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Breathing dynamics of the few-body Bose polaron in a species-selective harmonic trap

Maxim Pyzh and Peter Schmelcher
Phys. Rev. A 105, 043304 – Published 8 April 2022
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  • INTRODUCTION
  • SETUP AND HAMILTONIAN
  • COMPUTATIONAL APPROACH AND ANALYSIS
  • RESULTS
  • SUMMARY
  • CONCLUSIONS
  • ACKNOWLEDGMENTS
    • References

    Abstract

    We perform an extensive numerical study on the breathing dynamics of a few-body Bose polaron setup in a one-dimensional species-selective harmonic trap. The dynamics is triggered by a quench of the impurity trap. The excitation of the background majority atoms is mediated via the majority-impurity interaction. The breathing spectrum is obtained for different numbers of majority particles, several values of the majority-component interaction strengths, and trap ratios. It is further compared to the breathing spectrum of a particle-balanced few-body Bose-Bose mixture. In particular, for equal postquench traps the employed protocol allows to couple states of different center-of-mass parity in contrast to species-symmetric trap quenches. Among the participating eigenstates we identify one having odd center-of-mass parity and even global parity. The breathing frequency induced by this state is a monotonically decreasing function of the coupling parameter. Importantly, in order to be numerically observable, it requires the entanglement between the species to be taken into account. We demonstrate this by comparing the numerically exact results obtained by means of the multilayer multiconfiguration time-dependent Hartree method for mixtures to the ones of a species mean-field ansatz. The entanglement-sensitive breathing frequency persists also for unequal postquench traps where the center of mass cannot be decoupled. Finally, we analyze the impact of global parity symmetry on the breathing dynamics by initializing a state of odd global parity. We evidence a striking resemblance to the breathing spectrum of the ground state, but find also some additional modes.

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    • Received 20 July 2021
    • Accepted 22 March 2022

    DOI:https://doi.org/10.1103/PhysRevA.105.043304

    ©2022 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Cold atoms & matter wavesEntanglement in quantum gasesMixtures of atomic and/or molecular quantum gasesQuasiparticles & collective excitations
    1. Techniques
    Variational approach
    Condensed Matter, Materials & Applied Physics

    Authors & Affiliations

    Maxim Pyzh1,* and Peter Schmelcher1,2,†

    • 1Zentrum für Optische Quantentechnologien, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
    • 2The Hamburg Centre for Ultrafast Imaging, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany

    • *mpyzh@physnet.uni-hamburg.de
    • †pschmelc@physnet.uni-hamburg.de

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    Issue

    Vol. 105, Iss. 4 — April 2022

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    Images

    • Figure 1
      Figure 1

      The impurity (species B indicated by a red circle with a vertically dashed filling) in a harmonic trap of frequency ωB (red thin solid line) is immersed in a cloud of majority atoms (species A indicated by a blue blurred ellipse) subject to a different parabolic confinement of frequency ωA (blue broad solid line). The breathing dynamics is initiated by quenching the trap of the impurity (red dashed line) inducing thereby excitations (orange waves) in the composite system via the majority-impurity interaction.

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    • Figure 2
      Figure 2

      Fourier power spectrum X2(ω) (a) and Y2(ω) (b) obtained by applying a compressed sensing algorithm to the expectation values of the breathing observables 〈∑ix̂i2〉t and 〈ŷ2〉t (insets) evaluated with respect to a dynamical state |Ψ(t)〉 obtained by the multilayer multiconfiguration time-dependent Hartree method for mixtures. In the averaged power spectrum Σ(ω) (c) the dashed line indicates a threshold magnitude and only frequencies above it are accounted for in Sec. 4 (see text). The physical parameters are η=1, NA=5, NB=1, gAB=2.0, gA=0 (see Sec. 2) and the compressing sensing parameters are T=40, Δt=0.1, ωcut=20, and Δω=0.01 (see Sec. 3b). All quantities are given in harmonic units.

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    • Figure 3
      Figure 3

      Breathing mode frequencies (opaque lines) Ωk,l=Ek,l−E0,0 [see Eq. (11)] of two distinguishable particles NA=1 and NB=1 as a function of the coupling gAB≡g at equal trapping frequency ratio η=1. Population amplitudes (transparent lines) ck,l [see Eq. (13)] of eigenstates |ψk,l〉 upon quenching the ground state |E0〉η0 from η0=1.05 to η=1. Crosses stand for frequencies extracted from a laboratory frame mean-field ansatz. Inset: representative example of dynamical symmetry breaking in the one-body densities ρ1(x,t) (left) and ρ1(y,t) (right) at 2.5<g<6.5 occurring for a mean-field ansatz. All quantities are given in harmonic units.

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    • Figure 4
      Figure 4

      Decomposition of the one-body density ρ1(x): (a) time-independent background and time-dependent single-frequency modulations ρ1(k,l)(x) induced by (b) center-of-mass breathing ρ2(2,0), (c) interparticle distance breathing ρ2(0,2), and (d) hybrid sloshing ρ2(1,1) at different interaction strength g. All quantities are given in harmonic units.

