Parent Hamiltonian and intrinsic phase transition in non-Hermitian photonic systems

AI summary

Overview

Research area: Quantum physics — non-Hermitian quantum systems, matrix product states (MPS), and photonic quantum simulation.

Technical level: Advanced. The paper assumes familiarity with non-Hermitian Hamiltonians, biorthogonal quantum mechanics, tensor networks, and linear-optics experiments.

Scope (one sentence): The paper reports the first experimental realization of non-Hermitian parent Hamiltonians (NH-PHs) using single photons, constructing them from asymmetric Affleck–Kennedy–Lieb–Tasaki (AKLT) matrix product states for an N = 2 model and extending the framework to an N = 3 model that exhibits an intrinsic non-Hermitian phase transition.

What This Paper Is About

Most non-Hermitian experiments work by taking an established Hermitian Hamiltonian and adding gain, loss, or other dissipative terms. The authors instead ask whether a non-Hermitian Hamiltonian can be designed directly from a desired set of properties, without perturbing an existing Hermitian model. Using the recently proposed non-Hermitian parent Hamiltonian method, they build such Hamiltonians from a predesigned pair of matrix product states that serve as the left and right zero-energy ground states, then realize and characterize them with single photons.

Key Contributions

1. First experimental generation of NH-PHs. The authors report the first experimental realization of non-Hermitian parent Hamiltonians, constructed from MPSs representing asymmetric AKLT states, and validate the construction on a photonic platform.

2. Biorthogonal ground-state extraction via imaginary-time evolution. They implement imaginary-time evolution (with fixed evolution time τ = 10 in natural units) to project a maximally mixed initial state onto the right and left ground states of the constructed Hamiltonian, using a passive-optics mapping based on singular value decomposition so that non-unitary gain is converted into mode-selective loss.

3. Multi-observable characterization of the N = 2 model. They measure four order parameters — O_left, O_right, O_chiral, and O_AF — probing non-reciprocal correlations, chiral imbalance, and conventional antiferromagnetic correlations, plus the entanglement spectra of the reconstructed right and left ground states.

4. Observation of an intrinsic non-Hermitian phase transition in an N = 3 model. Extending the framework to a new spin-1/2 model with bond dimension d_v = 2, they observe an abrupt jump in a ferromagnetic order parameter coinciding with spectral crossings, and verify the transition persists for N = 4, 6, 10.

Main Findings

• Designed Hamiltonian reproduces target states. For the N = 2 model built from asymmetric AKLT MPSs, the constructed NH-PH has eigenvalues {0, 1, 1, 1}, with the designed |R⟩ and |L⟩ as the right and left zero-energy ground states.

• Order parameters match theory across the full μ range. Theory predicts ⟨O_AF⟩ = −1/4, ⟨O_left⟩ = −μ/4, and ⟨O_right⟩ = −1/(4μ); the experimentally extracted values agree with these predictions across the entire range of the asymmetry parameter μ.

• Chirality is directly observed. At μ = 1 (the Hermitian limit) the two directional correlators coincide at −1/4 and ⟨O_chiral⟩ = 0. Tuning μ away from 1 makes ⟨O_chiral⟩ change sign — positive for μ > 1 and negative for μ < 1 — directly demonstrating non-Hermitian chirality encoded in the MPS design.

• Entanglement spectra reflect the designed asymmetry. The two dominant Schmidt coefficients of both |Ψ_R^s⟩ and |Ψ_L^s⟩ split asymmetrically as μ deviates from 1, with the splitting increasing with |μ − 1|.

• Small system captures bulk physics. Despite N = 2, the observed trends align closely with theoretical predictions originally derived for infinitely large systems, indicating the MPS-based construction captures essential bulk physics even at the minimal two-site level.

• Intrinsic phase transition in the N = 3 model. As the phase parameter φ is varied, the real part of the ground-state energy becomes negative in the intervals π/6 < φ < π/3 and 2π/3 < φ < 5π/6. In these regions the designated zero-energy MPS is no longer the global ground state, because new eigenstates with lower negative energy emerge — a consequence of the breakdown of the non-Hermitian variational principle.

• Order parameter jumps at the transition. The ferromagnetic order parameter Õ = σ₁^z ⊗ σ₂^z ⊗ σ₃^0 shows abrupt jumps coinciding with the spectral crossings, confirming the phase transition.

• Transition is not a finite-size effect. The same transition is verified for N = 4, 6, 10.

• Independent cross-check. As a separate verification, the authors directly prepare the target AKLT states |R⟩ and |L⟩ by mapping the MPS tensors to quantum circuits with unitary operations U₁ and U₂, and the resulting order parameters agree with theory and with the imaginary-time-evolution results.

