⚛️ high-energy theory

Krylov Complexity of Time-Dependent Optical Hamiltonians

This paper establishes that Krylov complexity in time-dependent optical systems is experimentally accessible through specific observables, while demonstrating that its structure and solvability are fundamentally determined by the interplay between control history, state preparation, and the chosen reference frame across various Hamiltonian models.

Original authors: Abhishek Chowdhury, Aryabrat Mahapatra, Ajit Prasad Mahapatra

Published 2026-09-09

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CC BY 4.0

Original authors: Abhishek Chowdhury, Aryabrat Mahapatra, Ajit Prasad Mahapatra

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, the state of a system is not a single, static point but a vast, multidimensional landscape of possibilities. When a physicist prepares a quantum system—say, a collection of atoms or a pulse of light—they are placing it at a specific starting point on this landscape. As time passes and the system evolves under the influence of external forces, it does not simply move in a straight line; it spreads out, exploring new regions of its available space. Scientists have long sought a way to measure how far and how fast this exploration happens. One powerful tool for this is a concept called Krylov complexity. Think of it as a ruler that measures how much a quantum state has "spread out" from its original preparation as it evolves. This ruler is built by organizing the possible states the system can visit into a specific, ordered sequence, much like arranging books on a shelf by how closely they relate to the first book. The complexity is simply the average position of the system on this shelf at any given moment. Understanding this spread is crucial because it reveals the underlying geometry of the quantum world and how information flows within it, but measuring it has traditionally been difficult, especially when the forces driving the system are changing rapidly.

A team of researchers at the Indian Institute of Technology Bhubaneswar and the Indian Institute of Technology Kanpur has now turned this abstract mathematical ruler into a concrete tool for optical experiments. They focused on systems driven by light, such as lasers interacting with atoms or light trapped in a cavity. Their work demonstrates that the spread of a quantum state is not just a property of the light or the atoms alone, but a joint result of how the system was prepared, how the light was controlled, and the specific reference frame used to track the changes. By disentangling these ingredients, the researchers showed that for a wide variety of optical setups, the complexity can be directly read out using standard laboratory equipment. Instead of needing to reconstruct the entire quantum state, which is often impossible, they found that the complexity manifests as simple, measurable quantities: the number of excited atoms, the count of photons in a beam, or the interference patterns created by light waves.

The researchers discovered that the behavior of this complexity depends heavily on the type of optical system and the initial state of the light or atoms. For a collection of atoms driven by a laser, the complexity grows in a predictable, bounded way, resembling the way a population of excited atoms oscillates back and forth. For light in a cavity, the spread grows quadratically, like a square of time, until it hits a limit. In more complex systems involving the creation of pairs of light particles, the spread can grow exponentially, signaling a rapid expansion of the system's reach. Crucially, the team found that the way the system is prepared at the start changes the very definition of what is being measured. If you start with a specific type of light state, the "ruler" measures one thing; if you start with a different state, the ruler measures something else entirely. This means that the complexity is not a fixed number for a given experiment but is relative to the starting point chosen by the experimenter.

One of the most striking findings involves the geometry of the light's path through the quantum landscape. The researchers showed that the spread of the state and the phase of the light wave are deeply connected, acting as two sides of the same coin. As the system evolves, the complexity and the phase trace out a loop, and the area enclosed by this loop corresponds to a geometric phase, a fundamental property of the system's path. This connection allows scientists to measure the complexity by simply looking at the interference fringes of the light, turning a difficult quantum calculation into a visible pattern on a screen. The study also revealed that for certain periodic drives, the system's behavior repeats in a cycle, and the length of this cycle depends on whether the driving frequency is a simple fraction or an irrational number. If the frequency is a simple fraction, the system explores a finite number of states; if it is irrational, the exploration never repeats and covers an infinite number of states.

