Slow interaction quench in BCS superconductors: emergence of pre-formed pairs

TL;DR

This paper studies how slow changes in interaction strength affect BCS superconductors, revealing regimes where pairs survive without coherence, oscillations occur, or thermal distributions emerge, enabling control of superconducting properties.

Contribution

It introduces a detailed analysis of non-equilibrium dynamics in BCS superconductors during slow interaction ramps, highlighting the emergence of pre-formed pairs and regimes of coherence loss and retention.

Findings

Short ramps destroy phase coherence but preserve pairing.

Intermediate ramps sustain superconductivity with oscillations.

Long ramps maintain near-thermal pair distributions.

Abstract

We investigate the non-equilibrium behavior of BCS superconductors subjected to slow ramps of their internal interaction strength. We identify three dynamical regimes as a function of ramp duration. For short ramp times, these systems become non-superconducting; however, fermions with opposite momenta remain paired albeit with reduced amplitudes, and the associated pair amplitude distribution is non-thermal. In this first regime, the disappearance of superconductivity is due to the loss of phase coherence between pairs. By contrast, for intermediate ramp times, superconductivity survives but the magnitude of the order parameter is reduced and presents long-lived oscillations. Finally, for long ramp times, phase coherence is almost fully retained during the slow 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 in order to control the magnitude of the superconducting order parameter and even engineer a non-equilibrium state made of pre-formed pairs.

Equations

$$H_{\text{BCS}}$$

$$\Delta$$

$$P$$

$$\displaystyle\frac{1}{k_{F}a(t)}=\frac{1}{k_{F}a(t_{\text{f}})}+h(t,t_{\text{i}},t_{\text{f}})\Bigg{[}1-\sin^{2}\Big{(}\frac{\pi}{2}\frac{t-t_{\text{i}}}{t_{\text{ramp}}}\Big{)}\Bigg{]},$$

$$\hslash \frac{\partial }{\partial t}⟨c_{-\text{k},2}{c}_{\text{k},1}⟩$$

$$\hslash \frac{\partial }{\partial t}⟨n_{\text{k},1}⟩$$

$$\hslash \frac{\partial }{\partial t}⟨n_{-\text{k},2}⟩$$

Full text

The properties of quantum materials are extremely sensitive to external stimuli. In these systems, the interactions associated with, for example, the spin, charge, lattice and orbital degrees of freedom are often similar in magnitude with the electronic kinetic energy. The delicate balance between competing states can therefore be readily altered via external perturbations leading to the emergence of novel properties. Taking advantage of this distinctive characteristic of quantum materials and building on the tremendous technical progress achieved in the last decade, scientists can now dynamically engineer complex states and follow their non-equilibrium evolution. Phase transitions were photo-induced in strongly interacting solid state compounds using ultrafast optical pulses Basov et al. (2011); Orenstein (2012); Zhang and Averitt (2014); Giannetti et al. (2016), and similar achievements were also reported in ultracold atomic systems using time-dependent electromagnetic fields Bloch et al. (2008); Polkovnikov et al. (2011). For example, in a striped-order cuprate, a Josephson plasmon, a hallmark of the superconducting state, was activated above the critical temperature by the application of midinfrared femtosecond pulses Fausti et al. (2011). While these results are truly remarkable, the mechanisms underlying the non-equilibrium dynamics of strongly correlated matter are still being investigated.

Identifying the processes governing the evolution of order parameters when interactions are tuned over time remains an open question. As order parameters are global quantities often made up from sums of local or quasi-local (in position or momentum space) expectation values, one would like to understand how the time-dependent behavior of these different local components conspires to control the dynamics of the global order parameter. Superconductors described by the Bardeen-Cooper-Schrieffer (BCS) theory of superconductivity constitute an interesting example as in these systems the collective order parameter is built out of individual Cooper pair states labeled by their internal momentum. A similar situation also arises in magnets where the magnetization is a sum over all local spins.

Going back to our first example, superconductivity, in this case understanding the dynamics arising from the subtle interplay between the BCS collective mode and its constituting elements following a sudden quench of the pairing strength has been the focus of various works Barankov et al. (2004); Warner and Leggett (2005); Szymańska et al. (2005); Yuzbashyan et al. (2006); Barankov and Levitov (2006); Yuzbashyan and Dzero (2006); Papenkort et al. (2007); Dzero et al. (2007); Gurarie (2009); Dzero, M. et al. (2009); Galitski (2010); Scott et al. (2012); Yuzbashyan et al. (2015). The renewed interest for this problem has been triggered in part by the possibility in dilute fermionic gases cooled below degeneracy to control the interaction strength using Feshbach resonances Bloch et al. (2008). Using both analytical and numerical methods, three different dynamical regimes were unveiled in a space spanned by the ratio of the equilibrium superconducting order parameters of the initial state to the one of the final state. When this ratio is sufficiently small, the order parameter oscillates without damping, while for intermediate ratios, it is damped, exhibits decaying oscillations and saturates at an asymptotic value. In contrast, for larger ratios, the order parameter is overdamped following the sudden interaction quench and ultimately superconductivity is destroyed.

