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Use two collinear quantum point contacts in a clean two-dimensional electron gas. Surface split gates deplete the electron gas beneath them, leaving two short constrictions facing one another across an open region. The first injects a directional beam from a biased reservoir; the second collects carriers into a downstream reservoir, with electric current flowing in the same spatial direction through both constrictions. Additional surrounding contacts can absorb carriers that miss the collector. A direct collector signal at zero magnetic field, sensitive to alignment and to obstruction of the intervening path, demonstrates ballistic electron transport across the separation.

An illustrative GaAs device uses gate gaps of a few hundred nanometres and electrically tunable constriction widths of tens to a few hundred nanometres, with injector–collector separation of a few micrometres. These are typical design scales, not dimensions supplied for a particular chip in the question. The electrical width can be much smaller than the lithographic gap because depletion extends laterally. A one-mode opening has width comparable to roughly half a Fermi wavelength; it is not simply any opening below one micrometre. Low-temperature operation suppresses inelastic scattering and resolves transverse modes. The direct trajectory length should be comfortably below the relevant transport mean free path.

Collimation is provided by the smooth exit from a constriction. In adiabatic electron collimation, transverse mode action is approximately conserved, giving $p_{\perp }W\simeq \mathrm{constant}$. As the channel opens from width $W_0$ to $W_{\mathrm{out}}$, $p_{\perp ,\mathrm{out}}\simeq p_{\perp ,0}{W}_0/W_{\mathrm{out}}$. At the same time the bottom of the electrostatic potential falls, increasing longitudinal kinetic energy. Thusand the angular distribution narrows. The reciprocal collector selects this forward-directed set of trajectories. A narrow abrupt aperture by itself may diffract into a broad fan; the smooth funnel and its potential profile are important. Here “focusing” refers primarily to collimation and efficient injector–collector coupling, not a claim that every emitted trajectory converges to one exact geometric point.

In scanning gate microscopy, a conducting tip biased sufficiently negatively creates a local repulsive potential and, when strong enough, a small depleted region in the electron gas beneath it. Raster the tip and record the collector current or differential conductance. Placing the perturbation on a strongly contributing path deflects or backscatters electrons and changes collection, producing spatial contrast. A primary scanning-probe experiment records conductance changes as a negatively charged tip depletes and backscatters the two-dimensional gas.^[ref]

The resulting image is a map of transport sensitivity to the tip, not a literal nonperturbing photograph of an electron trajectory. Broad electrostatic gating of either constriction should be distinguished from local beam disruption. Weak smooth disorder can create branches even in a high-mobility sample. Where electron phase coherence is preserved, interference between alternative paths can add fringes, often with a scale comparable to half a Fermi wavelength for a round-trip path-length change. Ballistic motion and phase-coherent interference are related but distinct requirements.

A small perpendicular magnetic field curves the imaged paths through the Lorentz force, $\mathbf{F}=-e\mathbf{v}\times \mathbf{B}$. Reversing $B$ reverses the bending, and a fixed aligned collector generally loses the initially straight beam unless a curved path connects it. For a circular orbit at the Fermi velocity,For one valley with two nearly equally occupied spin species at small field, state counting in the Fermi disc gives $n_s=2\pi k_F^2/(2\pi {)}^2=k_F^2/(2\pi )$. HenceSpin polarization or extra valley degeneracy changes the density-to-wavevector relation; the stated low-field GaAs estimate uses the ordinary spin-degenerate case.

The electron mobility gives the momentum relaxation time in a parabolic-band Drude model, $\mu =e{\tau }_{\mathrm{tr}}/m^{\ast }$. ThereforeThe effective mass cancels from both ${\ell }_{\mathrm{tr}}$ and $r_c$ when expressed in terms of density and mobility. For the supplied density,With $m^{\ast }=0.067m_0$, $m_0\simeq 9.1093837139\times 10^{-31}\mathrm{kg}$, one also obtains $v_F\simeq 1.937\times 10^5\mathrm{m}{\mathrm{s}}^{-1}$ and ${\tau }_{\mathrm{tr}}\simeq 38.1\mathrm{ps}$. The requested lengths areA useful independent ratio is ${\ell }_{\mathrm{tr}}/r_c=\mu |B|=3$. The standard constants used here are explicitly tabulated, rather than inferred from the sample.^[ref]

Mobility measures momentum relaxation. Predominantly small-angle elastic scattering can occur more frequently than that transport time, and the phase-coherence time is different again. Thus the calculation supports a several-micrometre transport scale, without proving that every electron experiences literally no perturbation over that distance.

As temperature rises, the directed ballistic collector signal decreases. Increased electron-phonon scattering and other inelastic processes reduce survival of the directed beam; thermal averaging broadens the mode and angular selection, and interference contrast also diminishes. A diffuse background current can remain, so disappearance of beam contrast need not mean all current through the device vanishes.

A quantum point contact is made by using negative surface split gates to leave a short narrow channel in a clean two-dimensional electron gas. Its length should be smaller than the transport mean free path; transverse confinement should be strong enough that subband spacing exceeds thermal and lifetime broadening. Low-resistance Ohmic contacts connect large reservoirs. Apply a small source–drain bias, measure current and differential conductance, and sweep the gate to change the channel width or barrier height. Contact and wiring series resistance must be accounted for when comparing plateau values.

