Quantum Information Conic Solver

Updated 10 November 2025

Quantum Information Conic Solver is a computational tool that applies conic optimization to analyze and optimize quantum information processes.
It incorporates methodologies from semiconductor exciton research, utilizing concepts from quantum transport, many-body interactions, and moiré physics.
The solver enhances the design and tuning of quantum devices by efficiently modeling coherent states and collective excitonic phenomena.

Spatially indirect excitons (IXs) are neutral bosonic quasiparticles comprising an electron and a hole that are spatially separated in two distinct regions—most commonly, in adjacent quantum wells or atomic layers. This spatial separation dramatically modifies their binding energies, radiative lifetimes, quantum statistics, collective interactions, and transport properties compared to direct excitons. IXs can thus serve as tunable platforms for studying many-body bosonic phenomena, quantum transport, condensate phases, and fundamental dipolar interactions in solid-state systems.

1. Fundamental Properties and Models

In both semiconductor quantum wells (QWs) and atomically thin van der Waals (vdW) heterostructures, the intrinsic structure of IXs dictates their key physical features. For coupled quantum wells such as GaAs/AlGaAs or InGaAs/GaAs double QWs, the electron and hole are confined in different wells separated by a typical distance $d\sim 4$ –$17$ nm (Kuznetsova et al., 2016, Smallwood et al., 1 Apr 2025). In vdW bilayers (e.g., MoSe $_2$ /WSe $_2$ ), interlayer excitons form with $d\sim0.6$ –$1.3$ nm, set by van der Waals gaps (Fowler-Gerace et al., 2022, Zhou et al., 6 Jul 2025, Fowler-Gerace et al., 2023).

Key parameters:

Binding Energy: Hydrogenic model gives

$E_b = \frac{\mu e^4}{2(4\pi \varepsilon_0 \varepsilon)^2 \hbar^2}$

where $\mu$ is the reduced mass, $\varepsilon$ an effective dielectric constant. $E_b\sim 4$ –$20$ meV for GaAs QWs, and $E_b\sim 100$ –$300$ meV for TMD bilayers due to reduced screening and smaller effective masses (Fowler-Gerace et al., 2022, Zhou et al., 6 Jul 2025, Fiorentin et al., 2021).

Radiative Lifetime: Dramatically extended by electron–hole separation, with

$\tau_{\mathrm{IX}} \approx \tau_{\mathrm{DX}}\,\exp(2d/a_B)$

where $a_B$ is the in-plane Bohr radius. Lifetimes routinely reach tens–hundreds of nanoseconds for GaAs IXs and up to microseconds for high-quality TMD heterostructures (Kuznetsova et al., 2016, Fowler-Gerace et al., 2022, Zhou et al., 6 Jul 2025).

Permanent Dipole Moment: $p = e\,d\,\hat z$ leads to strong, long-range dipole–dipole interactions, governing blue-shifts in PL spectra, collective screening, and interaction-driven many-body physics (Laikhtman, 2018, Alloing et al., 2012).
Bosonic Statistics: IXs behave as composite 2D bosons; at high densities and low temperatures, quantum degeneracy and condensation regimes become accessible (Wu et al., 2015, Alloing et al., 2012).

2. Experimental Realizations and Spectroscopic Probes

Semiconductor Quantum Wells

GaAs/AlGaAs coupled wells: Generated via optical excitation with spatially and spectrally resolved photoluminescence; in high magnetic fields, transport and cooling are probed by imaging ring-like PL patterns (Kuznetsova et al., 2016, Dorow et al., 2017).
Gate-defined quantum dots: Electrostatic traps fabricated via gate lithography enable creation and detection of single and few IXs, with voltage and magnetic field tunability (Schinner et al., 2012).
Nonlinear optical probes: Despite suppressed oscillator strength, IX populations and states can be detected via nonlinear Kerr rotation and photoinduced reflectivity alterations that are sensitive to IX–DX interactions (Nalitov et al., 2013).

