Entanglement and U(D)-spin squeezing in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin–Meshkov–Glick D-level atom models

Quantum Information Processing (2021) 20:304 https://doi.org/10.1007/s11128-021-03218-6 Entanglement and U(D)-spin squeezing in symmetric multi-quDit systems and applications to quantum phase transitions in Lipkin–Meshkov–Glick D-level atom models Manuel Calixto1·Alberto Mayorgas1·Julio Guerrero2,3 Received: 23 April 2021 / Accepted: 11 August 2021 © The Author(s) 2021 Abstract Collective spin operators for symmetric multi-quDit (namely identical D-level atom) systems generate a U(D)symmetry. We explore generalizations to arbitrary Dof SU(2)-spin coherent states and their adaptation to parity (multi-component Schrödinger cats), together with multi-mode extensions of NOON states. We write level, oneand two-quDit reduced density matrices of symmetric N-quDit states, expressed in the last two cases in terms of collective U(D)-spin operator expectation values. Then, we evaluate level and particle entanglement for symmetric multi-quDit states with linear and von Neumann entropies of the corresponding reduced density matrices. In particular, we analyze the numerical and variational ground state of Lipkin–Meshkov–Glick models of 3-level identical atoms. We also propose an extension of the concept of SU(2)-spin squeezing to SU(D)and relate it to pairwise D-level atom entanglement. Squeezing parameters and entanglement entropies are good markers that characterize the different quantum phases, and their corresponding critical points, that take place in these interacting D-level atom models. Keywords Quantum phase transitions ·Many-body systems ·Symmetric quDits · Coherent states ·Parity ·Pairwise entanglement ·Spin squeezing BManuel Calixto [email protected] Alberto Mayorgas [email protected] Julio Guerrero [email protected] 1Department of Applied Mathematics and Institute Carlos I of Theoretical and Computational Physics (iC1), University of Granada, Fuentenueva s/n, 18071 Granada, Spain 2Department of Mathematics, University of Jaen, Campus Las Lagunillas s/n, 23071 Jaen, Spain 3Institute Carlos I of Theoretical and Computational Physics (iC1), University of Granada, Fuentenueva s/n, 18071 Granada, Spain 0123456789().: V,-vol 123 304 Page 2 of 27 M. Calixto et al. 1 Introduction The development of quantum technologies partially relies on the efficient preparation of nonclassical atomic states and the exploitation of many-body entanglement [1–3] and spin squeezing [4], specially to enhance the sensitivity of precision measurements like in quantum metrology. Such is the case of many-body entangled (and spin-squeezed) states of cold atoms generated for instance in atom–atom collisions in Bose–Einstein condensates (BECs) [2]. Indistinguishable particles are naturally correlated due to exchange symmetry and there has been a long-standing debate on whether identical particle entanglement is physical or merely a mathematical artifact (see, e.g., [5] and references therein). Recent work like [6] shows indeed entanglement between identical particles as a consistent quantum resource in some typical optical and cold atomic systems with immediate practical impact. It can also be extracted and used as a resource for standard quantum information tasks [7]. Moreover, multi-partite entanglement of symmetric multi-qubit systems can add robustness and stability against the loss of a small number of particles [5]. Understanding the role of the indistinguishableness of identical bosons and quantum entanglement has been the subject of many recent work (see, e.g., [8,9] and references therein). We know that, for N=2 particles, any quantum state is either separable or entangled. However, for N>2, one needs further classifications for multi-partite entanglement [10]. Many different measurements have been proposed to detect and quantify quantum correlations [3]. We shall restrict ourselves to bipartite entanglement of pure states, where necessary conditions for separability in arbitrary dimensions exist. In order to quantify entanglement between identical particles we shall follow Wang and Mølmer’s [11] procedure, who wrote the reduced density matrix (RDM) of oneand twoqubits, extracted at random from a symmetric multi-qubit state ψ, in terms of expectation values  Sψof collective spin operators  S. For pairwise entanglement, the concurrence C(an entanglement measure introduced by Wootters [12]) was calculated for spin coherent states (SCSs) [13,14], Dicke and Kitagawa-Ueda [15] spin squeezed states, together with mixed states of Heisenberg models. Kitagawa-Ueda [15]spin squeezed states are the spin version of traditional parity-adapted CSs, sometimes called “Schrödinger cat states” since they are a quantum superposition of weakly overlapping (macroscopically distinguishable) quasi-classical coherent wave packets. They were first introduced by Dodonov, Malkin and Man’ko [16] and later adapted to more general finite groups than the parity group Z2={1,−1}[17]. In this article, we shall introduce U(D)SCSs (denoted DSCSs for brevity) adapted to the parity symmetry Z2×D−1 ... ×Z2, which are a D-dimensional generalization of U(2)Schrödinger cats, and we shall refer to them as DCATs for short. In general, parity-adapted CSs are a special set of “nonclassical” states with interesting statistical properties (see [18–20] for several seminal papers). Parity-adapted DSCSs arise as variational states reproducing the energy and structure of ground states in Lipkin–Meshkov–Glick (LMG) D-level atom models (see [21] and later in Sect. 5). In Ref. [22], the concurrence Cwas related to the spin squeezing parameter ξ2= 4(Sn⊥)2/Nintroduced by [15], which measures spin fluctuations in an orthogonal 123 Entanglement and U(D)-spin squeezing in symmetric multi-quDit… Page 3 of 27 304 direction to the mean value n∝ Swith minimal variance. Spin squeezing means that ξ2<1, that is, when the variance (Sn⊥)2is smaller than the standard quantum limit S/2=N/4 (with Sthe spin) attained by (quasi-classical) SCSs. This study shows that spin squeezing is related to pairwise correlation for even and odd parity multiqubit states. Squeezing is in general a redistribution of quantum fluctuations between two noncommuting observables Aand Bwhile preserving the minimum uncertainty product ψAψB≥1 2|[A,B]ψ|. Roughly speaking, it means to partly cancel out fluctuations in one direction at the expense of those enhanced in the “conjugated” direction. For the standard radiation field, it implies the variance relation (q)2<1/4 for quadrature (position qand momentum p) operators. For general U(D)spin systems of identical D-level atoms or “quDits,” the situation is more complicated and we shall extend the D=2 definition of spin squeezing to general D. Spin squeezing can be created in atom systems by making them to interact with each other for a relatively short time in Kerr-like medium with “twisting” nonlinear Hamiltonians like H=λS2 x[15], generating entanglement between them [23,24]. This effective Hamiltonian can be realized in ion traps [25] and has produced fourparticle entangled states [26]. There are also some proposals for two-component BECs [23]. Likewise, the ground state at zero temperature of Hamiltonian critical many-body systems possessing discrete (parity) symmetries also exhibits a cat-like structure. The parity symmetry is spontaneously broken in the thermodynamic limit N→∞and degenerated ground states arise. Parity-adapted coherent states are then good variational states, reproducing the energy of the ground state of these quantum critical models in the thermodynamic limit N→∞, namely in matter-field interactions (Dicke model) of two-level [27,28] and three-level [29,30] atoms, BEC [31], U(3) vibron models of molecules [32,33], bilayer quantum Hall systems [34] and (LMG) models for two-level atoms [35–37]. Quantum information (fidelity, entropy, fluctuation, entanglement, etc.) measures have proved to be useful in the analysis of the highly correlated ground state structure of these many-body systems and the identification of critical points across the phase diagram. Special attention must be paid to the deep connection between entanglement, squeezing and quantum phase transitions (QPTs); see [1,4] and references therein. In this article, we want to explore squeezing and interparticle and interlevel quantum correlations in symmetric multi-quDit systems like the ones described by critical LMG models of identical D-level atoms (see, e.g., [21,38–40,40–42]forD=3level atom models). The literature mainly concentrates on two-level atoms, displaying a U(2)symmetry, which is justified when we make atoms to interact with an external monochromatic electromagnetic field. However, the possibility of polychromatic radiation requires the activation of more atom levels and increases the complexity and richness of the system (see, e.g., [29]). In any case, this also applies to general interacting boson models [43] and multi-mode BECs with two or more boson species. Collective operators generate a U(D)spin symmetry for the case of D-level identical atoms or Dboson species (quDits). Recently [21], we have calculated the phase diagram of a three-level LMG atom model. Here, we want to explore the connection between entanglement and squeezing with QPTs for this symmetric multi-qutrit system. For this purpose, we extend to Dlevels the usual definition of Dicke, parityadapted SCSs and NOON states, and propose linear and von Neumann entropies of 123 304 Page 4 of 27 M. Calixto et al. certain reduced density matrices as a measure of interlevel and interparticle entanglement. We also introduce a generalization of SU(2)spin squeezing to SU(D). The organization of the paper is as follows. In Sect. 2, we introduce collective U(D) spin operators Sij, their boson realization, their matrix elements in Fock subspaces of Nsymmetric quDits, DSCSs and their adaptation to parity (“DCATs”), and a generalization of NOON states to D-level systems (“NODONs”). In Sect. 3,wegiveabrief overview on the concepts and measures of interlevel and interparticle entanglement, considering different bipartitions of the whole system, that we put in practice later in Sect. 6. As entanglement measures, we concentrate on linear (impurity) and von Neumann entropies. We compute entanglement between levels and atoms for DSCSs, DCAT and NODON states. In Sect. 4, we extend Kitagawa-Ueda’s definition of SU(2) spin squeezing to SU(D)and we also connect it with two-quDit entanglement introduced in the previous section. In Sect. 5, we introduce D-level Lipkin–Meshkov–Glick (LMG) atom models (we particularize to D=3 for simplicity) and study their phase diagram and critical points. In Sect. 6, we analyze the ground state structure of the three-level LMG atom model across the phase diagram with the quantum information measures of Sect. 3and the SU(3)-spin squeezing parameters of Sect. 4, thus revealing the role of entanglement and squeezing as signatures of quantum phase transitions and detectors of critical points. We only compute linear entropy in Sect. 6, since it is easier to compute than von Neumann entropy and eventually provides similar qualitative information for our purposes; the interested reader can consult Refs. [44,45]fora more general study on the relation between both entropies. Finally, Sect. 7is devoted to conclusions. 2 State space, symmetries and collective operator matrix elements Those readers acquainted with the boson realization of U(D)spin operators and coherent states can skim read all the way to equation (16), just to introduce essential notation and necessary formulas. We consider a system of Nidentical atoms of Dlevels (N quDits in the quantum information jargon). Let us denote by Eij =|ij|the Hubbard operator describing a transition from the single-atom level |jto the level |i, with i,j=1,...,D. These are a generalization of Pauli matrices for qubits (D=2), namely E12 =σ+,E21 =σ−,E11 −E22 =σ3and E11 +E22 =σ0(the 2 ×2 identity matrix). The expectation values of Eij account for complex polarizations or coherences between levels for i= jand occupation probability of the level i for i=j.TheEij represent the D2step operators of U(D), whose (Cartan–Weyl) matrices l|Eij|k=δilδjk fulfill the commutation relations Eij,Ekl=δjkEil −δil Ekj.(1) Let us denote by Eμ ij,μ=1,...,Nthe embedding of the single μ-th atom Eij operator into the N-atom Hilbert space; namely, E3 ij =1D⊗1D⊗Eij ⊗1Dfor N=4, with 1Dthe D×Didentity matrix. The collective U(D)-spin (D-spin for short) operators are 123 Entanglement and U(D)-spin squeezing in symmetric multi-quDit… Page 5 of 27 304 Sij = N  μ=1 Eμ ij,i,j=1,...,D.