The metric gradient force is the only purely geometric force term in the irreversible equation of motion of the Order Parameter Spacetime Theory. Its acceleration is proportional to the square of the velocity and independent of the velocity direction, making it exclusively distinguishable from all known dissipative forces (which are odd functions of velocity). We perform the first direct test of this prediction using Stern-Gerlach experimental data of Feshbach molecules from a ⁸⁷Rb Bose-Einstein condensate published by the MPQ group in 2003. By extracting the center-of-mass position data of the molecular cloud at 16 time slices from Figure 4 of the experiment, we find that the molecular cloud exhibits simple harmonic oscillation in a 100 G/cm magnetic field gradient, with an oscillation frequency of 56 Hz and an amplitude of 0.68 mm. Comparing the experimental acceleration with the theoretically predicted metric gradient force a_eff = (1/2)·(∂ ln g_ii/∂x)·ẋ², we verify the a ∝ v² scaling relation and deduce an order-parameter-space metric gradient ∂ ln g_ii/∂x ≈ 3.4 × 10³ m⁻¹. The left and right limits of the molecular cloud oscillation are approximately symmetric about the equilibrium position (deviation from center ≈ 0.68 mm), consistent with the prediction of the metric gradient force as an even function of velocity. The 2004 BEC-BCS crossover experiment by the Innsbruck group and the 2003 ⁶Li Fermi gas experiment by the ENS group provide qualitative support for this conclusion. Combined with previous tests of the theory in the cosmological weak-field (BAO and full-shape) and strong-field black hole (EHT) regimes, the Order Parameter Spacetime Theory has now passed observational tests across scales from microscopic cold-atom physics to macroscopic cosmology.
The metric gradient force is the only purely geometric force term in the irreversible equation of motion of the Order Parameter Spacetime Theory. Its acceleration is proportional to the square of the velocity and independent of the velocity direction, making it exclusively distinguishable from all known dissipative forces (which are odd functions of velocity). We perform the first direct test of this prediction using Stern-Gerlach experimental data of Feshbach molecules from a ⁸⁷Rb Bose-Einstein condensate published by the MPQ group in 2003. By extracting the center-of-mass position data of the molecular cloud at 16 time slices from Figure 4 of the experiment, we find that the molecular cloud exhibits simple harmonic oscillation in a 100 G/cm magnetic field gradient, with an oscillation frequency of 56 Hz and an amplitude of 0.68 mm. Comparing the experimental acceleration with the theoretically predicted metric gradient force a_eff = (1/2)·(∂ ln g_ii/∂x)·ẋ², we verify the a ∝ v² scaling relation and deduce an order-parameter-space metric gradient ∂ ln g_ii/∂x ≈ 3.4 × 10³ m⁻¹. The left and right limits of the molecular cloud oscillation are approximately symmetric about the equilibrium position (deviation from center ≈ 0.68 mm), consistent with the prediction of the metric gradient force as an even function of velocity. The 2004 BEC-BCS crossover experiment by the Innsbruck group and the 2003 ⁶Li Fermi gas experiment by the ENS group provide qualitative support for this conclusion. Combined with previous tests of the theory in the cosmological weak-field (BAO and full-shape) and strong-field black hole (EHT) regimes, the Order Parameter Spacetime Theory has now passed observational tests across scales from microscopic cold-atom physics to macroscopic cosmology.
The physical world sometimes manifests as force (irreversible energy transfer), sometimes as geometry (local phase constraints without energy transfer), and sometimes as entanglement (nonlocal phase correlations without energy transfer). This paper proves that these three representations are three gauge projections of a single underlying structure---the order parameter space metric---at different values of the entropy production rate. Starting from CTP (Closed Time Path) nonequilibrium field theory, we rigorously derive the universal form of the order parameter effective action in the infrared slow-dynamics limit. The metric g_ii measures the "irreversible distance" between adjacent points in order parameter space, and its singularity structure exhaustively classifies all physical effects into five projection types: dynamical projection, critical slowing-down, instantaneous equilibrium, frozen projection, and cyclic projection. The variational principle yields a unified irreversible equation of motion containing an exclusive dynamical prediction---the metric gradient force. The information bottleneck theorem rigorously proves that in the frozen projection, the quantum mutual information irreversibly written into the environment is identically zero, yielding the thermodynamic criterion for the quantum-classical boundary. The core contribution of this paper is a unified framework with "information-entropy production rate" as its criterion. It provides four exclusive quantitative predictions testable on platforms such as cold atoms and superconducting qubits. This paper is the third in a series. The microscopic foundation is provided in a companion paper (DOI: 10.5281/zenodo.21325754), and the axiomatic framework is established in a second companion paper (DOI: 10.5281/zenodo.21325460).
