Efficient variational approach to dynamics of a spatially extended bosonic Kondo model

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Efficient variational approach to dynamics of a spatially extended bosonic Kondo model

Yuto Ashida, Tao Shi, Richard Schmidt, H. R. Sadeghpour, J. Ignacio Cirac, and Eugene Demler

Phys. Rev. A 100, 043618 – Published 31 October 2019

Abstract

We develop an efficient variational approach to studying dynamics of a localized quantum spin coupled to a bath of mobile spinful bosons. We use parity symmetry to decouple the impurity spin from the environment via a canonical transformation and reduce the problem to a model of the interacting bosonic bath. We describe coherent time evolution of the latter using bosonic Gaussian states as a variational ansatz. We provide full analytical expressions for equations describing variational time evolution that can be applied to study in- and out-of-equilibrium phenomena in a wide class of quantum impurity problems. In the accompanying paper [Ashida et al., Phys. Rev. Lett. 123, 183001 (2019)], we present a concrete application of this general formalism to the analysis of the Rydberg central spin model, in which the spin-1/2 Rydberg impurity undergoes spin-changing collisions in a dense cloud of two-component ultracold bosons. To illustrate new features arising from orbital motion of the bath atoms, we compare our results to the Monte Carlo study of the model with spatially localized bosons in the bath, in which random positions of the atoms give rise to random couplings of the standard central spin model.

Received 18 June 2019

DOI:https://doi.org/10.1103/PhysRevA.100.043618

Physics Subject Headings (PhySH)

Electronic excitation & ionization Impurities Molecular spectra Open quantum systems & decoherence Polarons Ultracold collisions

Bose-Einstein condensates Rydberg atoms & molecules

Variational approach

Atomic, Molecular & OpticalCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Yuto Ashida^1,2,*, Tao Shi^3,†, Richard Schmidt^4,5, H. R. Sadeghpour⁶, J. Ignacio Cirac^4,5, and Eugene Demler⁷

¹Department of Applied Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
²Department of Physics, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan
³CAS Key Laboratory of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
⁴Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany
⁵Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, 80799 München, Germany
⁶ITAMP, Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts 02138, USA
⁷Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

^*ashida@ap.t.u-tokyo.ac.jp
^†tshi@itp.ac.cn

Quantum Rydberg Central Spin Model

Yuto Ashida, Tao Shi, Richard Schmidt, H. R. Sadeghpour, J. Ignacio Cirac, and Eugene Demler

Phys. Rev. Lett. 123, 183001 (2019)

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Vol. 100, Iss. 4 — October 2019

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Images

Figure 1
Applications of the variational approach to nonequilibrium dynamics of a Rydberg impurity interacting with surrounding bosons via the anisotropic central spin coupling. The obtained overlaps $S (t)$ (top panels) and the corresponding absorption spectra $A (ω)$ (bottom panels) are plotted at different densities. In the top panels, the blue solid curves (red dashed curves) show the real (imaginary) parts of the overlaps. The absolute values $| S (t) |$ are shown in the insets. In the bottom panels, the dashed lines indicate the mean-field (MF) shifts $Δ_{MF}$ [cf. Eqs. (55) and (56)].
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Figure 2
Comparison between the absorption spectra obtained from the variational approach for the full interacting Hamiltonian $\hat{H}$ (blue) and the results for the noninteracting (quadratic) Hamiltonian ${\hat{H}}_{∥}$ (red). The density is $ρ = 3 \times 10^{12} cm^{- 3}$ . We use the isotropic interaction as consistent with Fig. 3 in the accompanying paper [82] and rescale the overall factors of $A (ω)$ for the sake of comparison between the two cases.
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Figure 3
(a) Dynamics of the electron spin $m_{z} (t) = 〈 {\hat{σ}}_{e}^{z} (t) 〉$ of the Rydberg impurity interacting with environmental bosons via the anisotropic central spin coupling. The results are obtained from the variational approach and plotted at different densities $ρ$ . (b) The corresponding Fourier spectra ${\tilde{m}}_{z} (ω)$ . The dashed lines indicate the values of the square root scaling $ω \propto \sqrt{ρ}$ .
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Figure 4
(a) Fourier spectra ${\tilde{m}}_{z} (ω)$ of the time evolution of the Rydberg electron spin with the anisotropic central spin coupling at different magnetic fields $h_{z}$ . The dashed lines indicate the linear scaling $δ ω = - h_{z}$ . (b) The corresponding real-time dynamics of the Rydberg spin at different magnetic fields. In (a) and (b), we set $ρ = 6 \times 10^{11} {cm}^{- 3}$ .
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Figure 5
(a) Absorption spectrum $A (ω)$ and (b) the corresponding central spin dynamics $m_{z} (t)$ for environmental particles of infinite mass (i.e., localized environmental spins). The results are obtained from Eqs. (67) and (68) and by taking the ensemble average over $10^{5}$ different stochastic realizations of atomic configurations [cf. Eqs. (69) and (70)]. In (a), the dashed line indicates the mean-field shift $Δ_{MF}$ . In (b), the blue thick solid curve indicates the ensemble-averaged result, while the dashed thin curves show typical dynamics for single stochastic realizations. We set the density to $ρ = 6 \times 10^{12} {cm}^{- 3}$ .
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Figure 6
Radial wave functions of the single-particle energy eigenstates for the mean-field Rydberg potential $V_{mean}$ . Solution $ν = 1$ is the most dominant bound state having the largest overlap with the initial single-particle wave function. The black dashed curve indicates the spatial profile of the Rydberg potential. The bound-state energies for $ν = 1, 2, 3$ modes are $- 21.5, - 13.1$ , and $- 3.8$ (in the unit of kHz), respectively.
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