Nonequilibrium functional renormalization for driven-dissipative Bose-Einstein condensation

We present a comprehensive analysis of critical behavior in the driven-dissipative Bose condensation transition in three spatial dimensions. The starting point is a microscopic description of the system in terms of a many-body quantum master equation, where coherent and driven-dissipative dynamics occur on an equal footing. An equivalent Keldysh real-time functional integral reformulation opens up the problem to a practical evaluation using the tools of quantum field theory. In particular, we develop a functional renormalization group approach to quantitatively explore the universality class of this stationary nonequilibrium system. Key results comprise the emergence of an asymptotic thermalization of the distribution function, while manifest nonequilibrium properties are witnessed in the response properties in terms of a new, independent critical exponent. Thus, the driven-dissipative microscopic nature is seen to bear observable consequences on the largest length scales. The absence of two symmetries present in closed equilibrium systems—underlying particle number conservation and detailed balance, respectively—is identified as the root of this new nonequilibrium critical behavior. Our results are relevant for broad ranges of open quantum systems on the interface of quantum optics and many-body physics, from exciton-polariton condensates to cold atomic gases.

Figure 1

Emergence of universality. (a) We show the flow of the complex renormalized two-body coupling ${\tilde{u}}_2=\tilde{\lambda }+i\tilde{\kappa }$ (see Sec. 8a) for various initial values ${\tilde{u}}_{2\Lambda }$. As a result of fine tuning the initial values $w_{\Lambda }$ of the dimensionless mass parameter close to criticality, all flow trajectories approach the Wilson-Fisher fixed point ${\tilde{u}}_{2*}=i5.308$ (indicated by the black dot) before eventually bending away. Shown are numerical solutions to the flow equations for $r_{K\Lambda }=10,r_{u_3\Lambda }=1,{\tilde{\kappa }}_{3\Lambda }=0.01$ and values of ${\tilde{u}}_{2\Lambda }$ lying on a rectangle with sides $\tilde{\lambda }\in [0,10],\tilde{\kappa }=2,10$ and $\tilde{\lambda }=10,\tilde{\kappa }\in [2,10]$. (See Sec. 8a for definitions of the parameters.) (b) Flow of $\tilde{\kappa }$ as a function of the dimensionless infrared cutoff $t=ln(k/\Lambda )$ for various starting values ${\tilde{\kappa }}_{\Lambda }$. Dots on the lines indicate the extent of the critical domain, which is set by the Ginzburg scale Eq. (3). Initial values are the same as in (a), apart from ${\tilde{\kappa }}_{\Lambda }=0.1,1,2,...,10$ and $r_{u_2\Lambda }=10$.

Figure 2

Scale dependence at criticality of the effective temperature $T_{\mathrm{eff}}=\overline{\gamma }/\left(4\right|Z\left|\right)$, where $\overline{\gamma }$ denotes the Keldysh mass and $Z$ is the wave-function renormalization (see Secs. 5b and 8a). For $t\to -\infty$ the effective temperature saturates to a constant value. Initial values are the same as in Fig. 1.

Figure 3

Illustration of the observability of the drive exponent. The absorption peak for a measurement that observes the single-particle dynamical response. The drive exponent ${\eta }_r$ reveals itself in a different scaling of the peak location and peak width as a function of the momentum.

Figure 4

Dispersion relation of single-particle excitations in the ordered phase. Frequencies and momenta are measured in units of the healing length $\xi =1/\sqrt{\lambda {\rho }_0}$. (a),(b) The Goldstone and massive modes Eq. (30), obtained in mean-field approximation, are shown as, respectively, dashed and solid lines for $\kappa =\lambda /2$. For small momenta both modes are purely diffusive and nonpropagating. The dotted lines in (a) correspond to the usual Bogoliubov dispersion relations for $\kappa =0$. (c),(d) Dispersion relations Eq. (54) with gradient coefficients $A,D$ that are generated upon renormalization. (d) For finite $D$, the damping rate grows $\propto$$q^2$ for large $q$. The regularized dispersion relations, where $a{q}^2$ is replaced with $p_a\left(q^2\right)$ for $a=A,D$ [cf. Eq. (58)], are shown as a dash-dotted lines. Here we chose parameters $A=1,D=1/2,\kappa =\lambda ,k=1/\left(2\xi \right)$.

