Scaling behavior of electron decoherence in a graphene Mach-Zehnder interferometer

Electron quantum optics explores the analogy between electron propagation in quantum conductors and photons in optical experiments, leveraging one-dimensional, chiral, dissipationless quantum Hall edge channels as electron analogs of optical fibers. In GaAs/AlGaAs heterostructures, various decoherence sources have been identified, including edge reconstruction by disorder, intra-channel Coulomb interactions within single edge channels, and inter-channel Coulomb interactions between adjacent channels. The latter has been a principal barrier to complex quantum circuits, prompting engineering efforts to mitigate it. However, new dissipation mechanisms have been recognized and longstanding experimental observations remain debated. Investigating decoherence in quantum Hall edge channels in different 2D materials can clarify these issues by providing systems with similar phenomenology but different intrinsic parameters (e.g., electron velocity, capacitive coupling/screening, and geometry). This work probes decoherence in a graphene quantum Hall PN junction Mach-Zehnder interferometer where energy scales for decoherence are about an order of magnitude larger than GaAs, enabling observation of a universal temperature-length scaling of interference visibility and identification of intra-channel interactions as the dominant decoherence mechanism.

Prior work in GaAs/AlGaAs-based quantum Hall interferometers has attributed decoherence to edge reconstruction by disorder (e.g., Chamon et al.), intra-channel interactions causing charge-fluctuation-induced dephasing (Seelig & Büttiker; Youn et al.), and inter-channel interactions leading to mode fractionalization and dephasing (Roulleau et al.; Levkivskyi & Sukhorukov; Bocquillon et al.). Engineering approaches have been attempted to reduce inter-channel interactions (Huynh et al.; Duprez et al.), yet new dissipation channels (e.g., Auger-like processes and quasiparticle relaxation) have been reported (Krähenmann et al.; Rodriguez et al.), and key observations (e.g., lobe patterns and visibility suppression) remain contested (Roulleau et al.; Litvin et al.). In graphene, valley isospin physics at PN junctions enables beam splitting (Tworzydło et al.; Trifunovic & Brouwer), and recent STM studies suggest native defects influence valley mixing (Joucken et al.). Graphene Fabry-Pérot interferometers have shown exponential thermal decay without algebraic regimes, likely due to etch-defined edges, with coherence lengths smaller than those reported here for MZIs (Déprez et al.; Ronen et al.).

Device and interferometer formation: A monolayer graphene sheet encapsulated in hexagonal boron nitride is patterned with independently controlled bottom and top gates; the bottom gate covers the entire sample, while the top gate covers only the right half. Under perpendicular magnetic field, the left half is tuned to a P region (ν = −1) hosting a clockwise spin-up edge channel; the right half is an N region (ν_N = 2) with two counterclockwise channels of opposite spin. The PN interface thus supports three co-propagating channels; the two spin-up channels have opposite valley isospin. A Mach-Zehnder interferometer (MZI) forms along the PN interface by tuning side gates to create beam splitters via valley mixing. In the 'large' MZI (ν1 = ν2 = −1), beam splitters occur at the intersections of the PN interface with the graphene edge under the side gates, yielding Aharonov–Bohm (AB) oscillations with period ΔB ≈ 20 mT and arm length L ≈ 1.5 µm (from geometry). Tuning a side gate to ν = 0 eliminates the edge intersection beam splitter; AB oscillations reappear when a beam splitter forms at a bulk intersection of the PN interface with a ν = 0 region under the gate. Two smaller MZIs are realized: 'intermediate' (ν1 = −1, ν2 = 0) with ΔB ≈ 34.5 mT and L ≈ 1.05 µm; and 'small' (ν1 = 0, ν2 = −1 or ν1 = 0, ν2 = −1 with ΔB ≈ 81 mT) with L ≈ 0.62 µm. Length ratios agree with AB periods, indicating similar arm spacings; calculations and observed periods suggest arm spacings on the order of ~90–110 nm depending on gating. Measurement of transmission and visibility: Current I0 is injected from an ohmic contact; due to spin selection (Zeeman splitting ~1 meV at 9 T), the spin-down spectator channel is reflected. For (ν_L, ν_R) = (−1, +2), half of I0 (spin-up) participates, and transmission is defined as T_MZ = I_T/(I0/2); for (−1, +4), T_MZ = I_T/(I0/4). Visibility is defined as Vis = (T_MZ,max − T_MZ,min)/(T_MZ,max + T_MZ,min); normalized visibility Vis/V0 uses V0 at base temperature. Thermal characterization: Electron temperature is verified via Johnson–Nyquist noise using home-made cryogenic amplifiers and an LC tank at 2.2 MHz to avoid 1/f and mechanical noise. Voltage noise spectra are fitted to a circuit model (sample resistance in parallel with an RLC resonator) to extract gain and electron temperature. The current noise S = 4k_B T_e/R_H is linear in refrigerator temperature down to 25 mK, confirming good thermalization. Bias dependence: Non-equilibrium decoherence is probed via DC bias V_DC applied either to the upper right (biasing two N-region channels) or upper left (biasing single P-region channel) contacts. Lobe patterns in visibility versus V_DC are compared across injection configurations. Theoretical model: An intra-channel capacitive interaction model is used with Hamiltonian H = H0 + H_int + H_T, where H0 describes chiral electrons in left/right arms with drift velocity v; H_int = E_c ∑_{a=1,2} (Q_a/e − N_a)^2 with E_c = g v h/(2L), g dimensionless interaction strength; H_T describes tunneling at beam splitters. Using bosonization, the finite-temperature Green’s functions G^±(t) are computed, yielding a visibility integral invariant under scaling L → bL, T → T/b, demonstrating Vis depends on the product LT. Model parameters (e.g., v and g) are chosen to fit data; example fit uses v ≈ 4.4×10^4 m/s and g ≈ 3.3. Measurements setup: Experiments are performed in a dry dilution refrigerator (base 13 mK). Transmitted currents and Hall resistances are measured with lock-in amplifiers and low-noise preamplifiers; AC excitations 1–5 nA at 70–300 Hz are used. Buried ohmic contacts under top gates allow direct determination of regional filling factors.

