Backflow is the phenomenon that the probability current of a quantum particle on the line can flow in the direction opposite to its momentum. In this article, previous investigations of backflow, pertaining to interaction-free dynamics or purely kinematical aspects, are extended to scattering situations in short-range potentials. It is shown that backflow is a universal quantum effect which exists in any such potential, and is always of bounded spatial extent in a specific sense. The effects of reflection and transmission processes on backflow are investigated, both analytically for general potentials, and numerically in various concrete examples.
We analyse the quantum backflow effect and extend it, as a limiting constraint to its spatial extent, for scattering situations in the presence of a purely transmitting discontinuous jump-defect. Analytical and numerical comparisons are made with a different situation in which a defect is represented by a $δ$ function potential. Furthermore, we make the analysis compatible with conservation laws.
In this paper we discuss relativistic quantum backflow. The general theory of relativistic backflow is written down and it is shown that the backflow can be written as a function of a simple parameter which is defined in terms of fundamental constants and the backflow period. Backflow eigenfunctions are determined numerically for a range of values of and an explicit expression for the relativistic backflow eigenvalue in terms of the non-relativistic backflow constant is presented. Then backflow eigenvectors are fitted with some standard functions which lead to substantially higher backflow tha…
Measurable quantities that have positive values in classical dynamical systems need not to be positive in quantum theory. For example, consider a free quantum mechanical particle in one dimension. There are quantum states in which the particle's velocity is positive with probability 1, but where the probability flux for its position is locally negative; that is, while its velocity points to the right, the particle travels to the left. These effects are however small and limited in space and time by certain lower bounds, which are called "quantum inequalities". Similar effects also appear for a…
The decay of quasi-stable quantum system involves primarily an outgoing probability current density. However, during the transition from exponential to inverse-power-law decay there are time intervals during which this current, although small, is inward. In this paper this inward flow is associated with quantum backflow. Furthermore substantial backflow exists for time-evolving free wave packets which are initially confined in space.
In the dynamics of open quantum systems, the backflow of information to the reduced system under study has been suggested as the actual physical mechanism inducing memory and thus leading to non-Markovian quantum dynamics. To this aim, the trace-distance or Bures-distance revivals between distinct evolved system states have been shown to be subordinated to the establishment of system-environment correlations or changes in the environmental state. We show that this interpretation can be substantiated also for a class of entropic quantifiers. We exploit a suitably regularized version of Umegaki'…
In its standard formulation, quantum backflow is a classically impossible phenomenon in which a free quantum particle in a positive-momentum state exhibits a negative probability current. Recently, Miller et al. [Quantum 5, 379 (2021)] have put forward a new, "experiment-friendly" formulation of quantum backflow that aims at extending the notion of quantum backflow to situations in which the particle's state may have both positive and negative momenta. Here, we investigate how the experiment-friendly formulation of quantum backflow compares to the standard one when applied to a free particle i…
Quantum backflow is usually understood as a quantum interference phenomenon where probability current of a quantum particle points in the opposite direction to particle's momentum. Here, we quantify the amount of quantum backflow for arbitrary momentum distributions, paving the way towards its experimental verification. We give examples of backflow in gravitational and harmonic potential, and discuss experimental procedures required for the demonstration using atomic gravimeters. Such an experiment would show that the probability of finding a free falling particle above initial level could gro…
General quantum backflow in realistic wave packets
School of Mathematics and Physics, Xiamen University Malaysia（马来西亚厦门大学数学与物理学院）；Institute of Theoretical Physics and Astrophysics, Faculty of Mathematics, Physics and Informatics, University of Gdańsk（格但斯克大学数学、物理与信息学学院理论物理与天体物理研究所）；Section of Mathematics, University of Geneva（日内瓦大学数学系）；Quantum Physics Corner Ltd（量子物理角有限公司）
Free motion of a quantum particle with the wave function entirely comprised of plane waves with non-negative momenta may be accompanied by negative probability current, an effect called quantum backflow. The effect is weak and fragile, and has not yet been observed experimentally. Here we show that quantum backflow becomes significantly more pronounced and more amenable to experimental observation if, instead of letting the particle move along a straight line, one forces it to move in a circular ring.
