Caustics in quantum many-body dynamics

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 $X_9$. These results describe a hitherto unrecognized form of universality in quantum dynamics organized by singularities that manifest as strong fluctuations in mode population probabilities.

Figure 1

Real optical caustics made by shining a laser pointer through a water droplet of radius $\sim 1$ mm and photographed on a screen at a distance of several meters. The droplet has a triangular perimeter imposed by placing it in a triangle cut out of tape stuck on a microscope slide. This mimics the triangular Fock space found in the BH trimer model and leads to the same families of caustics. Panel (a): Hyperbolic umbilic. Panel (b): Elliptic umbilic. Panel (c): Elliptic umbilic near its most singular point. What we see in these photos are slices through three dimensional catastrophes that also contain lower catastrophes: the hyperbolic umbilic contains a single cusp (which is the only structurally stable catastrophe in 2D and is dressed by the Pearcey function wave catastrophe) and the elliptic umbilic contains three cusps. Taking a 1D slice across a cusp gives the simplest catastrophe of all, the fold catastrophe (whose wave pattern is the Airy function), and in fact folds and cusps are the basic elements in “light cones” in Ising and XY models, see Fig. 2 in Ref. [40]. These images and the methods used to make them were inspired by the experiments reported in reference [42]. The different colors arise from using three different color lasers.

Figure 2

A small region of Fock space for the BH trimer. It can be tiled by hexagonal cells in the coordinates $\{\delta n_2,n_X\}$ we use throughout this paper. Hopping terms shift $\delta n_2$ by only 1 unit, while $n_X\equiv n_1-n_3$ can change by 1 or 2 units. If the reader zooms in on the BH trimer Fock space figures in this paper they will see this underlying hexagonal lattice. This discretization is a true feature of second quantization not present in classical wave catastrophes (it is not a false effect due to pixilation of the images).

Figure 3

Time slices comparing mean-field and quantum dynamics in the triangular Fock space found in the BH trimer under the constraint of total number conservation. Panel (a): An ensemble of initial points are evolved in time using the classical equations of motion Eqs. (13, 14, 15, 16) and form a space-time version of the elliptic umbilic catastrophe, with one particular trajectory picked out for illustration. In the TWA the initial conditions are drawn from a quantum probability distribution and in this simple case the initial state is a phase state meaning that the relative phases are sharply defined (here taken to be ${\varphi }_X={\varphi }_C=0$) but with the consequence that their conjugate number differences take all possible values with equal probability, thereby uniformly populating the allowed triangular region of Fock space. Panel (b): The quantum dynamics are obtained by solving Eq. (11). The initial phase state appears as a discretized plane wave in Fock space (although here $N=150$ and the discreteness of Fock space is hard to see unless the reader zooms in). The correspondence between the quantum and classical dynamics is clear: the quantum wave function is brightest where the density of trajectories is highest. In both images, $K_L=K_R=K_X\equiv J, U=-0.005J$, and ${\epsilon }_i=0$.

Figure 4

The wave catastrophes associated with the fold and the cusp. Panel (a): The fold catastrophe is decorated by an Airy function, Eq. (20). The location of the classical caustic is indicated by a vertical dotted line. In the classical (ray) theory the intensity is divergent at this point and falls off as $1/\sqrt{-C}$ on the bright side. By contrast, the Airy function is finite at the caustic and two-wave interference gives rise to oscillations; these two waves coalesce at $C=0$ and become evanescent on the dark side. Panel (b): The cusp catastrophe is decorated by the Pearcey function, which is a complex-valued function (here we plot the modulus) given by Eq. (22), with the divergent classical cusp caustic shown as a solid black curve.