• Universal scaling of thermal decoherence: The interference visibility for three MZIs of different lengths (L ≈ 1.5 µm, 1.05 µm, 0.62 µm) collapses onto a single curve when plotted versus scaled temperature LT/L0, revealing a universal crossover from algebraic decay at low temperature to exponential decay at higher temperature. The crossover temperature scales as ~ħv/(k_B L) and is ≈ 350 mK for the large MZI (L ≈ 1.5 µm). - Suppressed decoherence at low temperature: Below ~1 K the visibility decays algebraically, indicating suppression of thermal decoherence; interference persists above 1.5 K. The coherence length extracted from the exponential regime is ~1.24 µm at 1 K. - Validation of electron thermalization: Johnson–Nyquist noise measurements show linear current noise versus temperature down to 25 mK, ruling out heating as the origin of the algebraic regime. - Inter-channel interactions are weak: Visibility decay versus temperature at various filling factor configurations, including (ν_n, ν_p) = (2, −1), (2, −2), and (4, −1), are similar, indicating inter-channel interactions are not the dominant dephasing source. Device geometry (gates 30–50 nm from graphene versus 50–60 nm channel spacing) enhances screening compared to GaAs. Theoretical analysis shows that symmetric coupling of an additional channel to both arms produces no decoherence; fitting data with inter-channel interactions alone would require unrealistic asymmetry. - Intra-channel interactions dominate: An intra-channel capacitive interaction model quantitatively reproduces the scaling collapse and crossover, with good agreement to experiment for reasonable parameters (e.g., v ≈ 4.4×10^4 m/s, g ≈ 3.3). - Bias (lobe) pattern: Visibility versus DC bias shows a dip near ~220 µV and a single side lobe at higher |V_DC|. The lobe pattern is nearly independent of which contact is biased (two-channel N-side vs single-channel P-side injection), contrasting GaAs v = 2 MZIs where inter-channel interactions cause strong source dependence. - Interferometer characterization: AB periods of ΔB ≈ 20 mT (large), 34.5 mT and 81 mT (smaller MZIs) correspond to arm lengths consistent with device geometry; side gates shift beam splitter positions from edges to bulk intersections (ν = 0 regions).

The observed scaling of visibility with the product of temperature and interferometer length, and the universal crossover from algebraic to exponential decay, directly address the central question of decoherence mechanisms in graphene quantum Hall MZIs. The collapse across a threefold variation in L indicates that decoherence arises from processes that scale with L or from length-independent mechanisms, ruling out bulk disorder or small charge puddles as dominant sources. Combined with the weak dependence on added co-propagating channels and the near-identical bias lobe patterns for different injection schemes, the data demonstrate that inter-channel Coulomb interactions are strongly screened and not the primary cause of dephasing in this platform. Instead, intra-channel capacitive interactions within each arm govern decoherence; their impact is suppressed when the electron thermal length exceeds the arm length, producing the algebraic regime. These findings contrast with GaAs devices, where inter-channel interactions and mode fractionalization dominate and no algebraic regime has been unequivocally observed. The ability to reach a regime of nearly frozen decoherence in graphene, due to favorable device geometry and screening, has significant implications for scalable electron quantum optics and quantum information processing using quantum Hall edge states.

This work demonstrates a graphene quantum Hall Mach-Zehnder interferometer with a universal thermal scaling of visibility and a crossover from exponential to algebraic decay, evidencing suppression of decoherence at low temperatures. The results establish intra-channel interactions as the dominant decoherence mechanism in this platform, with inter-channel interactions effectively screened by nearby gates. Interference persists above 1.5 K, and a coherence length of ~1.24 µm at 1 K is extracted. The ability to operate in a regime of reduced decoherence makes graphene a promising platform for flying qubits, two-particle interferometry and entanglement generation, and valleytronics. Future work can aim at optimizing visibility toward the theoretical maximum, extending coherence lengths and device sizes, engineering tunable interaction strengths, and integrating multiple interferometers for complex quantum circuits.

• Maximum visibility is limited to about 60% at base temperature, and the origin of not reaching 100% remains unclear (as noted by the authors). - While inter-channel interactions appear weak and are not dominant, they may not be completely suppressed in all configurations. - Small mismatches between estimated arm spacings/lengths and AB periods indicate possible device-specific variations (e.g., edge electrostatics and gating conditions across cooldowns). - The theoretical fits rely on parameter choices (e.g., drift velocity v and interaction parameter g) inferred from the data; independent determination of these parameters would further validate the model.