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Reproducing Kernel Hilbert Space Approach to Non-Markovian Quantum Stochastic Models

John E. Gough, Haijin Ding, Nina H. Amini

TL;DR

The paper derives the non-Markovian quantum state diffusion equation of Di{ó}si and Strunz from a microscopic system–bath model, clarifying that complex trajectories arise as a mathematical consequence of using the Bargmann-Segal bath representation. It develops a reproducing kernel Hilbert space (RKHS) framework for the bath auto-correlation kernel $K(t,s)$ and shows that the space of complex trajectories is a Hilbert subspace of the RKHS, with the bath feature map provided by the one-particle bath modes. An exact Jaynes-Cummings solution is presented that satisfies the non-Markovian diffusion equation, illustrating how the stochastic trajectory formalism maps to concrete dynamics. The work also discusses the physical interpretation, showing that complex trajectories are not directly observable and that Markovian limits recover standard quantum stochastic differential equations, thereby linking microscopic models to trajectory-based unravellings in open quantum systems.

Abstract

We give a derivation of the non-Markovian quantum state diffusion equation of Di{ó}si and Strunz starting from a model of a quantum mechanical system coupled to a bosonic bath. We show that the complex trajectories arises as a consequence of using the Bargmann-Segal (complex wave) representation of the bath. In particular, we construct a reproducing kernel Hilbert space for the bath auto-correlation and realize the space of complex trajectories as a Hilbert subspace. The reproducing kernel naturally arises from a feature space where the underlying feature space is the one-particle Hilbert space of the bath quanta. We exploit this to derive the unravelling of the open quantum system dynamics and show equivalence to the equation of Di{ó}si and Strunz. We also give an explicit expression for the reduced dynamics of a two-level system coupled to the bath via a Jaynes-Cummings interaction and show that this does indeed correspond to an exact solution of the Di{ó}si-Strunz equation. Finally, we discuss the physical interpretation of the complex trajectories and show that they are intrinsically unobservable.

Reproducing Kernel Hilbert Space Approach to Non-Markovian Quantum Stochastic Models

TL;DR

The paper derives the non-Markovian quantum state diffusion equation of Di{ó}si and Strunz from a microscopic system–bath model, clarifying that complex trajectories arise as a mathematical consequence of using the Bargmann-Segal bath representation. It develops a reproducing kernel Hilbert space (RKHS) framework for the bath auto-correlation kernel and shows that the space of complex trajectories is a Hilbert subspace of the RKHS, with the bath feature map provided by the one-particle bath modes. An exact Jaynes-Cummings solution is presented that satisfies the non-Markovian diffusion equation, illustrating how the stochastic trajectory formalism maps to concrete dynamics. The work also discusses the physical interpretation, showing that complex trajectories are not directly observable and that Markovian limits recover standard quantum stochastic differential equations, thereby linking microscopic models to trajectory-based unravellings in open quantum systems.

Abstract

We give a derivation of the non-Markovian quantum state diffusion equation of Di{ó}si and Strunz starting from a model of a quantum mechanical system coupled to a bosonic bath. We show that the complex trajectories arises as a consequence of using the Bargmann-Segal (complex wave) representation of the bath. In particular, we construct a reproducing kernel Hilbert space for the bath auto-correlation and realize the space of complex trajectories as a Hilbert subspace. The reproducing kernel naturally arises from a feature space where the underlying feature space is the one-particle Hilbert space of the bath quanta. We exploit this to derive the unravelling of the open quantum system dynamics and show equivalence to the equation of Di{ó}si and Strunz. We also give an explicit expression for the reduced dynamics of a two-level system coupled to the bath via a Jaynes-Cummings interaction and show that this does indeed correspond to an exact solution of the Di{ó}si-Strunz equation. Finally, we discuss the physical interpretation of the complex trajectories and show that they are intrinsically unobservable.
Paper Structure (19 sections, 9 theorems, 109 equations)

This paper contains 19 sections, 9 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. The Microscopic Model
  3. The Bath Fock space
  4. Bargmann-Segal (Complex Wave) Representation
  5. The System-Bath Interaction
  6. The Open System Dynamics
  7. Caldeira-Leggett Baths
  8. Complex Trajectories
  9. Reproducing Kernel Hilbert Space Formalism
  10. Hilbert Space of Complex Trajectories
  11. Quantum Karhunen-Loève Theorem
  12. Causal Form
  13. Complex Trajectories from The Bargmann Representation
  14. Complex Trajectory Unravellings
  15. Complex Wave Representation for the Open System
  16. ...and 4 more sections

Key Result

Proposition 4

For each system operator $\hat{X}$, the operator $j_t (\hat{X})$ commutes with both $\hat{Z}_{\mathrm{out}} (t)$ and $\hat{Z}_{\mathrm{out}} (t)^\ast$.

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Remark 3
  • Proposition 4: Equal time commutation relations
  • Proposition 5: Input-Output Relation
  • Proposition 6: Ehrenfest equation
  • Remark 7
  • Definition 8
  • Theorem 9
  • Remark 10
  • ...and 13 more