Bosonic and fermionic Gaussian states from Kähler structures
We show that bosonic and fermionic Gaussian states (also known as "squeezed coherent states") can be uniquely characterized by their linear complex structure $J$ which is a linear map on the classical phase space. This extends conventional Gaussian methods based on covariance matrices a...
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| Format: | Article |
| Language: | English |
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SciPost
2021-09-01
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| Series: | SciPost Physics Core |
| Online Access: | https://scipost.org/SciPostPhysCore.4.3.025 |
| _version_ | 1818738348097798144 |
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| author | Lucas Hackl, Eugenio Bianchi |
| author_facet | Lucas Hackl, Eugenio Bianchi |
| author_sort | Lucas Hackl, Eugenio Bianchi |
| collection | DOAJ |
| description | We show that bosonic and fermionic Gaussian states (also known as "squeezed
coherent states") can be uniquely characterized by their linear complex
structure $J$ which is a linear map on the classical phase space. This extends
conventional Gaussian methods based on covariance matrices and provides a
unified framework to treat bosons and fermions simultaneously. Pure Gaussian
states can be identified with the triple $(G,\Omega,J)$ of compatible K\"ahler
structures, consisting of a positive definite metric $G$, a symplectic form
$\Omega$ and a linear complex structure $J$ with $J^2=-1\!\!1$. Mixed Gaussian
states can also be identified with such a triple, but with $J^2\neq -1\!\!1$.
We apply these methods to show how computations involving Gaussian states can
be reduced to algebraic operations of these objects, leading to many known and
some unknown identities. We apply these methods to the study of (A)
entanglement and complexity, (B) dynamics of stable systems, (C) dynamics of
driven systems. From this, we compile a comprehensive list of mathematical
structures and formulas to compare bosonic and fermionic Gaussian states
side-by-side. |
| first_indexed | 2024-12-18T01:07:30Z |
| format | Article |
| id | doaj.art-8b257af72789442183201362cf102340 |
| institution | Directory Open Access Journal |
| issn | 2666-9366 |
| language | English |
| last_indexed | 2024-12-18T01:07:30Z |
| publishDate | 2021-09-01 |
| publisher | SciPost |
| record_format | Article |
| series | SciPost Physics Core |
| spelling | doaj.art-8b257af72789442183201362cf1023402022-12-21T21:26:12ZengSciPostSciPost Physics Core2666-93662021-09-014302510.21468/SciPostPhysCore.4.3.025Bosonic and fermionic Gaussian states from Kähler structuresLucas Hackl, Eugenio BianchiWe show that bosonic and fermionic Gaussian states (also known as "squeezed coherent states") can be uniquely characterized by their linear complex structure $J$ which is a linear map on the classical phase space. This extends conventional Gaussian methods based on covariance matrices and provides a unified framework to treat bosons and fermions simultaneously. Pure Gaussian states can be identified with the triple $(G,\Omega,J)$ of compatible K\"ahler structures, consisting of a positive definite metric $G$, a symplectic form $\Omega$ and a linear complex structure $J$ with $J^2=-1\!\!1$. Mixed Gaussian states can also be identified with such a triple, but with $J^2\neq -1\!\!1$. We apply these methods to show how computations involving Gaussian states can be reduced to algebraic operations of these objects, leading to many known and some unknown identities. We apply these methods to the study of (A) entanglement and complexity, (B) dynamics of stable systems, (C) dynamics of driven systems. From this, we compile a comprehensive list of mathematical structures and formulas to compare bosonic and fermionic Gaussian states side-by-side.https://scipost.org/SciPostPhysCore.4.3.025 |
| spellingShingle | Lucas Hackl, Eugenio Bianchi Bosonic and fermionic Gaussian states from Kähler structures SciPost Physics Core |
| title | Bosonic and fermionic Gaussian states from Kähler structures |
| title_full | Bosonic and fermionic Gaussian states from Kähler structures |
| title_fullStr | Bosonic and fermionic Gaussian states from Kähler structures |
| title_full_unstemmed | Bosonic and fermionic Gaussian states from Kähler structures |
| title_short | Bosonic and fermionic Gaussian states from Kähler structures |
| title_sort | bosonic and fermionic gaussian states from kahler structures |
| url | https://scipost.org/SciPostPhysCore.4.3.025 |
| work_keys_str_mv | AT lucashackleugeniobianchi bosonicandfermionicgaussianstatesfromkahlerstructures |