotically given by $\ensuremath{\delta}=\frac{{q}^{2}}{2}$ when the factor $q\ensuremath{\equiv}\frac{\ensuremath{\hbar}}{2m\ensuremath{\sigma}\sqrt{2gx}}\ensuremath{\ll}1$ (semiclassical regime) and by $\ensuremath{\delta}=\sqrt{{\frac{2}{\ensuremath{\pi}}}^{\phantom{\ensuremath{\int}}}}q$ when $q\ensuremath{\gg}1$ (quantum regime), where $\ensuremath{\sigma}$ is the width of the initial Gaussian wave packet and x is the mean distance to the detector. We also discuss experimental conditions under which these predictions can be tested.