In this paper, we present a continuous normalized gradient flow (CNGF) and prove its energy diminishing property, which provides a mathematical justification of the imaginary time method used in the physics literature to compute the ground state solution of Bose--Einstein condensates (BEC). We also investigate the energy diminishing property for the discretization of the CNGF. Two numerical methods are proposed for such discretizations: one is the backward Euler centered finite difference (BEFD) method, the other is an explicit time-splitting sine-spectral (TSSP) method. Energy diminishing for BEFD and TSSP for the linear case and monotonicity for BEFD for both linear and nonlinear cases are proven. Comparison between the two methods and existing methods, e.g., Crank--Nicolson finite difference (CNFD) or forward Euler finite difference (FEFD), shows that BEFD and TSSP are much better in terms of preserving the energy diminishing property of the CNGF. Numerical results in one, two, and three dimensions with magnetic trap confinement potential, as well as a potential of a stirrer corresponding to a far-blue detuned Gaussian laser beam, are reported to demonstrate the effectiveness of BEFD and TSSP methods. Furthermore we observe that the CNGF and its BEFD discretization can also be applied directly to compute the first excited state solution in BEC when the initial data is chosen as an odd function.
The normalized gradient flow, i.e., the gradient flow with discrete normalization (GFDN) introduced in [W. Bao and Q. Du, SIAM J. Sci. Comput., 25 (2004), pp. 1674--1697] or the imaginary time evolution method is one of the most popular techniques for computing the ground states of Bose--Einstein condensates (BECs). In this paper, we revisit the time discretizations for the GFDN and its generalization to the multicomponent BECs. Several widely used time discretizations are demonstrated to be not accurate for computing the ground state solution in the general case, especially for the multicomponent BECs with two or more constraints even for the most accepted linearized backward Euler schemes. More precisely, these schemes usually converge to a solution with an error depending on the time step size. To accurately and efficiently compute the ground state solution of BECs, we propose the gradient flow with Lagrange multiplier (GFLM) method which can be viewed as the modified GFDN by introducing the explicit Lagrange multiplier terms or an approximation of the continuous normalized gradient flow. Through analysis and numerical computation, we clarify that, in order to accurately compute the ground state solution, the GFDN method must be discretized in very special ways in time such as the linearized backward Euler scheme, while the GFLM can be discretized by various schemes and works for multicomponent BEC with multiple constraints. We propose a simple backward-forward Euler scheme to discretize the GFLM, where only a linear system with constant coefficients needs to be solved at each time step. Taking the spin-1 BEC as an example, we demonstrate how the traditional GFDN approach fails in computing the ground state solution of general multicomponent BECs accurately. The proposed GFLM with backward-forward Euler discretization is shown to be accurate and efficient. Extensive numerical results are provided to verify our claims.
Microscopic quantum laws are time-symmetric: nothing in the Schrödinger equation or its relativistic extensions distinguishes future from past. Yet measurements produce irreversible records, an apparently one-way causal flow, and the familiar notion that causes precede effects. Within the Quantum Time-Symmetric Interpretation (QTSI), this asymmetry is not fundamental but emergent. Isolated quantum systems are described by a two-component temporal state containing forward- and backward-propagating amplitudes. Their mixing, governed by a parameter
Δ(ϕ), defines a retrocausal coherence time
τRC(ϕ) beyond which advanced components are suppressed. As the system couples to amplifying environments characterized by a macroscopic parameter
ϕ,
Δ(ϕ) decreases and the backward component is dynamically eliminated, giving rise to classical causality and effective collapse. QTSI aligns naturally with time-symmetric approaches from Wheeler--Feynman, Aharonov, and Price, agrees with all weak-measurement and quantum eraser results in their operational regimes, and predicts specific signatures in temporal echoes and chaotic cavities. Detailed formal and experimental developments appear in the Supplementary Addenda.