Interaction of a single mode field cavity with the 1D XY model: Energy spectrum
IInteraction of a single mode field cavity with the 1D XYmodel: Energy spectrum
H Tonchev, A A Donkov, H Chamati
Institute of Solid State Physics, Bulgarian Academy of Sciences,72 Tzarigradsko Chaussée, 1784 Sofia, BulgariaE-mail: [email protected], [email protected], [email protected]
Abstract.
In this work we use the fundamental in quantum optics Jaynes-Cummings model to study theresponse of spin chain to a single mode of a laser light falling on one of the spins, a focused interactionmodel between the light and the spin chain. For the spin-spin interaction along the chain we use the XYmodel. We report here the exact analytical results, obtained with the help of a computer algebra system,for the energy spectrum in this model for chains of up to 4 spins with nearest neighbors interactions, eitherfor open or cyclic chain configurations. Varying the sign and magnitude of the spin exchange couplingrelative to the light-spin interaction we have investigated both cases of ferromagnetic or antiferromagneticspin chains.
1. Introduction
The Jaynes-Cummings [1] model involving the interaction of an atom with a quantized electromagneticfield describes electron transitions between the atom levels induced by the modes of the field. Since itsintroduction this interaction have found many applications. It has been suggested that an ion trap of atomsor ions could be a realization of a quantum computer. In this context the Jaynes-Cummings model wasused in [2] to investigate atom- or ion-field interactions in an ion trap and in [3] in quantum dots. Otheruses of this model are for one photon lasers [4] and for single-photon photo detectors [5]. For somepedagogical texts describing the model see, for example [6, 7].In solid state physics there are many models used to quantum-mechanically describe the interactionbetween the spins in a spin chain. Commonly used models are the quantum Ising [8] and theHeisenberg [9] ones. In the Ising model the spins are allowed the fewest degrees of freedom – only one. Ifthe spins have all the degrees of freedom – three, then the Heisenberg model is used. In the intermediateXY model [10] the spins interact through two degrees of freedom. The XY model is used to studyentanglement [10], spin glasses [11], phase transitions [11], and the W state in quantum computing [12].In the literature, there exist several models describing the collective interaction between theelectromagnetic field and a spin chain. One such model is the Dicke model, often used to study thesuperadiance phase transitions [13]. Another recent model [14] describes the collective interactionbetween classical monochromatic circularly polarized light and the spin chain. Here, we report a novel useof the Jaynes-Cummings model, namely, the study of the response of a spin chain to a single quantummode of a laser light focused on a particlar spin.The description of the XY-Jaynes-Cummings model for the interaction Hamiltonian follows in thenext Section 2, after that we report results obtained by solving the model for the particular case of a finitechain of four spins in Section 3, followed by concluding remarks. a r X i v : . [ qu a n t - ph ] M a y . The Hamiltonian The object of our study is a spin chain of N sites and one mode of a laser field interacting with one of thespins in the chain. The spins form either an open or a closed, cyclic chain, for which the last spin interactswith the first, see Figure 1. We consider chains with nearest neighbor interactions J , in the XY Model,which couples the S x and S y components of the neighboring i , j spins operators S i , j = ( S xi , j , S yi , j , S zi , j ) , and the field is coupled to the k − th spin.The interaction Hamiltonian (cid:98) H reads [15]: (cid:98) H = G (cid:128) aS + k + a † S − k (cid:138) − J N − (cid:88) i = (cid:128) S xi S xi + + S yi S yi + (cid:138) , (1)where the first term is nothing but the Jaynes-Cummings model. Here G incorporates information aboutthe interaction between the mode of the field and the k -th spin, and S + ( S − ) lift (lower) the spin projection,respectively, and are given by S ± = S x ± i S y , with “i” the imaginary unit. They would correspond tothe transitions between the lower and the upper atomic levels in the commonly used formulations of theJaynes-Cummings model. The operators a and a † are the annihilation and creation operators of the modeof the field. The expression with the spin exchange interaction J between the spin dipoles is the XY model.With the “ − ” sign in mind, in the classical limit of spin vectors, J > would mimic a mostly ferromagnetic(FM) alignment in the chains (the S z components being arbitrary), J < an antiferomagnetic (AFM)alignment, and J = would describe independent spins along the chain. Figure 1.
