Superradiant with cavity assisted dynamical -orbit coupling

Ying Lei1 and Shaoliang Zhang1, 2, ∗ 1School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China 2State Key Laboratory of and Quantum Optics Devices, Shanxi University, Taiyuan, 030006,China (Dated: December 21, 2020) Superradiant phase transition represents an important quantum phenomenon that shows the col- lective excitations based on the coupling between atoms and cavity modes. The spin-orbit coupling is another quantum effect which induced from the interaction of the atom internal degrees of free- dom and momentum of center-of-mass. In this work, we consider the cavity assisted dynamical spin-orbit coupling which comes from the combination of these two effects. It can induce a series of interesting quantum phenomena, such as the flat spectrum and the singularity of the excitation energy spectrum around the critical point of quantum phase transition. We further discuss the influence of atom decay and nonlinear coupling to the phase diagram. The atom decay suppresses the singularity of the phase diragram and the nonlinear coupling can break the symmetric properties of the phase transition. Our work provide the theoretical methods to research the rich quantum phenomena in this dynamic many-body systems.

I. INTRODUCTION such as the stripe phase [23] and algebra quantum liquids [24], etc. Recently, the cavity-assisted dynamical spin- The cavity quantum electrodynamics (CQED) has a orbit coupling which induced by a superradiant phase wide range of applications in quantum information and transition has been realized in experiment by replacing quantum simulations, because the interaction between one of the Raman with an optical cavity [25–27]. hyperfine spin states of atom and quantum field In this system, the cavity modes not only couple with has a great enhancement by the cavity. A famous the- the hyperfine spin states of atoms, but also interact with oretical model which used to describe this interaction is the atomic COM momentum. The interplay of cavity called Dicke model [1]. It is well known to undergo a modes, atomic internal and external degrees of freedom quantum phase transition from normal phase to superra- are expected to lead to novel quantum phenomena [28] diant phase called ”superradiant phase transition” [2,3]. and pave a way for the research of quantum simulation Over the last few years, based on the development of ex- in topological and many-body physics. perimentally techniques, there are a series of works focus Our propose in this work is to understand the interplay on researching the properties of this system [4–8] and of these degrees of freedom. In the system with cavity- superradiant phase transition has also be observed ex- assisted dynamical SOC, the SOC comes from the col- perimentally first by the group at ETH [9–11] then many lective excitation of cavity and all atoms — the other groups [12–14]. Many interesting quantum phe- , which is very different from the coupling nomena has been observed such as supersolid-like phase, between Raman laser and single atom in static SOC. Al- and research about their applications has also been de- though the translation invariance is still conserved, the veloped rapidly [15]. energy dispersion is very different and some exotic quan- The standard Dicke model only includes the coupling tum phenomena emerges, such as the dynamical SOC between cavity modes and atomic internal degrees of free- induced flat spectrum. In this system, the superradi- dom. But in ultracold atomic system, the coupling be- ant phase transition has some interesting properties be- tween the internal degrees of freedom and the atomic cause of the interplay among cavity modes, hyperfine spin center-of-mass(COM) motion, which is called the spin- states and the COM momentum of atoms, which is very orbit coupling (SOC) effect, has attracted a lot of inter- different from the standard Dicke model. There is a first ests in past two decades[16–18]. With the SOC effect, order phase transition and the energy spectrum of polari- a lot of interesting quantum phenomena, especially the tons has singularity at the critical point. In this work, phenomena related to the topological physics, such as we also discuss the influence of atom decay and nonlinear arXiv:2012.10098v1 [cond-mat.quant-gas] 18 Dec 2020 the synthetic gauge fields [19, 20], chiral edge currents coupling to the phase diagram. We find that the atom [21, 22], has been observed experimentally. In ultracold decay suppresses the singularity of the phase diagram atom systems, the SOC effect can be realized by coupling which result in the first order phase transition is hard to two different hyperfine spin states using two-photon Ra- be observed experimentally, and the nonlinear coupling man process which also accompanied by the transfer of can break the symmetric form of the superradiant phase COM momentum. In systems with static SOC, the trans- transition. lation invariance is conserved and the energy dispersion Our paper is organized as follows. In Sec. II we’ll give has some anomalous properties because of the SOC effect. a brief introduction about the cavity-assisted dynamical It is the source of many interesting quantum phenomena SOC model. Then we calculate the energy spectrum and 2