Quantum backflow is the classically-forbidden effect pertaining to the fact that a particle with a positive momentum may exhibit a negative probability current at some space-time point. We investigate how this peculiar phenomenon extends to many-particle systems. We give a general formulation of quantum backflow for systems formed of $N$ free nonrelativistic structureless particles, either identical or distinguishable. Restricting our attention to bosonic systems where the $N$ identical bosons are in the same one-particle state allows us in particular to analytically show that the maximum achi…
Closed-form, normalizable solutions of Dirac's equation propagating within a semi-infinite cylindrical waveguide are obtained in terms of ordinary and modified Bessel functions. These relativistic wave packets induce quantum backflow on a cross-section of the cylinder at practically any distance along the waveguide, becoming spin-polarized in the nonrelativistic limit. The predicted backflow is stable in time and is manifest regardless of the initial wave function.
We study the quantum backflow problem in the noncommutative plane. In particular, we have considered a charged particle with and without an oscillator interaction with noncommuting momentum operators and examined angular momentum backflow in each case and how they differ from each other. We also propose a probability associated with the occurence of angular momentum backflow and investigate whether or not the probability depends on a physical parameter, namely the magnetic field.
Quantum backflow is an interference effect in which a matter-wave packet comprised of only plane waves with non-negative momenta exhibits negative probability flux. Here we show that this effect is mathematically equivalent to the appearance of classically-forbidden probability flux when a matter-wave packet, initially confined to a semi-infinite line, expands in free space.
We analyze the relation between CP-divisibility and the lack of information backflow for an arbitrary -- not necessarily invertible -- dynamical map. It is well known that CP-divisibility always implies lack of information backflow. Moreover, these two notions are equivalent for invertible maps. In this letter it is shown that for a map which is not invertible the lack of information backflow always implies the existence of completely positive (CP) propagator which, however, needs not be trace-preserving. Interestingly, for a {\em wide class of image non-increasing dynamical maps} this propaga…
We analyze recent approaches to quantum Markovianity and how they relate to the proper definition of quantum memory. We point out that the well-known criterion of information backflow may not correctly report character of the memory falsely signaling its quantumness. Therefore, as a complement to the well-known criteria, we propose several concepts of elementary dynamical maps. Maps of this type do not increase distinguishability of states which are indistinguishable by von Neumann measurements in a given basis. Those notions and convexity allows us to define general classes of processes witho…
In this work, dissipative quantum backflow is studied for a superposition of two stretched Gaussian wave packets and two identical spinless particles within the Caldirola-Kanai framework. Backflow is mainly an interference process and dissipation is not able to suppress it in the first case. For two identical spinless particles, apart from interference terms, the symmetry of the wave function seems to be crucial in this dynamics. The combined properties make bosons display this effect, even for the dissipative regime but, for fermions, backflow is not exhibited in any regime, dissipative and n…
The quantum backflow effect is a counterintuitive behavior of the probability current of a free particle, which may be negative even for states with vanishing negative momentum component. Here we address the notion of nonclassicality arising from the backflow effect, i.e. from the negativity of the probability current, and analyze its relationships with the notion of nonclassicality based on the negativity of the Wigner function. Our results suggest that backflow is linked to a different, and in fact more restrictive, notion of nonclassicality, the negativity of the Wigner function being only …
Quantum backflow is a classically forbidden effect consisting in a negative flux for states with negligible negative momentum components. It has never been observed experimentally so far. We derive a general relation that connects backflow with a critical value of the particle density, paving the way for the detection of backflow by a density measurement. To this end, we propose an explicit scheme with Bose-Einstein condensates, at reach with current experimental technologies. Remarkably, the application of a positive momentum kick, via a Bragg pulse, to a condensate with a positive velocity m…
We present an introduction to the backflow effect in quantum mechanics -- the phenomenon in which a state consisting entirely of positive momenta may have negative current and the probability flows in the opposite direction to the momentum. We show that the effect is present even for simple states consisting of superpositions of gaussian wave packets, although the size of the effect is small. Inspired by the numerical results of Penz et al, we present a wave function whose current at any time may be computed analytically and which has periods of significant backflow, with a backwards flux equa…
It was known that a free, nonrelativistic particle in a superposition of positive momenta can, in certain cases, bear a negative probability current --- hence termed quantum backflow. Here, it is shown that more variations can be brought about for a free Dirac particle, particularly when negative-energy solutions are taken into account. Since any Dirac particle can be understood as an antiparticle that acts oppositely (and vice versa), quantum backflow is found to arise in the superposition (i) of a well-defined momentum but different signs of energies, or more remarkably (ii) of different sig…
