Driven-Dissipative Bosonic Condensation

• Driven-dissipative bosonic condensation is a non-equilibrium phase transition where weakly interacting bosons form macroscopic coherent states under competing gain and loss processes.
• The Keldysh field-theoretic formulation and driven–dissipative Gross–Pitaevskii equation reveal key static and dynamic correlations, distinguishing these states from equilibrium condensates.
• Experimental platforms such as exciton-polariton microcavities, Josephson arrays, and cold atoms validate theoretical predictions, including transitions from unique to fragmented condensates.

Driven-dissipative bosonic condensation describes the formation of macroscopically occupied, phase-coherent states in non-equilibrium quantum systems of weakly interacting bosons, where gain and loss processes fundamentally affect the nature of the condensed state, its correlations, and its superfluidity. In contrast to equilibrium Bose–Einstein condensates (BECs), here the competition between coherent Hamiltonian evolution and external reservoirs (pump, loss) leads to genuinely non-equilibrium steady states, with properties that can differ dramatically from their equilibrium analogs depending on spatial dimension, anisotropy, and the microscopic structure of driving and dissipation. Such condensation is realized in systems including exciton-polariton microcavities, photonic condensates, Josephson arrays, and ultracold atomic platforms with engineered reservoirs.

1. Microscopic Models and Field-Theoretic Formulation

The prototypical driven-dissipative bosonic model consists of a weakly interacting Bose field $\psi(\mathbf{r})$ of mass $m$, governed by the Hamiltonian

$$H \;=\; \int d^d r\;\left[\psi^\dagger(r)\left(-\frac{\nabla^2}{2m}\right)\psi(r) + \frac{U}{2}\psi^\dagger(r)\psi^\dagger(r)\psi(r)\psi(r)\right]$$

and coupled to external Markovian reservoirs, inducing single-particle loss at rate $\kappa$, incoherent single-particle pump at rate $\gamma$, and two-body losses at rate $\Gamma$. The full quantum dynamics is encoded by the Lindblad master equation: $$\partial_t\rho = -i[H,\rho] + \int d^dr \left\{ \frac{\kappa}{2} \mathcal{L}[\psi] + \frac{\gamma}{2} \mathcal{L}[\psi^\dagger] + \frac{\Gamma}{4}\mathcal{L}[\psi^2]\right\}$$ where $$\mathcal{L}[X]\rho = 2 X\rho X^\dagger - \{X^\dagger X, \rho\}$$ generates quantum jumps.

The Keldysh functional integral provides a powerful field-theoretical approach, with the partition function written as a path integral over classical ($\psi_C$) and quantum ($\psi_Q$) fields. The effective action has the form

$$S[\psi_C, \psi_Q] = \int dt\,d^dr \left\{ \begin{pmatrix} \bar\psi_C & \bar\psi_Q \end{pmatrix} \begin{pmatrix} 0 & [D^A_0]^{-1}\ [D^R_0]^{-1} & [D_0^{-1}]_K \end{pmatrix} \begin{pmatrix} \psi_C\ \psi_Q \end{pmatrix} -\left[ \left(\frac{U}{2} + i\frac{\Gamma}{4}\right)(\bar\psi_C^2+\bar\psi_Q^2)\psi_C\psi_Q + \mathrm{c.c.}\right] + i\Gamma \bar\psi_C\psi_C\bar\psi_Q\psi_Q \right\}$$

with non-Hermitian terms encoding pump, loss, and quantum noise (Keeling et al., 2016).

In the large-occupancy (semiclassical) and weakly nonlinear limit, the driven–dissipative Gross–Pitaevskii equation (DDGPE) emerges: $$i\,\partial_t\varphi = \left[ -\frac{\nabla^2}{2m} + U|\varphi|^2 + i\frac{\gamma-\kappa-\Gamma|\varphi|^2}{2} \right]\varphi$$ which provides the baseline for both mean-field and field-theoretic analyses of condensation.

2. Static and Dynamical Correlation Functions

The nature of order in driven-dissipative condensates is extracted from the two-point function and linear response properties. By parameterizing $\psi \approx \sqrt{\rho_0} e^{i\theta(r)}$ in the condensed regime, the spatial correlation decays at long distances ($r \gg \xi_0$) as

$$G(r) = \langle \psi^*(r) \psi(0)\rangle \sim r^{-(d-2+\eta)}$$

with $\eta = \Delta_{\text{eff}}/(4\pi D_{\text{eff}})$ determined by the renormalized effective noise and phase-diffusion constant. In three dimensions, $\eta \to 0$ at large scales, reflecting emergent equilibrium-like static order (Keeling et al., 2016).

The dynamical retarded response function exhibits poles at complex Bogoliubov modes

$$\omega_k = -i\frac{\gamma_{\rm net}}{2} \pm \sqrt{\xi_k^2 - \left(\frac{\gamma_{\rm net}}{2}\right)^2}$$

with $$\xi_k^2 = \frac{k^2}{2m}\left(\frac{k^2}{2m} + 2U\rho_0\right)$$, indicating that at low $k$, excitations become diffusive: $\omega \sim -i D k^2$. The dynamical exponent $z$ governs relaxation, with $z \approx 2$ in equilibrium and possible non-equilibrium values under drive.

