Cooper pair turbulence in atomic Fermi gases
Abstract
We investigate the stability of spatially uniform solutions for the collisionless dynamics of a fermionic superfluid. We demonstrate that, if the system size is larger than the superfluid coherence length, the solution characterized by a periodic in time order parameter is unstable with respect to spatial fluctuations. The instability is due to the parametric excitations of pairing modes with opposite momenta. The growth of spatial modulations is suppressed by nonlinear effects resulting in a state characterized by a random superposition of wave packets of the superfluid order parameter. We suggest that this state can be probed by spectroscopic noise measurements.
pacs:
05.30.Fk, 32.80.-t, 74.25.GzHow does a homogeneous interacting many-body system develop spatial modulations as a result of a uniform quench? In general, this can happen due to an interplay between the extra energy introduced by the quench and an intrinsic coupling of the degrees of freedom at various length scales (1); (2); (3); (4). By the very nature of the problem, the energy distribution of these degrees of freedom is initially far from a state of thermodynamic equilibrium. This situation is common in a nonlinear medium and can be described by the term ”wave turbulence” (5). In the case when one of the parameters of the medium or an applied field is periodically varied in time, wave turbulence due to a parametric instability can develop. Examples of such a phenomenon include the decay of high-frequency electric field into Lagmuir and ion-sound waves in plasma (7) and the spin-wave instability in an rf-magnetic field in dielectric ferromagnets (8). A natural question is then whether a quench can lead to a parametric excitation of spatial modes.
Oscillating homogeneous fields can be generated by a uniform quench without an external drive. One of the recent examples where this takes place is a fermionic superfluid quenched by a sudden change of the pairing strength (9); (10). There are two types of asymptotic states the system can reach depending on the strength of the initial perturbation (11); (12). The first type has a constant value , while the second one is characterized by periodic . This analysis does not take into account the pair breaking processes and is thus valid only for , where is the quasiparticle relaxation time. Moreover, the emerging asymptotic states are spatially uniform: order parameter evolution was obtained as a solution of an effectively zero dimensional problem. Results of Refs. (11); (12) are thus valid when the system size is smaller than the coherence length, . The description of the order parameter evolution for requires an additional investigation.
In this paper, we perform a stability analysis of the solutions for the wave function and order parameter obtained in Refs. (11); (12) with respect to spatial fluctuations. We find that the asymptotic states with constant order parameter remain stable, while the state with periodic does not. The physical origin of the instability lies in the possibility of parametric excitation of the spatially modulated pairing modes, Fig. 1. In a homogeneous medium, the periodic (in time) order parameter can be considered as an energy pump allowing a coupling between two spatial modes with opposite momenta, and, at the same time, providing enough energy to overcome the damping due to the scattering of Cooper pairs. Subsequent scattering effects of the resonant modes limit the initial exponential growth and result in a state with a spatially inhomogeneous order parameter. We demonstrate that as the process of the parametric instability develops, the energy of the homogeneous state is transferred into that of the pairing wave packets with the typical size of the order of the coherence length, , signaling the onset of the Cooper pair turbulence. Since the amplitudes of the wave packets are essentially random, we suggest that this state can be experimentally probed by the noise measurements.
Consider the state with a periodically varying order parameter. Analytically, is described by the Jacobi elliptic function dn (9). Here we assume that the amplitude of the oscillations is small. This allows us to keep only the first two terms in the Fourier series for dn:
(1) |
We note that the non-dissipative dynamics within the BCS model is described by the Bogoliubov-de Gennes equations, which can be cast into the form of equations of motion for classical vector variables : . Here plays the role of an external magnetic field. The periodic external field makes it possible for the parametric instabilities to develop. Given that the asymptotic state (1) is robust against homogeneous perturbations (9); (10), it seems natural to look for an instability with respect to spatial fluctuations. In particular, we will investigate the conditions of the parametric decay of the homogeneous pairing mode (1) with an energy into two pairing modes with energies and momenta and , Fig. 1. In a continuous medium the energy and momentum have to be conserved: and . These type of instabilities appear in various physical systems, see e.g. Ref. (5) for an extensive review.
To analyze the stability of the asymptotic state (1), we employ the Bogoliubov-de Gennes (BdG) equations(13):
(2) |
Here , is the chemical potential and the order parameter is
(3) |
where is the BCS coupling constant. We linearize Eqs. (2) with respect to the deviations and from the homogeneous solution and :
(4) |
The order parameter (3) also contains a small inhomogeneous part . Plugging (4) into (2) and Fourier transforming with respect to the spatial coordinates, we find
(5) |
The time dependance of and (see Ref. (14)) suggests that for linear corrections (4) we write and with and . We also write
(6) |
where , , determines the growth rate, and is a time scale when (1) is reached (in what follows we set ). In the expression (6) we have neglected the higher harmonics etc. This is justified for , since their inclusion yields higher order in corrections to the growth rate and to the order parameter amplitudes (6). In the linear corrections to the Bogoliubov amplitudes (see above) we also keep only the lowest harmonics , i.e. , etc.
Next, we express the Bogoliubov amplitudes in terms of and by equating the coefficients in front of in Eq. (5). The resulting amplitudes are substituted into the self-consistency equation (3). One obtains a linear system for the variables and . Expressing in terms of and , we derive
(7) |
where , , and are nonlinear functions of the growth rate . Eqs. (7) can be interpreted as the equations of motion for a classical field (5). Then, has the meaning of the excitation spectrum of this field and stands for the pumping amplitude, which gives rise to a parametric instability. Finally, describes the damping of the parametric modes due to the intrinsic relaxation processes.