Methodology in Plain English

The approach inverts the usual logic of non-Hermitian experiments. Instead of starting from a Hermitian model and adding dissipation, the authors start by writing down the quantum states they want — as matrix product states, a compact tensor-network representation of a many-body wavefunction — and then use the parent Hamiltonian construction to analytically obtain a Hamiltonian for which those states are the left and right zero-energy ground states. The chirality is built in through an asymmetry parameter μ in the virtual bond of the MPS.

To verify that the constructed Hamiltonian really produces those states, they perform imaginary-time evolution, which exponentially suppresses higher-energy eigenstates and leaves the ground state after a sufficiently long evolution time. Since true non-unitary evolution with gain is hard for single photons, they factor out a global amplification and map the operation onto a passive optical network with no gain, implemented via singular value decomposition into unitary operations and a non-unitary diagonal loss element. The loss is realized with beam displacers and half-wave plates, and the unitary parts by recombining photons into designated spatial modes and rotating polarization.

The single photons come from type-II spontaneous parametric down-conversion in a periodically poled potassium titanyl phosphate (PPKTP) crystal. States are encoded in polarization (|h⟩, |v⟩) and spatial (|r⟩, |l⟩) modes. Photons are prepared in a maximally mixed state using two unbalanced interferometers (path difference exceeding the coherence length) built from four polarizing beam splitters, three half-wave plates, and a beam displacer. The left and right ground states are reconstructed by quantum state tomography using a beam displacer, a polarizing beam splitter, three quarter-wave plates, and three half-wave plates, with detection by avalanche photodiodes and coincidence counting. For the N = 3 model, the realization uses three qubits — two path qubits and one polarization qubit.

Why This Matters

Impact on research. The work provides the first experimental realization and characterization of non-Hermitian Hamiltonians with controllable and customizable properties, offering a systematic blueprint for reverse-engineering non-Hermitian phases rather than perturbing Hermitian ones. It also demonstrates an intrinsic non-Hermitian phase transition rooted in the breakdown of the non-Hermitian variational principle — a mechanism distinct from conventional thermodynamic phase transitions. The imaginary-time-evolution technique is not restricted to non-Hermitian Hamiltonians and can also prepare ground states of Hermitian systems, making it broadly applicable to quantum simulation.

Potential applications (as discussed in the paper):

• Enhanced sensing, where small perturbations near a critical point can be significantly amplified.
• Non-Hermitian engineered single-mode lasing.
• Robust transport and chiral state transfer.
• Accelerated entanglement generation.

Industry relevance. The results are relevant to photonic hardware and integrated optics, where gain and loss are natural design parameters; to metrology and sensing; and to quantum simulation platforms including photonics, cold atoms, and superconducting circuits.

Future Directions

• Non-Hermitian topology and many-body dynamics. The authors propose extending the NH-PH framework to study non-Hermitian topology, many-body dynamics, and quantum metrology in regimes beyond existing methods.
• Larger systems and other platforms. The N = 3 construction is verified for N = 4, 6, 10; scaling further and porting the scheme to cold atoms and superconducting circuits are natural next steps.
• Connecting level crossings to open-system thermodynamics. The paper states that the conceptual connection between these non-Hermitian level crossings and thermodynamic phase transitions in open systems remains an evolving area for future study.
• Broader state-preparation methodology. The MPS-to-quantum-circuit mapping used as a cross-check is described as a general approach that can be extended to larger N systems for quantum simulation of many-body systems.

Target Audience

Researchers and graduate students in quantum physics working on non-Hermitian systems, non-Hermitian topology, and parity-time symmetry; experimentalists in linear optics and photonic quantum simulation; and theorists interested in tensor-network methods, parent Hamiltonians, and biorthogonal quantum mechanics. Readers without a background in non-Hermitian physics will need to consult the cited references on biorthogonal expectation values and the non-Hermitian variational principle to follow the phase-transition argument.

Authors’ abstract

Non-Hermitian systems host phenomena absent in Hermitian physics, but realizing Hamiltonians with intrinsic non-Hermitian properties remains challenging. The theoretical method of non-Hermitian parent Hamiltonian (NH-PH) enables the construction of a non-Hermitian system from a pair of matrix product states (MPSs) with tailored properties. Here, we report the first experimental generation of NH-PHs. This generation starts from MPSs that represent asymmetric Affleck--Kennedy--Lieb--Tasaki (AKLT) states. The construction is validated with single photons via imaginary-time evolution of the generated NH-PH to obtain its left and right ground states. We then characterize the properties of the system by measuring four different order parameters that probe non-reciprocal correlations, chiral imbalance, and conventional antiferromagnetic correlations. Furthermore, extending the framework to a larger system with a different model, we observe an intrinsic non-Hermitian phase transition, manifested by abrupt jumps of an order parameter when the designated zero-energy modes cease to be the globally lowest-energy states. Our work provides the first experimental realization and characterization of non-Hermitian Hamiltonians with controllable and customizable properties, opening new avenues for exploring intrinsic non-Hermitian phenomena across diverse physical platforms.