(2) They are the generators of the underlying U(D)dynamical symmetry with the same commutation relations as those of Eij in (1). When focusing on two levels i>j,we might prefer to use  J(ij)=(J(ij) x,J(ij) y,J(ij) z), J(ij) x=Sij +Sji 2, J(ij) y=iSij −Sji 2,J(ij) z=Sjj −Sii 2,(3) (roman i denotes the imaginary unit throughout the article) with commutation relations [J(ij) x,J(ij) y]=iJ(ij) z(and cyclic permutations of x,y,z), which is an embedding of D(D−1)/2SU(2)subalgebras into U(D). Although the form (3)forD-spin operators could be more convenient to extrapolate all the D=2 level machinery to arbitrary D, we shall still prefer the form (2) (at least in this paper), which allows for more compact formulas. The DN-dimensional Hilbert space is the N-fold tensor product [CD]⊗N.The tensor product representation of U(D)is reducible and decomposes into a Clebsch– Gordan direct sum of mixed symmetry invariant subspaces. Here, we shall restrict ourselves to the N+D−1 N-dimensional fully symmetric sector (see [21] for the role of other mixed symmetry sectors), which means that our Natoms are indistinguishable (bosons). Denoting by a† i(resp. ai) the creation (resp. annihilation) operator of an atom in the i-th level, the collective D-spin operators (2) can be expressed (in this fully symmetric case) as bilinear products of creation and annihilation operators as (Schwinger representation) Sij =a† iaj,i,j=1,...,D.(4) Sii is the operator number of atoms at level i, whereas Sij,i= jare raising and lowering operators. The fully symmetric representation space of U(D)is embedded into Fock space, with Bose–Einstein–Fock basis (| 0denotes the Fock vacuum) |n=|n1,...,nD=(a† 1)n1...(a† D)nD √n1!...nD!| 0,(5) when fixing n1+···+nD=N(the linear Casimir C1=S11 +···+SDD)tothe total number Nof atoms. There are other realizations of D-spin operators in terms of more than Dbosonic modes (e.g., when each level jcontains Mdegenerate orbitals), which describe mixed symmetries [46–48], but we shall not consider them here. 123 304 Page 6 of 27 M. Calixto et al. Collective D-spin operators (4) matrix elements are given by m|Sij|n=(ni+1)njδmi,ni+1δmj,nj−1 D  k=i,j δmk,nk,∀i= j, m|Sii|n=niδm,n.(6) The expansion of a general symmetric N-particle state ψin the Fock basis will be written as |ψ= n cn|n=  n1+···+nD=N cn1,...,nD|n1,...,nD,(7) where is a shorthand for the restricted sum. D-spin operator expectation values (EVs) can then be easily computed as Sijψ=ψ|Sij|ψ= n ¯cnijcn(ni+1)nj,i= j,Siiψ= n ni|cn|2, (8) where we have used (6) and where by nij we mean to replace ni→ni+1 and nj→nj−1inn. Among all symmetric multi-quDit states, we shall pay special attention to U(D) SCSs (DSCSs for short) |z=|(z1,z2,...,zD)= 1 √N!z1a† 1+z2a† 2+···+zDa† D |z1|2+|z2|2+···+|zD|2N | 0,(9) which are labeled by complex points z=(z1,...,zD)∈CD. To be more precise, there is an equivalence relation: |z∼|zif z=qzfor any complex number q= 0, which means that |zis actually labeled by class representatives of complex lines in CD, that is, by points of the complex projective phase space CPD−1=[CD/∼] = U(D)/[U(1)×U(D−1)]. A class/coset representative can be chosen as ˜ z=z/ziwhen zi= 0, which corresponds to a certain patch of the manifold CPD−1. This is equivalent to choose ias a reference level and set zi=1. For the moment, we shall allow redundancy in z to write general expressions, although we shall eventually take i=1 as a reference (lower energy) level and set z1=1 in Sect. 5. DSCSs (multi-nomial) can be seen as BECs of Dmodes, generalizing the spin U(2) (binomial) coherent states of two modes introduced by [13] and [14] long ago. For z=ei(the standard/canonical basis vectors of CD), the DSCS |ei=(a† i)N| 0/√N! 123 Entanglement and U(D)-spin squeezing in symmetric multi-quDit… Page 7 of 27 304 corresponds to a BEC of Natoms placed at level i.1If we order levels i=1,...,D from lower to higher energies, the state |e1would be the ground state, whereas general |zcould be seen as coherent excitations. Coherent states are sometimes called “quasiclassical” states and we shall see in Sect. 5that |zturns out to be a good variational state that reproduces the energy and wave function of the ground state of multi-level LMG atom models in the thermodynamic (classical) limit N→∞. Expanding the multi-nomial (9), we identify the coefficients cnof the expansion (7)oftheDSCS|zin the Fock basis as cn(z)=N! D i=1ni!D i=1zni i |z|N,(10) where we have written |z|=(z·z)1/2=(D i=1|zi|2)1/2for the length of z. Note that DSCS are not orthogonal (in general) since z|z= (z·z)N (z·z)N/2(z·z)N/2,z·z=¯z 1z1+···+¯zDzD.(11) However, contrary to the standard CSs, they can be orthogonal when z·z=0. EVs of D-spin operators Sij (coherences for i= jand mean level populations for i=j) in a DSCS are simply written as Sijz=z|Sij|z=N¯zizj/|z|2.(12) DSCS non-diagonal matrix elements of D-spin operators can also be compactly written as z|Sij|z=N¯z izj (z·z)N−1 |z|N|z|N.(13) Similarly, EVs of quadratic powers of D-spin operators in a DSCS state can be concisely written as z|SijSkl|z=¯zizl |z|4Nδjk|z|2+N(N−1)¯zkzj,(14) and their DSCS matrix elements as z|SijSkl|z= ¯z izl |z|N|z|NNδjk(z·z)N−1+N(N−1)¯z kzj(z·z)N−2.