3. Renormalization Group and Dimensional Crossover

Long-wavelength phase dynamics of driven-dissipative condensates are governed by a stochastic KPZ-type equation: $$\partial_t\theta = D_x \partial_x^2 \theta + D_y \partial_y^2 \theta + \frac{1}{2} (\lambda_x \partial_x\theta)^2 + \frac{1}{2} (\lambda_y \partial_y\theta)^2 + \eta$$ with noise correlator $$\langle \eta\eta \rangle = 2\Delta\delta(r)\delta(t)$$. The RG flow equations in 2D for the dimensionless nonlinearity $g$ and anisotropy $\Phi$ are

$$\frac{dg}{dl} = \frac{g^2}{32\pi}(\Phi^2 + 4\Phi - 1) \qquad \frac{d\Phi}{dl} = \frac{g\Phi}{32\pi}(1 - \Phi^2)$$

(Keeling et al., 2016). In 3D, fluctuations render $g$ irrelevant: the system flows to an equilibrium-like fixed point for static correlations, with non-equilibrium effects confined to dynamical criticality (emergent Model A exponents). In isotropic 2D ($\Phi > 0$), $g \to \infty$ and algebraic order is washed out, with only stretched-exponential or exponential decay remaining: the paradigmatic Berezinskii–Kosterlitz–Thouless (BKT) phase is destroyed by drive/dissipation.

Anisotropy ($\Phi < 0$) can stabilize an effective equilibrium line with $g \to 0$: under strong anisotropy, algebraic (quasi-long-range) order and BKT-like scaling are recovered, though the order parameter exponent $\eta(\Phi)$ is shifted by the effective noise and stiffness.

4. Superfluid Response and the Landau Criterion

The superfluid density is obtained from the longitudinal and transverse current–current response as

$\rho_s = m (\chi_L - \chi_T)$

In the driven-dissipative Keldysh action, $\rho_s$ remains finite within the validity of the linearized theory, due to persistent Goldstone-like stiffness (suppressed only by $q^2 D^R(q, 0) \sim \mathrm{const}/q^2$). The normal density never vanishes, reflecting excitations created by drive-noise.

The conventional Landau criterion is softened: in equilibrium, drag on a defect vanishes strictly for $v < c_s$ (sound velocity), while for driven-dissipative condensates, the diffusive nature of the Goldstone mode implies that drag is suppressed for $v < c_s$ but does not vanish entirely; above threshold, it increases sharply.

5. Non-Equilibrium Bose Selection and Fragmented Condensation

For ideal (noninteracting) bosons in a non-equilibrium steady state, the system may not simply macroscopically occupy a unique ground state, but instead exhibit generalized "Bose selection": an odd number $M_S$ of single-particle modes can acquire macroscopic populations. This is determined by a criterion on the rate-imbalance matrix $A_{ij} = R_{ij} - R_{ji}$: $$\sum_{j \in S} (R_{ij} - R_{ji}) \bar n_j = 0, \quad i \in S$$ where $S$ is the set of selected states, $|S|=M_S$ odd (Vorberg et al., 2013, Schnell et al., 2018, Vorberg et al., 2015). In many physical models, $M_S=1$ (conventional condensation), but with multi-bath or driven settings $M_S>1$ ("fragmented" non-equilibrium condensate), and in random-rate settings $M_S$ can scale extensively with system size.

Macroscopic population of multiple selected states (fragmented BEC) underlies phenomena such as heat conduction switches, where transitions between regimes with different $M_S$ lead to abrupt changes in transport properties.

6. Experimental Signatures and Physical Realizations

Driven-dissipative bosonic condensation is realized in platforms including semiconductor polariton microcavities, cold atomic gases with engineered loss/gain, Josephson junction arrays, and photonic microcavities. Key experimental observables:

• Spatial and temporal coherence, via interferometric measurements: $G(r) = \langle \psi^*(r) \psi(0)\rangle$ measured by real-space or Michelson interferometry, with algebraic/stretched-exponential/exponential decay indicating the underlying phase.
• Extraction of the exponent $\eta$ from fringe visibility $V(r) \sim r^{-\eta}$.
• Tuning pump power enables observation of crossovers from apparent algebraic order to its breakdown, reflecting the finite KPZ crossover length $L_* \sim \xi_0 \exp(8\pi/g_0)$.
• Synthetic rotation (Berry curvature engineering) and response measurements (e.g., four-wave mixing, pump–probe) provide direct access to current–current correlations and thus superfluid density.
• Measuring abrupt transitions in transport (e.g., heat flow in tailored tight-binding chains connected to multiple baths) confirms the predicted "Bose selection" scenario (Vorberg et al., 2013).

7. Summary: Dimensionality, Universality, and Open Questions

Driven-dissipative bosonic condensation exhibits a rich hierarchy of steady-state order and universality classes:

• In 3D, static correlations renormalize to equilibrium-like behavior, though dynamical scaling contains new non-equilibrium exponents.
• In 2D, isotropic drive/dissipation generically destroys algebraic order; strong spatial anisotropy can restore BKT-like scaling.
• Superfluidity persists at the linearized level, but with diffusive excitations softening the Landau critical velocity threshold.
• "Bose selection" can drive a transition from unique-mode condensation to fragmented condensation, with significant impact on transport.
• Realizations in microcavity polaritons, small-$N$ photon BECs, and Josephson junction arrays confirm theoretically predicted thresholds and fragmentation (Walker et al., 2017, Reeves et al., 2021).

Future challenges include full characterization of the fate of Bose selection under interactions and disorder, the impact of strong quantum fluctuations beyond semiclassical or mean-field regimes, and experimental mapping of the universal crossover between equilibrium, KPZ-dominated, and fully non-equilibrium universality classes (Keeling et al., 2016, Tauber et al., 2013, Sieberer et al., 2013, Sieberer et al., 2013).