Dynamical Quantum Phase Transitions

• Dynamical Quantum Phase Transitions are non-equilibrium phenomena characterized by a breakdown of adiabaticity and the sudden amplification of quantum fluctuations.
• They involve a universal sequence of stages—freezing, squeezing, and reheating—that emerge as the system is driven through a critical point.
• The study of DQPTs connects quantum criticality with macroscopic observables, impacting experimental condensed matter physics and quantum algorithm design.

Dynamical quantum phase transitions (DQPTs) describe sudden, universal changes in the non-equilibrium real-time evolution of a quantum many-body system, typically after sweeping an external parameter through an equilibrium quantum critical point or driving the system far from equilibrium. The haLLMark of these transitions is a breakdown of adiabaticity near a vanishing energy gap, resulting in pronounced non-analytic features in observables and in the temporal behavior of the evolving state. DQPTs reveal amplification of microscopic quantum fluctuations and entail characteristic stages—such as freezing, squeezing, and reheating—particularly in symmetry-breaking transitions. The subject establishes a deep connection between quantum criticality, universal dynamics, and observables accessible in both condensed matter experiments and quantum algorithm design (Schützhold, 2010).

1. Non-Equilibrium Dynamics and Breakdown of Adiabaticity

DQPTs are fundamentally triggered by time-dependent external parameters, such as pressure or an interaction tuning, that drive a quantum system through an equilibrium phase transition. Near the critical point, the system's energy gap $\Delta E$ between ground and excited states closes ($\Delta E \to 0$ in the thermodynamic limit), and thus the system's response time $\sim \hbar / \Delta E$ diverges. Even arbitrarily slow parameter sweeps become non-adiabatic: the corrections to adiabatic evolution, arising from admixture of excited states,

$(\psi_n | \dot{H} | \psi_0) \ll (E_n - E_0)^2$

can no longer be maintained as $(E_n - E_0) \rightarrow 0$. As a result, if a system is initially prepared in its ground state at zero temperature, it is unavoidably driven out of equilibrium when the critical point is traversed. The universal divergence of the internal response time and the collapse of the energy gap underlie the emergence of DQPT phenomena.

2. Amplification and Freezing of Quantum Fluctuations

Approaching the critical point, certain quasiparticle modes (“soft modes”) exhibit vanishing excitation energy. The dynamics of these modes “freeze” as their ability to track the external sweep fails when their characteristic frequency becomes less than the rate of change. This is captured in the adiabatic expansion of instantaneous eigenstates,

$$\Psi(t) \approx \Psi_0(t) + \sum_{n>0} \frac{(\psi_n | \dot{H} | \psi_0)}{(E_n - E_0)^2} e^{i\phi_n} \psi_n(t)$$

and in effective low-energy field theories for Goldstone modes, using a Lagrangian of the form

$$\mathcal{L}_{\text{eff}} = \frac{1}{2}\left[ \frac{1}{\alpha(g_{-}(t))} \dot{\Phi}_{-}^2 - \beta(g_{-}(t)) \Phi_{-}^2 \right]$$

where $\alpha$ and $\beta$ evolve with the control parameter, and the mode velocity $c(t) = \sqrt{\alpha \beta}$ vanishes at criticality. The maximum spatial extent over which quantum fluctuations remain correlated—the “horizon size”—is given by

$\Delta r(t) = \int_t^{\infty} c(t') dt'$

As $c(t)\rightarrow 0$, correlations can be established only up to a finite $\Delta r(t)$. Quantum fluctuations initially present in the ground state are strongly amplified (by squeezing) and become observable, especially post-criticality (“reheating”), when these frozen-in fluctuations dictate large-scale dynamics.

3. Symmetry-Breaking and Topological Defect Formation

In symmetry-breaking (or restoring) transitions—such as in spinor Bose–Einstein condensates with quadratic Zeeman shift, or general models with continuous symmetry—the system undergoes an instability in a symmetric state (zero order parameter) as the control parameter is swept through the critical point. The post-critical potential landscape exhibits degenerate minima (e.g., in magnetization space), and the direction of the emergent order parameter is selected by the amplified quantum fluctuations. The result is a macroscopic, random direction of symmetry breaking determined by these fluctuations, leading to the spontaneous formation of domains and topological defects (e.g., vortices). The density and scaling of defects can be predicted from the power spectrum of the amplified fluctuations, linking the dynamics to universal scaling laws as in the Kibble–Zurek scenario (Schützhold, 2010).

