Dicke model: entanglement as a finite size effect
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J u l Di ke model: entanglement as a (cid:28)nite size e(cid:27)e tOleksandr Tsyplyatyev and Daniel LossDepartment of Physi s, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland(Dated: De ember 4, 2018)We analyze the Di ke model at zero temperature by matrix diagonalization to determine theentanglement in the ground state. In the in(cid:28)nite system limit the mean (cid:28)eld approximation predi tsa quantum phase transition from a non-intera ting state to a Bose-Einstein ondensate at a threshold oupling. We show that in a (cid:28)nite system the spin part of the ground state is a bipartite entangledstate, whi h an be tested by probing two parts of the spin system separately, but only in a narrowregime around the threshold oupling. Around the resonan e, the size of this regime is inverselyproportional to the number of spins and shrinks down to zero for in(cid:28)nite systems. This spinentanglement is a non-perturbative e(cid:27)e t and is also missed by the mean-(cid:28)eld approximation.Coherent intera tion between ele tromagneti photon(cid:28)elds and matter attra ted interest a long time ago [1℄with renewed attention gained in the last de ade due tosigni(cid:28) ant developments in the experimental te hniquesin various areas of physi s. A hievement of Bose Ein-stein ondensation of old atomi gases in ele tromag-neti traps enabled the oherent oupling of hyper(cid:28)nestates of atoms to a single photon mode of an opti alresonator [2℄. Advan es in the semi ondu tor te hnologyallowed to obtain opti al mi ro avities where ele tron-hole ex itations inside the semi ondu tor quantum wellare strongly oupled to an eigenmode of the opti al res-onator [3℄. Strong oupling of a single mode of a trans-mission line resonator to a Cooper pair box [4℄ and aquantized mode of an opti al rystal avity to severalsemi ondu tor quantum dots [5℄ have been demonstratedas a possible way to a quantum omputing devi e [6℄.Theoreti al understanding of all these systems is basedon a model proposed by Di ke [7℄ whi h des ribes N spins1/2 (identi al two-level systems) with splitting energy ǫ oupled to a single mode of ele tromagneti (cid:28)eld ω .It was shown that this model is exa tly diagaonalisable[8℄. At zero temperature it undergoes a quantum phasetransition from a non intera ting state with unpopulatedbosoni mode to a ondensed state with with a highlypopulated bosoni mode [9℄ if oupling between the bo-son and a single spin g is greater than a threshold value.In the thermodynami limit a phase transition o urs inthe region of strong oupling if temperature is less thana riti al temperature whi h an be des ribed by a Bo-golyubov Hamiltonian similarly to the pairing model ofsuper ondu tivity [10℄. Re ently, a variational wave fun -tion approa h to the generalised Di ke model was used[11℄ to des ribe Bose Einstein ondensation of ex iton-polaritons in a semi ondu tor opti al avity.In this paper we analyse the Di ke model at zerotemperature for a (cid:28)nite N using matrix diagonalisa-tion methods. We (cid:28)nd that for the parti ular ouplingstrength, ωǫ < g . ωǫ (cid:18) N (cid:19) , (1)the ground state of the spin subsystem is a bipartite en-tangled state and it is not entangled outside of this re- gion. The lower bound of the inequality is onset of thequantum transition des ribed by the mean (cid:28)eld theory[12℄. The upper bound is the ondition to have onlysingly populated bosoni mode in the ground state. Theapproximated value is the result of /N expansion aroundthe resonan e.In the thermodynami limit the ground state pro-je ted onto the subspa e of N spins is not