Figure 5

Recurring cusp caustics in the two-mode BH model. Panel (a): Each curve is a mean-field configuration obtained from Eqs. (6, 7) starting from $n=0$ and with an initial phase sampled from the distribution $\varphi =[0...2\pi )$ corresponding to a definite (equal) number of bosons in each well. Panel (b): Modulus of the quantum wave function calculated using the two-mode RN equations Eq. (3) with $N=400$ and where the initial state is the single Fock state $|n=0\rangle$. In both cases we can identify a series of cusp caustics corresponding to partial revivals of the initial state. In the inset we see that in the immediate vicinity of each cusp the wave function resembles a Pearcey function [36, 39] (the wave function appears continuous but the discretization of Fock space can be seen if the reader zooms in).

Figure 6

Recurring cusps in the BH trimer model, starting from an equal superposition of Fock states, the three-mode version of a phase state. Panel (a): Classical trajectories along the plane of fixed $n_X=0$. Panel (b): Quantum dynamics in the same plane as (a) where the classical cusps are now dressed by interference fringes, locally approximated by the Pearcey function. The bright streaks are finite-size effects due to reflections off the Fock-space boundary in the $n_X$ direction, but structural stability ensures the cusps survive. In both panels, $K_L=K_R\equiv J=100U, K_X=0$ (linear configuration), while for the quantum dynamics we used $N=180$.

Figure 7

Comparison of BH trimer dynamics with the canonical elliptic umbilic catastrophe. Panel (a): Fock-space amplitudes starting from an even superposition of Fock states [Eq. (29)] for $K_L=K_R=K_X\equiv J, U=0.01J$ and $N=150$ at $Jt/\hslash =0.47$. Panel (b): Diffraction pattern amplitude in the plane around the elliptic umbilic focus. Panel (c): Same conditions as panel (a), but now at $Jt/\hslash =0.54$. Panel (d): Diffraction pattern for a slice of the elliptic umbilic at $C_3=3.8$. Panel (e): Caustic surface of the elliptic umbilic catastrophe. The triple-cusped intersection with a $C_3=\text{const.}$ plane is highlighted in red.

Figure 8

Comparison of BH trimer dynamics with the canonical hyperbolic umbilic catastrophe. Panel (a): Wave function amplitudes for hopping strengths $K_L=K_R\equiv J=4U, K_X=0$, and $N=150$ at $Jt/\hslash =0.24$. The initial state was $|\delta n_2,n_X\rangle =|0,0\rangle$. Panel (b): Hyperbolic umbilic diffraction pattern on the plane $C_3=0$. Panel (c): Same conditions as panel (a) now at $Jt/\hslash =0.34$. The bottom umbilic is now completely unfolded. Panel (d): Diffraction pattern for the hyperbolic umbilic in the plane $C_3=3$. Panel (e): Caustic surface of the elliptic umbilic catastrophe. A projection onto the plane $C_3=\text{const.}$ is highlighted in red.

Figure 9

6-cusped diffraction pattern surrounded by a fold line, typical of systems with 3-fold symmetry also organized by $X_9$. Panel (a): Fock space amplitudes after a quench starting from all particles evenly distributed in each well, $|\delta n_2,n_X\rangle =|0,0\rangle$. Here, $N=150, K_L=K_R=K_X\equiv J, U/J=0.04$ at $Jt/\hslash =0.475$. Panel (b): As panel (a), now with $K_R=J, K_L=1.2J$, and $K_X=0.8J$, showing how an absence of symmetry does not destroy the caustics.

Figure 10

Hyperbolic sections of $X_9$ in the classical and quantum BH trimer. Panel (a): Caustic surface for ${\mathrm{\Phi }}_{X_9}^H$, as given in Eq. (38). The red highlights mark the intersection with the plane $C_4=\sqrt{2}$. Panel (b): Distribution of points arising from classical trajectories of the $\delta$QPM Hamiltonian Eq. (28) (see Appendix pp1 for solutions of Hamilton's equations) for the linear trimer configuration $(K_X=0)$ at $t={\hslash }^2/\left(JN\tilde{U}\right)$ and $\alpha =C_4=\sqrt{2}$, starting in an equal spread of number differences, and with phase differences ${\varphi }_C\left(0\right)={\varphi }_X\left(0\right)=0$. The resulting caustic is a partial section through the hyperbolic unfolding of $X_9$, restricted by the triangular Fock space (black solid lines) of physical classical paths. Panel (c): Same snapshot as in panel (b), now including some unphysical paths (such as those which have started or ended with $n_X>N$), to demonstrate how the caustic is restricted by the shape of Fock space (number conservation). Panel (d): Wave function amplitude at the same moment as panels (b) and (c), starting from an equal superposition of Fock states (29) and evolved using the RN equations corresponding to the $\delta$QPM model (square root factors set to unity). Here, $N=150$, and $\tilde{U}/\hslash =0.02$.