Figure 5

Visualization of the renormalization with $Z$. (Left column) Bare couplings. (Right column) Renormalized couplings. The renormalization of a complex coupling $\overline{g}$ corresponds to a rescaling $\left|g\right|=|\overline{g}|/|Z|$ of the modulus and a rotation of the phase by the argument of $Z$, $argg=arg\overline{g}-argZ$. (a) When all bare couplings lie on a single ray that is perpendicular to $Z$, the renormalized couplings are purely imaginary as in MA. (b) Deviations from the right angle incorporate MA with compatible reversible mode couplings. (c) In a generic nonequilibrium situation there is no fixed relation between the arguments of the various couplings.

Figure 6

Spectral density, Eq. (128), in the scaling regime. The solid line corresponds to the position $\sim$$A_0{q}^{2-{\eta }_A}$ of the peak of the Lorentzian curve while its width $\sim$$D_0{q}^{2-{\eta }_D}$ is indicated by dashed lines. For comparison we also show peak position and width for canonical scaling $\sim$$q^2$ as thin solid and dashed lines, respectively. (Canonical and anomalous scaling forms are chosen to coincide at $q\Lambda =0.1$.) Parameters are $A_0{\Lambda }^{{\eta }_A}=Z_0{\Lambda }^{{\eta }_Z}=1$ and $D_0{\Lambda }^{{\eta }_D}=1/2$.

Figure 7

(a) The flow of $c_1$ (solid line) describes the vanishing of coherent dynamics. A fit with $ln{c}_1=a_{1}t+b_1$ in the region $t\in [-24,-20]$ (the points $t=-24$ and $t=-20$ are highlighted by dots on the trajectory) yields the slope $a_1=0.10$ in agreement with smallest eigenvalue $n_1=-{\eta }_r$ of the stability matrix, Eq. (110). We also show the evolution of the coefficient $c_2$ (dashed line). For the evolution of $c_2$, the slope of a linear fit is $a_2=0.14$ and reproduces the eigenvalue $n_2$. In the scaling region, the coefficient $c_3$ drops to very small values $\lesssim {10}^{-11}$ on a scale $1/n_3\approx 0.6$. The exponential decay of the components of $\mathbf{r}$ is in this range still dominated by the contribution stemming from $c_2$. (b) The couplings $10w$ (solid line), $\tilde{\kappa }$ (dashed line), and ${10}^{-1}{\tilde{\kappa }}_3$ (dot-dashed line) are close to their fixed point values in the range from $t\approx -5$ to $t\approx -25$. A measure for the extent of the universal domain is given by the Ginzburg scale, Eq. (136), which here takes the value $t_G\approx -3.4$. Initial conditions for both (a) and (b) are $r_{K\Lambda }=r_{u\Lambda }=10$, $r_{u_3\Lambda }=1$, $w_{\Lambda }\approx 0.05810,\tilde{\kappa }=0.5,$ and ${\tilde{\kappa }}_3=0.01$.

Figure 8

Anomalous dimensions ${\eta }_{\mathit{ZR}}$ (solid line), ${\eta }_{\mathit{ZI}}$ (dashed line), and $-{\eta }_{\gamma }$ (dot-dashed line) for the solution of Fig. 7. From $t\approx -5$ to $t\approx -25$, where the values of $\mathbf{s}$ are close to the scaling solution, ${\eta }_{\mathit{ZR}}$ takes the constant value, Eq. (114), while ${\eta }_{\mathit{ZI}}$ decays to zero. The value of $-{\eta }_{\gamma }$ approaches the one of ${\eta }_{\mathit{ZR}}$ so that Eq. (114) is satisfied at late “times” $t$. Eventually, as the trajectory is driven away from the fixed point and enters the symmetric phase with $w=0$, the anomalous dimensions drop to zero.

Figure 9

Equilibrium vs nonequilibrium flow. (a) As discussed in Sec. 6d, in thermodynamic equilibrium all couplings lie on a single ray in the complex plane. (b) This geometric constraint is absent out of equilibrium. We show $g=10K,\tilde{u},$ and ${10}^{-1}{\tilde{u}}_3$ as solid, dashed, and dot-dashed lines, respectively. Stages of the flow at $t=0,-8,$ and $-16$ are indicated with points on the trajectories. Initial values are (a) $r_{\Lambda }=10,w_{\Lambda }=0.01281$ and (b) $r_{K\Lambda }=10,r_{u\Lambda }=5,r_{u_3\Lambda }=1,w_{\Lambda }=0.01264$. In both cases we have ${\tilde{\kappa }}_{\Lambda }=0.1,{\tilde{\kappa }}_{3\Lambda }=0.01,$ and $K_{\Lambda }=1+i0.1$.