(15) 1Note the difference between Fock states |n1,...,nDand DSCSs |(z1,...,zD), which are placed inside parentheses to avoid confusion. For instance, |ei=|(0,...,1,...,0)=(a† i)N| 0/√N!= |0,...,N,...,0. 123 304 Page 8 of 27 M. Calixto et al. Note that, for large N, quantum fluctuations are negligible and we have z|SijSkl|z≃ z|Sij|zz|Skl|z. Otherwise stated, in the thermodynamical (classical) limit we have lim N→∞ z|SijSkl|z z|Sij|zz|Skl|z=1.(16) We shall use these ingredients when computing oneand two-quDit RDMs in the next section. We shall see that DSCSs are separable and exhibit no atom entanglement (although they do exhibit level entanglement). The situation changes when we deal with parity-adapted DSCSs, sometimes called “Schrödinger cat states” (commented at the introduction), since they are a quantum superposition of weakly overlapping (macroscopically distinguishable) quasi-classical coherent wave packets, as we shall explicitly see below. These kind of cat states arise in several interesting physical situations. As we have already mentioned, they can be generated via amplitude dispersion by evolving CSs in Kerr media, with a strong nonlinear interaction, like the already commented spin-squeezed states of [15]. They exhibit statistical properties similar to squeezed states, with deviations from Poissonian (CS) distributions. Squeezing and multi-particle entanglement are important quantum resources that make Schrödinger cats useful for quantum enhanced metrology [2]. They are also good variational states [14], reproducing the energy of the ground state of quantum critical models in the thermodynamic limit N→∞. To construct them, we require parity operators defined as j=exp(iπSjj), j=1,...,D.(17) They are conserved when the Hamiltonian scatters pairs of particles conserving the parity of the population njin each level j=1,...,D. It is easy to see that j(a† j)nj| 0=(−a† j)nj| 0, so that the effect of parity operations on number states (5)isj|n=(−1)nj|n. Likewise, using the multi-nomial expansion (9), it is easy to see that the effect of parity operators on symmetric DSCSs |zis then i|z=i|(z1,...,zi,...,zD)=|(z1,...,−zi,...,zD).(18) Note that −1 i=iand 1...D=(−1)N, a constraint that says that the parity group for symmetric quDits is not Z2×D ...×Z2but Z2×D−1 ... ×Z2instead. In order to define a projector on definite parity (even or odd), we have to choose a reference level, namely i=1. Doing so, the projector on even parity becomes even =21−D b∈{0,1}D−1 b2 2b3 3... bD D,(19) where we denote the binary string b=(b2,...,bD)∈{0,1}D−1. Likewise, the projection operator on odd parity is odd =1−even. Choosing level i=1asa reference level is equivalent to choose a patch on the manifold CPD−1where z1= 0; 123 Entanglement and U(D)-spin squeezing in symmetric multi-quDit… Page 9 of 27 304 in this way, any coherent state |zis equivalent to the class representative |z/z1, due to equivalence relation |z∼|zif z=qzwith q= 0. Let us simply denote by z=(1,z2,...,zD)the class representative in this case. It will be useful, for later use as variational ground states, to define the (unnormalized) generalized Schrödinger even cat state |DCAT}=even|z=21−D b|zb,(20) where zb=(1,(−1)b2z2,...,(−1)bDzD)and we are using bas a shorthand for b∈{0,1}D−1. It is just the projection of a DSCS on the even parity subspace. The state (20) is a generalization of the even cat state for D=2 in the literature [16], given by |2CAT}=1 2|(1,α)+|(1,−α)(21) for the class representative z=(z1,z2)=(1,α). The shorthand |α=|(1,α)is used in the literature when a class representative (related to highest |e1or lowest |e2 weight fiducial vectors) is implicitly chosen. The squared norm of |2CAT}is simply N (2CAT)2={2CAT|2CAT}=1 21+1−|α|2 1+|α|2N.(22) Note that the overlap (1,α)|(1,−α)=((1−|α|2)/(1+|α|2))NN→∞ −→ 0, which means that |(1,α)and |(1,−α)are macroscopically distinguishable wave packets for any α(they are orthogonal for |α|=1). Likewise, the unnormalized 3CAT is explicitly given by |3CAT}=1 4|(1,α,β)+|(1,−α, β)+|(1,α,−β)+|(1,−α, −β)(23) when setting z=(z1,z2,z3)=(1,α,β)as a class representative. The squared norm is now N (3CAT)2=1 41+(1−|α|2+|β|2)N+(1+|α|2−|β|2)N+(1−|α|2−|β|2)N (1+|α|2+|β|2)N. (24) These expressions can be generalized to arbitrary Das N (DCAT)2=21−Db(zb·z)N |z|2N.(25) We shall use (23) and (24) in Sects. 