We combine the Ramsey interferometry protocol, the Stern-Gerlach detection scheme, and the use of elongated geometry of a cloud of fully polarized cold cesium atoms to measure the selected component of the magnetic-field gradient along the atomic cloud in a single shot. In contrast to the standard method where the precession of two spatially separated atomic clouds is simultaneously measured to extract their phase difference, which is proportional to the magnetic-field gradient, we here demonstrate a gradiometer using a single image of an expanding atomic cloud with the phase difference imprinted along the cloud. Using resonant radio-frequency pulses and Stern-Gerlach imaging, we first demonstrate nutation and Larmor precession of atomic magnetization in an applied magnetic field. Next, we let the cold atom cloud expand in one dimension and apply the protocol for measuring the magnetic-field gradient. The resolution of our single-shot gradiometer is not limited by thermal motion of atoms and has an estimated absolute accuracy below $\ifmmode\pm\else\textpm\fi{}0.2$ mG/cm ($\ifmmode\pm\else\textpm\fi{}20$ nT/cm).
A detailed analysis of the classic Stern–Gerlach experiment is presented. An analytical simple solution is presented for the quantum description of the translational and spin dynamics of a silver atom in a magnetic field with a gradient along a single z-direction. This description is then used to obtain an approximate quantum description of the more realistic case with a magnetic field gradient also in a second y-direction. An explicit relation is derived for how an initial off center deviation in the y-direction affects the final result observed at the detector. This shows that the “mouth shape” pattern at the detector observed in the original Stern–Gerlach experiment is a generic consequence of the gradient in the y-direction. This is followed by a discussion of the spin dynamics during the entry of the silver atom into the magnet. An analytical relation is derived for a simplified case of a field only along the z-direction. A central question for the conceptual understanding of the Stern–Gerlach experiment has been how an initially unpolarized spin ends up in a polarized state at the detector. It is argued that this can be understood with the use of the adiabatic approximation. When the atoms first experience the magnetic field outside the magnet, there is in general a change in the spin state, which transforms from a degenerate eigenstate in the absence of a field into one of two possible non-degenerate states in the field. If the direction of the field changes during the passage through the device, there is a corresponding adiabatic change of the spin state. It is shown that an application of the adiabatic approximation in this way is consistent with the previously derived exact relations.
Abstract Ultra-cold atom experiment in space with microgravity allows for realization of dilute atomic-gas Bose-Einstein condensate (BEC) with macroscopically large occupation number and significantly long condensate lifetime, which allows for a precise measurement on the shape oscillation frequency by calibrating itself over numerous oscillation periods. In this paper, we propose to measure the Newtonian gravitational constant via ultra-cold atom BEC with shape oscillation, although it is experimentally challenging. We also make a preliminary perspective on constraining the modified Newtonian potential such as the power-law potential, Yukawa interaction, and fat graviton. A resolution of frequency measurement of (1−100)nHz ( 1 - 100 ) nHz at most for the occupation number 109 10 9 , just one order above experimentally achievable number N∼106−108 N ∼ 10 6 - 10 8 , is feasible to constrain the modified Newtonian potential with Yukawa interaction greatly beyond the current exclusion limits.
When a gas of bosonic atoms is cooled sufficiently near to absolute zero, it undergoes a startling transition from a collection of randomly-behaved individual atoms to a phase in which thousands, even millions, of atoms act in collective unison. In this new phase, known as a Bose-Einstein condensate (BEC), the atoms all occupy the exact same quantum mechanical ground state and can be collectively described by a single phase-coherent wavefunction of macroscopic size. But how exactly does this harmonious behaviour arise within a collection of incoherent atoms as their temperature decreases? Does the condensate wavefunction grow from a single central nucleus, or does it emerge simultaneously throughout the system? And how does the situation differ when the gas is cooled suddenly, away from thermodynamic equilibrium? Can non-equilibrium behaviour be inherited by the condensate that forms? In this thesis we aim to answer these and other questions by studying the formation dynamics of Bose-Einstein condensates and the development of phase coherence. The research described herein makes use of several theoretical tools, which we lay out on the workbench in Chapter 2: Semi-classical Hartree-Fock theory provides a mean-field description of a condensate and thermal cloud at equilibrium; Quantum kinetic theory is used to calculate the dynamics of condensate growth into an equilibrium state, from a non-equilibrium thermal cloud; The stochastic projected Gross-Pitaevskii equation (SPGPE) employs a classical-field description to encapsulate the non-equilibrium dynamics of the condensate and low-energy region of the atom cloud, coupled to an equilibrium thermal cloud. This thesis is composed of four principal studies, each constituting a separate chapter.In Chapter 3 we study the formation of a Bose-Einstein condensate in a cigar-shaped 3D harmonic trap, induced by the controlled addition of an attractive ``dimple'' potential along the weak axis in a recent experiment [Garrett et al., Phys. Rev. A 83, 013630 (2011)]. In this manner condensation is induced without cooling due to a localized increase in the phase space density. We perform a quantitative analysis of the thermodynamic