van der Waals Heterostructures

TMD Bilayers: Interlayer excitons observed in MoSe $_2$ /WSe $_2$ , WS $_2$ /MoS $_2$ , and black/blue phosphorene double-layers, using photoluminescence and time-resolved PL. Encapsulation with h-BN, dual-gate control for out-of-plane field tuning, and deterministic control of twist angle allow exploration of moiré potentials and field-tunable dipoles (Fowler-Gerace et al., 2022, Zhou et al., 6 Jul 2025, Fiorentin et al., 2021, Fiorentin et al., 2021).
Band Structure and Dipole Engineering: Bilayer WSe $_2$ supports distinct spatially indirect, intervalley excitons (e.g., Q–K, Q–Γ), whose vertical dipoles and oscillator strengths are modulated by electric field (Huang et al., 2021). In black/blue phosphorene, exciton charge separation is achieved even in a homo-elemental system by type-II band alignment (Fiorentin et al., 2021).

3. Quantum Transport Dynamics and Moiré Physics

Spatially indirect exciton transport is fundamentally set by the competition between long lifetime and localization effects due to disorder, moiré potentials, and exciton–exciton interactions. The central equation for the steady-state IX density $n$ under continuous wave pumping:

$D \nabla^2 n - \frac{n}{\tau_{\mathrm{IX}}} + A(\mathbf{r}) = 0$

gives a characteristic propagation length $d_{1/e} = \sqrt{D \tau_{\mathrm{IX}}}$ (Fowler-Gerace et al., 2022).

Key phenomena:

Suppressive Moiré Localization: Moiré patterns in TMD heterostructures create periodic lattice potentials with amplitudes of several meV and periods $\sim 10$ –$20$ nm (Fowler-Gerace et al., 2022, Fowler-Gerace et al., 2023).
Screening and Delocalization: At low excitation powers or non-resonant excitation, $d_{1/e}$ is limited to a few microns due to trapping. However, under resonant direct-exciton excitation, screening of moiré and disorder potentials by the dense IX ensemble enables ballistic, superfluid-like propagation up to $d_{1/e} \gtrsim 100\,\mu$ m at $T<10$ K (Fowler-Gerace et al., 2022, Zhou et al., 6 Jul 2025, Fowler-Gerace et al., 2023).
Non-monotonic Density Dependence: As exciton density is increased, IX transport transitions from localized (insulator) to delocalized (superfluid), and then re-enters localization (Mott insulator) at densities $N_\text{mott}\sim1$ IX per moiré site, consistent with the Bose–Hubbard model (Zhou et al., 6 Jul 2025, Fowler-Gerace et al., 2023).

Regime	Mechanism	$d_{1/e}$
Low density	Moiré/disorder localization (Bose glass)	$\sim$ few $\mu$ m
Intermediate	Collective screening, weak scattering (superfluid/ballistic)	$\gtrsim 100\,\mu$ m
High density	Filling-induced Mott localization	$\sim$ few $\mu$ m

4. Many-Body Effects and Collective Phases

Spatially indirect excitons are prominent platforms for exploring 2D Bose–Hubbard model physics:

$H = - t \sum_{\langle i,j \rangle} (b_i^\dagger b_j + h.c.) + \frac{U}{2} \sum_i n_i(n_i-1)$

with inter-site hopping $t$ set by moiré band structure and on-site repulsion $U$ set by dipolar and exchange interactions (Zhou et al., 6 Jul 2025, Fowler-Gerace et al., 2023, Wu et al., 2015).

Superfluid–Insulator Physics:

Quantum-coherent phases emerge at half-filling ( $N\sim1/2$ IX per moiré site, density $n\sim 2\times 10^{11}\,\mathrm{cm}^{-2}$ ), with enhanced transport, macroscopic coherence, and ballistic expansion (Fowler-Gerace et al., 2023, Alloing et al., 2012).
Bose glass or Mott-insulating phases dominate at low and high filling respectively due to localization, evidenced by collapse of the transport length and the emergence of spatially structured PL (Fowler-Gerace et al., 2023, Zhou et al., 6 Jul 2025).

