## Abstract

We study a cavity-QED setup consisting of a two-level system coupled to a single cavity mode with two-photon relaxation. The system dynamics is modeled via a Lindblad master equation consisting of the Rabi Hamiltonian and a two-photon dissipator. We show that an even-photon relaxation preserves the Z2 symmetry of the Rabi model, and provide a framework to study the corresponding non-Hermitian dynamics in the number-parity basis. We discuss the role of different terms in the two-photon dissipator and show how one can extend existing results for the closed Rabi spectrum to the open case. Furthermore, we characterize the role of the Z2 symmetry in the excitation-relaxation dynamics of the system as a function of light-matter coupling. Importantly, we observe that initial states with even-odd parity manifest qualitatively distinct transient and steady state behaviors, contrary to the Hermitian dynamics that is only sensitive to whether or not the initial state is parity invariant. Moreover, the parity-sensitive dynamical behavior is not a creature of ultrastrong coupling and is present even at weak coupling values.

Original language | English (US) |
---|---|

Article number | 043601 |

Journal | Physical review letters |

Volume | 122 |

Issue number | 4 |

DOIs | |

State | Published - 2019 |

## All Science Journal Classification (ASJC) codes

- Physics and Astronomy(all)

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*Physical review letters*,

*122*(4), [043601]. https://doi.org/10.1103/PhysRevLett.122.043601

**Quantum Rabi Model with Two-Photon Relaxation**. In: Physical review letters. 2019 ; Vol. 122, No. 4.

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*Physical review letters*, vol. 122, no. 4, 043601. https://doi.org/10.1103/PhysRevLett.122.043601

**Quantum Rabi Model with Two-Photon Relaxation.** / Malekakhlagh, Moein; Rodriguez, Alejandro W.

Research output: Contribution to journal › Article › peer-review

TY - JOUR

T1 - Quantum Rabi Model with Two-Photon Relaxation

AU - Malekakhlagh, Moein

AU - Rodriguez, Alejandro W.

N1 - Funding Information: We appreciate helpful discussions with Pengning Chao. This work was supported by the National Science Foundation under Grants No. DMR-1454836, No. DMR 1420541, and Award No. EFMA-1640986. Funding Information: Here, we study the dissipative dynamics of the system and discuss the role of Z 2 symmetry. For concreteness, we consider the situation in which the cavity is initially prepared with an even or odd number of photons, and describe the ensuing dynamics of the cavity photon and qubit population as a function of both time and g . In particular, we consider two scenarios of starting with two [ ρ ^ ( 0 ) = | 2 , g ⟩ ⟨ 2 , g | ] or three [ ρ ^ ( 0 ) = | 3 , g ⟩ ⟨ 3 , g | ] initial cavity photons and a qubit in the ground state, as representatives of the plus or minus parity subspaces. Because of pair exchange of photons with the environment, we intuitively expect states with even or odd initial cavity photons to exhibit different transient and steady state behavior. First, consider the simplest case of g = 0 . This choice of parameter decouples the qubit and hence corresponds to the problem of a single cavity mode with two-photon relaxation, which has been studied in detail using multiple methods [36–40] . In this case, initial states having even (odd) numbers of cavity photons end up with zero (one) cavity photons in the steady state [55] . Next, we move on to characterize the interplay of two-photon relaxation and the qubit for g ≠ 0 . Here, closed form analytical solutions of the evolution operator at arbitrary g seem intractable, and instead we employ numerical integration of the Lindblad Eq. (6a) . The time evolution of the cavity or qubit excitations as a function of g is studied in Figs. 3 and 4 for the cases of two and three initial cavity photons, correspondingly. In both cases, it is generally observed that as g is increased, more complex beatings between various normal modes emerge. Such beatings can be approximately understood from the mapping of the initial cavity state to the corresponding eigenmodes of the open Rabi model. This shows which modes are more active at a given value of g in each parity subspace [Figs. 3(c) and 4(c) ]. For example, for the case of ρ ^ ( 0 ) = | 2 , g ⟩ ⟨ 2 , g | , the initial probability is shared between states | 1 g , + ⟩ and | 2 g , + ⟩ up to intermediate values of g ( 0 < g ≲ 0.5 ν c ), beyond which | 1 g , + ⟩ and | 3 g , + ⟩ dominate. The corresponding frequency and decay rate of the modes can be obtained from Figs. 2(a)–2(b) . 3 10.1103/PhysRevLett.122.043601.f3 FIG. 3. Excitation-relaxation dynamics of the system of Fig. 2 when the system is prepared with two cavity photons and the qubit is in the ground state, i.e., ρ ^ ( 0 ) = | 2 , g ⟩ ⟨ 2 , g | = | 2 , + ⟩ ⟨ 2 , + | as a function of light-matter coupling g . (a) Cavity photon number, (b) qubit excitation number, and (c) mapping of the bare state | 2 , g ⟩ to the eigenmodes in the even ( + ) parity subspace. For convenience, we omit the parity index in the x axis. (d) Steady state populations. Model parameters are the same as in Fig. 2 . The time axis in (a) and (b) is normalized to half of the cavity round-trip time T c ≡ π / ν c . The two-photon relaxation time reads T κ 2 ≡ 1 / κ c 2 = 40 T c / π . The cavity mode Hilbert space cutoff is chosen as N c = 9 . 4 10.1103/PhysRevLett.122.043601.f4 FIG. 4. Excitation-relaxation dynamics when the system is prepared with three cavity photons and the qubit is in the ground state, i.e., ρ ^ ( 0 ) = | 3 , g ⟩ ⟨ 3 , g | = | 3 , - ⟩ ⟨ 3 , - | , as a function of light-matter coupling g . The figure follows the same format as Fig. 3 , except that the bare state | 3 , g ⟩ is instead mapped to eigenmodes in the odd ( − ) parity subspace. Other parameters are the same as in Fig. 3 . Despite this generic similarity, it is observed that due to the nontrivial interplay of light-matter coupling and two-photon relaxation, the two cases under consideration have different transient and steady state characteristics. For the case of two initial cavity photons, we observe that the system reaches steady state on a timescale that is more or less given by the two-photon relaxation rate κ c 2 [Figs. 3(a)–3(b) ]. On the other hand, in the case of three initial cavity photons, the transient dynamics has more features. Generally, at small g , the dynamics can be described as follows [Figs. 4(a)–4(b) ]: First, a fast depletion of the initial three cavity photons into one photon, with timescale roughly determined by κ c 2 . This can be seen by the sharp transition of the cavity excitation number from 3 to approximately 1 [red to blue in Fig. 4(a) ]. Second, a slower depletion of the remaining cavity photon after a large number of Rabi exchanges between the qubit and the cavity, with timescale roughly determined by the decay rate of state | 1 g , - ⟩ . Essentially, since two-photon relaxation only allows pairs of exchange with the environment, the quantum state | 1 g , - ⟩ acts like a dark state at g = 0 (i.e., | 1 , g ⟩ ). As g is increased, the decay rate of this state is barely modified up until g / ν c ≈ 0.5 [See Fig. 2(b) ], consistent with the observed long-lived excitations in the qubit or cavity dynamics [Figs. 4(a)–4(b) ]. Steady state excitations have also been studied as a function of g in [Figs. 3(d)–4(d) ]. In the case of two initial photons, we observe that the steady state populations of the cavity and qubit increase nonmonotonically with increasing g , exhibiting a local maximum close to g ≈ κ c 2 . The case of three initial photons is more complicated. For small g < κ c 2 , one observes fast relaxation of two photons, while the remaining photon energy is transferred to the qubit at steady state. At intermediate values of g , the excitation is shared between the cavity and the qubit while at very large g , the qubit excitation saturates and the cavity photon population increases linearly [Fig. 4(d) ]. The overall increase observed in the steady state populations arises from the fact that the coupling in Eq. (6b) appears effectively as an incoherent drive on the cavity. 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PY - 2019