Figure 11

The spun cusp caustic generated by the triangular BH trimer $(K_X=J)$ within the $\delta$QPM. Panel (a): Distribution of points arising from classical trajectories of the $\delta$QPM Hamiltonian Eq. (28) (see Appendix pp1 for solutions of Hamilton's equations) at time $t={\hslash }^2/\left(JN\tilde{U}\right)$ (twice $t_{\text{cusp}}$), starting from an equal spread of number differences, and with phase differences ${\varphi }_C\left(0\right)={\varphi }_X\left(0\right)=0$. The circular caustic from the $K=2$ section of $X_9$ is clearly visible. Panel (b): Quantum wave function amplitude at the same time as panel (a), starting from an equal superposition of Fock states (29) and evolved using linearized RN equations. $N=150$, and $\tilde{U}/\hslash =0.02$. Panel (c): Caustic surface for the spun cusp, for negative $C_5$. The 2D circular caustic and the axial caustic are highlighted in red.

Figure 12

A Gaussian initial state gives the same caustic as a “plane wave”. Panel (a): The amplitude of the ground state of the noninteracting trimer model for $K_L=K_X=K_R\equiv J$ and $N=120$ has a Gaussian form. Panel (b): Time evolution of the ground state using the kicked Hamiltonian with $\tilde{U}/\hslash =0.08$, at time $t=2{\hslash }^2/\left(JN\tilde{U}\right)$. Comparing with Fig. 11 we see a strong resemblance indicating that initial states, which differ significantly but have the same general form (flat near the center of Fock space) will give rise to qualitatively similar caustics.

Figure 13

Breaking circular symmetry by going beyond the QPM. Panel (a): The energy surface determining classical trajectories as provided by $H_{\delta \mathrm{QPM}}$ in Eq. (28) immediately following the $\delta$ kick at $t=0$. The initial values of the dynamical variables that generate this surface are ${\varphi }_X={\varphi }_C=0$ and an equal superposition of all number differences; the phase trajectories are bent by the $\delta$ kick as explained in Appendix pp1. The overall scale is arbitrary and lighter colours represent higher energy. Note the circular symmetry as seen from the contour lines. Panel (b): Same as panel (a) but now with the $\delta$-kicked mean-field Hamiltonian $H_{\delta \mathrm{MF}}$ given in Eq. (47) with all hopping amplitudes equal. The circular symmetry is broken by the square root factors and the resulting three-fold symmetric “valleys” will act to pinch the caustic away from the corners of Fock space. Panel (c): Same as panel (a) but now with $H_{▵}$ as given in Eq. (48). In this Hamiltonian the square root factors have been expanded to first order and this is enough to break the circular symmetry and replace it with a three-fold symmetric energy surface. Panel (d): Quantum wave function obtained using the full RN equations with $\delta$-kicked interactions (quantum equivalent of $H_{\delta \mathrm{MF}}$) and the same parameters as used in Fig. 11 (b). The chosen time is $t={\hslash }^2/\left(JN\tilde{U}\right)$, which is twice that of where the cusp point appears.