entangledas it is a produ t state. For a weak oupling belowthe quantum transition the ground state is a produ tof all unex ited single spin states. For a strong ou-pling above the transition threshold the ground stateis also a produ t state of all single spin states [11℄ asa result of the mean-(cid:28)eld approximation. In a (cid:28)nitesystem the mean-(cid:28)eld approximation is not appli ablein a small region above the transition threshold wherethe expe tation value of the boson is of the order of1 and its (cid:29)u tuations are also of the order of 1. Theground state in this region is a superposition of the un-ex ited state of all spins and a spin state with only onesingle spin-(cid:29)ip ex itation, the N spin W-state, | W i =( |↑↓↓ . . . i + |↓↑↓ . . . i + |↓↓↑ . . . i + . . . ) / √ N . The W-state an also be interpreted as 'magnon state' at van-ishing wave ve tor [13℄.Furthermore, the W-state an be onsidered as a bi-partite entangled state in the following sense. Dividingall spins into two groups [14℄ the W-state is a Bell statein the subspa e restri ted by only one spin-(cid:29)ip ex itationabove the unex ited states of ea h group of the spins, seeEq. (13). In the ourse of a bipartite measurement ifthe (cid:28)rst group is found in the ex ited state then the se -ond group is proje ted onto the unex ited state and ifthe (cid:28)rst group is found in the unex ited state then these ond group is proje ted onto the ex ited state.We diagonalize the Di ke model for N spins 1/2 ou-pled to a single bosoni mode H = ωb † b + ǫ X j S zj + g √ N X j (cid:0) S + j b + S − j b † (cid:1) , (2)where the sum runs over N spin-1/2 operators S j thatobey the ommutation relations [ S αi , S βj ] = ǫ αβγ δ ij S γi ,and b ( b † ) is standard bosoni annihilation ( reation) op-erator.The Di ke model possesses the following onservedquantities. One is the number of ex itations of the ou-pled spin-boson system, L = n + J z , (3)expressed in terms of the z- omponent of total spin oper-ator J α = P j S αj , α = x, y, z , and the o upation numberoperator n = b † b of the boson mode. Note that J α and n are not onserved separately. The eigenvalues of L arethe so- alled ooperation numbers c , given by the sumof expe tation values of n and J z . A se ond onservedquantity is the total spin, J = J z + ( J + J − + J − J + ) / with eigenvalues j ( j + 1) .We represent the Di ke Hamiltonian (2) in a basiswhere J and L are blo k-diagonal. Within a blo k ofgiven c and j the remaining degrees of freedom an belabeled by the eigenvalues m of J z . In the representation | c, j, m i ea h blo k has a tridiagonal form. The diagonalmatrix elements represent the energies of states ontain-ing c − m bosons and m ex ited spins, h m | H | m i = ω ( c − m ) + ǫm. (4)The (cid:28)rst above- and below-diagonal matrix elements aretransition amplitudes onne ting all pairs of states whi hdi(cid:27)er by just one (cid:29)ipped spin, h m | H | m + 1 i = g √ N p ( c − m ) ( j ( j + 1) − m ( m + 1)) . (5)The size of ea h blo k is limited by the fa t that − N/ ≤ m ≤ N/ for j = N/ . However,the upper bound on m isfurther onstrained by the ooperation number c whi hhas the lower bound − N/ for a blo k with no bosonspresent in the state.The ground state of Eq. (2) is the lowest energy stateof all blo ks Eqs. (4,5). For di(cid:27)erent values of the pa-rameters ω , ǫ , and g the ground state an have di(cid:27)erent c ′ s and j ′ s (see below).We onsider now the ase with the bosoni ( ω ) and spinex itation energies ( ǫ ) lose to ea h other. For the un- oupled ase, g = 0 , the ground state ontains no bosonsand no ex ited spins (i.e. all spins are, say, down). FromEqs. (3) and (4) we see that the ooperation number