Superfluid Stiffness of a Driven Dissipative Condensate with Disorder
Observations of macroscopic quantum coherence in driven systems, e.g. polariton condensates, have strongly stimulated experimental as well as theoretical efforts during the last decade. We address the question of whether a driven quantum condensate is a superfluid, allowing for the effects of disorder and its non-equilibrium nature. We predict that for spatial dimensions the superfluid stiffness vanishes once the condensate exceeds a critical size, and treat in detail the case . Thus a non-equilibrium condensate is not a superfluid in the thermodynamic limit, even for weak disorder, although superfluid behavior would persist in small systems.
Perhaps the most spectacular manifestation of Bose-Einstein condensation, and its associated macroscopic quantum coherence, is superfluidity. Recent experiments Kasprzak et al. (2006) have shown macroscopic quantum coherence in a population of mixed matter-light excitations, so-called polaritons (see Deng et al. (2010) for a review). Aspects of superfluid behavior, including quantized vortices Lagoudakis et al. (2008); Sanvitto et al. (2010) and suppression of scattering from defects Amo et al. (2009), have also been observed. However, unlike the constituents of conventional condensates, such as cold atoms, polaritons have a finite lifetime. Thus, the polariton condensate is a non-equilibrium steady-state, in which the losses are compensated by particles flowing in from an external source. This leads to the interesting possibility of new universal behavior, different from that found in equilibrium Sieberer et al. (2013). Many similarities, nonetheless, appear to remain, at least in the absence of disorder: perturbatively, the forms of the correlation functions are the same as in equilibrium Chiocchetta and Carusotto (2013); Roumpos et al. (2012) (long-range order in three dimensions, and quasi-long-range order in two); superfluidity is predicted to survive Keeling (2011) (); and the static behavior, in three dimensions, involves the standard O(2) critical exponents Sieberer et al. (2013). A new dynamical critical exponent has, however, recently been discovered Sieberer et al. (2013).
In equilibrium, as predicted in a seminal work Fisher et al. (1989), the presence of sufficiently strong disorder may suppress the superfluid state, and cause a transition to the Bose glass. Neglecting the gain and loss processes, a similar transition to a glass-like state was predicted Malpuech et al. (2007) for polaritons. Here we show, however, that these non-equilibrium processes play a fundamental role. We consider the experimentally relevant case of two dimensions, and show that, for a driven open condensate, static disorder destroys long-range order. Furthermore, the superfluid stiffness, as probed by the energy shift induced by twisted boundary conditions Fisher et al. (1973), vanishes in the thermodynamic limit. Thus a two-dimensional driven condensate is not formally a superfluid, except for zero disorder, although superfluid behavior would persist below a critical length scale. We identify this length scale, and the mechanism responsible for the destruction of superfluidity, below. Our results have implications both in the search for superfluidity in polariton condensates Lagoudakis et al. (2008); Sanvitto et al. (2010); Amo et al. (2009); Keeling (2011), and in the emerging study of non-equilibrium phase transitions in quantum many-body systems Sieberer et al. (2013); Chiocchetta and Carusotto (2013); Roumpos et al. (2012). Experiments on polariton condensates may involve a significant level of static disorder Franke et al. (2012), and it is therefore important to establish how disorder affects a driven condensate.
A phenomenological description of the macroscopic wave-function of a weakly interacting Bose condensate with gain and loss is the extended Gross-Pitaevskii Equation (eGPE) Wouters and Carusotto (2007); Keeling and Berloff (2008),
where , is a random potential, and the interaction strength. The second term on the right introduces driving and losses, with the net linear gain, i.e. the stimulated in-scattering rate minus the loss rate, and a nonlinearity with gain depletion parameter (see Keeling and Berloff (2008)). These terms balance for a condensate density . For we choose -correlated Gaussian disorder
with strength ; denotes the disorder average. It is convenient to introduce units of length, time, and energy, namely, the healing length , , and the blue shift , respectively. We define a dimensionless wave-function , a disorder potential with strength , and a non-equilibrium control parameter , such that in equilibrium. These parameters are
In the following we consider steady-state solutions of Eq. (1). Then, the polariton condensate emits coherent light of one frequency , and has a time-independent density (in contrast to a desynchronized regime Wouters (2008); Eastham (2008) with several frequencies). With the ansatz
we obtain coupled differential equations for the condensate density and current ,
Eq. (4) determines the condensate emission frequency (chemical potential), , while Eq. (5) is a non-equilibrium continuity equation, taking into account the coupling of the driving and losses to condensate currents. Thus, regions with and act as local sources and sinks, respectively. Since there is no net current through the boundary, the first term in Eq. (5) vanishes when integrated over space, while the second gives the constraint
where is the system volume (area).