find the flat spectrum at critical point of superradiant where ω, δ, λ and U are the effective cavity frequency, phase transition. We also draw the phase diagram and energy detuning between hyperfine spin states, coupling discuss the first order phase transition. In Sec. III we amplitude and dispersive cavity shift. In equation (1), discuss the energy spectrum of polaritons and find the pj is the momentum of the j-th atoms, Sα = singularity induced by the flat spectrum. In Sec. IV we PN j j=1 σα where α = x, y, z and S± = Sx ± iSy. In most discuss the influence of atom decay and cavity photon part of our work, we’ll set the nonlinear atom-photon loss to the phase diagram. We find that in this dissipa- interaction U = 0 to simplify our discussion, the influence tive system, the singularity of the phase diagram will dis- of nonzero U will be discussion in the Sec. V. appear because of the atom decay. In Sec. V we discuss In this work, we only consider the case all atoms are the influence of nonlinear coupling to the phase diagram. condensed with the same COM momentum p. So pj in The Sec. VI is devoted to our conclusions. equation (1) can be simplified as a number instead of an operator. To further simplify our discussion, we can set the spin-orbit coupling along y direction and ignore the II. CAVITY-ASSISTED SPIN-ORBIT influence of other direction, the above Hamiltonian can COUPLING further be simplified as

p2 pq  H = N + r + δ S + ωa†a eff 2m m z (2) λ † + √ (a + a)(S+ + S−) N

Different with the original Dicke model, the Hamil- tonian (2) is also dependent on the COM momentum of atoms. The SOC effect doesn’t break the transla- tion invariant symmetry, and the commutation relation [p, H ] = 0 is kept, which means the COM momen- FIG. 1: The schematic diagram of our proposal. (a) Experi- eff mental setup; (b) Level diagram of atom and configura- tum of atom p is still a good quantum number, and the tion. eigenstates of the atoms are still plane waves. Because of the inversion symmetry, the Hamiltonian (2) is conserved with translation of p → −p, δ → −δ and Sz → −Sz, We consider the atoms with two hyperfine spin states which means the ground state has the same energy for (denoted as | ↑i and | ↓i) confined in an optical cav- different sign of δ, so only δ ≥ 0 need to be considered. ity, as shown schematically in Fig. 1. The cavity sup- Define the effective energy detuning between | ↑i and | ↓i ports a single-mode quantum field with frequency ω, two pqr states as ω0 ≡ m + δ and ignores the influence of the counter-propogating external (red and blue arrows) kinetic energy term, the Eq.(2) becomes to the standard with frequency q+ = q− = qr produces additional clas- Dicke model as sical lights with different frequencies, which couple dif- ferent hyperfine spin states respectively. Each of these † λ † HDicke = ω0Sz + ωa a + √ (a + a)(S+ + S−) (3) two classical lights and cavity quantum field induce a N two-photon Raman transition between different hyper- fine spin states, and spontaneously transfer COM mo- With the analytical calculation in the work of C. Emary mentum of ±q along the direction perpendicular to the and T. Brandes[29], one can get the eigenenergy and cor- r responding wave functions for different phase, the critical cavity axis in average. In the experiment of R. M. Kroeze √ et. al[27], the discretized momentum distribution and the point of superradiant phase transition is at λcr = ωω0/2 spatial modulation along cavity axis have been observed. in thermodynamic limit. But in this work, we just focus on the properties perpen- Consider the Hamiltonian (2), for an arbitrary cou- dicular to the cavity axis and the spatial modulation can pling strength λ and energy detuning δ, one can get the be ignored along this direction. After an unitary trans- energy spectrum with different COM momentum p first. formation, the effective Hamiltonian can be written as Because of the relative amplitude of λ and ω0 at different [4, 27] p, the corresponding eigenstates should be superradiant states or normal states due to different COM momentum m 4λ2  m 4λ2  p. (i) For − q ω + δ < p < q ω − δ , the relation N 2 r r X  pj pj · qr  U λ > λcr can be satisfied, so the eigenstate is superradiant H = + σj + δS − a†aS eff 2m m z z N z state and the corresponding energy can be expressed as j=1 (1) 2 2 2 2 λ † † EG 8mλ − ωq ωδqr λ ωδ + √ (a + a)(S+ + S−) + ωa a = r p2 − p − − (4) N N 16m2λ2 8mλ2 ω 16λ2 3

