A potent theoretical tool for examining phase transitions in systems subject to open quantum dynamics is the Quantum Contact Process (QCP).
What is QCP?
Combining quantum coherent and classical incoherent processes is the basic description of the Quantum Contact Process (QCP). It functions as an absorbing state-containing, archetypal, non-equilibrium quantum system. The Contact Process, the classical counterpart, is a classical stochastic system that belongs to the Directed Percolation (DP) universality class and is renowned for exhibiting critical behaviour. The main focus of QCP is on how this transition is different from the classical limit when quantum fluctuations originating from coherent spin-flips are present.
Coherent quantum fluctuations and incoherent classical fluctuations interact to produce the phase transitions within QCP, which are referred to as quantum absorbing phase transitions (QAPT). There are parameters that enable the relative contributions of the quantum coherent effects against the classical incoherent effects to be adjusted in the Lindblad formalism, which characterises the system’s dissipative nature.
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The Quantum Absorbing Phase Transition (QAPT)
A phase transition to an absorbing state occurs in the QCP. Once the system enters an absorbing state, it cannot exit this fluctuationless arrangement. This transition is usually continuous and falls into the DP universality class in the classical regime. The character of this transition might alter, though, if quantum coherent effects are added.
For example, research indicates that Directed Percolation could not be the QCP’s universality class. The shift from second-order (continuous) to first-order (discontinuous) can be altered by the addition of quantum fluctuations. The existence and nature of the absorbing state phase transition have been the subject of much discussion. Researchers have used real-time numerical simulations to provide estimates for critical exponents and evidence that the transition is continuous in some regimes.
Dependence on Interaction Range and Geometry
One important discovery about the QCP is that the contact range of the components has a major impact on the type of Quantum Absorbing Phase Transition (QAPT).
Short-Range Interactions: The majority of QCP research has focused on scenarios in which the s-excited state of Rydberg atoms induces active states. In this case, short-range interactions create quantum coherence. This arrangement usually produces a second-order QAPT (a continuous transition) in one dimension.
Long-Range Interactions: Long-range interactions need to be taken into consideration when atoms’ d-excited states induce active states. Even in one dimension, a discontinuous transition (first-order) may transpire in this long-range interaction scenario. The mean-field phase diagram is comparable to the nearest-neighbor QCP in long-range QCP models, where branching and coagulation processes occur over vast distances. In the weak quantum regime, the transition is continuous, but in the strong quantum domain, it is discontinuous.
Embedded Structures: By examining the QCP model on scale-free (SF) networks, researchers have also looked into how the QAPT is influenced by the system’s embedded structure. The average distance between nodes in scale-free networks varies logarithmically with system size, indicating their extreme heterogeneity. Researchers may ascertain how the QAPT is influenced by both the interaction range and the heterogeneity of connections by examining the QCP on these networks. They discover that several QAPT kinds arise based on the network’s degree exponent.
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Dimensionality and Crossover Behavior
The behaviour of the QCP is also significantly influenced by the system’s dimensionality and initial configuration.
One Dimension (1D): In one dimension, the QCP displays intricate crossover events. A critical exponent constantly drops from a quantum value towards the classical Directed Percolation (DP) value along a critical line that emerges when the process begins in a homogeneous state (all active sites are initially active). This implies that there is still some influence from the quantum coherent effect close to the boundary condition.
On the other hand, the crucial exponents match the traditional DP values if the QCP in one dimension begins from a heterogeneous state (just one site initially active). When long-range interactions are present, this continuously changing exponent behaviour is comparable to what is seen in the classical contact process.
Two Dimensions (2D): This intricate, unusual crossover behaviour does not exist in two dimensions. Instead, regardless of the original configuration, classical DP behaviour is observed throughout the parameter domain. However, the system may still exhibit a unique type of non-equilibrium phase transition, depending on whether classical or quantum processes predominate in a 2D lattice.
Complex Universality Classes
Highly intricate universality of the QCP can be observed in intermediary regimes where quantum dynamics and classical dynamics compete. According to the semiclassical method, the quantum coherence effect is a process in which two successive atoms help to excite a nearby atom.
There may be a tricritical point in this situation as a result of both first-order and second-order transitions. The Tricritical Directed Percolation (TDP) class frequently includes this bicritical point. Additionally, a new universality class has been discovered at the tricritical point for the long-range QCP model, which is different from the tricritical point of the classical DP model with long-range interactions as well as the nearest-neighbor QCP.
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Study Methods and Physical Context
Because of its relative simplicity, the QCP is a study topic and a perfect benchmark problem to examine numerical techniques for open quantum non-equilibrium systems. Numerous sophisticated numerical simulation methods are used, such as:
- The Monte Carlo method for quantum jumps.
Methods for Tensor networks.
Product states in a matrix. - The algorithm for time-evolving block decimation.
- Neural network, machine learning, which is utilised to find important exponents and the crucial line.
The underlying physical background frequently has to do with highly adjustable systems of atoms stimulated to Rydberg states or interacting cold gases. The QCP and related models of population dynamics or disease propagation are physically realised through experiments in strongly interacting gases containing highly excited Rydberg atoms, where aided excitation competes with radiative decay.
