We observe power-law scaling of the temporal onset of excitations with quench speed in the neighborhood of the quantum phase transition between the polar and broken-axisymmetry phases in a small spin-1 ferromagnetic Bose-Einstein condensate. As the system is driven through the quantum critical point by tuning the Hamiltonian, the vanishing energy gap between the ground state and first excited state causes the reaction time scale of the system to diverge, preventing it from adiabatically following the ground state. We measure the temporal evolution of the spin populations for different quench speeds and determine the exponents characterizing the scaling of the onset of excitations, which are in good agreement with the predictions of the Kibble-Zurek mechanism.In a second-order (or continuous) quantum phase transition (QPT), a qualitative change in the system's ground state occurs at zero temperature when a parameter in the Hamiltonian is varied across a quantum critical point (QCP) [1]. Near the critical point of the transition, the time scale characterizing the dynamics of a system diverges, and the scaling of this divergence with respect to the quench speed through the phase transition is characterized by universal critical exponents. The Kibble-Zurek mechanism (KZM) as originally formulated characterizes the formation of topological defects when a system undergoes a continuous phase transition at a finite rate. This concept was first conceived by Kibble in his study on topology of cosmic domains and strings in the early universe [2,3], and it was later extended by Zurek [4][5][6] who suggested applying these symmetry breaking ideas to condensed matter systems, such as superconductors and superfluids. This seminal work was followed by many theoretical studies applying the KZM to cosmology, condensed matter, cold atoms and more [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24]. In parallel, the KZM has been studied experimentally and verified in a large variety of systems, including liquid crystals [25,26] [33]. There has also been significant interest in the KZM in the cold atom community. In recent years, it has been observed in ion chains [34][35][36][37], in atomic gases in optical lattices [38], and in Bose-Einstein condensates (BECs), through the formation of spatial domains during condensation [39,40], vortices [41,42], creation of solitons [43] and supercurrents [44]. Only a few experiments have explored the KZM using QPTs (i.e. at zero temperature), namely an investigation of the Mott insulator to superfluid transition [45] and, in a recent preprint [46], an ion chain cooled to the ground state. There has been related work investigating universal scaling in optical lattices [47] and recently in the miscible-immiscible transition in a two-component Bose gas [48].A ferromagnetic spin-1 ( 87 Rb) BEC exhibits a QPT between a symmetric polar (P) phase and a brokenaxisymmetry (BA) phase [20,49] due to the competition between magnetic and collisional spin interaction energies. There have be...

Spontaneous symmetry breaking occurs in a physical system whenever the ground state does not share the symmetry of the underlying theory, e.g., the Hamiltonian. This mechanism gives rise to massless Nambu-Goldstone modes and massive AndersonHiggs modes. These modes provide a fundamental understanding of matter in the Universe and appear as collective phase or amplitude excitations of an order parameter in a many-body system. The amplitude excitation plays a crucial role in determining the critical exponents governing universal nonequilibrium dynamics in the Kibble-Zurek mechanism (KZM). Here, we characterize the amplitude excitations in a spin-1 condensate and measure the energy gap for different phases of the quantum phase transition. At the quantum critical point of the transition, finite-size effects lead to a nonzero gap. Our measurements are consistent with this prediction, and furthermore, we demonstrate an adiabatic quench through the phase transition, which is forbidden at the mean field level. This work paves the way toward generating entanglement through an adiabatic phase transition.adiabatic quenches | amplitude excitations | quantum phase transition T he amplitude mode and phase mode describe two distinct excitation degrees of freedom of a complex order parameter ψ = Ae iϕ appearing in many quantum systems such as the order parameter of the Ginzburg-Laudau superconducting phase transition (1) and the two-component quantum field of the NambuGoldstone-Anderson-Higgs matter field model (2-5). In a zerodimensional system of an interacting spin-1 condensate, the transverse spin, S ⊥ , plays the role of an order parameter in the quantum phase transition (QPT) with S ⊥ being zero in the polar (P) phase and nonzero in the broken axisymmetry (BA) phase (Fig. 1A). Representing the transverse spin vector as a complex number, S ⊥ = S x + iS y , with the real and imaginary parts being expectation values of spin-1 operators, the amplitude mode corresponds to the amplitude oscillation of S ⊥ .The amplitude mode can be studied in different spinor phases by tuning the relative strengths of the quadratic Zeeman energy per particle q ∝ B 2 and spin interaction energy c of the condensate (6) by varying the magnetic field strength B (Fig. 1). In the P phase, both the effective spinor potential energy V and the ground state (GS) spin vector have SO(2) rotational symmetry about the vertical axis (Fig. 1A), and there are two degenerate collective amplitude modes along the radial directions about the GS located at the bottom of the parabolic bowl. These amplitude excitations are gapped modes, which vary both the amplitude of S ⊥ and the energy.In the BA phase, the effective spinor potential energy V acquires a Mexican-hat shape with the GS occupying the minimal energy ring of radius ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4c 2 − q 2 p =ð2jcjÞ. The GS spin vector, S ⊥ (orange arrow in Fig. 1A), spontaneously breaks the SO(2) symmetry and acquires a definite direction (7,8). This broken symmetry induces a massless Na...

Background Microcomputed Tomography (μCT) is an efficient method for quantifying the density and mineralization of mandibular microarchitecture. Conventional radiomorphometrics such as Bone and Tissue Mineral Density are useful in determining the average, overall mineral content of a scanned specimen; however, solely relying on these metrics has limitations. Utilizing Bone Mineral Density Distribution (BMDD), the complex array of mineralization densities within a bone sample can be portrayed. This information is particularly useful as a computational feature reflective of the rate of bone turnover. Here we demonstrate the utility of BMDD analyses in the rat mandible and generate a platform for further exploration of mandibular pathology and treatment. Methods Male Sprague Dawley rats (n=8) underwent μCT and histogram data was generated from a selected volume of interest. A standard curve was derived for each animal and reference criteria were defined. An average histogram was produced for the group and descriptive analyses including the means and standard deviations are reported for each of the normative metrics. Results Mpeak (3444 Hounsfield Units, SD =138) and Mwidth (2221 Hounsfield Units SD =628) are two metrics demonstrating reproducible parameters of BMDD with minimal variance. A total of eight valuable metrics quantifying biologically significant events concerning mineralization are reported. Conclusion Here we quantify the vast wealth of information depicted in the complete spectrum of mineralization established by the BMDD analysis. We demonstrate its potential in delivering mineralization data that encompasses and enhances conventional reporting of radiomorphometrics. Moreover, we explore its role and translational potential in craniofacial experimentation.

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