(ii) For the other p, we have λ < λcr, the corresponding in this system. This flat spectrum comes from the inter- eigenstate is normal state and the energy should be play of the cavity modes, atomic internal and external degrees of freedom. The coupling energy between cavity E p2 1pq  G = ∓ r + δ (5) photons and hyperfine spin states and the kinetic energy N 2m 2 m of atoms which both having the quadratic form of COM 2 momentum p cancel each other out at the critical point. where ∓ correspond to p ≥ m 4λ − δ and p ≤ qr ω 2 − m 4λ + δ respectively. qr ω

FIG. 2: The energy spectrum for different COM momentum. Different color lines correspond to different λ. (a) δ > 0, 2 2 λ < ωEr/4 for the red line, λ = ωEr/4 for the blue line, 2 ω  2 ω  λ = 4 Er + δ for the green line and λ > 4 Er + δ for the magenta line. The red dot is the energy minimum for the first three cases, which is at p = q /2, the magenta dot is the min r FIG. 3: Phase diagram. (a). The expectation value of cavity energy minimum for the last case, which is at p < q /2; min r photon number hn i. (b). The expectation value of hS i. (c). (b) δ = 0.λ2 < ωE /4 for the red line, λ2 = ωE /4 for the c z r r The expectation value of cavity photon number for different δ. blue line, λ2 > ωE /4 for the green and magenta lines. The r The red, blue, green, magenta line correspond to δ = 0, δ = red dot is the energy minimum for the first two cases, which 0.1, δ = 0.2 and δ = 0.3 respectively. (c). The expectation is at p = q /2, the green and magenta dots are the energy min r value of hS i for different δ. The red, blue, green, magenta minimum for the last two cases, which is at p = 0. The z min line correspond to δ = 0, δ = 0.05, δ = 0.1 and δ = 0.15 inset shows the expectation value of cavity photon number respectively. hnci/N√ for the eigenstates with different p where δ = 0 and λ = ωEr/2.

With the above discussion, one can also draw the phase The energy spectrum of the eigenstates with different diagram in thermodynamic limit, as shown in Fig. 3. Fig. COM momentum p is shown in Fig. 2. The ground 3(a) and (b) show the expectation of cavity photons num- † state is at the minimum of these energy spectrum and ber hnci = ha ai and magnetic quantum number hSzi is pointed out with fixed dot. We find that with the with different δ and λ respectively. From the phase di- increase of the coupling strength λ, the energy of su- agram, one can easily find there is a singularity where perradiant state is decrease until two state energy are both expectation values have discontinuity along δ = 0, the same, there is a superradiant phase transition in this as shown the red lines in Fig. 3(c) and Fig. 3(d), which 2  system at λc = ω Er + δ /4. The phase transition need means the system has the first order phase transition. to be discussed in two different cases. (i). δ > 0. In this All these novel phenomena come from the flat spectrum 2 2  2 case, for λ ≤ λc = ω Er + δ /4 (where Er = qr /(2m) at critical point. The flat spectrum means the system is the recoil energy), the ground state is normal state is highly degenerate at this critical point. But all these 2  at momentum pmin = qr/2. For λ > ω Er + δ /4, eigenstates have different number of cavity photons and the ground state is superradiant state and the corre- magnetic quantum number because the effective detuning ωδqr sponding momentum is pmin = 2 , which moves 2(4λ −ωEr ) ω0 are different in these states with different p, as shown asymptotically to the point p = 0 with the increase of in inset of Fig. 2(b). The expectation of cavity photon λ. (ii). δ = 0. In this extreme case, the ground state number is the minimum hnci = 0 at p = ±qr/2 but the 2 is also normal state when λ < ωEr/4 with momentum maximum hnci/N = Er/(4ω) at p = 0. After passing the 2 pmin = qr/2. If λ > ωEr/4, the ground state is su- critical point, the momentum of COM of ground state is perradiant state with momentum pmin = 0. At critical suddenly changed from pmin = ±qr/2 to pmin = 0, the 2 point where λ = ωEr/4, the energy spectrum is flat in expectation of cavity photon number increases from the the range of q ∈ [−qr/2, qr/2], as shown the blue line in finite value hnci/N = Er/(4ω) instead of zero. We should Fig. 2(b), which is one of the most interesting property also point out that when δ = 0, the ground state energy 4 has two fold degeneracy at p = ±qr/2 in normal phase because of the symmetry, which means that all spins oc- cupying | ↑i and | ↓i will have the same energy. In the red line in Fig. 3(d), we plot the value hSzi/N as −1/2 2 instead of 1/2 when λ < ωEr/4, it can be considered as a spontaneous symmetry breaking.