5and 6, when discussing a LMG model of atoms with D=3 levels. These 3CAT states have also been used in U(3)vibron 123 304 Page 16 of 27 M. Calixto et al. 3.2.2 Two-quDit reduced density matrices Likewise, any density matrix of two quDits can be written as ρ2= D  i,j,k,l=1 rijklEij ⊗Ekl,rijkl =tr(ρ2Eji ⊗Elk)=Eji ⊗Elk.(45) with ¯rijkl =rjilk complex parameters subject to tr[ρ2]=1 and 0 <tr[(ρ2)2]≤1. Now we need to express the RDM on a pair of particles, extracted at random from a symmetric state of ND-level atoms, in terms of EVs of bilinear products of collective D-spin operators S. In particular, we have SijSkl= N  μ,ν=1Eμ ijEν kl= N  μ=1 δjkEμ il+ N  μ=ν=1Eμ ijEν kl =δjkSil+N(N−1)E1 ijE2 kl,(46) due to indistinguishableness. Therefore, the two-particle RDM of a symmetric state ψof N>2 quDits is written as ρ2(ψ) =1 N(N−1) D  i,j,k,l=1 (Sji Slk−δilSjk)Eij ⊗Ekl.(47) Using the Casimir values (41), one can directly prove that tr[ρ2(ψ)]=1 for any normalized symmetric state ψ. The case D=2 was considered by Wang and Mølmer in [11]. The procedure is straightforwardly extended to ρMfor an arbitrary number M≤N/2 of quDits. The purity of ρ2(ψ) can be compactly written as P a 2(ψ) =trρ2(ψ)2=1 N2(N−1)2⎡ ⎣ D  i,j,k,l=1Sji SlkSijSkl−2 D  i,j,k=1Sji SkjSik+ D  i,j=1SiiSjj⎤ ⎦. (48) In order to construct the two-particle RDM of a DSCS (9), we need the EVs of quadratic powers (14). With these ingredients, we can easily compute the two-particle RDM of a DSCS (9) which, for large Nhas the following asymptotic expression ρ2(z)= D  i,j,k,l=1¯zjzi¯zlzk+O(1/N)Eij ⊗Ekl.(49) The purity of ρ2(z)is 1 since zis separable in the tensor product Hilbert space [CD]⊗N, as we have already commented. Moreover, one can see that ρ2(z)=ρ1(z)⊗ρ1(z). However, the Schrödinger cat (20) is non-separable and has an intrinsic pairwise 123 Entanglement and U(D)-spin squeezing in symmetric multi-quDit… Page 17 of 27 304 Fig. 3 3D plots of linear L a 1and von Neumann S a 1entanglement entropies of the one-qutrit RDM ρ1(3CAT) of a U(3)Schrödinger cat (23)forN=10 atoms, as a function of the phase-space coordinates α, β (they just depend on moduli). The meaning of the magenta curve is the same as in Fig. 2 Fig. 4 Contour plots of linear L a 2and von Neumann S a 2entanglement entropies of the two-qutrit RDM ρ2(3CAT)of a U(3)Schrödinger cat (23)forN=10 atoms, as a function of the phase-space coordinates α, β (they just depend on moduli). The meaning of the magenta curve is the same as in Fig. 2 entanglement. Taking into account the particular structure of linear (26) and quadratic (27)D-spin operator EVs, The general formula (48) becomes P a 2(DCAT)=1 N2(N−1)2⎡ ⎢ ⎢ ⎢ ⎣ D  i,j,k,l=1 j=k Sji SlkSijSkl+ D  i,j=1Sji SijSijSji−2Sii+ D  i,j=1SiiSjj⎤ ⎥ ⎥ ⎥ ⎦ . (50) In Fig. 4, we represent contour plots of normalized linear and von Neumann L a 2=D2 D2−1(1− P a 2), S a 2=−tr(ρ2logD2ρ2)(51) 123 304 Page 18 of 27 M. Calixto et al. entanglement entropies for the two-qutrit RDM ρ2(3CAT)of a U(3)Schrödinger cat (23) as a function of the phase-space CP2coordinates α, β [they just depend on the moduli]. As for the one-quDit case, they attain their maximum value at the phase-space point (α, β) =(1,1); however, unlike the one-quDit case, pairwise entanglement entropies do not attain the maximum value of 1 at this point, but L a 2=5/6 and S a 2≃ 0.623 for large N. As already commented, variational (spin coherent) approximations to the ground state of the LMG three-level atom model [discussed later in Sect. 5] recover this maximum entanglement point (α, β) =(1,1)at high interactions λ→∞ (as can be seen in the magenta curve). For NODON states (30), the two-quDit RDM is ρ2(NODON)=1 D D  k=1 Ekk ⊗Ekk,(52) and therefore ρ2(NODON)2=1 Dρ2(NODON), which means that the linear entropy is L a 2(NODON)=D/(D+1), indicating a high level of pairwise entanglement in a NODON state. 4 SU(D)-spin squeezing: a proposal As we have already commented in the introduction, Wang and Sanders [22]showed a direct relation between the concurrence C, extracted from the two-qubit RDM (47) for D=2, and the SU(2)spin  J=(Jx,Jy,Jz)squeezing parameter ξ2=4 Nmin θ(cos(θ)Jx+sin(θ)Jy)2 =2 N!J2 x+J2 y−J2 x−J2 y2+JxJy+JyJx2"(53) introduced by [15], which measures spin fluctuations in an orthogonal direction to the mean value  Jwith minimal variance. Actually, the definition (53) refers to even and odd symmetric multi-qubit states [remember the extension of this concept to multiquDits after (17)] for which  J=(0,0,Jz)and therefore the orthogonal direction lies in the plane XY. This definition can be extended to even and odd symmetric multiquDit states for which Sij∝δij [see, e.g., (26) for the case of the even DCAT state]. Using the embedding (3)ofD(D−1)/2SU(2)spin subalgebras into U(D), and mimicking (53), we can define D(D−1)/2 spin squeezing parameters ξij,i>jfor D-spin systems as: ξ2 ij =1 N(D−1)!SijSji +Sji Sij−2|S2 ij|",i>j=1,...,D−1.