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    • Figure 5
      Figure 5

      Energy gaps Ωj=Ej−E0 (with respect to the ground state |E0〉) of a few-body bosonic mixture NA=2 and NB=2 as a function of the intercomponent coupling gAB at equal trapping frequency ratio η=1, intracomponent interaction strength gB=0 for the second component and (a) gA=−0.5, (b) gA=0, (c) gA=0.5 for the first component. Whether the corresponding eigenstates are actually excited depends on the quench protocol. Different colors (line styles) refer to the center-of-mass (c.m.) quantum number in the eigenstate |Ej〉. The c.m. is a decoupled degree of freedom in this harmonic confinement. The insets represent a zoom-in on regions with avoided crossings which are indicated by circles and caused by gA≠gB asymmetry. Curves of different colors (line styles) may only cross. All quantities are given in harmonic units.

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    • Figure 6
      Figure 6

      Breathing oscillations of the one-body densities of the majority component ρ1A(x) (first row) and the impurity ρ1B(y) (second row) at a fixed majority-component interaction gA=0.5 for NA=5 and NB=1 particles initiated by preparation of the ground state of even global parity followed by an abrupt change of the trap ratio from η=1.05 to 1. Columns 1–3 correspond to different intercomponent couplings: (a) gAB=0.5, (b) gAB=1.0, and (c) gAB=1.5. Note that the initial state displays the onset of phase separation in (b) and a pronounced core-shell phase in (c). All quantities are given in harmonic units.

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    • Figure 7
      Figure 7

      Frequencies Ω of breathing modes excited by quenching the ground state |E0〉 of the Bose polaron for NA=5 and NB=1, meaning a change in the trap ratio from η=1.05 to 1.0, shown as a function of the intercomponent coupling gAB for a fixed majority component interaction (a) gA=−0.5, (b) gA=0, and (c) gA=0.5. Each frequency data point (full circle) is divided into two sectors (of different colors) representing the contribution of the breathing observables 〈∑ix̂i2〉t or 〈ŷ2〉t to the averaged power spectrum Σ at that frequency (see Sec. 3b). The corresponding color intensity indicates the relative strength with respect to the maximum amplitude Σmax in the averaged power spectrum for fixed gAB and only modes with contribution above 10% of Σmax are presented. Crosses stand for frequencies of modes excited within the SMF approximation. Black dashed line indicates the entanglement entropy SvN of the initial state. All quantities are given in harmonic units.

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    • Figure 8
      Figure 8

      Same as in Fig. 7 but for NA=10. All quantities are given in harmonic units.

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    • Figure 9
      Figure 9

      Frequencies Ω of breathing modes for NA=5 and NB=1 as a function of the intercomponent coupling gAB at fixed majority component interaction gA=0. The trap ratio is quenched in (a) from η=0.536 to 0.51 and in (b) from η=4.2 to 4. Color coding according to Fig. 7. Black dashed line indicates the entanglement entropy SvN of the initial state. All quantities are given in harmonic units of the majority component.

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    • Figure 10
      Figure 10

      Same as in Fig. 9 but for NA=10. All quantities are given in harmonic units of the majority component.

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    • Figure 11
      Figure 11

      Energy gaps Ωj=|Ej−Eref| with respect to lowest-energy reference eigenstates |Eref〉 of odd global parity and even (first column) or odd (second column) c.m. parity in a few-body bosonic mixture NA=NB=2. The gaps are functions of the intercomponent coupling gAB at equal trapping frequency ratio η=1, intracomponent interaction strength gB=0 for the second component, and (a) gA=−0.5, (b) gA=0, (c) gA=0.5 for the first component. Whether the corresponding eigenstates are actually excited depends on the quench protocol. Different colors (line styles) refer to the center-of-mass (c.m.) quantum number in the eigenstate |Ej〉. The c.m. is a decoupled degree of freedom in this harmonic confinement. The insets represent a zoom-in on regions with avoided crossings which are indicated by circles and caused by gA≠gB asymmetry. Curves of different colors (line styles) may only cross. All quantities are given in harmonic units.

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    • Figure 12
      Figure 12

      Breathing oscillations of the one-body densities of the majority component ρ1A(x) (first row) and the impurity ρ1B(y) (second row) at a fixed majority-component interaction gA=0.5 for NA=5 and NB=1 particles initiated by preparation of the first excited state of odd global parity followed by an abrupt change of the trap ratio from η=1.05 to 1. Columns 1–3 correspond to different intercomponent couplings: (a) gAB=0.5, (b) gAB=1.0, and (c) gAB=1.5. All quantities are given in harmonic units.

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    • Figure 13
      Figure 13

      Frequencies Ω of breathing modes excited by quenching the odd global parity ground state |E1〉 of a Bose polaron NA=5 and NB=1 from a trap ratio η=1.05 to 1.0 as a function of the intercomponent coupling gAB for a fixed majority component interaction (a) gA=−0.5, (b) gA=0, and (c) gA=0.5. Color coding as in Fig. 7. Black dashed line indicates the entanglement entropy SvN of the initial state. All quantities are given in harmonic units.

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