Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2403.19801 · Optica Quantum

Modeling local decoherence of a spin ensemble using a generalized Holstein–Primakoff mapping to a bosonic mode

Andrew Kolmer Forbes, Philip Daniel Blocher, Ivan Deutsch

Published Aug 8, 2024• 4 citations
arXivHTMLPDF

Abstract

We show how the decoherence that occurs in an entangling atomic spin–light interface can be simply modeled as the dynamics of a bosonic mode. Although one seeks to control the collective spin of the atomic system in the permutationally invariant (symmetric) subspace, diffuse scattering and optical pumping are local, making an exact description of the many-body state intractable. To overcome this issue we develop a generalized Holstein–Primakoff approximation for collective states which is valid when decoherence is uniform across a large atomic ensemble. In different applications the dynamics is conveniently treated as a Wigner function evolving according to a thermalizing diffusion equation, or by a Fokker–Planck equation for a bosonic mode decaying in a zero-temperature reservoir. We use our formalism to study the combined effect of Hamiltonian evolution, local and collective decoherence, and measurement backaction in preparing nonclassical spin states for application in quantum metrology.

DOI
10.1364/opticaq.528078

Related papers

Three-dimensional light-matter interface for collective spin squeezing in atomic ensembles

Ben Q. Baragiola, Leigh Norris, Enrique Montaño +3 more

We study the three-dimensional nature of the quantum interface between an ensemble of cold, trapped atomic spins and a paraxial laser beam, coupled through a dispersive interaction. To achieve strong entanglement between the collective atomic spin and the photons, one must match the spatial mode of the collective radiation of the ensemble with the mode of the laser beam while minimizing the effects of decoherence due to optical pumping. For ensembles coupling to a probe field that varies over the extent of the cloud, the set of atoms that indistinguishably radiates into a desired mode of the field defines an inhomogeneous spin wave. Strong coupling of a spin wave to the probe mode is not characterized by a single parameter, the optical density, but by a collection of different effective atom numbers that characterize the coherence and decoherence of the system. To model the dynamics of the system, we develop a full stochastic master equation, including coherent collective scattering into paraxial modes, decoherence by local inhomogeneous diffuse scattering, and backaction due to continuous measurement of the light entangled with the spin waves. This formalism is used to study the squeezing of a spin wave via continuous quantum nondemolition (QND) measurement. We find that the greatest squeezing occurs in parameter regimes where spatial inhomogeneities are significant, far from the limit in which the interface is well approximated by a one-dimensional, homogeneous model.

arXiv:1311.2328•Physical Review A•Mar 26, 2014•30 citations

Generalized Holstein-Primakoff mapping and $1/N$ expansion of collective spin systems undergoing single particle dissipation

Diego Barberena

We develop a generalization of the Schwinger boson and Holstein-Primakoff transformations that is applicable to ensembles of $N$ spin $1/2$'s with weak permutational symmetry. These generalized mappings are constructed by introducing two independent bosonic variables that describe fluctuations parallel and transverse to the collective Bloch vector built out of the original spin $1/2$'s. Using this representation, we develop a systematic $1/N$ expansion and write down explicitly leading and next-to-leading order terms. We then illustrate how to apply these techniques using four example systems: (i) an ensemble of atoms undergoing spontaneous emission, incoherent pumping and single particle dephasing; (ii) a superradiant laser above and in the vicinity of the upper lasing transition; (iii) the all-to-all transverse field Ising model subject to incoherent pumping in the vicinity of its ordering phase transition; and (iv) the Dicke model at finite temperature both away and in the vicinity of its thermal phase transition. Thus, these mappings provide a common, Bloch-sphere based, geometrical description of all-to-all systems subject to single particle dissipation or at finite temperature, including their phase transitions.

arXiv:2508.05751•Aug 7, 2025•0 citations

Collective processes of an ensemble of spin- 1 ∕ 2 particles

Bradley A. Chase, J. M. Geremia

When the dynamics of a spin ensemble are expressible solely in terms of symmetric processes and collective spin operators, the symmetric collective states of the ensemble are preserved. These many-body states, which are invariant under particle relabeling, can be efficiently simulated since they span a subspace whose dimension is linear in the number of spins. However, many open system dynamics break this symmetry, most notably when ensemble members undergo identical, but local, decoherence. In this paper, we extend the definition of symmetric collective states of an ensemble of spin-$1∕2$ particles in order to efficiently describe these more general collective processes. The corresponding collective states span a subspace that grows quadratically with the number of spins. We also derive explicit formulas for expressing arbitrary, identical, local decoherence in terms of these states. We then investigate the open system dynamics of experimentally relevant nonclassical collective atomic states, including superposition and spin squeezed states, subject to various symmetric but local decoherence models.

