Materials Beyond Hamiltonian Limits -- Quantum Measurement as a Resource for Material Design

Abstract

Recent studies have identified materials and devices whose behavior lies beyond the scope of conventional electronic-structure theory. Such theories are formulated entirely in terms of Hamiltonian evolution and therefore describe only unitary dynamics and thus only a restricted class of quantum systems. In contrast, electron systems that incorporate quantum measurement as an intrinsic dynamical element undergo Hamiltonian evolution interleaved with projection-induced state updates. This unitary-projective dynamics breaks constraints imposed by purely unitary evolution and permits stochastic population transfer between symmetry-related transport channels, thereby enabling fundamentally new material functionalities. This insight motivates the deliberate design of materials and devices that harness unitary-projective dynamics. This article explores the foundations of unitary-projective electron dynamics and charts the resulting landscape of quantum materials and their functionalities. Model calculations demonstrate passive mesoscopic structures with intrinsic nonreciprocal single-electron transmission, materials exhibiting a novel category of magnetism, and possible platforms for energy harvesting and conversion with efficiencies that exceed the standard Carnot limit.

Figures (9)

• Figure 1: Descriptions of quantum states at finite temperatures. (A) Schematic illustration of a conventional Hamiltonian-based description of electronic systems at a finite temperature. This framework applies in regimes where electronic coherence lengths and mean free paths are large compared to microscopic length scales, such that electronic states are well described by stationary energy eigenstates of an underlying Hamiltonian. Solving the Schrödinger or Dirac equation yields energy eigenstates $|n\rangle$ and eigenvalues $E_n$ of an underlying Hamiltonian. Finite-temperature effects are incorporated by assigning thermal occupation probabilities (indicated by the saturation of the blue color representing the eigenstates in the figure) to these eigenstates or to corresponding quasiparticle states, resulting in a thermal ensemble described by a density matrix $\rho$. In more elaborate treatments, temperature is incorporated through thermally induced scattering and lifetime broadening encoded in self-energies, or through thermal sampling of ionic configurations. In all cases, the description remains grounded in an underlying Hamiltonian eigenstate or quasiparticle structure, which provides the basis states populated by the electrons. (B) Illustration of unitary–projective evolution of an electron trajectory encountering a trap site (an inelastic scattering center), sketched in a quantum-trajectory setting. This description is appropriate in the regime where inelastic scattering events occur on timescales comparable to the electron’s traversal time through the system. An electron incident from the left propagates ballistically under unitary Hamiltonian dynamics until it encounters a trap. There, coupling to a phonon bath enables a projective event: the electron is either captured by the trap (downward branch) or it continues propagating in a null-measurement branch (upward arrow). In both cases, the subsequent evolution is again unitary until the electron encounters another trap or is released from the trap by a thermal fluctuation, allowing the process to repeat. Here, temperature governs the trap occupancy, the detrapping rate, and the energy distribution of the states occupied after release. The resulting trajectories are not energy eigenstates, and the associated thermal ensemble differs qualitatively from the thermal ensemble of Hamiltonian eigenstates as illustrated in (A).
• Figure 2: Design principle of unitary–projective devices with emergent non-equilibrium functionality. The central idea is to combine coherent (Hamiltonian) but symmetry-broken propagation with localized projective state updates. This converts internal time asymmetry into directional transport and non-equilibrium order. Unitary Hamiltonian evolution (top left): Although coherent unitary dynamics in a spatially asymmetric structure can produce direction-dependent dwell times $\tau_{L \to R}$ and $\tau_{R \to L}$, the absence of projections and explicit time-reversal-symmetry breaking ensures that the scattering matrix $S$ remains unitary. Consequently, the forward and backward transmission probabilities $T_{L \to R}$ and $T_{R \to L}$ are identical, in accordance with microreversibility and Onsager reciprocity. Projective, stochastic state updates (bottom left): Projection-like processes export phase information from the device (subsystem) to the environment, giving rise to nonunitary reduced dynamics and, in non-equilibrium settings, the breakdown of detailed balance. Such behavior is not captured within conventional linear-response theory or purely unitary scattering models. Combined unitary–projective evolution (right): When symmetry-broken Hamiltonian propagation is combined with localized projections, qualitatively new functionalities can emerge, including directional transmission probabilities (single-particle valve behavior), nonreciprocal transport of coherence, and, at finite temperatures $T$, non-equilibrium steady-state circulating currents $I$ generating magnetic moments $M$. Directionality arises from the asymmetric exposure of counter-propagating trajectories to projective updates.
• Figure 3: Asymmetric nanoscale conductors. From left to right, the figure shows (i) a nominally symmetric one‑dimensional conductor containing two tunnel barriers of different heights (pink and purple, respectively), (ii) a conductor with transverse asymmetry, (iii) a conductor with longitudinal asymmetry, and (iv) an asymmetric Aharonov–Bohm ring. To generate directional asymmetry in ballistic electron propagation along the transport direction, the transversely asymmetric conductor (ii) and the quantum ring (iv) must be subjected to a perpendicular magnetic field $H$. In contrast, conductors (i) and (iii) possess the necessary spatial asymmetry without a magnetic field. Figure adapted from Bredol2021b.
• Figure 4: Ballistic transport in an asymmetric quantum ring. Time evolution of unitary electron propagation through the ring from the left contact $L$ to the right contact $R$ (green) and from $R$ to $L$ (blue). The schematic on the left depicts the ring, threaded by an applied magnetic flux $\Phi$, with an ion $A$ (white circle) embedded at the center of its upper arm. The panels on the right show quantitative results for the time-dependent probability density of the electron wave function at the position of ion $A$, obtained from tight-binding calculations Bredol2021b. For propagation from $L$ to $R$ (top panel), the ion is encountered once, whereas for propagation from $R$ to $L$ (bottom panel), the electron passes the ion three times. The oscillatory “ringing” in the probability density predominantly arises from the finite number of tight-binding sites. Figure adapted from Bredol2021b.
• Figure 5: Unitary–projective transport in asymmetric structures. (A) Schematic of an asymmetric quantum ring connecting contacts $L$ and $R$, threaded by a magnetic flux $\Phi$. The ring contains an inelastic scattering center (yellow dot), such as an electron trap, at the center of its upper arm. This center couples the electronic degrees of freedom of the ring to the phonon system of the environment (e.g., a substrate, lila) at temperature $T$, which may or may not be thermally coupled to a further external bath (e.g., the atmosphere). (B) Difference between the transmission probabilities $P_{L\to R}$ and $P_{R\to L}$ calculated as a function of the average number of inelastic scattering events per electron passage. The calculations are based on quantum-trajectory simulations Bredol2021b for the structure sketched at the top, where the dots represent the sites of the tight-binding model. Yellow sites mark electron traps. Nonreciprocal transmission arises for an intermediate range of inelastic scattering, approximately 0.05–50 scattering events per passage. In the purely unitary quantum limit with negligible scattering and in the classical limit with a large number of scattering events leading to diffusive transport, the transmission is reciprocal, in accordance with Onsager’s reciprocity theorem. Figure adapted from Bredol2021b.