Theory of Two-Qubit $T_2$ Spectroscopy of Quantum Many-Body Systems

Abstract

Multi-qubit quantum sensors are rapidly emerging as platforms that extend the capabilities of conventional single-qubit sensing. In this work we show how suitable pulse sequences applied to a two-qubit sensor enable separate extraction of the response and noise of a probed environment within a $T_2$ spectroscopy framework. By resorting to representative examples, we demonstrate that this approach can resolve the spatio-temporal spreading of correlations in a many-body system. In particular, the resulting correlated dephasing signal captures features such as the dispersion of low-energy excitations, which manifest as light-cone-like profiles in the propagation of correlations. We further show that non-equilibrium conditions, for instance those induced by external driving, can modify this profile by producing additional fringes outside the light-cone. As a complementary application, we demonstrate that the method clearly distinguishes between different transport regimes in the system, including ballistic spreading, diffusive broadening, and the crossover between them.

Figures (7)

• Figure 1: (a) A conceptual representation of the general setup for two-qubit $T_2$ spectroscopy. Two qubits, indicated by arrows, evolve in time under the influence of correlated noise originating from a many-body system. Correlations within the many-body system are then reflected in the outcomes of measurements performed on the qubits. (b) The protocol for response correlation sensing. The probed qubit (blue) is allowed to dephase while being affected by the presence of a bystander qubit (red), through the modification of the many-body environment induced by the latter. (c) Protocol for fluctuation correlation sensing, where both qubits are allowed to undergo dephasing for a period of time before being projected along their initial directions.
• Figure 2: Space-time profile of the correlated dephasing induced by an environment with a gapped spectrum. (Left) At thermal equilibrium, the correlated dephasing ($\mathcal{N}_{12}$) exhibits a light-cone structure, vanishing for distances larger than $c\,t$, where $t$ is the elapsed time since the start of the protocol. This behavior reflects the propagation of correlations within the many-body system. (Middle) Driving low-momentum modes modifies the noise profile by introducing long-range correlation fringes. (Right) Driving finite-momentum modes produces fringes with well-defined spatial periodicity proportional to $k_\mathrm{dr}^{-1}$, accompanied by an overall decay set by the length scale $\sigma_\mathrm{dr}^{-1}$. Other parameters are $T=\omega_0$, $\sigma_\mathrm{dr}=0.1$, and the noise magnitude is shown in arbitrary units.
• Figure 3: Spatiotemporal profile of the correlated dephasing induced by an environment with gapless spectrum. The normalized scaling function $f(r^z/\alpha t)$ is shown for $D=3$ and $z=1$ (left) and $z=2$ (right) at thermal equilibrium. Dashed lines indicate correlation fronts defined by fixed values of $\mathcal{N}^\mathrm{Ram}$. Both time and distance are shown in arbitrary units due to the scale invariance of the noise.
• Figure 4: Momentum-frequency space representation of two-qubit $T_2$ spectroscopy. (Left) At any instance of time after initializing the Ramsey sequence, the correlated $T_2$ noise probes excitations in the MBS with energies up to the inverse time and momenta up to the inverse distance between the qubits. For a gapped MBS, the noise is suppressed once $t$ exceeds the inverse gap, $\Delta^{-1}$, whereas for a gapless system the noise persists indefinitely. (Right) Consequently, the dephasing induced by a gapped MBS saturates, while it continues to grow for a gapless system, provided that the density of excitations vanishes sufficiently slowly at low energies.
• Figure 5: Correlated dephasing generated by diffusive dynamics. Deep in the ballistic regime ($t \ll \tau_D$), the dephasing spreads in a light-cone-like fashion (left), similar to the behavior in a coherent MBS. At longer times and larger distances, the signal displays a crossover from ballistic ($r\propto t$) to diffusive ($r\propto \sqrt{t}$) transport (right), characterized by the sublinear spreading of the correlated dephasing.