Fock-state lattices (FSLs) are becoming an emerging research hotspot in quantum physics, not only because the FSLs provide a new perspective for studying atom-field interactions, but also because they build a connection between quantum optics and condensed matter physics. Because of the multiple transition paths in the lattices, an inherent quantum interference effect exists in these systems, and hence finding new quantum-coherent phenomena and exploiting their applications become a significant and desired task in this field. In this work, we study the dark-state effect in the FSLs generated by the multimode Jaynes-Cummings (JC) models. By considering the FSLs in certain excitation-number subspaces, we study the dark states with respect to the states associated with the atomic excited state using the arrowhead-matrix method. We find that dark-state subspaces exist with dimensions determined by the number of orthogonal dark states. When the dimension is greater than one, the forms of these dark-state bases are not unique. Further, we obtain the number and form of the orthogonal dark states in the two-, three-, and four-mode JC models. In addition, we find that for a general N -mode JC model, there are C N + n − 2 N − 2 orthogonal dark states in the n -excitation subspace. We also build the relationship between the dark states and dark modes. Our work paves the way for exploring quantum optical effects and quantum information processing based on the FSLs.
Fock-state lattices (FSLs) are becoming an emerging research hotspot in quantum physics, not only because the FSLs provide a new perspective for studying atom-field interactions, but also because they build a connection between quantum optics and condensed matter physics. Because of the multiple transition paths in the lattices, an inherent quantum interference effect exists in these systems, and hence finding new quantum-coherent phenomena and exploiting their applications become a significant and desired task in this field. In this work, we study the dark-state effect in the FSLs generated by the multimode Jaynes-Cummings (JC) models. By considering the FSLs in certain excitation-number subspaces, we study the dark states with respect to the states associated with the atomic excited state using the arrowhead-matrix method. We find that dark-state subspaces exist with dimensions determined by the number of orthogonal dark states. When the dimension is greater than one, the forms of these dark-state bases are not unique. Further, we obtain the number and form of the orthogonal dark states in the two-, three-, and four-mode JC models. In addition, we find that for a general N -mode JC model, there are C N + n − 2 N − 2 orthogonal dark states in the n -excitation subspace. We also build the relationship between the dark states and dark modes. Our work paves the way for exploring quantum optical effects and quantum information processing based on the FSLs.
We analyze a set of models frequently appearing in quantum optical settings by expressing their Hamiltonians in terms of Fock-state lattices (FSLs). The few degrees-of-freedom of such models, together with the system symmetries, make the emerging FSLs relatively simple such that they can be linked to known lattice models from the condensed-matter community. Thus, the FSLs may shed new light on known quantum optical systems. While we provide a rather long list of models and their corresponding FSLs, we pick a few to demonstrate the method's strength. The three-mode boson model, for example, is shown to display a fractal spectrum and chiral evolution in the FSL characterized by localized distributions traversing along symmetric trajectories. In a second example, we consider the central spin model, which generates an FSL reminiscent of the Su-Schrieffer-Heeger model hosting topological edge states. We further demonstrate how the phenomenon of flat bands in lattice models can manifest in related FSLs, which can be linked to so-called dark states.
The dark-state effect, caused by destructive interference, not only is an important fundamental research topic in atomic physics and quantum optics, but also has wide potential application in quantum physics and quantum information science. Using the arrowhead-matrix method, here we study the dark-state effect in a coupled cavity-Rydberg-atom system, in which N Rydberg atoms with the dipole-dipole interactions are coupled to a single-mode cavity field. We obtain the numbers and form of the dark states in certain excitation-number subspaces for the two-, three-, and four-atom cases, as well as in the single-excitation subspace for a general N-atom case. We also suggest to characterize the dark states by inspecting the populations of some specific quantum states, which can be detected in experiments. Furthermore, we analyze the dark-state effect in a realistic case, where both the atomic dipole-dipole interaction strengths and the atom-cavity-field coupling strengths depend on the position of the atoms. Our findings pave the way for studying dark-state physics and applications in the cavity-Rydberg-atom platform.