ofthis state is then c = − N/ with orresponding groundstate energy E = − N ǫ/ . For (cid:28)nite but still small ou-pling, g ≪ , we an use a perturbative approa h toremove the boson mode via a S hrie(cid:27)er-Wol(cid:27) transfor-mation to obtain an e(cid:27)e tive spin-Hamiltonian [16℄, H = ωb † b + (cid:18) ǫ + 2 g b † bN ( ǫ − ω ) (cid:19) X j S zj + g N ( ǫ − ω ) X i,j S + i S − j , (6)where the spin-boson mixing is eliminated up to the se -ond order in g [15℄, introdu ing an e(cid:27)e tive XY oupling Figure 1: Cooperation number c of the ground state obtainedby matrix diagonalization of the Di ke model for N = 3 . The(cid:28)rst solid line from the left indi ates the quantum transitionto the strongly orrelated non-perturbative regime, and these ond solid line is determined by Eq. (9). The dashed line orresponds to the resonan e ǫ = ω .between the spins within a subband of given boson o - upation number n (with n being onserved under thee(cid:27)e tive Hamiltonian Eq. (6)). Su h a perturbative ap-proa h gives a qualitatively orre t des ription of theenergies and wave fun tions until g rosses a thresholdvalue g c where a transition to a strongly orrelated non-perturbative regime takes pla e (see below). The ooper-ation number c of the ground state for the Di ke Hamilto-nian Eq. (2) is plotted in Fig.1 for the solution for N = 3 .There are several regimes: for small g the ground state ofthe system is de(cid:28)ned by a regime (bla k) with c = − N/ and h n i = 0 . Then, with in reasing oupling a quan-tum phase transition at g c takes pla e to a new regimewith c > − N/ where h n i > , and where subbands withsharp bosoni o upation numbers no longer exist.In reasing g further above the transition threshold asequen e of states with c = 1 − N/ , c = 2 − N/ , andso on be omes subsequently the ground state due to theintera tion energy. The ground state with c greater but lose to − N/ an not already be approximated with helpfrom the perturbation theory as oupling is too strongand an not yet be approximated using the mean (cid:28)eldapproa h as the (cid:29)u tuations of the order parameter aretoo large ompared to its mean. Here we diagonalizethe Hamiltonian Eq. (2) using the matri es Eqs. (4,5).Sizes of the matri es in this regime are limited by the ooperation number c . In the presentation | c, j, m i thematri es are not larger then × and × for c = 1 − N/ and c = 2 − N/ respe tively. The lowest eigenenergiesof these matri es are E = − ( N − ǫ ω − s g + (cid:18) ω − ǫ (cid:19) (7)for c = 1 − N/ and E = − ( N − ǫ ω − g r − N + ω − ǫ N − . (8)for c = 2 − N/ . The last expression is a result of ex-pansion at the resonan e in powers of ω − ǫ . Both E and E belong to subblo ks with j = N/ in a ordan ewith a theorem from [10℄. Comparing E and E we(cid:28)nd g c = √ ωǫ . The same ondition to have a non zeropopulation of the bosoni mode in the ground state wasestablished in [9℄.Comparing E and E near the resonan e ω = ǫ we(cid:28)nd that a state with c = 2 − N/ be omes the groundstate when g ex eeds some value g given by g = ǫ + ω + ( ω − ǫ ) / (2 N − (cid:16) p − / N − (cid:17) . (9)Thus, g < g together with g > g c de(cid:28)ne the regimeof the model parameters where matrix diagonalization isthe only way to study the Di ke model. The upper boundof Eq. (1) oin ides with Eq. (9) for ǫ ≈ ω and N ≫ .In reasing g further, we an determine the boundariesbetween the ground states with di(cid:27)erent values of c ( i.e. c = 3 − N/ , − N/ , . . . ) by numeri al diagonalization,as shown in Fig.1 for N = 3 . A result similar to Eqs. (7,8, 9) was obtained in [17℄ but the des reete jumps in the ooperation number of the ground state were interpretedas an in(cid:28)nite