Efficient basis for the Dicke Model II: wave function convergence and excited states
EEfficient basis for the Dicke Model II: wave functionconvergence and excited states
Jorge G. Hirsch and Miguel A. Bastarrachea-Magnani
Instituto de Ciencias Nucleares, Universidad Nacional Aut´onoma de M´exico, Apdo.Postal 70-543, Mexico D. F., C.P. 04510E-mail: [email protected]
Abstract.
An extended bosonic coherent basis has been shown by Chen et al [1] toprovide numerically exact solutions of the finite-size Dicke model. The advantages inemploying this basis, as compared with the photon number (Fock) basis, are exhibitedto be valid for a large region of the Hamiltonian parameter space and many excitedstates by analyzing the convergence in the wave functions.
PACS numbers: 3.65.Fd, 42.50.Ct, 64.70.Tg
1. Introduction
The Dicke Hamlitonian describes a system of N two-level atoms interacting with asingle monochromatic electromagnetic radiation mode within a cavity. It is describedin the accompanying article [2]. The purpose of this second part is to show that thebenefits to employ the coherent basis are valid for a large region of the Hamiltonianparameter space, not only to obtain converged values of the energy, but also for thewave function, for the ground state and for a significative part of the energy spectra.It can be particularly useful to study the presence of chaos [3, 4] and of excited statesphase transitions [5, 6] in this model.The interaction between a system of N two-level atoms and a single mode of aradiation field can be described by the Dicke Hamiltonian: H D = ωa † a + ω J (cid:48) z + γ √N (cid:16) a + a † (cid:17) (cid:16) J (cid:48) + + J (cid:48)− (cid:17) . (1)The frequency of the radiation mode is ω , which has an associated number operator a † a .For the atomic part ω is the excitation energy, meanwhile J (cid:48) z , J (cid:48) + , J (cid:48)− , are collectiveatomic pseudo-spin operators which obey the SU(2) algebra. It holds that if j ( j +1) is theeigenvalue of J = J (cid:48) x + J (cid:48) y + J (cid:48) z , then j = N / γ depends principally onthe atomic dipolar moment. a r X i v : . [ qu a n t - ph ] D ec fficient basis for the Dicke Model II
2. Numerical Diagonalization
We compare the minimal truncation needed to obtain convergence of the solution, usingthe two basis defined in Ref. [2]: the coherent basis | N ; j, m (cid:105) and the Fock basis | n ; j, m (cid:105) .The wave functions, expanded in the truncated Fock (F) and coherent (C) basisare, for a given j = N / | Ψ kX (cid:105) = x max (cid:88) x =0 j (cid:88) m = − j C k,Xm,x | x ; j, m (cid:105) , (2)where x = n for X = F , and x = N for X = C , and k = 1 , ..., ( x max + 1)(2 j + 1)enumerates the eigenstates ordered by their energies E kX with k = 1 assigned to theground state. The probability P n of having n photons in the k -th state in the Fock basis, or P N ofhaving N excitations in the coherent basis is: P k,x = |(cid:104) x | Ψ kX (cid:105)| = (cid:88) m | C ,Xm,x | , (3)where x = n, N for X = F, C , respectively. The ground state probability distribution P x = P ,x is shown as a function of n or N up to n max or N max , for γ = 0 . . j = 10 in Figs. 1 and 2. Both wave functions were calculated with the truncationnecessary to have the energy converged with ∆ E < (cid:15) = 1 × − . n P n j (cid:61) Γ(cid:61) n P n j (cid:61) Γ(cid:61)
Figure 1. P ,n as function of n in the Fock basis, for j = 10 , γ = 0 . , n max = 15(left); and j = 10 , γ = 1 . , n max = 50 (right). From Fig. 1 and 2 it is clear that many components which contributes very little tothe wave function must be included in the calculations to obtain the desired precisionin the ground state energy. It can also be observed in the figures that for γ = 0 .
5, whichis γ c in this case, the largest probability is to have no photons in the Fock basis, or noexcitations in the coherent basis. The situation is different in the superradiant region, γ = 1, where in the Fock basis the distribution of photons resembles a Gaussian curve, fficient basis for the Dicke Model II N P N j (cid:61) Γ(cid:61) N P N j (cid:61) Γ(cid:61)
Figure 2. P ,N as function of N in the coherent basis, for j = 10 , γ = 0 . , N max = 7(left) and j = 10 , γ = 1 . , N max = 8 (right). with its maximum at a photon number proportional to the number of atoms, while inthe coherent basis the probability of having zero excitations remains dominant. This isthe power of the coherent basis, which allows to obtain numerically exact ground statewave functions for numbers of atoms which are intractable in the Fock basis.To study the convergence in the wave function we define its precision ∆ P X [7] as∆ P X ≤ j (cid:88) m = − j (cid:12)(cid:12)(cid:12) C ,Xx max +1 ,m ( x max + 1) (cid:12)(cid:12)(cid:12) . (4)where x = n, N for X = F, C , respectively. This ∆ P criteria demands less computingresources than the ∆ E criteria [7, 2], because it requires only the information about onetruncation value ( x max ) instead of two.Fig. 3 displays the plots of − Log (∆ P F ) as a function of n max , and of − Log (∆ P C )as a function of N max . n max (cid:45) Log (cid:72) (cid:68) P F (cid:76) N max (cid:45) Log (cid:72) (cid:68) P C (cid:76) Figure 3. (Color online). ∆ P as function of n max (left) and N max (right). From leftto right j = 1 (blue), 5, 10, 20, 30 and 40 (green). For γ = 0 . A linear fit, for j = 40 give us the following relation between N max and ∆ P C : − Log (∆ P C ) = 1 .