arXiv:0805.2911•Physical Review A•Nov 4, 2008•81 citations

Light mediated non-Gaussian atomic ensemble entanglement

Olov Pettersson, Tim Byrnes

We analyze a similar scheme for producing light-mediated entanglement between atomic ensembles, as first realized by Julsgaard, Kozhekin and Polzik [Nature {\bf 413}, 400 (2001)]. In the standard approach to modeling the scheme, a Holstein-Primakoff approximation is made, where the atomic ensembles are treated as bosonic modes, and is only valid for short interaction times. In this paper, we solve the time evolution without this approximation, which extends the region of validity of the interaction time. For short entangling times, we find this produces a state with similar characteristics as a two-mode squeezed state, in agreement with standard predictions. For long entangling times, the state evolves into a non-Gaussian form, and the two-mode squeezed state characteristics start to diminish. This is attributed to more exotic types of entangled states being generated. We characterize the states by examining the Fock state probability distributions, Husimi $Q$ distributions, and non-local entanglement between the ensembles. We compare and connect several quantities obtained using the Holstein-Primakoff approach and our exact time evolution methods.

arXiv:1703.07974•Mar 23, 2017•0 citations

Light-mediated non-Gaussian entanglement of atomic ensembles

Olov Pettersson, Tim Byrnes

We analyze a similar scheme for producing light-mediated entanglement between atomic ensembles, as first realized by Julsgaard, Kozhekin, and Polzik [Nature (London) 413, 400 (2001)]. In the standard approach to modeling the scheme, a Holstein--Primakoff approximation is made, where the atomic ensembles are treated as bosonic modes, and is only valid for short interaction times. In this paper, we solve the time evolution without this approximation, which extends the region of validity of the interaction time. For short entangling times, we find that this produces a state with characteristics similar to those of a two-mode squeezed state, in agreement with standard predictions. For long entangling times, the state evolves into a non-Gaussian form, and the characteristics of the two-mode squeezed state start to diminish. This is attributed to more exotic types of entangled states being generated. We characterize the states by examining the Fock-state probability distributions, Husimi $Q$ distributions, and nonlocal entanglement between the ensembles. We compare and connect several quantities obtained by using the Holstein--Primakoff approach and our exact time evolution methods.

Physical Review A•Apr 12, 2017•17 citations

Collective States and Symmetric Local Decoherence in Large Ensembles of Qubits

Bradley A. Chase, J. M. Geremia

The symmetric collective states of an atomic spin ensemble (i.e., many-body states that are invariant under particle exchange) are not preserved by decoherence that acts identically but individually on members of the ensemble. We develop a class of collective states in an ensemble of N spin-1/2 particles that is invariant under symmetric local decoherence and find that the dimension of the Hilbert space spanned by these collective states scales only as N^2. We then investigate the open system dynamics of experimentally relevant non-classical collective atomic states, including Schroedinger cat and spin squeezed states, subject to various symmetric but local decoherence models.

