We present a generalized variational method to analytically obtain the ground-state properties of the unsolvable Jaynes-Cummings model with the ultrastrong coupling. An explicit expression for the ground-state energy, which agrees well with the numerical simulation in a wide range of the experimental parameters, is given. In particular, the introduced method can successfully solve this Jaynes-Cummings model with the positive detuning (the atomic resonant level is larger than the photon frequency), which can not be treated in the adiabatical approximation and the generalized rotating-wave approximation. Finally, we also demonstrate analytically how to control the mean photon number by means of the current experimental parameters including the photon frequency, the coupling strength, and especially the atomic resonant level. The Jaynes-Cummings model, which describes the important interaction between the atom and the photon of a quantized electromagnetic field, is a fundamental model in quantum optics and condensed-matter physics as well as in quantum information science. In the optical cavity quantum electrodynamics, the atom-photon coupling strength is far smaller than the photon frequency. As a result, the system dynamics can be well governed by the Jaynes-Cummings model with the rotating-wave approximation (RWA) [1]. In the case of the RWA, its energy spectrum and wavefunctions can be solved exactly [2]. With the rapid development of fabricated technique in solid-state systems, the Jaynes-Cummings model can be realized in semiconducting dots [3][4][5][6] and superconducting Josephson junctions [7][8][9][10][11][12]. More importantly, recent experiment has reported the existence of the ultrastrong coupling with the ratio 0.12 between the coupling strength and the microwave photon frequency [13]. Moreover, this ratio maybe approach unit due to the current efforts [14,15].However, in this ultrastrong coupling regime the wellknown RWA breaks down and the whole Hamiltonian is written aswhere a † and a are creation and annihilation operators for photon with frequency ω, σ ± are the raising and lowering operators of the two-level atom in the basis of σ z , Ω is the atomic resonant frequency and g is the atomphoton coupling strength. Due to the existence of the counter-rotating terms (σ + a + and σ − a), Hamiltonian (1) is very difficult to be solved analytically except for Ω = 0. Although the energy spectrum of Hamiltonian (1) has been obtained perfectly by means of the numerical simu- * Corresponding author: chengang971@163.com lation [16][17][18], the analytical solutions are very necessary for extracting the fundamental physics as well as in processing quantum information [19][20][21]. In the negative detuning Ω < ω, the adiabatic approximation method that the second term of Hamiltonian (1) is treated as a small perturbation has been considered [22]. Recently, a generalized rotating-wave approximation (GRWA) has also been proposed to solve Hamiltonian (1) in the displaced oscillator basis states [23]. This method ...

In recent pioneer experiment, a strong spin-orbit coupling, with equal Rashba and Dresselhaus strengths, has been created in a trapped Bose-Einstein condensate. Moreover, many exotic superfluid phenomena induced by this strong spin-orbit coupling have been predicted. In this report, we show that this novel spin-orbit coupling has important applications in quantum metrology, such as spin squeezing. We first demonstrate that an effective spin-spin interaction, which is the heart for producing spin squeezing, can be generated by controlling the orbital degree of freedom (i.e., the momentum) of the ultracold atoms. Compared with previous schemes, this realized spin-spin interaction has advantages of no dissipation, high tunability, and strong coupling. More importantly, a giant squeezing factor (lower than −30 dB) can be achieved by tuning a pair of Raman lasers in current experimental setup. Finally, we find numerically that the phase factor of the prepared initial state affects dramatically on spin squeezing.

Node line (NL) band-touchings protected by mirror symmetry (named as m-NL), the product of inversion and time reversal symmetry S = PT (s-NL), or nonsymmorphic symmetry are nontrivial topological objects of topological semimetals (SMs) in the Brillouin Zone (BZ). In this work, we screened the family of MgSrSi-type crystals using first principles calculations, and discovered that more than 70 members are NLSMs. A new type of multi-loop structure was found in AsRhTi that a s-NL touches robustly with a m-NL at some "nexus point", and in the meanwhile a second m-NL crosses with the s-NL to form a Hopf-link. Unlike the previously proposed Hopf-link formed by two s-NLs or two m-NLs, a Hopf-link formed by a s-NL and a m-NL requires a minimal three-band model to characterize its essential electronic structure. The associated topological surface states on different surfaces of AsRhTi crystal were also obtained. Even more complicated and exotic multiloop structure of NLs were predicted in AsFeNa and PNiNb. Our work may shed light on search for exotic multi-loop NLSMs in real materials.