(54) 123 Entanglement and U(D)-spin squeezing in symmetric multi-quDit… Page 19 of 27 304 Fig. 5 Contour plots of the squeezing parameter ξ2 D=3of a U(3)Schrödinger cat for N=10 atoms, as a function of the phase-space coordinates α, β (it just depends on moduli). The meaning of the magenta curve is the same as in previous figures We have chosen the normalization factor 1 N(D−1)so that (54) reduces to (53)forD=2 and so that the total D-spin squeezing parameter ξ2 D= D  i>j=1 ξ2 ij (55) is one (no squeezing) for the DSCSs |zin (9). Actually, for DSCSs we have that ξ2 ij =(|zi|2+|zj|2)/(|z|2(D−1)) is written in terms of average level populations z|Sii|z=N|zi|2/|z|2of levels iand j, according to (12). Therefore, the presence of D-spin squeezing means in general that ξ2 D<1. Using the EVs (31)forNODON states (30), the corresponding spin squeezing parameters are ξij =2/[D(D−1)], which gives ξ2 D=1, thus implying that NODON states do not exhibit spin squeezing. Note that D-spin squeezing parameters ξij are constructed in terms of D-spin quadratic EVs, as the two-quDit RDM (47) and its purity (48) do. Therefore, the deep relation between pairwise entanglement and spin squeezing revealed by Wang-Sanders in [22] for symmetric multi-qubit systems is extensible to symmetric multi-quDits in the sense proposed here. In Fig. 5, we show a contour plot of the total D-spin squeezing parameter ξ2 Dfor the 3CAT. As in previous figures, the magenta curve represents the trajectory in phase space of the stationary points (59) of the energy surface (57)ofthe three-level LMG Hamiltonian (56) as a function of the control parameter λ. See later around Fig. 9in Sect. 6for further discussion. 123 304 Page 20 of 27 M. Calixto et al. 5 LMG model for three-level atoms and its quantum phase diagram In this section, we apply the previous mathematical machinery to the study and characterization of the phase diagram of quantum critical D-level Lipkin–Meshkov–Glick atom models. The standard case of D=2 level atoms has already been studied in the literature (see, e.g., [37]). We shall restrict ourselves to D=3 level atoms for practical calculations, although the procedure can be easily extended to general D.In particular, we propose the following LMG-type Hamiltonian H= N(S33 −S11)−λ N(N−1) 3  i=j=1 S2 ij,(56) written in terms of collective U(3)-spin operators Sij. Hamiltonians of this kind have already been proposed in the literature [38–42] (see also [21] for the role of mixed symmetry sectors in QPTs of multi-quDit LMG systems). We place levels symmetrically about i=2, with intensive energy splitting per particle /N. For simplicity, we consider equal interactions, with coupling constant λ, for atoms in different levels, and vanishing interactions for atoms in the same level (i.e., we discard interactions of the form SijSji). Therefore, His invariant under parity transformations jin (17), since the interaction term scatters pairs of particles conserving the parity of the population njin each level j=1,...,D. Energy levels have good parity, the ground state being an even state. We divide the two-body interaction in (56) by the number of atom pairs N(N−1)to make Han intensive quantity, since we are interested in the thermodynamic limit N→∞. We shall see that parity symmetry is spontaneously broken in this limit. As already pointed long ago by Gilmore and coworkers [14,55], coherent states constitute in general a powerful tool for rigorously studying the ground state and thermodynamic critical properties of some physical systems. The energy surface associated with a Hamiltonian density His defined in general as the coherent state expectation value of the Hamiltonian density in the thermodynamic limit. In our case, the energy surface acquires the following form E(α,β)(, λ) =lim N→∞z|H|z=β¯ β−1 α¯α+β¯ β+1 −λα2¯ β2+1+β2+1¯α2+¯ β2+β2 α¯α+β¯ β+12,(57) where we have used DSCS EVs of linear (12) and quadratic (14)powersofD-spin operators Sij [actually, linear powers are enough due to the lack of quantum spin fluctuations in the thermodynamic limit (16)], and we have used the parametrization z=(1,α,β),asinEq.