Quantum simulation of gravitational-like waves in minisuperspace with an artificial qubit

On the background of the Born-Oppenheimer (adiabatic) approximation, we investigate the geometrical and topological structure in the theory of quantum gravity by using the path integral method and half-classical approximation. As we know, Berry curvature can be extracted from the linear response of a driven two-level system to nonadiabatic manipulations of its Hamiltonian. In parameter space of the Hamiltonian, magnetic monopoles can be artificially simulated. Ripples occur in Hilbert space when the monopole travels from inside to outside the surface of energy manifold spanned by system parameters. From this point of view, we set up the connection between the ripples characterized by the fidelity of quantum states in Hilbert space and gravitational-like waves in minisuperspace. This might open a window for the study of geometrical and topological properties of quantum gravity with the help of quantum physical systems.

Figure 1

Illustration of a spatial lattice. The elementary building blocks for three-dimensional space-time are simplices of dimension three. A one-simplex is an edge ($\lambda$, the blue line), a two-simplex is a triangle (${\lambda }^2$, the yellow lines), and a three-simplex is a tetrahedron (${\lambda }^3$, the green lines). On a random simplicial lattice there are in general no preferred directions, except for the special path (the maroon part) we choose here.

Figure 2

Polyhedral approximation to a sphere. The green spatial lattice in the left is a closed three-dimensional manifold with all three-space (sum over all points of a spatial lattice), which is homeomorphic to a connected polyhedron in piecewise linear space, and its points possess neighborhoods that are homeomorphic to the interior of the three-dimensional sphere (in the right). According to the Gauss-Bonnet theorem [32], the topological structure between the polyhedra and the sphere is equivalent (both have the same Euler characteristic). The light blue ball represents the matter states associated with the closed red trajectories $C$ of an adiabatic evolution. The dashed red line is the simplification of the red cyclic polyhedron (all possible paths with the same quantum number corresponding to a net at each time, not a point), which is homeomorphic to the red torus in the right.

Figure 3

Quasilattice points in space-time. (a) The trajectory of lattice points depicts the evolution history of the matter state in the gravitational field. The simultaneously created lattice points constitute a spatial net at every moment. Points on every spatial net can be viewed as equivalent to a quasilattice point on the trajectory of evolution (dashed circles). So the position of an ensemble of lattice points could be described by a set of quasilattice vectors. The quasilattice point from one to another reads ${\overrightarrow{K}}_i$ to ${\overrightarrow{K}}_{i+1}$. (b) Quasilattice function $\mathcal{K}$ in vector space. For example, we set ${\overrightarrow{X}}_n={\overrightarrow{R}}_n$, ${\overrightarrow{Y}}_n={\overrightarrow{\mathrm{\Theta }}}_n$, ${\overrightarrow{Z}}_n={\overrightarrow{\mathrm{\Phi }}}_n$, and $x_n^n=r_n^n$, $y_n^n={\theta }_n^n$, $z_n^n={\varphi }_n^n$, (here the subscript $n=2$) [34].

Figure 4

A sphere constructed by the quasilattice points of the space-time. The red closed loop $C$ represents the evolution trajectory of the matter state. The blue dashed circle depicts that $C$ is formed by all quasilattice points during the whole evolution time, as shown in Fig. 3. During this excursion along $C$ in the external parameter space that is spanned by $\{{\overrightarrow{K}}_{\mathcal{S}}^2\}$ and $\{{\overrightarrow{K}}_{\mathcal{S}}^3\}$, the adiabatic change of the matter state gains a geometric phase in addition to the conventional dynamical phase.

Figure 5

Energy spectrum of the transmon qubit. Here we assume the qubit that with a transition frequency of ${\omega }_q=4.395\mathrm{GHz}$ is effectively a nonlinear resonator, and the anharmonicity of 280 MHz ensures that the qubit transition only occurs between the two lowest energy levels [36].

Figure 6

Quantum simulation of artificial gravitational-like waves in minisuperspace with a two-level system. (a) The gravitational-like waves in microscopic scales emitted by the inspiral and merger of two micro black holes correspond to the process of population inversion of the two eigenstates and the process of the transfer into the superposition state, respectively. The frequency of the inspiral of micro black holes is analogues to the frequency of the population inversion of the two eigenstates. (b) Ripples of the wave function of the superposition state in Hilbert space. Here we set ${\mathrm{\Delta }}_1=6\pi \times 10\mathrm{MHz}$ and $\mathrm{\Omega }=3\pi \times 10\mathrm{MHz}$, and manipulate ${\mathrm{\Delta }}_2$ from 0 to $2{\mathrm{\Delta }}_1$ at $\theta =\pi$. The fidelity oscillates faster and faster with the increasing of ${\mathrm{\Delta }}_2/{\mathrm{\Delta }}_1$ before the topological transition [21] (the blue real line represents the first Chern number $C_1=\frac{1}{2\pi }{\int }_0^{\pi }d\theta {\int }_0^{2\pi }d\varphi F_{\theta \varphi }={\int }_0^{\pi }{F}_{\theta \varphi }d\theta$, which turns from 1 to 0 at ${\mathrm{\Delta }}_2/{\mathrm{\Delta }}_1$ in the inserted figure). This corresponds to the period of inspiral of the two micro black holes becoming shorter and shorter.

Figure 7

Illustration of the lapse function and the shift vector. These two quantities depict the lapse of proper time ($N$) between two infinitesimally close hypersurfaces, and the corresponding shift in spatial coordinate $(N^i)$.