Hidden Single‐Qubit Topological Phase Transition without Gap Closing in Anisotropic Light‐Matter Interactions

Abstract

Conventionally the occurrence of topological phase transitions (TPTs) requires gap closing, whereas there are also unconventional cases without need of gap closing. Although traditionally TPTs lie in many-body systems in condensed matter, both cases of TPTs may find analogs in few-body systems. Indeed, the ground-state node number provides a topological classification for single-qubit systems. While the no-node theorem of spinless systems is shown to restrict the fundamental quantum Rabi model in light-matter interactions, it is demonstrated that the limitation of the no-node theorem can be broken not only in small counter-rotating terms (CRTs) but also in the large-CRT regime, which striates a rich phase diagram with different TPTs. While these transitions are mostly accompanied with gap closing and parity reversal, a hidden node-phase transition is revealed that has neither gap closing nor parity change, which turns out to be an analog of the unconventional TPT in condensed matter. A hysteresis sign for the unconventional TPT is unveiled via the transition from phase squeezing to amplitude squeezing in the gapped phase. The imprints in the Wigner function are also addressed. The clarified mechanisms provide some special insights for the subtle role of the CRTs.

Critical Quantum Metrology in the Non-Linear Quantum Rabi Model

“…The model has a phase transition in the low-frequency limit at the critical point [ 65 , 68 ] with [ 17 , 19 ], and is the critical value of beyond which the Hamiltonian ( 1 ) is no longer self-adjoined and becomes unphysical [ 59 , 69 , 70 , 71 , 72 , 73 ]. The transition in this limit is second-order-like at [ 17 , 18 , 19 , 20 , 26 , 27 , 28 ] and first-order-like at finite [ 65 , 68 ]. The precision (signal-to-noise ratio) of any experimental estimation of one of the parameters in ( 1 ) is bound by [ 74 ], where is the quantum Fisher information [ 14 , 74 , 75 ], which takes the following form for pure states where denotes the derivative of the ground state (GS) ...…”

“…Indeed, under standard assumptions, critical protocols can achieve the Heisenberg scaling—a quadratic growth of parameter-estimation precision—both with respect to the number of probes and with respect to the measurement time. Furthermore, a recent theoretical work [ 15 ] demonstrated that the optimal limits of precision can be achieved using finite-component phase transitions [ 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 , 28 ], which are criticalities that take place in quantum optical systems where the thermodynamic limit is replaced by a scaling of the system parameters [ 20 , 29 , 30 , 31 , 32 , 33 , 34 ]. Critical quantum sensors can then also be implemented with controllable small-scale quantum devices, without requiring the control of complex many-body systems.…”

“…When considering finite-component phase transition, the most widely studied case is the quantum Rabi model (QRM) [ 55 , 56 , 57 ], composed of a two-level atom coupled to a single quantum harmonic mode. This model undergoes a second-order critical phase transition in the slow-resonator limit [ 16 , 17 , 18 , 19 , 20 , 26 , 27 , 28 ], where the frequency of the mode and the coupling strength are sent to zero with a given scaling law. Focusing on the scaling limit, one can obtain interesting results on the growth of the estimation precision in terms of fundamental resources such as the size of probe systems or photon number, however, to assess the actual precision of practical protocols, finite values of the parameters must be considered.…”

Scaling Relations and Topological Quadruple Points in Light‐Matter Interactions with Anisotropy and Nonlinear Stark Coupling

“…It should be noted that neither the anisotropy nor the nonlinear Stark coupling breaks the parity so that the model preserves the parity symmetry, with H commuting with the parity operator P = 𝜎 x (−1) a † a . The parity symmetry is relevant for symmetryprotected TPTs [27,28] as in condensed matter, [78][79][80][81] while there is a hidden symmetry breaking of spin reversion or space inversion for the symmetry-breaking QPT in the GS. [27] Changing to the quadrature representation by a…”

“…On the other hand, the milestone work of Braak revealing the integrability [3] of DOI: 10.1002/qute.202200068 the quantum Rabi model (QRM), [17] which is a most fundamental model of light-interactions, has triggered an intensive dialogue between mathematics and physics [18] and leads to a boom of theoretical developments. [4,5, Without mentioning the ubiquitous role of lightmatter interaction and its broad relevance to quantum optics, quantum information and quantum computation, [1,[64][65][66][67] quantum metrology, [36][37][38]68] condensed matter, [2,27,28] and relativistic systems, [69] the explosively-growing investigations have yielded abundant findings in the QRM and its extensions, such as hidden symmetry, [60][61][62][63] various patterns of symmetry breaking, [26,27,29] few-body quantum phase transitions, [5,[22][23][24][25][26][27][28][29]70] multicriticalities and multiple points, [26][27][28] universality classification, [24,25,27,52] spectral collapse, [33]…”