Nonzero solutions of Eqs. (7) for exist provided . Thus, the stability analysis is reduced to the solution of the nonlinear equation for . We have analyzed this equation numerically and present the results on Fig. 2. We find that the instability region is centered around and has a width , where is the coherence wave vector. From (7) it follows that for a fixed the parametric growth will be suppressed as soon as the energy pumped into the system fully goes into dissipation. This condition determines the maximum growth rate , i.e. . Our estimate yields . Lastly, we have also verified that asymptotic states with constant order parameter remain stable with respect to the spatial fluctuations of the above type.
The initial growth of the parametric instability (11) will be limited by nonlinear effects which lead to the transient behavior with subsequent transition into a post-threshold state. The latter is defined as a state in which Fourier components of the order parameter (6) are time independent, and . Below we focus on finding the resulting post-threshold state of the condensate. From the linear analysis we have seen that the fastest growing modes are the ones with a certain magnitude of the momentum. Thus, in the BdG equations (2) among the nonlinear in powers of terms we keep the resonant ones with frequencies and momenta , where is a new post-threshold state momentum to be determined below. The resulting set of nonlinear in equations for the order parameter amplitudes can be written as Eq. (7) with renormalized coefficients
(8) |
where and are the scattering matrix elements. They vary slowly on a scale of and are almost independent of the angle between and . In what follows we neglect the -dependence in the scattering matrix elements, and . Note that each contribution in (8) is either phase independent or depends on a sum of the two phases of and . This can be interpreted as follows. There are two physical processes which limit the parametric excitations: one has to do with the reduction of the absolute value of the amplitudes, while the other is related to the phase decoherence of the two pairing modes with opposite momenta. In its spirit approximation (8) is similar to the mean-field BCS model where only diagonal in momentum terms are kept in the interaction. The inclusion of off-diagonal terms in Eq. (8) is expected to cause a broadening of the post-threshold state momentum , see Fig. 2.
To determine the parameters of our post-threshold state, we insert , and Eqs. (8) for and into (7). The post-threshold state momentum is determined by the condition that the magnitude of the pumping field does not exceed the damping for any . As a result we have . Phase and amplitude are given by and (the corresponding expressions for and can be derived similarly). We obtain
(9) |
where , and for . Note that the post-threshold momentum , i.e. the energy cascades to smaller length scales, as expected of turbulent behavior.
The individual phases and cannot be determined within the diagonal approximation (8). In a continuous medium, one can treat them as random variables. For the correlators we take , and , where stands for averaging over the phase distribution.
To get further insight into the nature of the post-threshold state, consider the following choice for the phases and . It leads to a spherically symmetric wave packet, a ”bubble”, with a periodic amplitude
(10) |
In general we obtain a linear combination of these bubbles (10) by writing and similarly for . Then, Eq. (9) can be viewed as a superposition of wave packets of the form (10) centered at different with random amplitudes . This suggests that the parametric instability results in a random distribution of the wave packets.
It is instructive to compare (10) with at the linear stage of the parametric instability, which can be derived using our result for (see Fig. 2) and Eqs. (6,7). Taking and , we obtain
(11) |
where , , is a constant, and . In deriving Eq. (11), we also assumed and replaced a slowly varying function . Expression (11) describes the initial formation of a wave packet (10). Note that on a time scale at which the order parameter deviation is of the order , the width of the packet is .
The observations above help to identify features of the post-threshold state (9) relevant for the experimental verification of our theory. To be more specific, let us compute the correlator , where and are taken in the post-threshold stage. characterizes the spatial and time distribution of the parametric noise in the system. Using the correlators for the random functions (see above), we obtain from Eq. (9)
(12) |
where . This means that the spatial noise spectrum has a peak at the wave vector and decays as for .
Formation of an isolated wave packet (10) induces an oscillating supercurrent , where is the phase of the order parameter. Setting we find that only the radial component of the current is nonzero. To the lowest order in , . This implies a spatial re-distribution of Cooper pairs similar to the Friedel oscillations in the density of a degenerate Fermi gas induced by a weak scattering potential.
In our discussion so far we treated the pairing mode (1) giving rise to the parametric instability as an independent external field. Inclusion of the feedback on this mode as weak turbulence () develops may modify the post-threshold state (9). We leave a detailed analysis of possible feedback effects for future studies.
In the post-threshold state (9) the Fourier components of the order parameter are . Inelastic scattering or thermal effects generally leads to a broadening in the momentum and frequency distributions of (8). The latter might cause a damping of the temporal oscillations in Eq. (9). On a time scale dissipation due to quasi-particle scattering processes ultimately forces the system to reach an equilibrium state. Finally, we comment that in the transient regime leading to an asymptotic state with constant the order parameter is (cf. (1)) (10). Oscillatory behavior suggests that this asymptotic state might never be attained owing to the development of the parametric instability of the type considered above.
In conclusion, we have investigated the stability of the nonequilibrium asymptotic states of a fermionic superfluid, which can be generated e.g. by a uniform quench of the pairing strength. We have demonstrated that in a system of size larger than the coherence length the asymptotic state (1) with periodic in time order parameter is unstable with respect to spatial fluctuations. The instability is due to the parametric excitation of two pairing modes with opposite momenta. The initial exponential growth of deviations from the homogeneous state (11) is suppressed by nonlinear effects eventually leading to a spatially nonuniform post-threshold state described by Eq. (9). This state can be interpreted as a superposition of bubbles of the superfluid order parameter (10) with random amplitudes. The parametric instability of the uniform oscillations can be experimentally probed by spectroscopic noise measurements.
M. D.’s research was financially supported by the Department of Energy, grant DE-FE02-00ER45790. E.A.Y. acknowledges the financial support by a David and Lucille Packard Foundation Fellowship for Science and Engineering, NSF award NSF-DMR-0547769, and Alfred P. Sloan Research Fellowship. B.L.A. thanks The US-Israel Binational Science Foundation for the financial support.
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