III. THE PROPERTIES OF EXCITATIONS

FIG. 4: The excitation energies as the function of λ. The To explore the properties of the superradiant phase parameters are all scaled to ω = m = qr = 1. The brown transition further, we estimate the energy of the ground solid line and the blue dotted line correspond to − and + state. When δ 6= 0, the ground state energy can be writ- respectively. (a) δ = 0, the excitation energy is discontinuous. ten as (b). δ = 0.6, the excitation energies is similar with standard Dicke model. ( Er δ − 4 − 2 (λ ≤ λc) 2 2 2 2 (δ, λ) = ω δ Er λ ωδ (6) − 2 2 − − 2 (λ > λc) 16λ (4λ −ωEr ) ω 16λ on the side of superradiant phase in this extreme case as ω   2 2 + = 4λ /ω and − = ω, which means the atomic branch where λc = 4 Er + δ . With a direct calculation, one can easily estimate the second order derivation of the and photon branch are decoupled. The reason of this energy with δ and λ and find that there is a discontinuity decoupling is the COM momentum in the superradiant around the phase boundary, so the system has a second phase side is p = 0 when δ = 0, then the effective energy order phase transition when δ 6= 0, as the standard Dicke detuning of different hyperfine spin states is zero. model. In superradiant phase, the collective excitations of the IV. THE INFLUENCE OF ATOM DECAY atoms and cavity photons are called polaritons, which can be effectively equivalent to two independent har- In standard Dicke model (3), the critical point of su- monic oscillator modes [29]. In standard Dicke model, perradiant phase transition is severely dependent on the one of oscillator modes approaches to zero from both dissipation of cavity photons and atoms [43]. By con- sides with the asymptotic form ∝ |λ − λ |1/2, where √ cr sidering the atomic decay and cavity loss, the master λ = ωω /2 is the critical point of standard Dicke cr 0 equation of the density matrix can be written as model. In the system with Hamiltonian (2), for an ar- bitrary δ, the excitation energy can be expressed as the N   X ` ` 1  ` ` function of λ as[30] ρ˙ = −i[Heff , ρ] + γ σ ρσ − σ σ , ρ − + 2 + − (9) s `=1 ω2  ω2 2  †  †  22 = ω2 + 0 ± ω2 − 0 + (4λ)2ωω µ (7) + κ 2aρa − a a, ρ ± µ2 µ2 0 The Lindblad operators include two parts, the first part pminqr describes the atom decay with rate γ and the second part where ω0 = m +δ is the effective energy detuning be- tween two hyperfine spin states, µ = 1 in normal phase describes the cavity photon loss with decay rate κ. We 2 2 and µ = λc /λ < 1 in superradiant phase. The en- should point out that the atom decay rate in above equa- ergy of polaritons has the same asymptotic form with tion (9) is not the rate of spontaneous emission from ex- the standard Dicke model at the normal phase side when cited state to ground state, because both of the hyperfine λ ≤ λc(δ). But at the superradiant phase side, the ω0 is spin states are the ground states. This decay rate comes the function of δ instead of a constant in standard Dicke from the difference of spontaneous emission rate to these model, the asymptotic form will change with δ as two ground states from excited states which have been eliminated adiabatically, such as the |+i and |−i states v u   in Fig. 1(b). This type of decay will not induce the atom u64λ3 1 + Er u 0 δ 1  ∼ |λ − λ | 2 (8) number loss but only the spin decoherence. If we ig- − t 16λ4 c 0 2 nore the momentum transfer in the spontaneous emission ω2 + ω process in average, the COM momentum p can still be With decrease of δ, the energy spectrum becomes more a number. Then with the standard method in quantum and more steep. When δ = 0, the energy of polaritons optics, the Heisenberg-Langevin equations for the atomic (7) has a singularity at the critical point as shown in and cavity photon operators can be written down and the Fig.4(a). This is the result of first order phase transition. steady state solutions can be estimated as shown in [43]. We can also estimate the excitation energy analytically Then with the similar method in the Section II, one can 5 also estimate the steady state with p which has the low- any more for an arbitrary nonzero γ, as shown in Fig. est energy expectation and draw the phase diagram. The 5(c). The expectation value of cavity photon number critical point of the superradiant phase transition has be has also been suppressed around δ = 0. These new phe- changed into: nomena can be simply explained by the similar results in standard Dicke model. In the cavity-assisted SOC sys- s 2 2 2 2 1 (γ + 4ω0)(κ + ω ) tem, δ ≈ 0 means the effective energy detuning between λc = (10) 4 ω0ω two hyperfine spin states is very small, then the atom de- cay will dominate the physical properties of in this case. 2 qr To observe the first order phase transition in this system, where ω0 = + |δ|.The expectation value of operators 2m one should either reduce the spontaneous emission rate hai, hσ+i and hσzi in superradiant phase can be expressed as as far as possible or use two ground states with the same spontaneous emission rate from the excited state. r q 2 2  2 2 2 2  κ +ω (γ +4ω0 )(κ +ω ) ω0 1 − 2 ω 16λ ωω0 hai = ± √ 2 (−ω + iκ) V. THE INFLUENCE OF NONLINEAR TERMS r  2 2 2  λ ω γ +4ω0 2 2 ω0 2 2 − (κ + ω ) † κ +ω 16ω0 1 iγ  In real experiment, Ua aSz in equation (1) which com- hσ+i = ± √ − ing from mutual feedback between cavity and atoms is 2λ2ω 2 4ω0 always very small but not zero. If the influence of this (γ2 + 4ω2)(κ2 + ω2) z 0 nonlinear coupling is inavoidable, we should consider the hσ i = − 2 16λ ωω0 following effective Hamiltonian (11) p2 pq  H = N + r + δ S + ωa†a eff 2m m z (12) λ † U † + √ (a + a)(S+ + S−) + a aSz N N