Figure 14

Stability of cusp formation in the $n_X=0$ plane against inclusion of beyond QPM effects. Panel (a): Set of classical trajectories for the $\delta$QPM when $n_X=0$ starting from a range of $\delta n_2$ values and kicked with $\tilde{U}/\hslash =0.02$. The cusp point here is $Jt/\hslash \approx 0.167$. Panel (b): Exact trajectories (without approximation to the square root factors) for the triangular trimer with a kicked interaction term as given by $H_{\delta \mathrm{MF}}$. The cusp structure remains intact and is only curved near the edge of Fock space, showing that the $\delta$QPM is a good approximation for this subcatastrophe and that caustics have structural stability against perturbations.

Figure 15

Bordering (or abutment) diagram for the catastrophe $X_9$, adapted from Nye [49] and obtained using theorems of catastrophe projection. Each catastrophe of higher order contains (many, not necessarily all) catastrophes of lower order, shown by the direction of arrows. The subfamilies of $X_9$ lead to two different subcatastrophe sets. ${}^4{X}_9$ contains the singularities $E_7$ and $D_6$, shown by the red dashed arrows. ${}^0{X}_9$ does not contain $E_7$ and will only contain $E_6$, shown by the blue dotted arrow.

Figure 16

Most singular sections of the diffraction patterns for $D_4^{-}$ and $D_6^{-}$ showing how the foci of the two subfamilies of $X_9$ differ. Panel (a): The focal plane of $D_4^{-}$ revealing an elliptic umbilic focus. Panel (b): The focal plane of $D_6^{-}$ revealing another, but subtly different type of elliptic umbilic-like structure. In the case of $D_4^{-}$ the caustic in its canonical form is threefold symmetric with the brightest ribs (traced with green dashed lines) meeting in straight lines at angles of $2\pi /3$. For $D_6^{-}$ the bright central ribs meet along curved lines obeying $27C_2^4=64C_1^3$. The presence of a $D_6^{-}$ focus can in principle be a diagnostic tool for determining the presence of ${}^4{X}_9$.

Figure 17

Distorted elliptic umbilic due to unequal hopping strengths using the full trimer Hamiltonian. $K_R=1.1K_L, K_X=1.2K_L, U/K_L=0.01$ and $N=150$, starting from an equal superposition of Fock states. Panel (a): Distorted focus at $K_{L}t/\hslash =0.431$. Panel (b): Unfolded at $K_{L}t/\hslash =0.513$

Figure 18

Instability of perfect focusing events to the introduction of interactions. Panel (a): Set of classical trajectories starting from an equal spread of Fock states, with $K_L=K_R, U=K_X=0$. Focusing events are isolated, meaning that all trajectories meet at a point. Panel (b): Same initial state as panel (a), but now with weakly attractive interactions, $K_L=K_R\equiv J$ and $U/J=-0.03$. A similar result was found for repulsive interactions in panel (a) of Fig. 6, except that the cusps open in the opposite direction for attractive interactions.

Figure 19

Formation of fold lines around cusps due to attractive interactions. Panel (a): Same as panel (c) of Fig. 7, now with $U/J=-0.01$. The elliptic umbilic diffraction pattern is now surrounded by bright fringes, which consist of small fold lines. Panels (b) and (c): Same as panel (a) but at $Jt/\hslash =0.641$ and $Jt/\hslash =0.818$. As the elliptic umbilic focus is approached, the fold lines around each cusp extend and form a ring. Panel (d): Schematic of a section of the elliptic umbilic caustic, with “lips” surrounding each cusp point.

Figure 20

Catastrophe formation at high interaction strength. In this figure, we show exact (numerical) dynamics deep in the chaotic regime with $U=J$ in the linear configuration of the BH trimer where $K_L=K_R\equiv J$, and $K_X=0$. Caustic formation follows similarly to previously discussed cases with weaker interactions, however stronger interactions compress the dynamics to a smaller region in Fock space. Panel (a): Formation of a fourfold diffraction pattern at $Jt/\hslash =0.13$, similar to the one seen in Fig. 8, but its largest extent is smaller than the weaker interaction case. Panel (b): At $Jt/\hslash =0.36$, the wavefront has become highly distorted, although strong focusing (fluctuations) remains.