Y1 - 2019

N2 - We study a cavity-QED setup consisting of a two-level system coupled to a single cavity mode with two-photon relaxation. The system dynamics is modeled via a Lindblad master equation consisting of the Rabi Hamiltonian and a two-photon dissipator. We show that an even-photon relaxation preserves the Z2 symmetry of the Rabi model, and provide a framework to study the corresponding non-Hermitian dynamics in the number-parity basis. We discuss the role of different terms in the two-photon dissipator and show how one can extend existing results for the closed Rabi spectrum to the open case. Furthermore, we characterize the role of the Z2 symmetry in the excitation-relaxation dynamics of the system as a function of light-matter coupling. Importantly, we observe that initial states with even-odd parity manifest qualitatively distinct transient and steady state behaviors, contrary to the Hermitian dynamics that is only sensitive to whether or not the initial state is parity invariant. Moreover, the parity-sensitive dynamical behavior is not a creature of ultrastrong coupling and is present even at weak coupling values.

AB - We study a cavity-QED setup consisting of a two-level system coupled to a single cavity mode with two-photon relaxation. The system dynamics is modeled via a Lindblad master equation consisting of the Rabi Hamiltonian and a two-photon dissipator. We show that an even-photon relaxation preserves the Z2 symmetry of the Rabi model, and provide a framework to study the corresponding non-Hermitian dynamics in the number-parity basis. We discuss the role of different terms in the two-photon dissipator and show how one can extend existing results for the closed Rabi spectrum to the open case. Furthermore, we characterize the role of the Z2 symmetry in the excitation-relaxation dynamics of the system as a function of light-matter coupling. Importantly, we observe that initial states with even-odd parity manifest qualitatively distinct transient and steady state behaviors, contrary to the Hermitian dynamics that is only sensitive to whether or not the initial state is parity invariant. Moreover, the parity-sensitive dynamical behavior is not a creature of ultrastrong coupling and is present even at weak coupling values.

UR - http://www.scopus.com/inward/record.url?scp=85061003405&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85061003405&partnerID=8YFLogxK

U2 - 10.1103/PhysRevLett.122.043601

DO - 10.1103/PhysRevLett.122.043601

M3 - Article

C2 - 30768294

AN - SCOPUS:85061003405

VL - 122

JO - Physical Review Letters

JF - Physical Review Letters

SN - 0031-9007

IS - 4

M1 - 043601

ER -