Comparing with the original Hamiltonian (2), the COM momentum p is still a good quantum number because the external trap is absent here. Although the inversion symmetry in Hamiltonian (2) has been broken because of the nonlinear coupling, the more general inversion sym- metry with translation p → −p, δ → −δ, Sz → −Sz and U → −U is still conserved, so we just need to consider the case of U > 0. The analytical solution is hard to approach because of the nonlinear term, then we use the mean-field approximation to estimate the scale energy of Hamiltonian (12). With Holstein-Primakoff√ † † † FIG. 5: The expectation value of cavity photon number transformation√ Sz = b b − N/2, S+ = b N − b b † † hnci/N for different δ and λ in the system with (a). small and S− = N − b bb where b (b) is the creation (an- atom decay rate γ = 0.1 and (b). lager atom decay rate γ = 2. 2 nihilation) operator with boson commutation relation (c). The expectation value of photon number α /N for dif- [b, b†] = 1, the Hamiltonian (12) can be expressed as ferent atomic emission rate γ. γ = 0 for blue line, γ = 0.1 for red line, γ = 1 for green line. (d). Phase boundry λ for a Two-mode bosonic Hamiltonian.√ Then using two√ shift- c † † † † different parameters. κ = 0.1 and γ = 0 for blue line, κ = 0.5 ing boson operators a = as + Nα, b = bs − Nβ, and γ = 0.1 for red soild line, κ = 1 and γ = 0.1 for green then using relation hasi = hbsi = 0, ignore the quan- soild line. All parameters are scaled to ω = qr = m = 1.0. tum fluctuation, the scale energy of Hamiltonian (12) (p, α, β) = E(p, α, β)/N can be estimated by

