Physics:Out-of-time-ordered correlator

In quantum physics, the out-of-time-ordered correlator (OTOC) serves as a powerful diagnostic tool for characterizing quantum chaos, information scrambling, and other aspects of many-body dynamics. In addition, it provides a quantum mechanical analog to the Lyapunov exponent, often used to characterize the sensitivity of variables to initial conditions in classical chaos. The OTOC thus provides a natural extension of classical chaos theory to the quantum realm, and can be calculated both numerically and experimentally .

Definition

For two observable $V$ and $W$ in Heisenberg picture, the out-of-time-order correlator (OTOC) is typically defined in two different but physically closely related ways:

1. Based on commutator of $W(t)$ and $V(0)$:$$C(t)=⟨[W(t),V(0){]}^{†}[W(t),V(0)]⟩$$direct calculation gives $C(t)=⟨V(0{)}^{†}W(t{)}^{†}W(t)V(0)⟩+⟨W(t{)}^{†}V(0{)}^{†}V(0)W(t)⟩-⟨V(0{)}^{†}W(t{)}^{†}V(0)W(t)⟩-⟨W(t{)}^{†}V(0{)}^{†}W(t)V(0)⟩.$
2. More directly $$F(t)=⟨W(t{)}^{†}V(0{)}^{†}W(t)V(0)⟩$$Generally $C(t)=⟨V(0{)}^{†}W(t{)}^{†}W(t)V(0)⟩+⟨W(t{)}^{†}V(0{)}^{†}V(0)W(t)⟩-2\mathrm{R}\mathrm{e}F(t).$ When $V$ and $W$ are unitaries, we have $C(t)=2(1-\mathrm{R}\mathrm{e}F(t)).$

where the expectation value $⟨\bullet ⟩=\mathrm{Tr}[\rho \bullet ]$ is usually taken over some thermal state $\rho =\exp (-\beta H)/Z$ with $\beta =1/k_{B}T$ ($k_B$ is Boltzmann constant, $T$ is temperature) and $H$ is Hamiltonian, $Z=\mathrm{Tr}\exp (-\beta H)$ is canonical partition function.

Physically, the growth of this commutator measured by $C(t)$ tracks scrambling. And from chaos theory perspective, we have $C(t)\simeq e^{{\lambda }_{L}t}$ where ${\lambda }_L$ is the quantum Lyapunov exponent. This has a similar form as the classical dependence of initial pertuvation in classical chaos theory. Thus OTOC can be regarded as an indicator of quantum chaos.

See also

• Quantum chaos
• SYK model
• Chaos theory