Theorist’s rendering of open (a), or cyclic (b) configurations of four spin chains. The wiggleline represents the photon.For the calculations we choose a basis set of states by first ordering the field states, ( | 〉 , | 〉 ) , where and stand for the number of photons, then for each spin site we have the spin states ( | ↑〉 , | ↓〉 ). Thisgives a set of N + basis states. The spin operators in this basis are expressed through the Pauli matricesby the standard relations, S x , y , z = ħ h σ x , y , z . Since the spin-spin term S xi S xi + + S yi S yi + can be written as ( S + i S − i + + S − i S + i + ) , it useful to define additionally σ ± = ( σ x ± i σ y ) , so by definition S ± = ħ h σ ± . Tosummarize for spin we have the expressions: σ x = (cid:18) (cid:19) , σ y = (cid:18) -ii (cid:19) , σ z = (cid:18) − (cid:19) , and (cid:73) = (cid:18) (cid:19) , (2a) | ↑〉 = (cid:18) (cid:19) , | ↓〉 = (cid:18) (cid:19) , (2b)and for the photon we have a = (cid:18) (cid:19) , a † = (cid:18) (cid:19) , (3a) 〉 = (cid:18) (cid:19) , | 〉 = (cid:18) (cid:19) . (3b)Since the field and the spin operators on different sites act independently, the matrix form for thesubscripted σ i is a tensor product of N + , × matrices as follows (e.g. § σ xi = (cid:73) (cid:124)(cid:123)(cid:122)(cid:125) Field mode ⊗ (cid:73) ⊗ (cid:73) ⊗ · · · ⊗ (cid:73) (cid:124) (cid:123)(cid:122) (cid:125) i − ⊗ σ x (cid:124)(cid:123)(cid:122)(cid:125) i th spin ⊗ (cid:73) ⊗ (cid:73) ⊗ · · · ⊗ (cid:73) (cid:124) (cid:123)(cid:122) (cid:125) N − i terms , (4)and similarly for σ yi , σ ± i . This gives for the field-spin parts of the Hamiltonian: a † σ yj = a † (cid:124)(cid:123)(cid:122)(cid:125) Field mode ⊗ (cid:73) ⊗ (cid:73) ⊗ ... ⊗ (cid:73) (cid:124) (cid:123)(cid:122) (cid:125) j − ⊗ σ y (cid:124)(cid:123)(cid:122)(cid:125) j th spin ⊗ (cid:73) ⊗ (cid:73) ⊗ ... ⊗ (cid:73) (cid:124) (cid:123)(cid:122) (cid:125) N − j terms , (5)and for the spin-spin terms: σ xi σ yj = (cid:73) (cid:124)(cid:123)(cid:122)(cid:125) Field mode ⊗ (cid:73) ⊗ · · · ⊗ (cid:73) (cid:124) (cid:123)(cid:122) (cid:125) i − ⊗ σ x (cid:124)(cid:123)(cid:122)(cid:125) ith spin ⊗ (cid:73) ⊗ · · · ⊗ (cid:73) (cid:124) (cid:123)(cid:122) (cid:125) j − i − ⊗ σ y (cid:124)(cid:123)(cid:122)(cid:125) jth spin ⊗ (cid:73) ⊗ · · · ⊗ (cid:73) (cid:124) (cid:123)(cid:122) (cid:125) N − j terms . (6)
3. The energy spectra for 4 spin chain
We diagonalize equation (1), expressed in matrix form using (5-6) for the case of four spins, N = toobtain the exact solutions for the energy levels. To this end we solve the characteristic equation for E i.e. det (cid:128) (cid:98) H − E (cid:73) × (cid:138) = The Hamiltonian is a × matrix, and accordingly, there are 32 (some multiply degenerate) energylevels, which we found with help of a computer algebra system. For the open spin chain, the results dependon whether the photon falls on the first or the second spin, while the addition of the cyclic interactionsterm would appear to make this distinction irrelevant for the closed chain, as the results show as well.In order to check the computer derived formulae, it is useful to compare the limit J = with the knownsolutions of the two level Jaynes-Cummings model, which are ± G and in units of ħ h . The spectrum in this case consists of 9 distinct energy branches shown in Figure 2. The analyticexpressions , where the superscripts D , Q , or O stand for the degeneracy of each level
2, 4, or times,listed in decreasing order at G = are: E D ( G , J ) = (cid:113) G + J + (cid:112) G J + J , (7a) E Q ( G , J ) = (cid:198) G + J + (cid:112) G − G J + J (cid:112) (7b) E D ( G , J ) = (cid:113) G + J − (cid:112) G J + J , (7c) E Q ( G , J ) = (cid:198) G + J − (cid:112) G − G J + J (cid:112) (7d) E O ( G , J ) = (7e)and E Q = − E Q , E D = − E D , E Q = − E Q , E D = − E D . The (at least) double degeneracy of the levelsfollows from the Kramers theorem (we do not have an explicit magnetic field in the Hamiltonian), wherethe photon acts as an additional spin in the chain for the purposes of the theorem (see e.g. §
60 of [17]). Toetter show the behavior of the levels with predominantly light-spin interaction ( J (cid:28) G ) or predominantlyspin-spin interaction ( G (cid:28) J ) we have used the substitution G = C cos ϕ , J = C sin ϕ , where in theFigure 2(b), C = With this notation, when ϕ ∈ ( − π , 0 ) the chain is an AFM chain, and when ϕ ∈ ( π ) it is a FM chain. Additionally shown is the region ϕ > π , where G formally takes negative values. Thiswould mean an overall change of the sign of the Hamiltonian, and accordingly, we see an energy spectrumthat consists of the same levels, just arranged with opposite signs, so solid lines go to the correspondingsolid lines at the line at ϕ = π . G > < b ) - π / π / π π / φ - - E Figure 2. (Color online) The energy spectra given by equations (7) for a 4 spin chain coupled to a photonat the edge of the chain. (a) Energy surfaces at various finite strengths G and J . (b) The