Experimentally, quenches are typically realized within a finite ramp time. This can be achieved, for example, by exciting phononic modes in solids Sentef et al. (2016) or in cold gases by ramping a magnetic field. We study in this article the dynamics of the BCS order parameter when the interaction between fermions is slowly changed in time. Focusing on the dynamical region where a sudden change of the interaction strength would obliterate the superconducting order parameter, we find that when the same interaction change is carried out at a slower rate three different regimes emerge. As shown in Fig. 1, for sufficiently short ramp times, the system becomes non-superconducting as the interaction strength is lowered to its new value. However, in contrast to the usual non-superconduting thermal state above the critical temperature, Cooper pairs are still present in this dynamically generated non-superconducting state, but with reduced amplitudes. This is signalled by the finite value taken by the sum over the magnitude of the momentum-dependent pair amplitudes. Hence, the loss of phase coherence between these pairs is responsible here for the disappearance of superconductivity. Consequently, for short ramp times, dynamically engineering a state made of incoherent pre-formed pairs, the so-called phase disordered superconductor Emery and Kivelson (1995), is possible. With increasing ramp time, partial phase coherence is restored and the superconducting order parameter acquires a finite value. Whereas in the incoherent pairing regime, the sum over the magnitude of pair amplitudes decreases, this quantity increases again once superconductivity is re-established in the second dynamical regime displaying a surprising non-monotonic behavior. Finally, for long ramp times, full phase coherence is almost retained during the slow interaction change, and the amplitude distribution of the individual pairs becomes thermal-like. In the remaining of this article, we present in more details the subtle mechanism responsible for the dynamics of the superconducting order parameter during and after a slow ramp down of the interaction strength.

To study this dynamics, we consider a situation applicable to both solid state systems and cold atom gases: a three-dimensional gas made of two species of fermions described by the BCS $s$-wave Hamiltonian

[TABLE]

where $c_{\textbf{k},\sigma}^{({\dagger})}$ are the fermionic annihilation (creation) operators, $n_{\textbf{k},\sigma}$ is the particle number operator of momentum k and species $\sigma=\{1,2\}$, and $\epsilon_{\textbf{k}}=\hbar^{2}\textbf{k}^{2}/(2m)$ is the single-particle dispersion. The superconducting order parameter enters this Hamiltonian as

[TABLE]

with $V$ the system volume and $g$ the interaction strength. Here, the expectation value $P_{\textbf{k}}=\langle c_{-\textbf{k},2}c_{\textbf{k},1}\rangle$ relates to individual Cooper pairs. Additionally, to assess the individual pairing strength, we introduce a second quantity corresponding to the sum over the magnitude of the momentum-dependent pair amplitudes

[TABLE]

For ultracold gases, the strength of the interaction between the fermions of two different hyperfine states can be tuned via Feshbach resonances Bloch et al. (2008), and at sufficiently low temperatures the $s$-wave scattering is the dominant contribution. In this situation, the interaction can be parametrized by a single parameter, the $s$-wave scattering length $a$, via $\frac{1}{k_{\text{F}}a}=\frac{8\pi E_{\text{F}}}{gk_{\text{F}}^{3}}+\frac{2}{\pi}\sqrt{\frac{E_{\text{C}}}{E_{\text{F}}}}$ with $k_{\text{F}}$ and $E_{\text{F}}$ the Fermi momentum and energy, respectively, and $E_{\text{C}}$ a suitably chosen energy cutoff.

The equilibrium phase diagram for a system described by this Hamiltonian has been thoroughly studied (see Ketterle and Zwierlein (2008) and references therein). For $\frac{1}{k_{F}a}<0$, the interaction is attractive and below the critical temperature, $T_{c}$, this system arranges into a superfluid of Cooper pairs. In this situation, the value of the superconducting order parameter decreases with increasing temperature as the thermal generation of single-particle excitations leads to the breaking of Cooper pairs.