The paper further explored three-level atoms, where a ground state is connected to two excited states by light. In this setup, the researchers identified a special "bright" state that interacts with the light and a "dark" state that does not. The complexity of the system is determined by how the light drives the population between these states. They found that by carefully tuning the difference in the frequencies of the driving light, they could either close off the path to the dark state, keeping the system simple, or open it up to allow full transfer of population. This control allows for precise manipulation of the system's complexity, enabling complete transfer of energy or the creation of specific quantum superpositions. The study also looked at how the system behaves when the driving forces change over time, showing that even in these complex, non-stationary situations, the spread can be predicted and measured if the initial state and the control history are known.

Ultimately, this work provides a clear dictionary for translating the abstract concept of quantum complexity into the language of experimental optics. It shows that the spread of a quantum state is not an elusive theoretical quantity but a tangible feature that can be observed through photon counts, parity measurements, and phase shifts. By understanding how the preparation of the system and the control of the light work together, scientists can now design experiments that directly probe the geometry of quantum evolution. This opens the door to testing fundamental theories about how quantum systems explore their space and how information is processed in optical devices, all without needing to solve the full, intractable equations of the system. The results confirm that while the quantum world is vast and complex, its behavior under controlled conditions follows precise, measurable rules that can be uncovered by looking at the right quantities in the right way.

Technical Summary: Krylov Complexity of Time-Dependent Optical Hamiltonians

Problem and Context
The paper addresses the experimental concreteness of state Krylov complexity in driven quantum-optical systems. While Krylov complexity ( $C(t)$ ) is defined as the mean position of a state on a chain generated by the Lanczos algorithm, its physical interpretation in time-dependent settings has been ambiguous. The authors identify that for time-dependent Hamiltonians, the definition of the complexity chain depends jointly on the control history, the prepared initial state (seed), and the reference path used to handle explicit time dependence. The central problem is to disentangle these ingredients for optical systems governed by finite-dimensional Lie algebras (specifically oscillator, SU(2), and SU(1, 1)) to determine when preparation dependence preserves exact solvability and to identify the corresponding optical observables.

Methodology
The authors employ the state-Krylov construction using the extended generator $G = H(t) - i\partial_t$ . They analyze systems where the Hamiltonian is linear in a finite set of optical generators:
$H(t) = \sum_\alpha \lambda_\alpha(t) L_\alpha$
where $\{L_\alpha\}$ closes a finite-dimensional Lie algebra. The methodology involves:

Lanczos Recursion: Constructing an orthonormal Krylov basis $\{|K_n(t)\rangle\}$ via the Lanczos algorithm applied to the extended generator, ensuring positive hopping coefficients $b_n(t)$ .
Geometric Analysis: Utilizing the geometry of coherent-state orbits (spheres for SU(2), planes for Heisenberg-Weyl, and hyperboloids for SU(1, 1)) to relate the spread complexity to geodesic distances.
Exact Reductions: Identifying specific classes of seeds and controls where the full dynamics reduce to solvable scalar equations or known spectral measures. Key techniques include:
- Group Displacement: Using time-independent unitary group elements ( $W_s$ ) to map extremal seeds to displaced states, conjugating the control coefficients while preserving the algebraic structure.
- Christoffel Transformations: Analyzing polynomial preparations $Q(X)|\chi\rangle$ , which transform the spectral measure of a fixed operator $X$ and generate new chains from old Krylov rungs.
- Affine Moving Frames: Constructing Hamiltonians of the form $H(t) = i\dot{W}W^\dagger + \dot{\tau}WXW^\dagger + \dot{\gamma}$ , which transport a fixed spectral measure even when laboratory Hamiltonians at different times do not commute.
- Parity and Floquet Analysis: Examining seeds with definite photon-number parity and periodically forced systems to derive stroboscopic Arnoldi chains.