arXiv:0805.2910•May 19, 2008•0 citations
We show how the decoherence that occurs in an entangling atomic spin–light interface can be simply modeled as the dynamics of a bosonic mode. Although one seeks to control the collective spin of the atomic system in the permutationally invariant (symmetric) subspace, diffuse scattering and optical pumping are local, making an exact description of the many-body state intractable. To overcome this issue we develop a generalized Holstein–Primakoff approximation for collective states which is valid when decoherence is uniform across a large atomic ensemble. In different applications the dynamics is conveniently treated as a Wigner function evolving according to a thermalizing diffusion equation, or by a Fokker–Planck equation for a bosonic mode decaying in a zero-temperature reservoir. We use our formalism to study the combined effect of Hamiltonian evolution, local and collective decoherence, and measurement backaction in preparing nonclassical spin states for application in quantum metrology.
We study the three-dimensional nature of the quantum interface between an ensemble of cold, trapped atomic spins and a paraxial laser beam, coupled through a dispersive interaction. To achieve strong entanglement between the collective atomic spin and the photons, one must match the spatial mode of the collective radiation of the ensemble with the mode of the laser beam while minimizing the effects of decoherence due to optical pumping. For ensembles coupling to a probe field that varies over the extent of the cloud, the set of atoms that indistinguishably radiates into a desired mode of the field defines an inhomogeneous spin wave. Strong coupling of a spin wave to the probe mode is not characterized by a single parameter, the optical density, but by a collection of different effective atom numbers that characterize the coherence and decoherence of the system. To model the dynamics of the system, we develop a full stochastic master equation, including coherent collective scattering into paraxial modes, decoherence by local inhomogeneous diffuse scattering, and backaction due to continuous measurement of the light entangled with the spin waves. This formalism is used to study the squeezing of a spin wave via continuous quantum nondemolition (QND) measurement. We find that the greatest squeezing occurs in parameter regimes where spatial inhomogeneities are significant, far from the limit in which the interface is well approximated by a one-dimensional, homogeneous model.
Generalized Holstein-Primakoff mapping and 1/N1/N1/N expansion of collective spin systems undergoing single particle dissipation
We develop a generalization of the Schwinger boson and Holstein-Primakoff transformations that is applicable to ensembles of NNN spin 1/21/21/2's with weak permutational symmetry. These generalized mappings are constructed by introducing two independent bosonic variables that describe fluctuations parallel and transverse to the collective Bloch vector built out of the original spin 1/21/21/2's. Using this representation, we develop a systematic 1/N1/N1/N expansion and write down explicitly leading and next-to-leading order terms. We then illustrate how to apply these techniques using four example systems: (i) an ensemble of atoms undergoing spontaneous emission, incoherent pumping and single particle dephasing; (ii) a superradiant laser above and in the vicinity of the upper lasing transition; (iii) the all-to-all transverse field Ising model subject to incoherent pumping in the vicinity of its ordering phase transition; and (iv) the Dicke model at finite temperature both away and in the vicinity of its thermal phase transition. Thus, these mappings provide a common, Bloch-sphere based, geometrical description of all-to-all systems subject to single particle dissipation or at finite temperature, including their phase transitions.
When the dynamics of a spin ensemble are expressible solely in terms of symmetric processes and collective spin operators, the symmetric collective states of the ensemble are preserved. These many-body states, which are invariant under particle relabeling, can be efficiently simulated since they span a subspace whose dimension is linear in the number of spins. However, many open system dynamics break this symmetry, most notably when ensemble members undergo identical, but local, decoherence. In this paper, we extend the definition of symmetric collective states of an ensemble of spin-1∕21∕21∕2 particles in order to efficiently describe these more general collective processes. The corresponding collective states span a subspace that grows quadratically with the number of spins. We also derive explicit formulas for expressing arbitrary, identical, local decoherence in terms of these states. We then investigate the open system dynamics of experimentally relevant nonclassical collective atomic states, including superposition and spin squeezed states, subject to various symmetric but local decoherence models.
We analyze a similar scheme for producing light-mediated entanglement between atomic ensembles, as first realized by Julsgaard, Kozhekin and Polzik [Nature {\bf 413}, 400 (2001)]. In the standard approach to modeling the scheme, a Holstein-Primakoff approximation is made, where the atomic ensembles are treated as bosonic modes, and is only valid for short interaction times. In this paper, we solve the time evolution without this approximation, which extends the region of validity of the interaction time. For short entangling times, we find this produces a state with similar characteristics as a two-mode squeezed state, in agreement with standard predictions. For long entangling times, the state evolves into a non-Gaussian form, and the two-mode squeezed state characteristics start to diminish. This is attributed to more exotic types of entangled states being generated. We characterize the states by examining the Fock state probability distributions, Husimi QQQ distributions, and non-local entanglement between the ensembles. We compare and connect several quantities obtained using the Holstein-Primakoff approach and our exact time evolution methods.
We analyze a similar scheme for producing light-mediated entanglement between atomic ensembles, as first realized by Julsgaard, Kozhekin, and Polzik [Nature (London) 413, 400 (2001)]. In the standard approach to modeling the scheme, a Holstein--Primakoff approximation is made, where the atomic ensembles are treated as bosonic modes, and is only valid for short interaction times. In this paper, we solve the time evolution without this approximation, which extends the region of validity of the interaction time. For short entangling times, we find that this produces a state with characteristics similar to those of a two-mode squeezed state, in agreement with standard predictions. For long entangling times, the state evolves into a non-Gaussian form, and the characteristics of the two-mode squeezed state start to diminish. This is attributed to more exotic types of entangled states being generated. We characterize the states by examining the Fock-state probability distributions, Husimi QQQ distributions, and nonlocal entanglement between the ensembles. We compare and connect several quantities obtained by using the Holstein--Primakoff approach and our exact time evolution methods.
The symmetric collective states of an atomic spin ensemble (i.e., many-body states that are invariant under particle exchange) are not preserved by decoherence that acts identically but individually on members of the ensemble. We develop a class of collective states in an ensemble of N spin-1/2 particles that is invariant under symmetric local decoherence and find that the dimension of the Hilbert space spanned by these collective states scales only as N^2. We then investigate the open system dynamics of experimentally relevant non-classical collective atomic states, including Schroedinger cat and spin squeezed states, subject to various symmetric but local decoherence models.