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arXiv:2201.07611 · Physical Review A

Permutational symmetry for identical multilevel systems: A second-quantized approach

Rui E. F. Silva, Johannes Feist

Published Apr 4, 2022• 20 citations
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Abstract

We develop a framework that provides a straightforward approach to fully exploit the permutational symmetry of identical multilevel systems. By taking into account the permutational symmetry, we outline a simple scheme that allows us to map the dynamics of $N$ identical $d$-level systems to the dynamics of $d$ bosonic modes with $N$ particles, achieving an exponential reduction on the dimensionality of the problem in a simple and straightforward way. In particular, we consider the Lindblad dynamics of several identical multilevel systems interacting with a common subsystem under the action of collective dissipation terms.

DOI
10.1103/physreva.105.043704

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We develop a framework that provides a straightforward approach to fully exploit the permutational symmetry of identical multilevel systems. By taking into account the permutational symmetry, we outline a simple scheme that allows us to map the dynamics of NNN identical ddd-level systems to the dynamics of ddd bosonic modes with NNN particles, achieving an exponential reduction on the dimensionality of the problem in a simple and straightforward way. In particular, we consider the Lindblad dynamics of several identical multilevel systems interacting with a common subsystem under the action of collective dissipation terms.
We consider the Lindblad equation for a collection of multilevel systems coupled to independent environments.The equation is symmetric under the exchange of the labels associated with each system and thus the open-system dynamics takes place in the permutation-symmetric subspace of the operator space.The dimension of this space grows polynomially with the number of systems.We construct a basis of this space and a set of superoperators whose action on this basis is easily specified.For a given number of levels, M, these superoperators are written in terms of a bosonic realization of the generators of the Lie algebra M . ( ) slIn some cases, these results enable finding an analytic solution of the master equation using known Lie-algebraic methods.To demonstrate this, we obtain an analytic expression for the state operator of a collection of three-level atoms coupled to independent radiation baths.When analytic solutions are difficult to find, the basis and the superoperators can be used to considerably reduce the computational resources required for simulations.
Quantum systems of indistinguishable particles are commonly described using the formalism of second quantization, which relies on the assumption that any admissible quantum state must be either symmetric or antisymmetric under particle permutations. Coherence-induced many-body effects such as superradiance, however, can arise even in systems whose constituents are not fundamentally indistinguishable as long as all relevant dynamical observables are permutation-invariant. Such systems are not confined to symmetric or antisymmetric states and therefore require a different theoretical approach. Focusing on noninteracting systems, here we combine tools from representation theory and thermodynamically consistent master equations to develop such a framework. We characterize the structure and properties of the steady states emerging in permutation-invariant ensembles of arbitrary multilevel systems that are collectively weakly coupled to a thermal environment. As an application of our general theory, we further explore how these states can in principle be used to enhance the performance of quantum thermal machines. Our group-theoretical framework thereby makes it possible to analyze various limiting cases that would not be accessible otherwise. In addition, it allows us to show that the properties of multilevel ensembles differ qualitatively from those of spin ensembles, which have been investigated earlier using the standard Clebsch-Gordan theory. Our results have a large scope for future generalizations and pave the way for systematic investigations of collective effects arising from permutation invariance in quantum thermodynamics.
In a recent publication we showed that permutation symmetry reduces the numerical complexity of Lindblad quantum master equations for identical multi-level systems from exponential to polynomial scaling. This is important for open system dynamics including realistic system bath interactions and dephasing in, for instance, the Dicke model, multi-Λ system setups etc. Here we present an object-oriented C++ library that allows to setup and solve arbitrary quantum optical Lindblad master equations, especially those that are permutationally symmetric in the multi-level systems. PsiQuaSP (Permutation symmetry for identical Quantum Systems Package) uses the PETSc package for sparse linear algebra methods and differential equations as basis. The aim of PsiQuaSP is to provide flexible, storage efficient and scalable code while being as user friendly as possible. It is easily applied to many quantum optical or quantum information systems with more than one multi-level system. We first review the basics of the permutation symmetry for multi-level systems in quantum master equations. The application of PsiQuaSP to quantum dynamical problems is illustrated with several typical, simple examples of open quantum optical systems.
We show how one may perform a stochastic unravelling which respects weak permutation symmetry for general models of NNN emitters coupled to a common system (e.g. a cavity mode). Our work extends prior work by Zhang et al. [2018 New J. Phys. 20 112001] which provided an efficient quantum jump approach for a restricted subset of such problems. For problems involving 222-level emitters, such an unravelling reduces the computational cost from O(N5)\mathcal{O}(N^5)O(N5) to O(N2)\mathcal{O}(N^2)O(N2), and with additional refinements, allows reduction to O(N)\mathcal{O}(N)O(N). This significantly increases the range of system sizes for which one can model exact quantum dynamics of such systems. We further show how the method can also be applied to ddd-level systems, with computational effort scaling as O(Nd(d−1)/2)\mathcal{O}(N^{d(d-1)/2})O(Nd(d−1)/2), and we show it allows large-NNN simulations for d=3d=3d=3.
The symmetry properties of the N-particle multilevel system are studied. For indistinguishable particles, the total wave function must be either symmetric or antisymmetric in the exchange of any two particles. The internal states of the multilevel system are combined with suitable spatial wave functions to produce a totally symmetric or antisymmetric wave function. The problem of degeneracy and the method of combining a given permutational symmetry of the internal wave function with the same or the conjugate symmetry for the spatial part are described. It is found that Young’s diagram corresponding to the given irreducible representation of the permutation group plays an important role.