Study Rigorously Proves Noisy Quantum Quench Dynamical Phase Transitions Ruled Out for Mixed States

The behaviour of quantum systems undergoing rapid change, and whether these changes exhibit predictable patterns, remains a central question in modern physics. J. Sirker rigorously challenges recent claims regarding dynamical quantum phase transitions, specifically those found in noisy systems, by demonstrating a fundamental constraint on the conditions under which these transitions can occur. The research proves that the existence of these transitions relies on a system being in a pure quantum state, a condition incompatible with the averaging over noise typically used in these investigations. This finding fundamentally alters the understanding of dynamical transitions in noisy environments and demonstrates that previously reported transitions are, in fact, smoothed out by the presence of noise, offering a crucial correction to the field.

Ramped Field Reveals Smoothed Quantum Transitions

Scientists rigorously investigated dynamical quantum phase transitions (DQPTs) within a two-band model subject to noise, challenging previously reported findings. The research focused on the Loschmidt echo, a measure of how a quantum state overlaps with itself over time, and its sensitivity to noise-induced changes in quantum systems. The team developed a theoretical framework demonstrating that noise fundamentally alters the system’s evolution, smoothing out any potential abrupt changes indicative of DQPTs. This work involved a detailed analysis of the Lindblad master equation, a mathematical tool describing how density matrices, which represent quantum states, evolve in open quantum systems.

The study began by preparing the system in a pure ground state with an initial magnetic field, then subjecting it to a linearly increasing magnetic field accompanied by classical Gaussian white noise. This noise, characterized by its random fluctuations, was incorporated into the master equation to model the system’s interaction with its environment. The team then calculated the noise-averaged density matrix using the master equation, effectively simulating the system’s evolution under the combined influence of the changing magnetic field and the environmental noise. This calculation formed the basis for examining the Loschmidt echo between the noise-averaged state and the state evolving under a constant magnetic field.

Crucially, the team proved two key theorems. The first theorem demonstrates that the solution to the Lindblad master equation is invariably a mixed state, meaning the system loses coherence due to the noise. This is established by showing that the purity of the state, a measure of its coherence, decreases over time, ultimately leading to a completely mixed state. The second theorem rigorously proves that in a two-dimensional quantum system, the Loschmidt echo can only be zero if both the initial and final states are pure, effectively ruling out the existence of DQPTs in the presence of non-zero noise. This is because a zero Loschmidt echo requires the initial and final states to be orthogonal, a condition only met when both states are pure. The research conclusively demonstrates that noise fundamentally alters the system’s dynamics, preventing the emergence of the previously reported DQPTs and establishing a new understanding of quantum phase transitions in noisy environments.

Noise Destroys Dynamical Quantum Phase Transitions

This work presents a rigorous analysis of dynamical quantum phase transitions (DQPTs), challenging recent findings that suggest DQPTs persist even in the presence of noise. Researchers proved a critical theorem demonstrating that the solution to the Lindblad master equation, used to describe the system’s evolution under noise, invariably results in a mixed state, unless trivial conditions are met. This finding directly contradicts the claim that noise-averaged density matrices can exhibit the purity required for DQPTs to occur. The study focuses on a system subjected to a linear increase in a magnetic field, coupled with classical Gaussian white noise.

By examining the time evolution of the system’s density matrix, scientists derived a key equation demonstrating that the purity of the state, a measure of its mixed or pure character, decreases over time due to the influence of the noise. This decrease in purity is mathematically proven, establishing that the noise-averaged state is fundamentally mixed, precluding the possibility of DQPTs. Further analysis revealed that even when using alternative methods to average over noise realizations, the abrupt changes indicative of DQPTs are consistently smoothed out. The team rigorously demonstrated that the existence of DQPTs requires a pure state, a condition that is demonstrably not met when noise is present and the system is governed by the Lindblad master equation. These findings effectively rule out the existence of the novel dynamical phases, identified in previous work, that were proposed to be created by noise, and invalidate the associated dynamical phase diagram. The work establishes a firm theoretical foundation for understanding DQPTs and clarifies the conditions under which they can, and cannot, occur in noisy quantum systems.

Noise Averaging Eliminates Dynamical Phase Transitions

This research rigorously demonstrates that dynamical quantum phase transitions (DQPTs), previously suggested to persist even with noise, are in fact smoothed out by noise averaging in two-band models. Scientists proved that a properly evaluated Loschmidt return rate, a key indicator of DQPTs, cannot exhibit abrupt changes when noise is present, effectively disproving the earlier claim. The team established general theorems showing that noise averaging replaces a pure state with a mixed state, fundamentally altering the expected behaviour and eliminating the possibility of DQPTs. The investigation considered various noise averaging protocols, consistently finding that while individual noise realizations might show DQPT-like features, these are not coherent and therefore disappear upon averaging.

This work confirms that previous understandings of DQPTs in the presence of finite temperatures or dissipation remain valid, reinforcing the principle that averaging over noise consistently leads to smoother, more predictable quantum behaviour. While the research focuses on two-band models, scientists note that DQPTs may be possible in more complex, multi-band systems, though likely only under specific, finely tuned conditions. The authors acknowledge that their findings correct a potentially misleading claim regarding the robustness of DQPTs in noisy environments. This clarification is particularly important given the potential implications for quantum information applications, where understanding the effects of noise is crucial for developing stable and reliable quantum technologies. Future research may explore the conditions under which DQPTs could emerge in more complex systems, but this work establishes a firm theoretical foundation for understanding their behaviour in the presence of noise.