transformation in both the sudden and adiabatic regimes for a range of dimple widths and depths. We find good agreement with equilibrium calculations based on self-consistent semi-classical Hartree-Fock theory describing the condensate and thermal cloud. We observe there is an optimal dimple depth that results in a maximum in the condensate fraction. We also study the non-equilibrium dynamics of condensate formation in the sudden turn-on regime, finding good agreement for the observed time dependence of the condensate fraction with calculations based on quantum kinetic theory. In Chapter 4 we study the onset of phase coherence during the formation of a BEC following a sudden quench to below the transition temperature. Using the SPGPE, we simulate a recent experiment performed at ANU in which a sudden RF quench is applied to a dilute gas of metastable Helium atoms confined in an elongated 3D harmonic magnetic trap. At regularly-spaced intervals following the quench, broadband RF pulses are applied to homogeneously outcouple small (<5%) pulses of atoms, which expand and fall to a microchannel plate detector. The second-order correlation function at the detector, postulated to provide a measure of phase coherence of the in-situ cloud from which they were outcoupled, is measured versus time and compared with the condensate growth curve. A delay was observed in the relaxation of the second-order correlation function, relative to the peak in the number of condensate atoms. In our simulations we incorporate the effects of periodic outcoupling both the condensate and thermal cloud, and monitor the time-dependence of first- and second-order correlation functions in both position and momentum space. We observe long-range phase coherence in the final equilibrium state, though mild persistent phase fluctuations are present. We also observe the curious phenomenon of ``superbunching'' - an enhancement of density fluctuations - at specific values of the momentum, which we relate to anomalous correlations of fluctuations. Despite obtaining a good quantitative fit to the experimentally-measured condensate growth curve in our SPGPE simulations, we do not observe any evidence of the delay in coherence observed in the experiment, which remains unexplained. In Chapter 5 we use the SPGPE to study the crossover from a true BEC with long-range phase coherence to a quasicondensate dominated by phase fluctuations, for the case of a Bose gas in an highly-elongated 3D harmonic trap. We use parameters corresponding to an experiment in which a sudden quench resulted in the formation of a non-equilibrium condensate, the dynamics of which we study in the chapter that follows. Sweeping through a range of temperatures and chemical potentials within the crossover region, we reveal the behaviour of phase fluctuations through analysis of correlation functions. We compare the ideal-gas approximation for the transition temperature with the more accurate value obtained via the Binder cumulant. By comparing the phase coherence length with the spatial extent across which density fluctuations are suppressed, we identify the BEC/quasicondensate crossover point. We compare directly with predictions of mean-field theory [Petrov et al., Phys. Rev. Lett. 87, 050404 (2001)] and find that phase fluctuations persist to lower temperatures in our SPGPE simulations. We calculate anomalous correlations of the fluctuation field and show that the resulting correction to the mean-field condensate self-interaction energy is significant within the quasicondensate regime. In Chapter 6 we study the formation of a BEC into a non-equilibrium state by simulating an experiment in which a dilute gas of Rubidium-87 atoms in a highly-elongated 3D harmonic magnetic trap is rapidly quenched to below the transition temperature. Following the quench, a condensate is observed to form with an excessively elongated shape, and subsequently undergo damped quadrupole shape oscillations [Shvarchuck et al., Phys. Rev. Lett. 89, 270404 (2002)]. We simulate this experiment by modifying the SPGPE to incorporate a hydrodynamic thermal cloud, parametrized by axially and temporally-dependent temperature and chemical potential. Comparison against simulations without this axial dependence suggests the hydrodynamic behaviour of the thermal cloud plays a crucial role in the formation of a non-equilibrium condensate. Analyses of the first-order correlation function and the condensate wavefunction reveal the spontaneous formation of quantum vortices, as well as strong phase fluctuations at equilibrium indicating quasicondensate behaviour.
The Irreversible Equivalence Principle of the Order Parameter Spacetime Theory proposes a central assertion: dissipation can be geometrized as the imaginary part of a complex metric. But is this geometrization a physical reality or merely a mathematical trick? This paper answers this question by designing an experimental scheme on a cold-atom interferometer platform to simultaneously measure the geometric offset of interference fringes and the visibility decay, testing whether the real and imaginary parts of the complex metric exist as two components of the same complex number. Furthermore, we predict that when the external driving frequency equals the local relaxation rate, the system enters an irreversible geometric resonance state, and the energy absorption rate exhibits an anomalously enhanced resonance peak. If these two predictions are experimentally confirmed, they will constitute irrefutable evidence that dissipation is not just "geometric bending in appearance", but is physically complex geometric bending. We also systematically comb through traces in existing cold-atom experiments that narrowly missed these signals. If both predictions are verified, it directly proves that the Irreversible Equivalence Principle possesses physical reality. If falsified, the principle must be weakened from "dissipation = complex geometric bending" to "dissipation can be described by complex geometry".