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arXiv:1609.04502 · Journal of Physics B Atomic Molecular and Optical Physics

Catastrophes in non-equilibrium many-particle wave functions: universality and critical scaling

J. Mumford, W. Kirkby, Duncan H J O’Dell

Published Jan 4, 2017• 15 citations
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Abstract

Abstract As part of the quest to uncover universal features of quantum dynamics, we study catastrophes that form in simple many-particle wave functions following a quench, focusing on two-mode systems that include the two-site Bose–Hubbard model, and under some circumstances optomechanical systems and the Dicke model. When the wave function is plotted in Fock space certain characteristic shapes, that we identify as cusp catastrophes, appear under generic conditions. In the vicinity of a cusp the wave function takes on a universal structure described by the Pearcey function and obeys scaling relations which depend on the total number of particles N . In the thermodynamic limit ( N → ∞ ) the cusp becomes singular, but at finite N it is decorated by an interference pattern. This pattern contains an intricate network of vortex–antivortex pairs, initiating a theory of topological structures in Fock space. In the case where the quench is a δ -kick the problem can be solved analytically and we obtain scaling exponents for the size and position of the cusp, as well as those for the amplitude and characteristic length scales of its interference pattern. Finally, we use these scalings to describe the wave function in the critical regime of a Z 2 symmetry-breaking dynamical phase transition.

DOI
10.1088/1361-6455/aa56af

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As part of the quest to uncover universal features of quantum dynamics, we study catastrophes that form in simple many-body wave functions after a quench, focusing on two-mode systems that include the two-site Bose Hubbard model, and under some circumstances optomechanical systems and the Dicke model. When the wave function is plotted in Fock space plus time certain characteristic structures generically appear that we identify as cusp caustics. In the vicinity of such a catastrophe the wave function takes on a universal form described by the Pearcey function and obeys scaling relations which depend on the total number of particles $N$. In the thermodynamic limit ($N \rightarrow \infty$) the cusp becomes singular, but at finite $N$ it is decorated by an interference pattern. This pattern contains an intricate network of vortex-antivortex pairs, initiating a theory of topological structures in Fock space. In the case where the quench takes the form of a $\delta$-kick we show how to analytically map the wave function onto the Pearcey function and hence obtain the scaling exponents for the size and position of the cusp, as well as those for the amplitude and characteristic length scales of its interference pattern.