Here, we consider a slower interaction change of the parameter $\frac{1}{k_{F}a}<0$ using the schedule

[TABLE]

where h(t,t_{\text{i}},t_{\text{f}})=\Theta(t_{\text{f}}-t)\Big{[}1/(k_{F}a(t_{\text{i}}))-1/(k_{F}a(t_{\text{f}}))\Big{]}, $t_{\text{ramp}}=t_{\text{f}}-t_{\text{i}}$ with $t_{\text{i}}$ and $t_{\text{f}}$ the times at which the interaction ramp begins and ends, and $\Theta$ is the Heaviside function. $t_{\text{ramp}}\rightarrow 0$ corresponds to a sudden interaction change while $t_{\text{ramp}}\rightarrow\infty$ would correspond to an adiabatic interaction change. We focus on the situation where the interaction strength is ramped down such that the initial equilibrium value of the superconducting order parameter, $\Delta_{0,\text{i}}$, is larger than the final equilibrium value, $\Delta_{0,\text{f}}$.

To understand the dynamics of the order parameter, we obtain a set of coupled differential equations connecting the superconducting order parameter to the expectation values of individual pairs and atom densities:

[TABLE]

Solving numerically this set of equations together with the self-consistency condition for $\Delta$, Eq. 2, we compare and contrast the non-equilibrium evolution due to a sudden interaction change, $t_{\text{ramp}}\rightarrow 0$, to ones due to longer ramp times.

The main results are summarized in Fig. 1 comparing the time-averaged superconducting order parameter, $\langle\!\langle|\Delta|\rangle\!\rangle$, and the time-averaged sum over the magnitude of the pair amplitudes, $\langle\!\langle|P|\rangle\!\rangle$. For a sudden interaction change and for ramp times up to $3\hbar/E_{\text{F}}$, the superconducting order parameter is found to average to zero whereas, for slower interaction ramps, this order parameter retains a finite value. The precise ramp duration at which the crossover occurs depends on the interaction strength and the cutoff.

In contrast, we find that the sum over the magnitude of the pair amplitudes, remains finite for all ramp times. This important difference in behavior between these two quantities signals that the evolution of the relative phase between individual pairs plays a crucial role in the dynamics. As the amplitude of pairs is reduced but remains finite, the destruction of superconductivity for fast ramps is associated with the loss of phase coherence between pairs. Therefore, phase unlocking is the main mechanism responsible for the suppression of superconductivity. This is in stark contrast to the finite-temperature equilibrium scenario where superconductivity is suppressed by an increase in thermal fluctuations resulting in pair breaking. Interestingly, this result implies that stabilizing a state made of pre-formed pairs is possible via a fast ramp. Within the scope of the BCS model, this state is long-lived; however, in real materials the presence of various coupling mechanisms could likely affect the long-time stability of this state.

Unexpectedly, $\langle\!\langle|P|\rangle\!\rangle$, the sum over the magnitude of the pair amplitudes, is non-monotonous as a function of ramp time: it first decreases with increasing ramp times and then increases again. For a sudden quench, this quantity has a large value due to the freezing of the initial state which is then projected onto the new Cooper pairs. Within this new basis, this frozen state contains excited quasiparticle pairs which contribute to the sum over the magnitude of pair amplitudes. As these quasiparticle pairs are not coherent, their contributions to the total order parameter dephase after a short time resulting in the suppression of the superconducting order parameter. For short but finite ramp times, the same mechanism persists until the sum over the magnitude of the pair amplitudes reaches a minimum. For longer ramp times, this quantity rises again and the dephasing becomes less important such that the value of the superconducting order parameter is finite at longer times.

In the following, we analyze carefully the behavior of the excitations responsible for the emergence of the non-equilibrium states detailed above. Useful information can be obtained by analyzing the momentum distribution of the Cooper pair amplitudes. As displayed in Fig. 2, initially the distribution of pair amplitudes follows the zero temperature and interaction-dependent expression $P_{\textbf{k}}=\frac{1}{2}\sqrt{1-\xi_{\textbf{k}}^{2}/E_{\textbf{k}}^{2}}$ with $\xi_{\textbf{k}}=\epsilon_{\textbf{k}}-\mu_{\text{i}}$ where $\mu_{\text{i}}$ is the chemical potential at the initial interaction strength. For $\frac{1}{k_{\text{F}}a}=-0.1072$, this distribution has a maximum close to the Fermi momentum and drops down for larger momentum values. In contrast, the distribution corresponding to the final ground state of an adiabatic quench to $\frac{1}{k_{\text{F}}a}=-1.3493$ is strongly peaked around the Fermi momentum and is much lower in value than the intial distribution.