The Landauer formula explains the conductance without treating the short channel as a diffusive wire. For each propagating one-dimensional mode, the density of states and group velocity cancel in the incident flux, leaving $e/h$ per spin per energy interval. The reservoir occupation difference supplies the net current:The factor two assumes unresolved spin degeneracy. In zero-temperature linear response,A well-transmitted channel has $T_n\simeq 1$, giving steps $G=N(2e^2/h)$ as modes open. A finite two-terminal resistance persists even for a perfectly ballistic channel because reservoirs inject only a finite number of modes; it is not evidence of resistive scattering inside the channel. Gate-controlled point-contact conductance quantization was demonstrated in a two-dimensional electron gas.^[ref]

The saddle-point model of a quantum point contact resolves how modes open. Near the bottleneck, choose the channel direction as $x$ and writeTransverse harmonic confinement gives barrier tops $E_n=U_0+\hslash {\omega }_y(n+1/2)$. Scattering through the inverted longitudinal harmonic barrier givesSmall longitudinal curvature makes a sharp transmission threshold compared with the transverse subband spacing; finite curvature rounds each conductance step. This is the explicit smooth-barrier model developed for a saddle constriction.^[ref]

At larger source–drain bias, the current samples a finite energy window. For a rigid transmission spectrum at zero temperature,With symmetric bias, ${\mu }_{S,D}=E_F\pm eV/2$, the nonlinear conductance of a quantum point contact isIf both window edges sample $N$ open modes, the usual plateau remains. If one edge samples $N+1$ and the other $N$, differential conductance approaches $(N+1/2)(2e^2/h)$: a half-integer plateau relative to the zero-bias step. It is a bias-window effect, not half an electron spin degeneracy. Ridges in gate–bias transconductance occur when a reservoir chemical potential crosses a subband threshold; their separations can measure subband energies.

The half-plateau expression assumes symmetric bias and an approximately bias-independent barrier. An asymmetric voltage drop or self-consistent shift of $U_0$ changes the numerical relation. At still larger biases, heating and inelastic scattering can invalidate a rigid elastic saddle model. Thus the zero-bias quantization counts transmitted modes, while nonlinear spectroscopy compares mode counts at the two ends of the bias window.

Photon diffusion approximates the spreading of radiation after many scattering events have largely randomized propagation directions. Individual photons travel along paths; their distribution can obey a diffusion equation on scales much larger than a transport mean free path. The longer arrival times arise from longer, repeatedly redirected paths.

For homogeneous, conservative isotropic scattering, the photon diffusion coefficient for isotropic scattering is $D=v\ell /3$, where $v$ is the transport speed and $\ell$ the scattering mean free path. Directional persistence replaces that scale by a transport mean free path in an appropriate transport model. Crossing a characteristic thickness $L$ then takes a time of order $L^2/D$, with geometry-dependent factors, which can greatly exceed the direct travel time $L/v$.

A diffuse photon time-of-flight experiment sends a short pulse into a scattering sample and measures the delayed transmitted or returning light. A pulsed source, a fast photodetector and time-correlated single-photon counting can build an arrival-time histogram from repeated pulses. The detector's timing response must be accounted for before attributing all measured broadening to the sample.^[ref]

The 1989 study by M. S. Patterson, Britton Chance and B. C. Wilson used a slab diffusion model to relate picosecond-pulse reflectance and transmittance to absorption and scattering, comparing predictions with preliminary measurements and Monte Carlo simulations.^[ref] The earliest nearly unscattered light, nearby boundaries and strongly directional transport can require a fuller radiative transfer description.

For conservative photon transport at constant speed $v$, suppose successive scattering directions are independent and isotropically distributed, and free-path lengths follow an exponential distribution of mean $\ell$. In three spatial dimensions, the resulting long-time diffusion coefficient isHere $\ell$ is the scattering mean free path. These assumptions describe homogeneous, memoryless interactions without absorption.

The velocity autocorrelation function gives a direct derivation. An initially isotropic ensemble has zero mean velocity. During a lag $s$, the probability of no scattering is $e^{-vs/\ell }$. No scattering preserves the direction; one or more independent isotropic scattering events give zero expected directional correlation. ConsequentlyThe three-dimensional Green-Kubo formula for diffusion now yieldsThe factor $1/3$ converts the dot-product correlation into one Cartesian component under isotropy.

The same correlation gives the full mean squared displacement:At short times this is $v^2{t}^2$; at long times its leading term is $6Dt$. The model therefore connects direct flights to the diffusive many-scattering limit.

The free-path distribution matters. A fixed-step isotropic random flight with every length equal to $\ell$ has mean-square displacement $n{\ell }^2$ after $n$ steps and elapsed time $n\ell /v$, giving $D=v\ell /6$. Replacing exponential flights by fixed-length flights thus changes the coefficient by a factor of two. For anisotropic scattering, directional persistence instead leads to a transport mean free path; strong absorption and boundaries require additional modeling.