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Abstract As part of the quest to uncover universal features of quantum dynamics, we study catastrophes that form in simple many-particle wave functions following a quench, focusing on two-mode systems that include the two-site Bose–Hubbard model, and under some circumstances optomechanical systems and the Dicke model. When the wave function is plotted in Fock space certain characteristic shapes, that we identify as cusp catastrophes, appear under generic conditions. In the vicinity of a cusp the wave function takes on a universal structure described by the Pearcey function and obeys scaling relations which depend on the total number of particles N . In the thermodynamic limit ( N → ∞ ) the cusp becomes singular, but at finite N it is decorated by an interference pattern. This pattern contains an intricate network of vortex–antivortex pairs, initiating a theory of topological structures in Fock space. In the case where the quench is a δ -kick the problem can be solved analytically and we obtain scaling exponents for the size and position of the cusp, as well as those for the amplitude and characteristic length scales of its interference pattern. Finally, we use these scalings to describe the wave function in the critical regime of a Z 2 symmetry-breaking dynamical phase transition.
As part of the quest to uncover universal features of quantum dynamics, we study catastrophes that form in simple many-body wave functions after a quench, focusing on two-mode systems that include the two-site Bose Hubbard model, and under some circumstances optomechanical systems and the Dicke model. When the wave function is plotted in Fock space plus time certain characteristic structures generically appear that we identify as cusp caustics. In the vicinity of such a catastrophe the wave function takes on a universal form described by the Pearcey function and obeys scaling relations which depend on the total number of particles NNN. In the thermodynamic limit (N→∞N \rightarrow \inftyN→∞) the cusp becomes singular, but at finite NNN it is decorated by an interference pattern. This pattern contains an intricate network of vortex-antivortex pairs, initiating a theory of topological structures in Fock space. In the case where the quench takes the form of a δ\deltaδ-kick we show how to analytically map the wave function onto the Pearcey function and hence obtain the scaling exponents for the size and position of the cusp, as well as those for the amplitude and characteristic length scales of its interference pattern.
This thesis explores the effects singularities have on stationary and dynamical properties of many-body quantum systems. In papers I and II we find that the ground state suffers a Z2 symmetry breaking phase transition (PT) when a single impurity is added to a Bose-Einstein condensate (BEC) in a double well (bosonic Josephson junction). The PT occurs for a certain value of the BEC-impurity interaction energy, Λc . A result of the PT is the mean-field dynamics undergo chaotic motion in phase space once the symmetry is broken. We determine the critical scaling exponents that characterize the divergence of the correlation length and fidelity susceptibility at the PT, finding that the BEC-impurity system belongs to the same universality class as the Dicke and Lipkin-Meshkov-Glick models (which also describe symmetry breaking PTs in systems of bosons). In paper III we study the dynamics of a generic two-mode quantum field following a quench where one of the terms in the Hamiltonian is flashed on and off. This model is relevant to BECs in double wells as well as other simple many-particle systems found in quantum optics and optomechanics. We find that when plotted in Fock-space plus time, the semiclassical wave function develops prominent cusp-shaped structures after the quench. These structures are singular in the classical limit and we identify them as catastrophes (as described by the Thom-Arnold catastrophe theory) and show that they arise from the coalescence of classical (mean-field) trajectories in a path integral description. Furthermore, close to the cusp the wave function obeys a remarkable set of scaling relations signifying these structures as examples of universality in quantum dynamics. Within the cusp we find a network of vortex-antivortex pairs which are phase singularities caused by interference. When the mean-field Hamiltonian displays a Z2 symmetry breaking PT modelled by the Landau theory of PTs we calculate scaling exponents describing how the separation distance between the members of each pair diverges as the PT is approached. We also find that the cusp becomes infinitely stretched out at the PT due to critical slowing down. In paper IV we investigate in greater detail the morphology of the vortex network found within cusp catastrophes in many-body wave functions following a quench. In contrast to the cusp catastrophes studied so far in the literature, these structures live in Fock space which is fundamentally granular. As such, these cusps represent a new iii type of catastrophe, which we term a ‘quantum catastrophe’. The granularity of Fock space introduces a new length scale, the quantum length lq = N −1 which effectively removes the vortex cores. Nevertheless, a subset of the vortices persist as phase singularities as can be shown by integrating the phase of the wave function around circuits in Fock-space plus time. Whether or not the vortices survive in a quantum catastrophe is governed by the separation of the vortex-antivortex pairs lv ∝ N −3/4 in comparison to lq , i.e. they survive if lv lq . When particle numbers are reached such that lq ≈ lv the vortices annihilate in pairs.
Caustics occur widely in dynamics and take on shapes classified by catastrophe theory. At finite wavelengths they produce interference patterns containing networks of vortices (phase singularities). Here we investigate caustics in quantized fields, focusing on the collective dynamics of quantum spins. We show that, following a quench, caustics are generated in the Fock space amplitudes specifying the many-body configuration and which are accessible in experiments with cold atoms, ions, or photons. The granularity of quantum fields removes all singularities, including phase singularities, converting point vortices into nonlocal vortices that annihilate in pairs as the quantization scale is increased. Furthermore, the continuous scaling laws of wave catastrophes are replaced by discrete versions. Such "quantum catastrophes" are expected to be universal dynamical features of quantized fields.
We describe a new class of nonequilibrium quantum many-body phenomena in the form of networks of caustics that dominate the many-body wave function in the semiclassical regime following a sudden quench. It includes the light cone-like propagation of correlations as a particular case. Caustics are singularities formed by the birth and death of waves and form a hierarchy of universal patterns whose natural mathematical description is via catastrophe theory. Examples in classical waves range from rainbows and gravitational lensing in optics to tidal bores and rogue waves in hydrodynamics. Quantum many-body caustics are discretized by second quantization (``quantum catastrophes'') and live in Fock space, which can potentially have many dimensions. We illustrate these ideas using the Bose Hubbard dimer and trimer models, which are simple enough that the caustic structure can be elucidated from first principles and yet run the full range from integrable to nonintegrable dynamics. The dimer gives rise to discretized versions of fold and cusp catastrophes whereas the trimer allows for higher catastrophes including the codimension-3 hyperbolic and elliptic umbilics, which are organized by, and projections of, an 8-dimensional corank-2 catastrophe known as X9{X}_{9}X9​. These results describe a hitherto unrecognized form of universality in quantum dynamics organized by singularities that manifest as strong fluctuations in mode population probabilities.
One of the most powerful and prized concepts in physics is that of universal behaviour. Universality allows us to make predictions for entire classes of systems without requiring knowledge of the microscopics, and can be found in classical and quantum systems in both equilibrium and in their dynamics. Often in many-body systems, this universal behaviour is found in regimes where effects at macroscopic length scales dominate over microscopic fluctuations, which is particularly true at a phase transition. In this thesis, we will address universality in quantum many-body physics, and its connection to the branch of mathematics known as catastrophe theory (CT). In CT, singularities in a theory take on several universal forms, known as catastrophes, which can be shown to manifest themselves in classical mechanical trajectories. We extend the concept of catastrophes to their wave variants, known as diffraction integrals, and identify how these universal features appear in many-body wavefunctions and observables. Specifically, in Chapter 2, we examine how the wavefunction of a δ\deltaδ-kicked Hamiltonian can be mapped exactly onto the Pearcey function, along with the effects of a phase transition on the diffraction. In Chapter 3, we examine the free-Fermion representation of the one-dimensional transverse-field Ising model in a similar vein, and identify the presence of catastrophes away from and through the critical point.
Uncovering and understanding universal dynamics in matter far from equilibrium remains a key challenge. In this work, we identify a so far unrecognized form of universal behavior that emerges after a sudden symmetry-breaking quench at continuous phase transitions. Our key observation is that the order-parameter fluctuations in Ising models exhibit a compelling temporal collapse across a wide range of system sizes and quench strengths, indicative of an emergent single-variable scaling form. This phenomenon can be explained by introducing a so far unknown dynamical critical exponent for the underlying continuous phase transition. We find evidence for a lower critical effective dimension of this universal regime: it is observed in the 2D quantum and 3D and 4D classical Ising models, but not in the 1D quantum or 2D classical cases. Our results suggest that our observed universal far-from-equilibrium scaling may extend beyond the Ising models studied here and could more broadly characterize systems with non-conserved order parameters, opening new avenues for exploring universal dynamics both theoretically and in current experimental platforms.