We investigate the backflow effect in elementary quantum mechanics - the phenomenon in which a state consisting entirely of positive momenta may have negative current and the probability flows in the opposite direction to the momentum. We compute the current and flux for states consisting of superpositions of gaussian wave packets. These are experimentally realizable but the amount of backflow is small. Inspired by the numerical results of Penz et al (M.Penz, G.Grübl, S.Kreidl and P.Wagner, J.Phys. A39, 423 (2006)), we find two non-trivial wave functions whose current at any time may be comput…
Dissipative backflow is studied in the context of open quantum systems. This theoretical analysis is carried out within two frameworks, the effective time-dependent Hamiltonian due to Caldirola-Kanai (CK) and the Caldeira-Leggett (CL) one where a master equation is used to describe the reduced density matrix in presence of dissipation and temperature of the environment. Two examples are considered, the free evolution of one and two Gaussian wave packets as well as the time evolution under a constant field. Backflow is shown to be reduced with dissipation and temperature but never suppressed. I…
We present an exhaustive class of states with quantum backflow -- the phenomenon in which a state consisting entirely of positive momenta may have negative current and the probability flows in the opposite direction to the momentum. They are characterized by a general function of momenta subject to very weak conditions. Such a family of states is of interest in the light of a recent experimental proposal to measure backflow. We find one particularly simple state which has surprisingly large backflow -- about 41 percent of the lower bound on flux derived by Bracken and Melloy. We study the eige…
We consider a two-level open quantum system undergoing either pure dephasing, dissipative, or multiply decohering dynamics and show that, whenever the dynamics is non-Markovian, the initial speed of evolution is a monotonic function of the relevant physical parameter driving the transition between the Markovian and non-Markovian behaviour of the dynamics. In particular, within the considered models, a speed increase can only be observed in the presence of backflow of information from the environment to the system.
We show that, contrary to the statements made by many authors, the backflow is not a nonclassical effect. The backflow is a characteristic feature of solutions of the wave equations: quantum and classical. We present simple solutions of the Dirac equation, Maxwell equations and equations of linearized gravity where the backflow phenomenon is clearly seen. In this work we describe backflow in relativistic theories but this phenomenon can occur in the solutions of all kinds of wave equations: quantum and classical.
We address the dynamics of quantum correlations for a bipartite continuous-variable quantum system interacting with its fluctuating environment. In particular, we consider two independent quantum oscillators initially prepared in a Gaussian state, e.g. a squeezed thermal state, and compare the dynamics resulting from local noise, i.e. oscillators coupled to two independent external fields, to that originating from common noise, i.e. oscillators interacting with a single common field. We prove non-Markovianity (non-divisibility) of the dynamics in both regimes and analyze the connections betwee…
It is known that for a non-relativistic quantum particle traveling freely on the $x$-axis, the positional probability can flow in the opposite direction to the particle's velocity. The maximum possible amount of such backflow that can occur over any time interval has been determined previously as the largest positive eigenvalue of a certain hermitian observable, with the value $0.0384517\dots$, or about $4\%$ of the total probability on the line. The eigenvalue problem is now considered numerically in the more general case of states with momentum restricted to the range $p_0<p<\infty$, for any…
Magnetic tuning of quantum backflow: Schrödinger vs Pauli system
Faculty of Mathematics and Natural Sciences, College of Sciences, Institute of Physical Sciences, Cardinal Stefan Wyszynski University in Warsaw（华沙卡罗尔·斯特凡·维希尼茨基大学理学院自然科学系物理科学研究所）
We present and motivate an efficient way to include orbital dependent many--body correlations in trial wave function of real--space Quantum Monte Carlo methods for use in electronic structure calculations. We apply our new orbital--dependent backflow wave function to calculate ground state energies of the first row atoms using variational and diffusion Monte Carlo methods. The systematic overall gain of correlation energy with respect to single determinant Jastrow-Slater wave functions is competitive with the best single determinant trial wave functions currently available. The computational c…
In its original formulation, quantum backflow (QB) is an interference effect that manifests itself as a negative probability transfer for free-particle states comprised of plane waves with only positive momenta. Quantum reentry (QR) is another interference effect in which a wave packet expanding from a spatial region of its initial confinement partially returns to the region in the absence of any external forces. Here we show that both QB and QR are special cases of a more general classically-forbidden probability flow for quantum states with certain position-momentum correlations. We further …
Spin-induced and orbital quantum backflow in Pauli theory