The dark-state effect, caused by destructive quantum interference, is an important physical effect in atomic physics and quantum optics. It not only deepens the understanding of light-atom interactions, but also has wide applications in quantum physics and quantum information. Therefore, how to efficiently and conveniently determine the number and form of the dark states in multilevel quantum systems with complex transitions is an important and interesting topic in this field. In this work, we present a general theory for determining the dark states in multilevel quantum systems with any coupling configuration using the arrowhead-matrix method. To confirm the dark states in a multilevel system, we first define the upper- and lower-state subspaces, and then diagonalize the Hamiltonians restricted within the two subspaces to obtain the dressed upper and lower states. By further expressing the transitions between the dressed upper and lower states, we can map the multilevel system to a bipartite-graph network, in which the nodes and links are acted by the dressed states and transitions, respectively. Based on the coupling configurations of the network, we can determine the lower dark states with respect to the upper-state subspace. As examples, we analyze the dark states in three-, four-, and five-level quantum systems, for all possible configurations through the classification of the numbers of upper and lower states. Furthermore, we extend the framework to multilevel quantum systems and discuss the existence of dark states in some typical configurations. We also recover the results of the dark-state polaritons in driven three-level systems with the arrowhead-matrix method. Our theory paves the way for manipulating and utilizing the dark states of multilevel quantum systems in atomic physics and quantum optics.
We analyze a set of models frequently appearing in quantum optical settings by expressing their Hamiltonians in terms of Fock-state lattices. The few degrees-of-freedom of such models, together with the system symmetries, make the emerging Fock-state lattices rather simple such that they can be linked to known lattice models from the condensed matter community. This sheds new light on known quantum optical systems. While we provide a rather long list of models and their corresponding Fock-state lattices, we pick a few ones in order to demonstrate the strength of the method. The three-mode boson model, for example, is shown to display a fractal spectrum, and chiral evolution in the Fock-state lattice characterized by localized distributions traversing along symmetric trajectories. In a second example we consider the central spin model which generates a Fock-state lattice reminiscent of the SSH-model hosting topological edge states. We further demonstrate how the phenomena of flat bands in lattice models can manifest in related Fock-state lattices, which can be linked to so called dark states.
The multiphoton Jaynes-Cummings model is investigated and applications in quantum information science are explored. Considering the strong atom-field coupling regime and an N-photon interaction, a nonlinear driving field can perform an arbitrary rotation in the Fock space of a bosonic mode involving the vacuum and an M-Fock state, with M<N. In addition, driving a bosonic mode with a linear coherent field (superposition of many Fock states), only the cavity states within the Fock subspace {|0⟩,|1⟩,…,|N-1⟩} can be populated; i.e., we show how to implement a Fock state filter, or quantum scissor, that restricts the dynamics of a given bosonic mode to a limited Hilbert space. Such a device can be employed as a generator of finite-dimensional quantum-optical states and also as a quantum-optical intensity limiter, allowing as a special case the generation of single-photon pulses. On the other hand, our system also provides a very rich physics in the weak atom-field coupling regime, in particular, multiphoton electromagnetically induced transparencylike phenomena, inducing a narrow (controllable) reflectivity window for nonlinear probe fields. These results are useful for applications in quantum information processing and also motivate further investigations, e.g., the use of an N-photon Jaynes-Cummings system as a qudit with harmonic spectrum and the exploration of multiphoton quantum interference.
Activating transitions between internal states of physical systems has emerged as an appealing approach to create lattices and complex networks. In such a scheme, the internal states or modes of a physical system are regarded as lattice sites or network nodes in an abstract space whose dimensionality may exceed the systems’ apparent (geometric) dimensionality. This introduces the notion of synthetic dimensions, thus providing entirely novel pathways for fundamental research and applications. Here, we analytically show that the propagation of multiphoton states through multiport waveguide arrays gives rise to synthetic dimensions where a single waveguide system generates a multitude of synthetic lattices. Since these synthetic lattices exist in photon-number space, we introduce the concept of pseudo-energy and demonstrate its utility for studying multiphoton interference processes. Specifically, the spectrum of the associated pseudo-energy operator generates a unique ordering of the relevant states. Together with generalized pseudo-energy ladder operators, this allows for representing the dynamics of multiphoton states by way of pseudo-energy term diagrams that are associated with a synthetic atom. As a result, the pseudo-energy representation leads to concise analytical expressions for the eigensystem of N photons propagating through M nearest-neighbor coupled waveguides. In the regime where N ≥ 2 and M ≥ 3 , nonlocal coupling in Fock space gives rise to hitherto unknown all-optical dark states that display intriguing nontrivial dynamics.