45 + 0 . N max ⇒ ∆ P C = 0 . − . N max (5) fficient basis for the Dicke Model II
500 1000 1500 k (cid:68) P F
20 40 60 80 100 120 140 k (cid:45) (cid:45) (cid:45) (cid:45) (cid:45) Log (cid:72) (cid:68) P F (cid:76)
500 1000 1500 k (cid:68) P C
20 40 60 80 100 120 140 k (cid:45) (cid:45) (cid:45) Log (cid:72) (cid:68) P C (cid:76) Figure 4. ∆ P of all states as a function of the state number. Details are given in thetext.
3. Numerically exact results for Excited States
In this section we extend the analysis to the excited states. To accurately evaluatea significative part of the energy spectrum is a necessary ingredient in the study ofquantum chaos [3] and of excited state quantum phase transitions (ESQPT) [5, 6].In figure 4 we display plots of ∆ P as a function of the state k , for j = 40, γ = 0 . ω = 1 . N max = 20 and (cid:15) = 1x10 − . In the upper figures we show the ∆ P F and in thelower ones ∆ P C . On the left the vertical scale is linear and all states are listed in thehorizontal axis, while on the right hand side the vertical scale is logarithmic and onlythe 150 states with lower energies are included. The horizontal green line depicts thetolerance (cid:15) .It is indeed remarkable to observe in Fig. 4 that a few hundred states calculated inthe coherent basis have their wave function converged, and ∆ P C , for these states, growsin a smooth and nearly monotonous way as a function of the k index. This is not thecase in the Fock basis, where ∆ P F fluctuates by orders of magnitude between a givenstate and the following one. It is worth to compare the convergence criteria based inthe wave function and described above, with the more standard convergence in energy,which was described in the previous article [2]. In figure 5 we show ∆ P C versus ∆ E C for the first 250 excited states, k , whose energies converged in the coherent basis with∆ E < × − .A linear fit of these data results in − Log [∆ P C ( k )] = 0 . − . Log [∆ E C ( k )] (6) fficient basis for the Dicke Model II (cid:45) Log (cid:72) (cid:68) E C (cid:76) (cid:45) Log (cid:72) (cid:68) P C (cid:76) Figure 5. ∆ P C vs ∆ E C for the first 250 states, which convergence is under (cid:15) = 1 × − . ⇒ ∆ P C = 0 . E C ( k )] . The number of states whose ∆ P is smaller than a tolerance (cid:15) for j and n max givenfor the Fock basis and N max for the coherent basis is presented in Table 1. The twotolerances selected are (cid:15) = 1 × − and (cid:15) = 1 × − , with γ = 0 . ω = 1. (cid:15) (cid:15) j n max /N max Fock coherent Fock coherent10 10 1 18 4 3710 15 7 55 15 9110 20 20 112 39 16620 10 0 21 2 4320 15 3 65 8 10620 20 8 136 20 19340 10 0 23 0 4840 15 1 70 4 13140 20 4 154 12 241
Table 1.
Number of states whose ∆ P is less than a tolerance (cid:15) for j and n max givenfor the Fock basis and N max for the coherent basis. Tolerances (cid:15) = 1x10 − and (cid:15) = 1x10 − , γ = 0.5, ω = 1. The advantages associated with the use of the coherent basis are even more clearin this case, because the number of states whose wave function has converged withthe selected tolerance is larger than those whose energies have converged. It should bementioned, however, that the tolerances in ∆ P are absolute, because its best case valueof a fully converged state is zero, and the worst situation, for completely different wavefunctions, is one. On the contrary, the energy scale is arbitrary, and can have positiveand negative values, even some levels with energies very close to zero. It makes the useof the relative error employed in Ref [1, 7] dangerous when the reference energy is very fficient basis for the Dicke Model II (cid:15) implies the need of more precise digitsin the calculated energy, making more difficult the convergence for higher energies. Forthis reason our ∆ E criteria is more stringent than the ∆ P one: every excited state withconverged energy has guaranteed the convergence of its wave function. As the coherentbasis provides many converged states with a single truncation value, it is promising tostudy the presence of Excited States Quantum Phase Transitions (ESQPT), predictedin Dicke-like systems and spin systems for γ values deeply in the superradiant phase[5, 6].
4. Conclusions
To obtain the eigenvalues and eigenvectors of the Dicke Hamiltonian for a finite numberof atoms it is necessary to perform a numerical diagonalization, employing a truncatedboson number space. Two basis, associated with the two integrable limits of theHamiltonian, are used along this work. In the present article we have shown that,in most of the Hamiltonian’s parameter regions including the QPT, the coherent basisrequires a significative smaller truncation. We extended the analysis to the convergencein the wave function, exhibiting both convergence criteria as equivalent, and presentedthe numerical relationships between them. The study of the probability distributionsof the number of bosons was helpful in understanding the differences between the twobasis, and the advantages of the coherent basis. The convergence of the energies and thewave functions was also investigated for the excited states, showing that the coherentbasis is very powerful also in this case, allowing to obtain hundreds of converged stateswith a single truncation value. This findings can be very useful in order to observe thepresence of quantum chaos around the phase transition, as well as to study the excitedstates quantum phase transitions.We thank O. Casta˜nos, R. L´opez-Pe˜na and E. Nahmad for many useful andinteresting conversations.This work was partially supported by CONACyT-M´exico andPAPIIT-UNAM 102811.
References [1] Chen Q H, Zhang Y Y, Liu T and Wang K L 2008
Phys. Rev. A Phys. Rev. A Phys. Rev. E Phys. Rev. Lett. Phys. Rev. Lett. Phys.Rev. A Phys.Rev. E Rev. Mex. Fis. S ibid AIP Conf.Proc. ibid