To understand the time evolution of the distribution of pair amplitudes, we consider both snapshots of $|P_{k}|$ at particular times and values denoted by $\langle\!\langle|P_{k}|\rangle\!\rangle$ that are time-averaged between $100\hbar/E_{\text{F}}$ and $400\hbar/E_{\text{F}}$. As shown in Fig. 2, for a sudden ramp, the magnitudes of the pair amplitudes settle quickly as the snapshot distribution at $E_{\text{F}}t=3\hbar$ already agrees approximately with the one obtained via time-averaging. However, even at long times, this distribution takes much larger values than the ones expected in equilibrium at the final interaction strength $\frac{1}{k_{\text{F}}a}=-1.3493$. This result explains the large finite value of the sum over the magnitude of the pair amplitudes presented in Fig. 1 signalling that Cooper pairs survive through the quench (even though with a smaller amplitude than in the initial state).

As shown in the central panel of Fig. 2, for the sudden ramp, each pair rapidly acquires a particular phase proportional to $2E_{\textbf{k}}$ leading to complete dephasing such that already at $E_{\text{F}}t=10\hbar$ the superconducting order parameter is totally obliterated.

The evolution of the phases can be understood via the Fourier transform of the pair amplitudes, $|\mathcal{F}[\text{Im}[P_{\textbf{k}}(t)]]|$, and as $\text{Im}[P_{\textbf{k}}(t)]$ and $\text{Re}[P_{\textbf{k}}(t)]$ provide the same information about the phase evolution, we only consider the former without loss of generality. From this quantity, we see that the sudden ramp generates quasiparticle pairs at $-2\mu_{\text{f}}\pm 2E_{\textbf{k}}$ (see lower panel of Fig. 2) with $\mu_{\text{f}}$ the chemical potential at the final interaction strength. These quasiparticle pairs are at the origin of the parabolic distribution of the phases (central panel). This result indicates that the system dynamically organizes into a non-thermal state made of pre-formed but dephased Cooper pairs.

The pair amplitude distributions are also non-trivially affected when the interaction strength is slowly ramped down. For the ramp time $E_{\text{F}}t_{\text{ramp}}=6\hbar$, both the snapshot and time-averaged distributions are clearly finite and non-thermal. Only the small and large momentum tails of the time-averaged distribution agree with the thermal equilibrium distribution at $\frac{1}{k_{\text{F}}a}=-1.3493$. The temperature $T$ used in Fig. 3 is found by solving the finite-temperature gap equation Tinkham (1996) assuming that $\langle\!\langle|\Delta|\rangle\!\rangle=\Delta(T)$.

As we see in the lower panel of Fig. 3, the $E_{\text{F}}t_{\text{ramp}}=6\hbar$ ramp creates fewer quasiparticle pairs at $-2\mu_{\text{f}}\pm 2E_{\textbf{k}}$ and all $P_{\textbf{k}}$ signals have a strong component at $-2\mu_{\text{f}}$. At short times compared to the ramp duration, the phase remains fully locked, then as the evolution goes on, in the momentum interval where most of the quasiparticle pairs are generated, each Cooper pair begins accumulating a particular phase. This process leads to a partial loss of phase coherence, but, as shown in Fig. 1, the Cooper pairs are still sufficiently synchronized for superconductivity to survive at the considered times.

Finally, for $E_{\text{F}}t_{\text{ramp}}=30\hbar$, we find that the dynamics enters a different regime as the distribution of pair amplitudes becomes thermal. As one sees from Fig. 4, the distribution obtained at the end of the ramp strongly resembles the one expected for a superconducting system in equilibrium at $T=0.35T_{c}$ for an interaction strength of $\frac{1}{k_{\text{F}}a}=-1.3493$. For this ramp schedule, quasiparticle pairs are solely generated in a small momentum region around $k_{\text{F}}$ (see lower panel of Fig. 4). The phase coherence remains for the most part undisturbed by the interaction ramps. As illustrated in Fig. 4, during the ramp the phase starts to ripple around $k_{\text{F}}$, the region where quasiparticle pairs are generated, but phase locking is for the most part maintained throughout the system.

To summarize, we analyzed the non-equilibrium dynamics of a BCS superconductor when the interaction strength is slowly ramped down. We identified three different dynamical regimes and, in particular, we demonstrated the dynamical creation of a steady state of pre-formed pairs without global phase coherence. The insights gained from this study will likely pave the way to employ slow quenches to create other steady states with novel properties absent in thermal equilibrium.

Acknowledgments: We thank Kuiyi Gao for useful discussions. We acknowledge funding from the European Research Council (ERC) under the Horizon 2020 research and innovation programme, grant agreement No. 648166 (Phonton) and No. 616082 (UpFermi), and from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) project number 277625399 - TRR 185 project B4 and project number 277146847 - project C05.