Adiabatic quenches and characterization of amplitude excitations in a continuous quantum phase transition

Abstract

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...

Entanglement and extreme spin squeezing of unpolarized states

“…In this paper, we present a general condition that: (i) provides a lower bound on the entanglement depth, (ii) is applicable to spin-j systems, for any j, (iii) works both for spin-squeezed states and Dicke states, and, (iv) is close to provide a tight bound in the sense mentioned above in the large particle number limit. Such a criterion can be applied immediately, for instance, in experiments producing Dicke states in spinor condensates [41].…”

Higgs mode in a strongly interacting fermionic superfluid

“…However, experimental detections have been solely indirect as the Higgs mode does not couple directly to electromagnetic fields owing to the gauge invariance required for its existence. The far-reaching importance of the Higgs mode is further illustrated by its observation in a variety of specially tuned systems such as antiferromagnets 13 , liquid 3 He 14 , ultracold bosonic atoms near the superfluid/Mott-insulator transition 15,16 , spinor Bose gases 17 , and Bose gases strongly coupled to optical fields 18 . In contrast, weakly-interacting Bose-Einstein condensates do not exhibit a stable Higgs mode 6,10,11 .…”

Entanglement certification from theory to experiment

“…In particular, let us emphasise Dicke states in this context, which are attracting increasing attention and are produced (up to the ten of dB of squeezing with respect to a SQL) in experiments with Bose-Einstein condensates [285,301,334,335], with atomic spin-mixing dynamics resembling parametric down-conversion of photons to some extent. A depth of entanglement of several hundreds has been inferred with collective measurements also for these generalised spin-squeezed states [109,285,[336][337][338]. Finally, current experimental efforts have been oriented towards demonstrating entanglement between spatially separated parts of BECs (still in localized traps) [100,302,339].…”