Observation of Coherence:

First-order spatial coherence measured by Mach–Zehnder interferometry shows coherence lengths $\xi\gtrsim 1\,\mu$ m at sub-Kelvin temperature, far exceeding thermal wavelengths and demonstrating macroscopic phase coherence in the fragmented state (Alloing et al., 2012).
Temporal coherence (characterized by narrowing of spectral PL linewidths and longer coherence time $\tau_c$ ) is limited by residual free carriers at the $\sim 4$ ps level for high-purity traps, but can double across Bose–Einstein condensation thresholds (Anankine et al., 2016).

5. Quantum Transport Through Constrictions

IXs in engineered nanostructures (quantum point contacts, slits) exhibit quantum transport phenomena analogous to mesoscopic electrons:

Conductance quantization: Each transverse bosonic subband contributes a conductance quantum $N/h$ (spin/valley degenerate) (Xu et al., 2019).
Diffraction and interference: Single- and double-slit experiments reveal the bosonic de Broglie wavelength, observable directly in PL (Xu et al., 2019).
Talbot effect: Near-field self-imaging and multimode quantum interference accessible over mean free paths of tens of microns are feasible due to long IX lifetimes.

6. IX Mixtures, Band-Structure Engineering, and Materials Diversity

Mixed Direct–Indirect Exciton Phases: In double QWs, a density-induced blue-shift of the IX line brings it into resonance with DXs, yielding a mixture phase with strong van der Waals clustering and phase-separated emission features (Laikhtman, 2018).
Band Structure Control: IX properties can be manipulated by electric fields, strain (in black/blue phosphorene), or stacking order, leading to field-tunable dipole moments, charge separation, and multi-species intervalley excitons with switchable ordering (Huang et al., 2021, Fiorentin et al., 2021).
Interfacial IXs in ZnO/GaN: Spatially indirect excitons form at sharp epitaxial junctions, with binding energies higher than the parent bulk excitons under bias, but become unsustainable above critical field or temperature due to carrier leakage (Arora et al., 2023).

7. Applications, Open Problems, and Outlook

The tunable, long-lived, and strongly interacting nature of spatially indirect excitons renders them central to several research directions:

Cold Bose gases and dipolar quantum fluids: IXs provide a platform to paper quantum condensation, superfluidity, and quantum phase transitions in 2D and moiré lattices (Wu et al., 2015, Alloing et al., 2012, Zhou et al., 6 Jul 2025).
Quantum devices: Prospects include excitonic transistors, interconnects, and quantum photonic circuits exploiting long-range ballistic transport, macroscopic coherence, and voltage-controlled emission (Schinner et al., 2012, Zhou et al., 6 Jul 2025).
Many-body quantum optics: Nonlinear optical responses, strong light–matter coupling, and cross-coupling to direct excitons offer routes to probe dark exciton phases and quantum dynamics inaccessible to linear spectroscopy (Nalitov et al., 2013, Smallwood et al., 1 Apr 2025).
Valleytronics and opto-valley devices: Intervalley IXs in TMD bilayers with field-tunable circular polarization and long lifetimes open prospects for valley-based information processing (Huang et al., 2021).
Transport in moiré superlattices: Moiré potentials allow simulation of Bose–Hubbard physics, Mott transitions, and superfluid–insulator boundaries in solid-state systems using IX gases (Zhou et al., 6 Jul 2025, Fowler-Gerace et al., 2023).
Quantum simulation: Controlled double-layer graphene and TMD bilayer structures can realize rich phase diagrams distinguished by layer, spin, and flavor orders, accessible via capacitance, transport, and coherent optics (Su et al., 2016, Wu et al., 2015).

These advances have caused spatially indirect excitons to emerge as a core element in the paper of collective phenomena, quantum transport, and low-dimensional excitonics across a broad class of solid-state systems.