Divergent dissipative susceptibility at the many-body localization transition: disentangling dephasing, band excitation, and loss in a controlled open quantum system

A landmark controlled-open-system experiment (Lüschen et al., Phys. Rev. X 7, 011034 (2017)) drove a many-body localized gas of interacting fermions in a quasiperiodic optical lattice with tunable photon scattering and found that an initial charge-density wave decays via a stretched exponential whose rate grows linearly with the scattering rate, defining a dissipative susceptibility χ. Deep in the localized phase χ is small and essentially interaction-independent, but it rises strongly as the many-body localization (MBL) transition is approached, and the authors speculated that interactions might drive χ to diverge at the critical point, so that even infinitesimal environmental coupling would dominate the dynamics. Whether this divergence is real remains undetermined, because the experiment could not independently vary the microscopic relaxation channels (ground-band dephasing versus band excitation and particle loss, with the trapped first excited band acting as an internal small bath), and available numerical methods could not include loss and higher-band dynamics on the long times required near criticality. The open problem is to establish, experimentally and theoretically, how weak coupling to an environment destabilizes MBL near its transition: to quantify and independently control the decay channels, to characterize the dynamics and backaction of the trapped excited-band population as a finite bath, and to decide whether χ diverges at the transition or crosses over, including the ergodic Griffiths side of the transition.

Background

Many-body localization (MBL) is a phenomenon of interacting quantum systems with sufficiently strong quenched disorder in which thermalization fails: the system fails to act as a bath for itself, and local memory of an initial state, such as an imprinted charge-density wave, persists indefinitely in the ideal closed limit. It was predicted theoretically for disordered electrons and is studied experimentally in ultracold atoms in optical lattices, where the interacting Aubry-André model — fermions hopping with amplitude $J$ in a quasiperiodic potential $V_i = \Delta\cos(2\pi\alpha i + \varphi)$ with onsite interaction $U$ — is a standard realization; for $\Delta \gtrsim 2J$ (about $4J$ with interactions) the system enters the MBL phase. The standard experimental observable is the imbalance $I(t) = (N_{\mathrm{even}} - N_{\mathrm{odd}})/(N_{\mathrm{even}} + N_{\mathrm{odd}})$ between the populations of even and odd lattice sites, which stays finite in the localized phase and decays to zero in the ergodic phase. No real experiment is perfectly closed, however, and theory predicts that any weak coupling to an environment eventually thermalizes the system, broadening the sharp closed-system transition into a coupling-controlled crossover. On the ergodic side of the transition, rare insulating regions — so-called Griffiths effects — dominate the dynamics close to criticality and are expected to slow the imbalance decay from exponential to power-law form. Prior theory had shown that dephasing — the loss of local phase coherence, acting like a continuous measurement of the local density — leads to only very slow, ultimately stretched-exponential relaxation of localization proxies (Levi, Heyl, Lesanovsky, and Garrahan, Phys. Rev. Lett. 116, 237203 (2016)), and that when an MBL system is coupled to a bath through its local integrals of motion (the quasi-local conserved quantities that characterize the MBL phase), particle loss produces pronounced interaction effects whereas pure dephasing largely does not (Fischer, Maksymenko, and Altman, Phys. Rev. Lett. 116, 160401 (2016)). The experiment of Lüschen et al. (Phys. Rev. X 7, 011034 (2017)) realized this physics in a controlled way: near-resonant photon scattering from a dedicated beam imposed dissipation of tunable rate $\gamma$ on a many-body localized gas. Each scattering event either dephases an atom within the lowest band (rate $\gamma_{\mathrm{dp}} = p_{\mathrm{dp}}\gamma$, incoherent position measurement), or excites it to a higher band ($\gamma_{\mathrm{ex}}$); excited atoms can be lost from the trap, but a significant fraction remains trapped in the first longitudinally excited band, which has a larger tunneling rate and can be delocalized, so this population forms an effective internal small bath whose properties may in turn be modified by backaction from the localized ground-band system. The measured imbalance decayed as a stretched exponential $I(t) \propto \exp[-(\Gamma_I t)^\beta]$, and the decay rate $\Gamma_I$ grew linearly in $\gamma$, motivating the definition of a susceptibility $\chi = d\Gamma_I/d\gamma$ (after subtracting a background decay) that quantifies how strongly the localization responds to weak coupling. In the noninteracting limit an upper bound $\chi \le p_{\mathrm{dp}}$ holds; with interactions, measured values at disorder $\Delta = 4J$ rose strongly with $U$, in contrast to number-conserving numerical simulations that omit loss. Comparisons with theory relied on time-evolving block decimation (TEBD) simulations of the Lindblad master equation — the standard Markovian equation of motion for the density matrix of a system subject to decoherence and loss — using matrix-product operators (a tensor-network representation of open-system density matrices that limits entanglement growth), which could not include the number-nonconserving loss process, nor higher-band dynamics, on the long times needed near the transition. The interplay of a finite bath and a localized system is a topic of separate theoretical and experimental interest: a bath of tunable finite size can leave localization intact on experimental timescales while a large bath destroys it, and avalanche-type arguments — in which a rare locally thermal region grows and absorbs the surrounding localized phase — suggest that in the thermodynamic limit even a small embedded bath can nucleate thermal regions that destabilize MBL at any nonzero coupling.

Problem Statement

Determine how weak coupling to an environment destabilizes many-body localization near its transition, with the single overarching objective of establishing whether the dissipative susceptibility $\chi$ (the linear response of the imbalance relaxation rate to the photon-scattering rate) truly diverges at the MBL transition — so that arbitrarily weak couplings dominate the dynamics near criticality — or instead saturates/crosses over, by resolving the following jointly required, scientifically linked components: (1) experimentally and quantitatively disentangle the contributions of ground-band dephasing ($\gamma_{\mathrm{dp}}$) versus band excitation plus particle loss ($\gamma_{\mathrm{ex}}$) to imbalance relaxation in the interacting regime, requiring an apparatus in which the ratio $\gamma_{\mathrm{dp}}/\gamma_{\mathrm{ex}}$ can be varied independently rather than being fixed by lattice geometry and scattering wavelength; (2) characterize the dynamics and backaction of the finite population trapped in the first longitudinally excited band, establishing whether it acts as an effective small bath that thermalizes or dephases the ground-band MBL system and how strongly it is in turn modified by the localized system; (3) verify or refute the divergence of $\chi$ at the transition, which demands simultaneously smaller background decoherence and remaining in the weak-scattering limit (photon scattering slow compared to intrinsic relaxation) while the intrinsic critical slowing down lengthens relaxation times close to the critical point, and includes the complementary measurement of power-law imbalance decays expected on the ergodic Griffiths side of the transition; and (4) develop the numerical methodology that can include fermion-number nonconservation and higher-band dynamics on the long time scales needed to extract susceptibilities near the transition, enabling quantitative theory-experiment comparison in the regimes of components (1)-(3).

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