sequen e of instabilities.In the strong oupling regime, g ≫ g c , the mean (cid:28)eldapproa h provides a good approximation to the exa tground state. Indeed, introdu ing the expe tation valueof the bosoni operator B = h b i and negle ting quantum(cid:29)u tuations around it, the Di ke Hamiltonian Eq. (2)be omes H = ω |B| + X j (cid:18) ǫS zj + g √ N (cid:0) S + j B + S − j B ∗ (cid:1)(cid:19) . (10)Eigenstates of this Hamiltonian are produ t states of theN spins, and thus are manifestly not entangled. Diago-nalization of the × - matri es for ea h spin and subse-quent minimization of the sum of the lowest eigenenergiesover |B| gives the following self- onsisten y (mean-(cid:28)eld)equation ω = g q |B| g /N + ǫ , (11)whi h des ribes a quantum phase transition at a thresh-old value of the oupling strength g c that we have alreadyfound from the matrix diagonalization. Figure 2: Expe tation value of n al ulated by matrix diago-nalization of Eqs. (4,5) - solid line and the mean (cid:28)eld result- dashed line. Plots are for N = 3 spins and at resonan e ǫ = ω . The grey area is the regime de(cid:28)ned by Eq.(1) wherethe ooperation number of the ground state is c = 1 − N/ .The mean (cid:28)eld approximation is satisfa tory even fora system of only a few spins. To see this, we om-pare for N = 3 spins in Fig.2 the expe tation valueof n taken with respe t to the exa t ground state ofEqs. (4,5) with |B| given in Eq. (11) at the resonan e ω = ǫ . There we see that the largest deviation is in theregion of intermediate oupling strength where the oop-eration number of the exa t ground state is c = 1 − N/ ,grey area in Fig.2. For weak oupling g < g c theground state oin ides with the non-intera ting one and h n i = |B| = 0 . The quantum (cid:29)u tuations of n omparedto its mean value are already small for the oupling or-responding to the exa t ground state with c = 4 − N/ , (cid:16)(cid:10) n (cid:11) − h n i (cid:17) / h n i ≃ . . Thus, in the strong ouplingregime the approximation of negle ting these (cid:29)u tuationsin Eq. (10) is already good for g & g irrespe tive of N .The ground state an be hara terized in termsof entanglement between di(cid:27)erent parts of the spin-subsystem. In the weak oupling regime, the groundstate, being a dire t produ t of (unex ited) spin states,is not entangled. In the regime of strong oupling,where the approximate Hamiltonian Eq. (10) is valid,the ground state is also a produ t of the individual spinstates whi h, thus, also has no entanglement of any pairof spins. The ground state in the intermediate regionEq. (1) has to be found by matrix diagonalization andwill be analyzed below.Diagonalization of the matrix Eq. (4,5) for c = 1 − N/ and g = √ ωǫ gives a ground state in the regime de(cid:28)nedby Eq. (1). We hange the mixed spin-boson representa-tion | c, l, m i to a separate representation of spins and theboson | n , l, m i , where n is the bosoni o upation num-ber. Then, tra ing out the bosoni degree of freedom n we obtain the redu ed density matrix of the spins only,whi h in the representation of J z eigenstates | m i is givenby ˆ ρ = ǫǫ + ω (cid:12)(cid:12)(cid:12)(cid:12) − N (cid:29) (cid:28) − N (cid:12)(cid:12)(cid:12)(cid:12) + ωǫ + ω (cid:12)(cid:12)(cid:12)(cid:12) − N (cid:29) (cid:28) − N (cid:12)(cid:12)(cid:12)(cid:12) . (12)There is a (cid:28)nite probability to (cid:28)nd either all spins in the ompletely polarized state or in the W-state |− N/ i .The W-state is a Bell state for N = 2 . For N > it is a bipartite entangled state. The set of spins anbe divided into two equal groups onsisting of N/ spinsea h (assuming N even). In the basis | m i | m i , m and m are the eigenvalues of the operators J z belongingto the (cid:28)rst (se ond) group, the state |− N/ i is (cid:18)(cid:12)(cid:12)(cid:12)(cid:12) − N (cid:29) (cid:12)(cid:12)(cid:12)(cid:12) − N (cid:29) + (cid:12)(cid:12)(cid:12)(cid:12) − N (cid:29) (cid:12)(cid:12)(cid:12)(cid:12) − N (cid:29)(cid:19) / √ , (13)i.e. a measurement of one group proje ts another grouponto the de(cid:28)nite state. Therefore in the intermediateregion Eq.