4. Universal Dynamical Features: Freezing, Squeezing, Reheating

Multiple quantum systems, regardless of microscopic details, exhibit a sequence of universal dynamical stages across the transition:

• Cooling: Initial adiabatic decrease of excitation.
• Freezing: Soft modes fail to follow external sweep; their amplitudes become fixed over domains of size $\Delta r(t)$.
• Squeezing: Frozen fluctuations are amplified, leading to macroscopic observability.
• Reheating: As the gap re-opens post-transition, non-adiabatic excitations “defrost”, and the energy initially stored in quantum fluctuations is released.

These signatures are observed in a broad class of second-order quantum phase transitions, including both symmetry-breaking and certain non-symmetry-breaking transitions. The underlying mechanism—freezing due to critical slowing down—draws a formal analogy with horizon formation during cosmic inflation, where quantum fluctuations are also frozen at super-horizon scales.

5. Representative Physical Examples

The general principles are exemplified by diverse systems:

• Two-component Bose–Einstein condensates: Driving the inter-component coupling from miscible to immiscible leads to instability of the soft “spin” mode and triggers phase separation. The evolution follows the universal cooling–freezing–squeezing–reheating sequence, with frozen spin fluctuations setting the structure of domains.
• Bose–Hubbard model: Sweeping the tunneling rate from superfluid to Mott insulator induces a second-order transition. The phase mode (Goldstone) freezes due to a shrinking horizon, leading to “number squeezing”; the final magnitude of density fluctuations is set by the quench rate.
• Extended Bose–Hubbard model: Introducing long-range interactions creates a “roton” minimum in the excitation spectrum, producing instability against density wave (supersolid) order. The dominant length scale for modulations is fixed by the wavevector of the roton minimum.
• Spinor condensates (S=1): Under a sweep of quadratic Zeeman energy, a transition from a paramagnetic to a ferromagnetic phase occurs. The random direction and scaling of magnetization and winding number are consequences of amplified vacuum fluctuations.

The principles transcend condensed matter, influencing quantum computation via adiabatic algorithms, where bottlenecks analogous to critical points can result in non-adiabatic excitations and limit performance.

6. Scaling Theory and Characteristic Length/Time Scales

DQPT phenomena are governed by scaling arguments rooted in the vanishing gap and divergent correlation length. The characteristic correlation length $\xi$ and associated timescale $\tau$ are set by the time-dependence of the control parameter and dispersion relation $\omega^2(k) = m^2 + c^2 k^2 + O(k^4)$, where instabilities may be driven by the mass term $m^2$ or stiffness $c^2$. The final domain size or defect density often exhibits a universal dependence on the sweep rate, in agreement with Kibble–Zurek predictions.

7. Broader Impact and Applications

The amplification of quantum fluctuations and the associated universal non-equilibrium phenomena have significant implication for:

• Creation of topological defects: Predicting and controlling defect density based on sweep protocols.
• Initiating macroscopic order from microscopic quantum noise: Relevant for quantum simulation and “quantum seeding” of symmetry breaking.
• Adiabatic quantum computation: Understanding critical slowing down and non-adiabatic corrections helps in optimizing quantum annealing or adiabatic optimization algorithms, where quantum phase transitions set fundamental speed limits.
• Design of measurement protocols: As the amplified fluctuations become large-scale, they can be probed experimentally even in cold atom and condensed matter setups.

In summary, dynamical quantum phase transitions constitute a universal class of non-equilibrium behavior in quantum many-body systems subjected to parameter sweeps through criticality. They are characterized by breakdown of adiabaticity, subsequent amplification of quantum fluctuations, and universal features such as freezing, squeezing, and the emergence of an effective horizon scale. These phenomena play a crucial role in a variety of contexts, from condensed matter to quantum information processing (Schützhold, 2010).