cross sectionsof the energy surfaces at the red (solid line) circle shown in (a). See text for further discussion of thenotations and the symmetries of the spectrum. Close to ϕ = J (cid:28) G , and close to ϕ = π , G (cid:28) J . Due to the extra neighbor to the spin impacted by light, the qualitative change in the spectrum, is that thequadruply degenerate bands in (7) split into doubly degenerate bands, and thus all levels, except the zeroenergy, are now doubly degenerate. The eigenvalues in explicit form look rather cumbersome, so we firstgive the characteristic equation. It could be written in factored form as: det (cid:128) (cid:98) H − E (cid:73) × (cid:138) = E (cid:148) E − ( G + J ) E + ( G + J ) J (cid:151) × (cid:166) E − ( G + J ) E + ( G + G J + J ) E − ( G + G J + G J + J ) E + G J + G J + G J + J (cid:169) = (8)The solutions of this equation are shown in Figure 3, and there are 6 branches in the upper half-plane,with 13 overall distinct solutions. From the first factor we get the octuply degenerate zero level, E O ( G , J ) = (9)from the second factor (expression in the square brakets) and the third factor (curly brackets) the remainingdoubly degenerated eigenvalues. The expressions from the second factor: E Da , b ( G , J ) = − E Dc , d ( G , J ) = (cid:198) G + J ± (cid:112) G + G J + J (cid:112) (10)ook similar to the E Q ( G , J ) (and E Q ( G , J ) ) in equation (7), with the only change in sign in the term inthe second square root: − G J → + G J . The expressions for those from the third factor (the eightremaining lines on the figure), as solutions of fourth order equation for E , look involved and we onlynote that the topmost band, which is given by one of the roots of this fourth order equation, looks similar,with a slight decrease in the initial slope at J → to the topmost band E D in (7). G > < b ) - π / π / π π / φ - - E Figure 3. (Color online) The energy spectra given by equations (8) for a 4 spin chain coupled to a photonat second spin in the chain. Notations are similar to Figure 2.
The 4-spin cyclic configuration is described by the following characteristic equation: det (cid:128) (cid:98) H C − E (cid:73) × (cid:138) = E (cid:128) E − G (cid:138) (cid:148) E − ( G + J ) E + G J (cid:151) × (cid:166) E − ( G + J ) E + ( G + G J + J ) E − G J − G J (cid:169) = (11)whose 13 distinct solutions are shown in the Figure 4. (cid:98) H C compared to (1), has an additional spin-spin interaction term between the spins on sites N and ; thus closing the chain. Except for the octuplydegenerate zero level, E O ( G , J ) = (12)the energy spectrum consists of doubly degenerate levels. The second factor gives: E DA , B ( G , J ) = ± G , (13)which is independent of J . In Figure 4(b) it appears as a cosine (cyan) line. We have checked, that for a3-spin chain there also is an energy level that is linear in G, but has an additional, also linear, dependenceon J . Here such levels come from a mixture of different basic spin configurations, for which the magneticpart of the Hamiltonian (1) produces canceling contributions. The third, quadratic in E , factor gives: E Da , b ( G , J ) = − E Dc , d ( G , J ) = (cid:198) G + J ± (cid:112) G + J (cid:112) (14)The last factor, curly brackets in (11), is cubic in E and gives the remaining 12 roots (counting thedegeneracy), or 6 lines on the figure, and the topmost band is one of these roots. The explicit formulaeare again somewhat cumbersome, and we skip writing them explicitly. > < b ) - π / π / π π / φ - - E Figure 4. (Color online) The energy spectra from equation (11) for a 4 spin cyclic chain coupled to aphoton. Notations are similar to Figure 2.
4. Conclusions
In this report we have discussed some analytical results of applying the Jaynes-Cummings model to anXY- AFM or FM spin chain. Because we limited ourselves to a single mode photon field, the resultingHamiltonian in matrix form resembles the case of a pure spin chain with one extra spin site inducinganisotropy. Nevertheless the light-spin interaction term does bring a novel effect: it does not commutewith the total spin S total z = (cid:80) Ni = σ zi . This means that the eigenstates of the XY-Jaynes-CummingsHamiltonian consist of some mixtures of the product basis states, which have definite spin moment value.For that reason only for G = ( ϕ = π point in figures 2, 3, 4) it is possible to assign definite spinmoment to the energy levels. As can be seen from the equations, the 4-spin chain seems to be the limit forthe analytical formulae, since more sites would require solving polynomial equations of more than 5-thpower, which may not have a closed form. It was possible to obtain numerical data for the energy levelsof a chain made of more than 4 spins, this would be reported in some future work [18]. Acknowledgments
This work was supported by EU FP7 INERA project grant agreement number 316309. We thankN. Tonchev for bringing our attention to [19].
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