Measurement of many-body chaos using a quantum clock

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 $\langle O_2\left(t\right)O_1\left(0\right)O_2\left(t\right)O_1\left(0\right)\rangle$ 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 that controls the direction of time. Specifically, by coupling the state of the 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. We discuss the implementation of our protocol with current circuit-QED technology for a class of interacting Hamiltonians. Our protocol is immune to errors that could occur when the direction of time evolution is externally controlled by a classical switch.

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 (quantum clock). The time evolution conditioned by clock state $|0_a\rangle$ ($|1_a\rangle$) is forward (backward) in the beginning. After applying the ${\tau }^x$ operations, the clock 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 clock in the $x$ and $y$ basis gives the real and imaginary parts of the OTO correlator. We emphasize that the actual experimental time always goes from left to right. (b) 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 (red circles), a coupling cavity (blue bar) serving as a passive quantum bus, and an ancilla cavity (green square) serving as a quantum clock. (b) When there is no photon in the clock, the bus frequency ${\omega }_b$ is above the qubit frequency $\epsilon$, with a negative detuning ${\mathrm{\Delta }}_b<0$. (c) When there is one photon in the clock, the bus frequency ${\omega }_b-\eta$ is pushed down below the qubit frequency $\epsilon$ by a distance $|{\mathrm{\Delta }}_b|$, 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 squares), qubits (red circles) 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 clock, the qubit energy $\epsilon$ is above the local cavity frequency ${\omega }_b$, with a positive detuning ${\mathrm{\Delta }}_b$. (c) When a single photon is present in the clock, the qubit level spacing ${\epsilon }^{\prime }\equiv \epsilon -2\chi$ is pushed down below the local cavity frequency ${\omega }_b$ by a distance ${\mathrm{\Delta }}_b$, which inverts the sign of the detuning and hence the sign of the controlled Hamiltonian. We note that, in (b) and (c), the level diagram represents the level spacings rather than absolute energy levels.

Figure 4

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 splits the evolution into two branches in order to show 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.

Figure 5

Effect of imperfect sign change via classical switch for a spin model. The model considered here is $H={\sum }_i({\vec{\sigma }}_i·{\vec{\sigma }}_{i+1}+h_i{\sigma }_i^z)$, where the $h_i$ are chosen randomly from a uniform distribution in the interval $[-0.5,0.5]$. The correlator $\langle O_2\left(t\right)O_1\left(t\varepsilon \right)O_2\left(t\right)O_1\left(0\right)\rangle$ is shown, with $O_1={\sigma }_2^z$ and $O_2={\sigma }_{L-1}^z$, where $L=12$ is the total number of sites. We take $\varepsilon$ to be a random Gaussian variable with variance $\delta$ and averaging in $\overline{\langle O_2\left(t\right)O_1\left(t\varepsilon \right)O_2\left(t\right)O_1\left(0\right)\rangle }$ is performed over this ensemble. The inset shows the relative error $\langle O_2\left(t\right)O_1\left(t\varepsilon \right)O_2\left(t\right)O_1\left(0\right)\rangle /\langle O_2\left(t\right)O_1\left(0\right)O_2\left(t\right)O_1\left(0\right)\rangle -1$.

Figure 6

A 2D generalization of the cavity-QED implementation. Two types of multilevel atoms (qudits), represented by blue squares and red circles, form a checkerboard lattice that 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 7

Cavity- or circuit-QED architecture, which realizes the model described by Eq. (14) and illustrated in Fig. 3. (a) A 2D on-chip circuit-QED network. The setup consists of a global transmission-line resonator serving as the quantum clock; local transmission-line resonators, which play the role of active degrees of freedom; and qubits, which are passive degrees of freedom that mediate interactions between local resonators and are controlled by the global resonator. Alternatively, one can have an additional ancilla qubit coupled to the global resonator, which can either be used to manipulate the cavity photon state or be dispersively coupled to the local qubits mediated by the cavity bus and hence serves as the clock. (b) A 3D cavity QED with superconducting qubit array, with qubits of two different frequencies (represented as red and blue). The blue qubits play the role of active degrees of freedom, while the red qubits are passive couplers that mediate interactions between the blue qubits. The dipoles of the qubits are facing different directions to enable selective coupling to the global cavity.

Figure 8

Numerical comparison of the original and second-order effective Hamiltonian for a dimer. The parameters are ${\mathrm{\Delta }}_b=50\text{MHz}, {\mathrm{\Delta }}_a=800\text{MHz}, \chi =50\text{MHz}$ (or equivalently $g_a=200\text{MHz}$), an on-site photon cutoff $n_b^{\text{max}}=3$. (a) The setup for numerical simulations contains two local cavities, one qubit, and one global cavity. (b) Comparison of the spectrum between the exact (blue circles) and effective (yellow squares) Hamiltonians obtained from numerical exact diagonalization. The spectrum is separated into two clock sectors. The red circle show states in the one-photon manifold (${\sum }_j\langle b_j^{†}{b}_j\rangle \approx 0$ and $\langle {\sigma }^z\rangle \approx 0$). (c) Relative error between the exact and effective spectra for $g_b/{\mathrm{\Delta }}_b=0.1$. (d) Average photon and qubit excitation numbers for the low-lying states in both clock sectors, obtained from exact (blue circles) and effective (yellow squares) Hamiltonians. The red circles show the states in the one-photon manifold. (e) Energy splitting in the one-photon manifold $\delta$ for both clock sectors obtained from exact diagonalization of the original Hamiltonian and the prediction $2g_b^2/{\mathrm{\Delta }}_b$ from second-order perturbation theory.

Figure 9

Numerical results for a three-site ring. (a) The setup for numerical simulations contains three local cavities, three qubits, and one global cavity, which form a periodic ring. (b)–(e) Average photon and qubit excitation numbers for the low-lying states in both clock sectors, obtained from exact (blue circles) and effective (yellow squares) Hamiltonians. The red circles show the states in the one-photon manifold and the insets show the zoomed-in spectrum in that manifold. (f) Splitting in the one-photon manifold for both clock sectors obtained from exact diagonalization of the original Hamiltonian and the prediction $3g_b^2/{\mathrm{\Delta }}_b$ from second-order perturbation theory.