As pointed out elsewhere Wouters and Carusotto (2010), the application of the Landau criterion to a driven condensate gives a vanishing critical velocity. Nonetheless, for the clean system superfluidity has been shown to survive Keeling (2011), if it is defined by the irrotational current response at long wavelengths Noziéres and Pines (1966). We therefore probe superfluidity in the disordered case by applying a twist of the phase between two boundaries of the condensate separated by its size in the direction . This is equivalent to a local transformation where is the twist current and satisfies periodic boundary conditions. The superfluid stiffness is then Leggett (1970); Fisher et al. (1973)
In the limit of weak disorder, we perturbatively solve Eqs. (4,5) by expanding the fields , and the frequency in powers of : and with . All disorder contributions for the frequency are of even order in . This approach does not, in general, account for vortex formation Keeling and Berloff (2008); Wouters and Savona (2010). To confirm that vortices can indeed be neglected, we have performed direct numerical simulations of Eq. (1) starting from initial conditions both with and without vortices. We find that dynamically stable, well separated vortex-antivortex pairs do, in some parameter regimes, occur, but they always significantly increase the frequency of the condensate. We consider the low-energy sector, which will be selected by thermalization processes at low temperature, and focus on solutions without vortices where the circulation, , vanishes around any closed path. The leading order solution of Eqs. (4,5) with in momentum space is
and response function . We point out that this steady-state is a stable fixed point of the dynamical system, since the excitation spectrum of a driven condensate is diffusive Wouters and Carusotto (2006); Szymańska et al. (2006); Wouters and Carusotto (2009), i.e., has both real and imaginary parts. The latter leads to an exponential decay in time for any excitation. The condensate frequency, up to quadratic order in , is
Here the second order density fluctuations were calculated using Eq. (6). Since the condensate phase is a massless mode for , its propagator behaves like at long wavelengths, leading to infrared divergences of the momentum integrals which we regularize by a finite-size cut-off at the wavevector . Note that any non-zero average of the disorder potential, , can be compensated by a shift of the frequency , see Eq. (4). Thus we may take , implying .
In the following we consider dimensions and, first, discuss the ground state properties, . Using Eqs. (8) and (10), one finds that the correlation function for density fluctuations decays exponentially, with the healing length as the decay length. Thus, density fluctuations tend to screen the disorder potential, largely uninfluenced by the driving mechanism. As discussed in Ref. Nattermann and Pokrovsky (2008), also significant is the density Larkin length , at which the energy cost of density fluctuations balances the energy gained from collective pinning in the random potential; in equilibrium, superfluidity occurs for Nattermann and Pokrovsky (2008), see Eq. (13). A strong effect of the driving appears through the result for the phase correlation function, Eqs. (9) and (11). In particular, density fluctuations generate random sources and sinks, and hence random currents, causing the phase to fluctuate and destroying long-range order in the wavefunction
Here, sub-leading contributions from density fluctuations and logarithmic finite-size corrections were neglected. The phase correlation length is , defined such that the typical phase variation over this distance is of order . This scale can also be obtained by a generalized Imry-Ma analysis Imry and Ma (1975). We integrate Eq. (5) over a region of linear size : the first term becomes the current through the region’s boundary, of order , which accounts for the non-equilibrium current generated according to the second term, of order (since at scale ). As was recently also found for the driven Jaynes-Cummings-Hubbard model Kulaitis et al. (2013), driving and potential disorder combine to act as phase disorder, destroying long-range order according to an Imry-Ma analysis. In the next step we calculate the condensate stiffness using Eq. (7), perturbatively to order ,
where we have omitted finite-size corrections vanishing for . The coefficients in this expansion are
In the equilibrium limit, , Eq. (13) reproduces previous findings Huang and Meng (1992); Meng (1994); Giorgini et al. (1994). As the disorder strength, , increases from zero the stiffness continuously reduces, before vanishing at the critical strength . In contrast, for a driven condensate, the perturbative result breaks down in the thermodynamic limit , for any non-zero disorder strength. We observe that the fastest divergence is controlled by the length scale , and below this scale the perturbative result remains finite and physical. Thus, for systems smaller than we expect superfluid behavior; however a driven disordered condensate is not a superfluid in the thermodynamic limit. Generalizing Eq. (13) to arbitrary dimensions , we find a suppression of superfluidity proportional to and, thus, expect that superfluidity is destroyed for all .