Faculty of Mathematics and Natural Sciences, College of Sciences, Institute of Physical Sciences, Cardinal Stefan Wyszynski University in Warsaw（华沙卡罗尔·斯特凡·维希尼茨基大学理学院自然科学系物理科学研究所）
The phase synchronization of a single qubit in a dissipative bath in the absence of driving field is demonstrated. Using the Husimi $Q$-function we show that the phase preference is present in the long time limit only during non-Markovian evolution with a finite detuning. This happens due to the information backflow signifying that non-Markovianity is a resource for quantum synchronization. To quantify synchronization we use the shifted phase distribution as well as its maximal value. From the maximal value of the shifted phase distribution we observe the signatures of quantum synchronization …
The interpretation of non-Markovian effects as due to the information exchange between an open quantum system and its environment has been recently formulated in terms of properly regularized entropic quantities, as their revivals in time can be upper bounded by means of quantities describing the storage of information outside the open system [Phys. Rev. Lett. 127, 030401 (2021)]. Here, we elaborate on the wider mathematical framework of the theory, specifying the key properties that allow us to associate distinguishability quantifiers with the information flow from and towards the open system…
The degree of non-Markovianity allows to characterizing quantum evolutions that depart from a Markovian regime in a similar way as Schmidt number measures the degree of entanglement of pure states. Maximally non-Markovian dynamics are the analogous of maximally entangled states [D. Chruściński and S. Maniscalco, Phys. Rev. Lett. 112, 120404 (2014)]. Here, we demonstrate that there exists a class of maximally non-Markovian quantum evolutions where the associated environment (degrees of freedom not belonging to the system) obeys a Markovian (memoryless) dynamics, which in turn is unperturbed by …
Quantum memory effects can be qualitatively understood as a consequence of an environment-to-system backflow of information. Here, we analyze and compare how this concept is interpreted and implemented in different approaches to quantum non-Markovianity. We study a non-operational approach, defined by the istinguishability between two system states characterized by different initial conditions, and an operational approach, which is defined by the correlation between different outcomes associated to successive measurement processes performed over the system of interest. The differences, limitat…
Any realistic quantum system is inevitably subject to an external environment. This environment makes the open-system dynamics significant for many quantum tech-nologies, such as entangled-state engineering, quantum simulation, and quantum sensing. The information flow of a system to its environment usually induces a Markovian process, while the backflow of information from the environment exhibits non-Markovianity. The practical environment, usually consisting of a large number of degrees of freedom, is hard to control, despite some attempts on controllable transitions from Markovian to non-M…
Backflow is a counter-intuitive phenomenon in which a forward propagating quantum particle propagates locally backwards. The actual counter-propagation property associated with this delicate interference phenomenon has not been observed to date in any field of physics. Here, we report the observation of transverse optical backflow where a beam of light propagating to a specific transverse direction is measured locally to propagate in the opposite direction. This observation is relevant to any physical system supporting coherent waves and might lead to unique applications.
We experimentally emulate, in a controlled fashion, the non-Markovian dynamics of a pure dephasing spin-boson model at zero temperature. Specifically, we use a randomized set of external radio-frequency fields to engineer a desired noise power-spectrum to effectively realize a non-Markovian environment for a single NMR qubit. The information backflow, characteristic to the non-Markovianity, is captured in the nonmonotonicity of the decoherence function and von Neumann entropy of the system. Using such emulated non-Markovian environments, we experimentally study the efficiency of the Carr-Purce…
A Markovian process of a system is defined classically as a process in which the future state of the system is fully determined by only its present state, not by its previous history. There have been several measures of non-Markovianity to quantify the degrees of non-Markovian effect in a process of an open quantum system based on information backflow from the environment to the system. However, the condition for the witness of the system information backflow does not coincide with the classical definition of a Markovian process. Recently, a new measure with a condition that coincides with the…
By employing the full counting statistics formalism, we characterize the first moment of energy that is exchanged during a generally non-Markovian evolution in non-driven continuous variables systems. In particular, we focus on the evaluation of the energy flowing back from the environment into the open quantum system. We apply these results to the quantum Brownian motion, where these quantities are calculated both analytically, under the weak coupling assumption, and numerically also in the strong coupling regime. Finally, we characterize the non-Markovianity of the reduced dynamics through a…
Transient energy backflow enhanced ergotropy in an open qubit quantum battery assisted by an auxiliary oscillator
Center for Quantum Materials and Computational Condensed Matter Physics, Faculty of Science, Kunming University of Science and Technology（昆明理工大学理学院量子材料与计算凝聚态物理中心）