The expectation of cavity photon number is shown in 2 p 2  pqr  2 1 Fig. 5. There are some features of the phase diagram (p, α, β) = + ωα + δ + β − 2m m 2 (13) in the dissipative system. First, the loss of cavity pho- p  1 − 4λαβ 1 − β2 + Uα2 β2 − ton has no qualitative influence to the phase diagram 2 as shown in Fig. 5(d), because we just consider the case where ω  κ. Second, the atom decay completely The minimum of the scale energy can be estimated nu- changes the properties around the line δ = 0. The first merically with the same method in above section.√ One order phase transition along δ = 0 can not be observed of typical dispersion with δ = 0 and λc = ωEr/2 is 6

pqr shown in Fig. 6(a). If U = 0, there is a flat spec- where ω0 = δ + m which is dependent with the COM trum which inducing a series of novel critical phenom- momentum p. Taking this expression into (13) and ex- ena. But in the case of U > 0, the energy spectrum panding the scale energy with U, we can get dispersion becomes asymmetric because of the inversion symmetry as breaking. Then the energy minimum is at some position 2 2 2 2 2 8mλ − ωq  λ ω ω Uω0 with 0 < p < q /2. The mean-field phase diagram (p) ≈p2 r − + 0 − 1 min r 16λ2m2 ω 16g4 8ω (15) with U > 0 is shown in Fig. 6(b) and two important in- fluences of nonlinear term can be observed. First, there is no first order phase transition anymore because of the which is a cubic function of COM√ momentum and asym- disappearance of the flat spectrum for arbitrary param- metric. When δ = 0 and λ0 = ωEr/2, the dispersion eters. The expectation of cavity photons and magnetic 2  ω2q2p2  will becomes (p) = − λ + r − 1 Uqr p which is quantum number of atom system changes continuously ω 16g4m2 8ωm with both parameters. Second, the phase boundary is shown as the red line in Fig. 6(a). asymmetric, as the red line shown in Fig. 6(b). This asymmetry can be briefly explained as follow. Consider VI. DISCUSSION AND CONCLUSION two points on phase diagram with δ = ±δ0 (δ0 > 0) and λ = λc. Without the influence of U, the phase bound- In conclusion, we discussed the influence of cavity in- ary has the COM momentum pmin = ±qr/2 with same ground state energy respectively because of the inversion duced dynamical spin-orbit coupling to ultracold atom symmetry. If U > 0, because the effective energy de- systems. The phase diagram in thermodynamic limit shows some interesting performance. When two hyper- tuning ω0 = δ + pqr/m has opposite signs at p = ±qr/2, † fine spin states are degenerate, the energy spectrum at the influence of Ua aSz has opposite contributions to the 2 the phase boundary are flat and the excitation spectrum mean-field energy of above two cases as Uα hSzi, which increase (decrease) the ground state energy of superra- has the singularity. But from our discussion, this singu- larity is hard to be observed experimentally because of diant state for hSzi < 0 (> 0) respectively. So on the the atom decay in two-photon Raman process. We also right side of phase diagram with δ = δ0 > 0, the nonlin- ear term will enhance the superradiant phenomena and discussed the influence of the nonlinear coupling which decrease the phase boundary, but vice versa on the left breaking the inversion symmetry makes the phase dia- gram asymmetric. side with δ = −δ0 < 0. The dynamical SOC effects induced by superrandiance opens a new way to quantum simulation in nonequilib- rium systems. It has a wide application in the explor- ing the novel topological states in dynamical systems [34–37]; the combination of the cavity-assisted SOC and the atomic interaction induces many interesting quan- tum phenomena in dynamic many-body systems [38–41]; the dissipation of the cavity photons can also be utilized to explore the properties of open quantum systems[42– 45]. All these applications can be discussed in the future works and we hope the methods and results in this work FIG. 6: Phase diagram with the influence of nonlinear cou- can give some help to the following researches in this pling U. (a). The energy spectrum with different COM mo- area. mentum for δ = 0. the blue line corresponds to U = 0 and the red line corresponds to U > 0; (b). The expectation value of cavity photon number hnci when U > 0, the red line represents the phase boundary, the white line represents ACKNOWLEDGEMENTS phase boundary with U = 0. All parameters are scaled to ω = qr = m = 1.0. We thanks Han Pu and Xiang-Fa Zhou for their valu- able suggestions. This research is supported by start-up funding of Huazhong University of Science and Tech- To further explain the influence of the nonlinear term, nology and the Program of State Key Laboratory of we can estimate the energy spectrum analytically using Quantum Optics and Quantum Optics Devices (No: perturbation theory when U is smaller than any other KF201903). energy scales. In superradiant state, the expectation of excited atoms β2 has the following form[31] s 2 2 2 1 ω ω 4λ  U  β = − + 2 1 − 2 (14) ∗ 2 U U 4λ + Uω0 4ω Electronic address: [email protected] 7

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