Key Contributions and Results

Experimental Dictionary: The paper establishes a direct mapping between Krylov complexity and measurable optical quantities for extremal seeds:
- Collective atomic excitation: $C(t) = \langle S_z \rangle + j$ .
- Cavity photons: $C(t) = \langle a^\dagger a \rangle$ .
- Signal-idler pair number: $C(t) = \langle n_a \rangle = \langle n_b \rangle$ (in the zero-difference sector).
- For interior seeds, $C(t)$ becomes a coherent combination of populations and coherences, not just a simple excitation count.
Exact Solvability and Reductions:
- Extremal Seeds: For arbitrary controls in oscillator, SU(2), and SU(1, 1) algebras, the complexity is exactly determined by a single scalar function (Gauss coordinate $x(t)$ ) solving a linear or Riccati equation.
- Displaced Extremal Seeds: A fixed group displacement of an extremal seed ( $W_s |\ell, 0\rangle$ ) preserves the exact solvability of arbitrary controls via time-independent conjugation.
- Polynomial Seeds: A degree- $d$ polynomial preparation $Q(X)|\chi\rangle$ creates a new seed that is a superposition of the first $d+1$ old Krylov rungs. If the Hamiltonian is affine in $X$ within a moving frame, the complexity evolves exactly according to the Christoffel-transformed chain.
- Periodic Forcing: For a periodically forced oscillator starting from a Fock state $|r\rangle$ , the cyclic dimension of the Krylov space is finite ( $q_F$ ) if the rotation number is rational ( $p/q_F$ ) and infinite if irrational. At a half-turn ( $\pi$ rotation), the complexity is exactly the survival deficit, related to Laguerre polynomials.
Geometric and Phase Relations:
- The spread complexity $C(t)$ and the coherent-state phase $\Phi$ form a canonical pair. The geometric phase (Berry phase) acquired over a closed trajectory is given by $\gamma_g = -\oint C d\Phi$ .
- The variance of the Krylov site index $V_K$ and the mean $C$ obey a universal relation dependent on the curvature $\kappa$ of the orbit: $V_K = C(1 - \kappa C / 2\ell)$ . This links sub-Poissonian, Poissonian, and super-Poissonian statistics to the algebraic curvature.
Three-Level Systems (V and $\Lambda$ ):
- V-System: The Krylov recursion naturally reconstructs the Morris-Shore bright-dark basis. The first hopping $b_1$ is the total coupling amplitude $G$ , and the second hopping $b_2$ measures the motion of the bright state away from the dark direction (geometric and dynamical contributions).
- STIRAP ( $\Lambda$ -system): In the instantaneous dark-state frame, the first Krylov hopping is exactly the non-adiabatic coupling rate $|\dot{\theta}|$ . The complexity $C(t)$ resolves diabatic leakage, distinguishing between population in the bright state and the excited state.
Parity and Wigner Readouts: For seeds with definite photon-number parity, the odd-site occupation of the Krylov chain is an affine-displaced parity readout of the seed's Wigner function. This allows the direct observation of Wigner negativity (e.g., in photon-added states) and interference fringes (in cat states) via Krylov complexity measurements.

Significance and Claims
The paper claims to resolve the "joint dependence" of Krylov complexity on control history and state preparation, turning abstract complexity measures into concrete optical observables (populations, parity fringes, phases). It demonstrates that:

Preparation defines the observable: Changing the seed changes the physical operator whose expectation value corresponds to the complexity.
Exactness is achievable: Despite time dependence, exact solutions exist for broad classes of seeds (extremal, displaced, polynomial) and controls, provided the system remains within a specific algebraic orbit.
Geometric unity: The spread of the state, its fluctuations, and the geometric phase are unified through the metric and symplectic geometry of the coherent-state orbit.
Diagnostic utility: The framework provides specific diagnostics for non-adiabaticity (via $b_1$ in STIRAP), bright-dark decoupling (via $b_2$ in V-systems), and the nature of the driving (elliptic vs. hyperbolic in parametric amplification).

The authors emphasize that their scope is limited to closed unitary dynamics with prescribed classical drives. They note that extensions to open systems, pump depletion, or general interior weights require further development (e.g., Liouvillian constructions or block Lanczos methods), but the current work establishes the exact "coherent skeleton" for these optical systems.

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