(1) the ground state of the Hamiltonian Eq. (2)is a bipartite entangled state. Note that N spin W-statedoes not belong to the lass of N spin entangled statesprepared by squeezing [18℄. The squeezing parameter is not de(cid:28)ned for the N = 2 W-state and is greater thanone for
N > W-states.Let the spins have di(cid:27)erent splitting energies ǫ j inEq.(2) instead of the ase ǫ j ≡ ǫ whi h we have onsidered in the paper. A small variation h ǫ j i − ǫ j ≪ ǫ j , h ǫ j i being the average over splitting ener-gies of all spins, an be treated using perturbation the-ory and will not a(cid:27)e t our results, Eqs.(1,13) mu h.For instan e, the N = 2 W-state in Eq.(13) be omes (cid:0) (1 − ( ǫ − ǫ ) / √ g ) |↑↓i + |↓↑i (cid:1) / √ at the resonan e ǫ = ω . The amplitudes of di(cid:27)erent omponents in the sin-glet will alter slightly while the W-state will not hangequalitatively. For a more detailed analysis of the inho-mogeneous Di ke model see [19℄.In on lusion, we analyzed the Di ke model for a (cid:28)nite-size system by matrix diagonalization. We found thatthe ground state is a bipartite entangled state only in anarrow regime of parameters next to the quantum phasetransition at zero temperature. The ground state in thisregime annot be obtained from mean-(cid:28)eld theory whi happroximates the ground state as a produ t state. Alsoperturbation theory is not valid in this regime. For anin(cid:28)nite system width of the orresponding parametri re-gion vanishes as /N if system is lose the resonan e.We a knowledge (cid:28)nan ial support from the Swiss NF,NCCR Nanos ien e Basel, and JST ICORP.[1℄ C. W. Gardiner and P. Zoller, Quantum Noise, Springer,2004.[2℄ F. Brenne ke, T. Donner, S. Ritter, T. Bourdel, M. Köhl,and T. Esslinger, Nature 450, 268 (2007).[3℄ J. Kasprzak, M. Ri hard, S. Kundermann, A. Baas, P.Jeambrun, J. M. J. Keeling, F. M. Mar hetti, M. H.Szyma(cid:0)nska, R. Andrè, J. L. Staehli, V. Savona, P. B.Littlewood, B. Deveaud, and Le Si Dang, Nature 443,409 (2006).[4℄ A. Wallra(cid:27), D. I. S huster, A. Blais, L. Frunzio, R.- S.Huang, J. Majer, S. Kumar, S. M. Girvin and R. J.S hoelkopf, Nature 431, 162 (2004) .[5℄ K. Hennessy, A. Badolato, M. Winger, D. Gera e, M.Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. (cid:157)mamo§lu,Nature 445, 896 (2007).[6℄ A. (cid:157)mamo§lu, D. D. Aws halom, G. Burkard, D. P. Di-Vin enzo, D. Loss, M. Sherwin, and A. Small, Phys. Rev.Lett. 83, 4204 (1999).[7℄ R. H. Di ke, Phys. Rev. 93, 99 (1954).[8℄ M. Tavis and F. W. Cummings, Phys. Rev. 170, 379(1968).[9℄ G. S harf, Helv. Phys. A ta 43, 806(1970).[10℄ K. Hepp and E. H. Lieb, Ann. Phys. (N.Y.) 76, 360(1973).[11℄ P. R. Eastham and P. B. Littlewood, Phys. Rev. B 64, 235101 (2001).[12℄ R. Bonifa io and G. Preparata, Phys. Rev. A 2, 336(1970).[13℄ J. S. Pratt, Phys. Rev. B 73, 184413 (2006).[14℄ Independent probe of di(cid:27)erent spins is possible as theyare spatially separated. For instan e, size of the ex iton-polariton ondensates in [3℄ is few µm ; loud of oldatoms in [4℄ has igar shape of length that an be 100 µm .[15℄ The Hamiltonian Eq.(6) was obtained using perturbationtheory for ǫ ≈ ω ≫ g / ( ǫ − ω ) . Expressing XY term fromthe total spin onseravtion law the Hamiltonian Eq.(6) is H = ωb † b + “ ǫ + g (2 b † b +1) N ( ǫ − ω ) ” P j S zj − g N ( ǫ − ω ) “P j S zj ”2