Finite-duration interaction quench in dilute attractively interacting Fermi gases: Emergence of preformed pairs

Johannes Kombe, Jean-Sébastien Bernier, Michael Köhl, and Corinna Kollath

Phys. Rev. A 100, 013604 – Published 8 July 2019

Abstract

We investigate the nonequilibrium behavior of dilute, attractively interacting Fermi gases subjected to finite-duration ramps of their internal interaction strength. We identify three dynamical regimes as a function of ramp duration using a time-dependent version of the Bardeen-Cooper-Schrieffer theory of superconductivity to model these systems. For short ramp durations, these systems become nonsuperconducting; however, fermions with opposite momenta remain paired albeit with reduced amplitudes, and the associated pair amplitude distribution is nonthermal. In this first regime, the disappearance of superconductivity is due to the loss of phase coherence between pairs. By contrast, for intermediate ramp durations, superconductivity survives but the magnitude of the order parameter is reduced and presents long-lived oscillations. Finally, for long ramp durations, phase coherence is almost fully retained during the finite-duration interaction quench, and the steady state is characterized by a thermal-like pair amplitude distribution. Using this approach, one can therefore dynamically tune the coherence between pairs to control the magnitude of the superconducting order parameter and even engineer a nonequilibrium state made of preformed pairs.

Received 19 February 2019

DOI:https://doi.org/10.1103/PhysRevA.100.013604

Physics Subject Headings (PhySH)

Cooper pairs Dynamical phase transitions Quantum phase transitions Superconducting fluctuations Superconducting order parameter

Collective dynamics Fermi gases Nonequilibrium systems Ultracold gases

BCS theory Mean field theory

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Johannes Kombe, Jean-Sébastien Bernier, Michael Köhl, and Corinna Kollath

Physikalisches Institut, University of Bonn, Nussallee 12, 53115 Bonn, Germany

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Vol. 100, Iss. 1 — July 2019

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Figure 1
Time-averaged value of the superconducting order parameter $〈 〈 | Δ | 〉 〉$ and of the sum over the magnitude of the pair amplitudes $〈 〈 | P | 〉 〉$ as a function of ramp duration. The interaction strength is ramped down from $1 / (k_{F} a) = - 0.1072$ ( $| Δ_{0, i} | = 0.60 E_{F}$ ) to $1 / (k_{F} a) = - 1.3493$ ( $| Δ_{0, f} | = 0.11 E_{F}$ ), and the time-average is taken between $100 ℏ / E_{F}$ and $400 ℏ / E_{F}$ . These two quantities signal three different dynamical regimes (highlighted by shaded regions). For short quenches, the system is characterized by preformed pairs (incoherent pairing state). For intermediate quench durations, superconductivity is maintained but with only partial phase coherence; while, for longer ramp durations, phase coherence is mostly unaffected and the order parameter asymptotes to $Δ_{0, f}$ (value marked by the arrow). The boundary between the partial phase coherence and BCS superconductor regimes is located at $| 〈 〈 | Δ | 〉 〉 - 〈 〈 | P | 〉 〉 | \sim 10^{- 4} E_{F}$ . Inset: Evolution of the order parameter as a function of time for three different ramp durations.
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Figure 2
Sudden quench of the interaction strength from $1 / (k_{F} a) = - 0.1072$ to $1 / (k_{F} a) = - 1.3493$ . (a) Distribution of the magnitude of the pair amplitude as a function of momentum: (i, blue) ground state at $1 / (k_{F} a) = - 0.1072$ ; (ii, red) ground state at $1 / (k_{F} a) = - 1.3493$ ; (iii, pink) snapshot at $3 ℏ / E_{F}$ ; (iv, orange) snapshot at $10 ℏ / E_{F}$ ; and (v, green) time-average between $100 ℏ / E_{F}$ and $400 ℏ / E_{F}$ . This distribution is already hardly distinguishable form its steady-state configuration at $10 ℏ / E_{F}$ . It is nonthermal and signals the presence of preformed pairs. Note that lines (iv) and (v) are on top of each other. (b) Phase of the pair amplitude as a function of momentum. Rapid phase unlocking is responsible for the destruction of superconductivity. Note: The legend is in descending order. (c) Fourier transform of the momentum-dependent pair amplitude $| F [Im [P_{k} (t)]] |$ . The sudden quench generates quasiparticle pair excitations along the parabolic line $\pm 2 E_{k} - 2 μ_{f}$ (marked by the red crosses and stars, respectively; red circles mark the coherent evolution at $- 2 μ_{f}$ ).
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Figure 3
Slow quench performed in $E_{F} δ t_{ramp} = 6 ℏ$ (same interaction strengths as in Fig. 2). (a) Distribution of the magnitude of the pair amplitude as a function of momentum: (i, blue) and (ii, red) same as in Fig. 2; (iii, black dashed) thermal distribution at $1 / (k_{F} a) = - 1.3493$ , $T = 0.89 T_{c}$ is chosen such that $〈 〈 | Δ | 〉 〉 = Δ (T)$ ; (iv, pink) snapshot at $3 ℏ / E_{F}$ ; (v, green) time-average between $100 ℏ / E_{F}$ and $400 ℏ / E_{F}$ . The time-averaged distribution is nonthermal, and $| P_{k} |$ exhibit strong oscillations represented, together with (v), by vertical bars (peak-to-peak amplitude of the oscillations). (b) Phase of the pair amplitude as a function of momentum. Note: The legend is in descending order. (c) Fourier transform of the momentum-dependent pair amplitude $| F [Im [P_{k} (t)]] |$ . While this quench generates quasiparticle pair excitations, all $P_{k}$ signals have a strong in-phase component at $- 2 μ_{f}$ .
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Figure 4
Slow quench performed in $E_{F} δ t_{ramp} = 30 ℏ$ (same interaction strengths as in Fig. 2). (a) Distribution of the magnitude of the pair amplitude as a function of momentum: (i, blue) and (ii, red) same as in Fig. 2; (iii, black dashed) thermal distribution at $1 / (k_{F} a) = - 1.3493$ , $T = 0.35 T_{c}$ is chosen such that $〈 〈 | Δ | 〉 〉 = Δ (T)$ ; (iv, pink) snapshot at $10 ℏ / E_{F}$ ; (v, orange) snapshot at $30 ℏ / E_{F}$ ; (vi, green) time-average between $100 ℏ / E_{F}$ and $400 ℏ / E_{F}$ . The steady-state distribution is thermal. Note that lines (ii), (iii), (v), (vi) are on top of each other. (b) Phase of the pair amplitude as a function of momentum. Phase coherence is only slightly lost near $k_{F}$ . Note: The legend is in descending order. (c) Fourier transform of the momentum-dependent pair amplitude $| F [Im [P_{k} (t)]] |$ . Quasiparticle pairs are only generated near $k_{F}$ (red crosses and stars). $P_{k}$ are dominated by the coherent phase evolution at $- 2 μ_{f}$ .
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