To go beyond perturbation theory, we solve the eGPE numerically on a discrete lattice of spacing . At each site, the potential is independently drawn from a Gaussian distribution of variance by using a Mersenne Twister generator. Starting from a spatially constant density and phase we evolve the eGPE until a steady-state is reached. In the parameter range studied the steady-state is stable against perturbing the initial state. For each disorder realization, we then apply twisted boundary conditions, and increasing in steps of , and evolve the eGPE to find the perturbed steady-state. The resulting frequency response fits to a quadratic function of , allowing us to extract the stiffness from Eq. (7), and we finally average over disorder realizations.
Fig. 1 shows how the stiffness obtained numerically compares with Eq. (13), for different system sizes and non-equilibrium parameters. We see that when the condensate remains stiff, , the perturbative result agrees both qualitatively and quantitatively with simulations. However, in the regime where the stiffness is strongly suppressed, the decay of deviates from the analytical prediction, even if . The suppression of superfluidity in this strong fluctuation regime is thus not accurately described by perturbation theory. Nonetheless, the divergent perturbative result suggests a mechanism controlled by which we will confirm in the following by further numerical investigations.
To this end we have studied two limiting cases. First, if the response to the twist is almost homogeneous, , and well described by perturbation theory, c.f. the discussion above. Second, if the phase response occurs in two domains, with and , separated by a randomly pinned domain wall of thickness . This behavior is shown in Fig. 2. The associated density response (not shown) involves the left (right) edge of the domain wall forming a source (sink), as described by Eq. (5). This allows a current response that is localized inside the domain wall. These results motivate the ansatz
where denotes the domain wall center, the domain wall size, and the amplitude is fixed by the twist angle. We extract from simulations by fitting to Eq. (14) in each disorder realization before averaging. As discussed above, in perturbation theory is the relevant length scale, suggesting that . This scaling is confirmed by our simulations, as shown in Fig. 3. In the parameter range used we have observed single domain walls, only. However, the formation of several walls might be possible. Nevertheless, the identified mechanism relies on the localization of the response, which remains present for several domain walls.
We also propose a scaling ansatz for the stiffness
to generalize the perturbative result into the regime of vanishing stiffness. This reproduces Eq. (13) when the exponential is expanded to first order, and takes into account that the dominant mechanism suppressing the stiffness is controlled by . The function includes logarithmic corrections and the equilibrium result, and at lowest order in perturbation theory is , c.f. Eq. (13). The simulation results, shown in Fig 4, confirm a clear data collapse with . The exponential behavior in the regime is in very good agreement with the scaling form incorporating the perturbative results for , as shown in the inset. Note that to compare with simulations we calculate the perturbative form retaining the sums over discrete wavevectors; with this infrared regularization .
Finally we propose an experiment, illustrated in Fig. 5, to measure the superfluid stiffness of the non-equilibrium polariton condensate. We note that both the emission frequency and phase profile of the condensate can be measured Kasprzak et al. (2006); Roumpos et al. (2012) while a phase twist could be imposed by driving with two coherent beams, resonant with the condensate, along either edge. In the limit of zero effective temperature, considered here, phase-locking Wouters (2008); Eastham (2008); Pikovsky et al. (2003) will pin the condensate phases at the boundaries to these beams and hence enforce a phase difference, , across the condensate. Measuring the condensate emission frequency for various twists , retuning the locking lasers appropriately, could allow the stiffness to be determined via Eq. (7). Alternatively, the phase map with the imposed phase twist , obtainable interferometrically, would show the characteristic formation of a domain wall, as in Fig. 2, when compared with the untwisted case.
In conclusion, we have found that the superfluid stiffness of a driven quantum condensate in a random potential vanishes in the thermodynamic limit for any non-zero disorder strength. In a finite system, it decays exponentially with size, , with the length scale controlling the decay. As decreases when moving away from equilibrium or the clean limit, our work shows that the universal properties of driven condensates are completely different from those of equilibrium ones, if there is any static disorder. These predictions could be tested by measuring the phase profiles and emission frequency of a polariton condensate in the presence of an imposed phase twist.
We thank A. Amo, S. Richter, R. Schmidt-Grund and M. Thunert for stimulating and helpful discussions concerning experiments. AJ is supported by the Leipzig School of Natural Sciences BuildMoNa, TH by the Dutch Science Foundation NWO/FOM and PRE by Science Foundation Ireland (09/SIRG/I1592).
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