Measurement of many-body chaos using a quantum clock

TL;DR

This work addresses measuring quantum chaos in many-body systems by introducing a quantum-clock ancilla that coherently controls the direction of time evolution through a total Hamiltonian $H_ ext{tot}=\tau^z\otimes H$. The authors present two realizations (non-local all-to-all and local lattice) in circuit-QED/cavity-QED settings, derive the ancilla-dependent effective Hamiltonians, and outline a robust measurement protocol for the out-of-time-ordered correlator (OTOC) using Ramsey interferometry. They analyze imperfections arising from clock-pulse errors and couplings, showing the protocol’s resilience compared to classical-switch approaches, and extend the local model to soft-core photons, extended Hubbard physics, disorder, and higher dimensions. The proposed framework enables practical, scalable probing of scrambling, chaos, and localization phenomena in diverse quantum platforms, with potential applications to Loschmidt echoes and beyond.

Abstract

There has been recent progress in understanding chaotic features in many-body quantum systems. Motivated by the scrambling of information in black holes, it has been suggested that the time dependence of out-of-time-ordered (OTO) correlation functions such as $\langle O_2(t) O_1(0) O_2(t) O_1(0) \rangle $ is a faithful measure of quantum chaos. Experimentally, these correlators are challenging to access since they apparently require access to both forward and backward time evolution with the system Hamiltonian. Here, we propose a protocol to measure such OTO correlators using an ancilla which controls the direction of time. Specifically, by coupling the state of ancilla to the system Hamiltonian of interest, we can emulate the forward and backward time propagation, where the ancilla plays the role of a 'quantum clock'. Within this scheme, the continuous evolution of the entire system (the system of interest and the ancilla) is governed by a time-independent Hamiltonian. Our protocol is immune to errors that could occur when the direction of time evolution is externally controlled by a classical switch.

Figures (9)

• Figure 1: (a) Illustration of the Ramsey interferometry protocol. The interferometry starts from the left, with the initial state $|\, \psi \, \rangle_S \otimes |\, 0_a \, \rangle$. The Hadamard rotation splits the time evolution of the many-body state $|\, \psi \, \rangle_S$ into two branches, conditioned by the ancilla. The time evolution conditioned by ancilla state $|\, 0_a \, \rangle$ ($|\, 1_a \, \rangle$) is forward (backward) in the beginning. After applying the $\tau^x$ operations, the ancilla states on the two branches interchange, and so are the directions of time evolution. The red dashed lines show the canceled time evolution. Conditional operations $O_1$ and $O_2$ on either branch are applied. A final measurement of the ancilla in the x- and y-basis gives the real and imaginary part of the OTO correlator. We emphasize that the actual experimental time always goes from left to right. (b) The quantum circuit description of the same protocol.
• Figure 2: Schematic diagrams of a cavity-QED implementation of an all-to-all coupled spin model. (a) Illustration of the many-body system, consisting of system qubits/spins (red circle), a coupling cavity (blue bar) serving as a passive quantum bus, and an ancilla cavity (green box) serving as a quantum clock. (b) When there is no photon in the ancilla cavity, the coupling cavity frequency $\omega_b$ is above the qubit frequency $\epsilon$, with a negative detuning $\Delta_b < 0$. (c) When there is one photon in the ancilla cavity, the coupling cavity frequency $\omega+\eta$ (where $\eta <0$) is pushed down below the qubit frequency $\epsilon$ by a distance $|\Delta_b|$, which inverts the sign of the detuning and hence the sign of the controlled Hamiltonian.
• Figure 3: Schematic diagrams of the cavity-QED implementation of a local model. (a) Illustration of the many-body system, consisting of local cavities (blue box), qubits (red circle) mediating interactions between the cavities, and a global control cavity (green bar) serving as a quantum clock. (b) When no photon is present in the global control cavity, the qubit energy $\epsilon$ is above the local cavity frequency $\omega_b$, with a positive detuning $\Delta_b$. (c) When a single photon is present in the global control cavity, the qubit energy $\epsilon' \equiv \epsilon+2\chi$ (where $\chi <0$) is pushed down below the local cavity frequency $\omega_b$ by a distance $\Delta_b$, which inverts the sign of the detuning and hence the sign of the controlled Hamiltonian.
• Figure 4: A 2D generalization of the cavity-QED implementation. Two types of multi-level atoms (qudits), represented by blue boxes and red circles, form a checkerboard lattice which is placed in a 3D cavity. The blue atoms play the role of active degrees of freedom, while the red atoms are passive coupler mediating interactions between red atoms. The two types of atoms are coupled by nearest-neighbor flip-flop interactions. The cavity is selectively coupled to only the red atoms with dispersive interaction to shift their frequencies.
• Figure 5: Measurement protocol using a classical switch to control the 'arrow of time'. An ancilla qubit is initialized as the superposition of $|\, 0_a \, \rangle$ and $|\, 1_a \, \rangle$ and hence split the evolution into two branches in order to do the Ramsey interference. The ancilla enables conditional-$O_1$ operation but does not control the sign of the Hamiltonian. Another classical switch (such as the detuning) is used to change the sign of the Hamiltonian and hence flip the 'arrow of time'.