(23), for U(3)SCSs. Note that this energy surface is invariant under α→−αand β→−β, which is a consequence of the inherent parity symmetry of the Hamiltonian (56). 123 Entanglement and U(D)-spin squeezing in symmetric multi-quDit… Page 21 of 27 304 The minimum energy E0(, λ) =minα,β∈CE(α,β)(, λ) (58) is attained at the stationary (real) phase-space values α± 0=±α0and β± 0=±β0with α0(, λ) =⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 0,0≤λ≤ 2, 2λ− 2λ+, 2≤λ≤3 2, 2λ 2λ+3,λ≥3 2, β0(, λ) ='0,0≤λ≤3 2, 2λ−3 2λ+3,λ≥3 2.(59) In Figs. 2,3,4and 5, we plotted (in magenta color) the stationary-point curve (α0(λ), β0(λ)) on top of level, oneand two-qutrit entanglement entropies and squeezing parameter, noting that (α0(λ), β0(λ)) →(1,1)for high λ→∞interactions. We will come to this later in Sect. 6. Inserting (59)into(57) gives the ground state energy density at the thermodynamic limit E0(, λ) =⎧ ⎪ ⎨ ⎪ ⎩ −, 0≤λ≤ 2,(I) −(2λ+)2 8λ, 2≤λ≤3 2,(II) −4λ2+32 6λ,λ≥3 2.(III) (60) Here, we clearly distinguish three different phases: I, II and III, and two second-order QPTs at λ(0) I↔II =/2 and λ(0) II↔III =3/2, respectively, where ∂2E0(,λ) ∂λ2are discontinuous. In the stationary (magenta) curve (α0(λ), β0(λ)), the phase I corresponds to the origin (α0,β 0)=(0,0)(squared point), phase II corresponds to the horizontal part β0=0 up to the star point, and phase III corresponds to β0= 0. Note that the ground state is fourfold degenerated in the thermodynamic limit since the four U(3)SCSs |z±± 0=|1,±α0,±β0have the same energy density E0. These four U(3)SCSs are related by parity transformations jin (17) and, therefore, parity symmetry is spontaneously broken in the thermodynamic limit. In order to have good variational states for finite N, to compare with numerical calculations, we have two possibilities: 1) either we use the 3CAT (23) as an ansatz for the ground state, minimizing 3CAT|H|3CAT, or 2) we restore the parity symmetry of the U(3) SCS |1,α 0,β 0for finite Nby projecting on the even parity sector. Although the first possibility offers a more accurate variational approximation to the ground state, it entails a more tedious numerical minimization than the one already obtained in (58) for N→∞. Therefore, we shall use the second possibility which, despite being less accurate, it is straightforward and good enough for our purposes. That is, we shall use the 3CAT (23), evaluated at α=α0and β=β0and conveniently normalized (24), as a variational approximation |3CAT0to the numerical (exact) ground state |ψ0for finite N. 123 304 Page 22 of 27 M. Calixto et al. Fig. 6 Contour plot of the energy surface (57) for real αand β, in the vicinity of the critical points λ=1/2 and λ=3/2(inunits). Degenerate minima are perceived in phases II (1 2≤λ≤3 2)and III (λ ≥3 2) 6 Entanglement and squeezing as signatures of QPTs The objective in this section is to use level and particle entanglement and squeezing measures as signatures of QPTs in these LMG models, playing the role of order parameters that characterize the different phases or markers of the corresponding critical points. We restrict ourselves to linear entropy which, as already shown, gives qualitative information similar to von Neumann entropy for this study, with the advantage that it requires less computational resources. As already commented, Refs. [44,45] contain more general information about the relation between both entropies. Linear entropies of oneand two-qutrit RDMs turn also to provide similar qualitative information, although pairwise entanglement shows a more direct relation to spin squeezing. We have numerically diagonalized the Hamiltonian (56)forN=50 three-level atoms, and several values of λ(in units), and we have calculated level, oneand twoqutrit entanglement linear entropies for the ground state |ψ0=ncn|n, plugging the coefficients cninto (8,33,48). We have also calculated level and atom entanglement linear entropies for the variational approximation |3CAT0to the ground state |ψ0 discussed in the previous section for N=50. In Fig. 7, we compare numerical with variational ground state entanglement measures between levels i=1,2,3. According to Fig. 7, we see that, in phase I, 0 ≤λ≤1/2, variational results indicate that there is no entanglement between levels, whereas numerical results show a small (but nonzero) entanglement for N=50. In phase II, 1/2≤λ≤3/2, levels i=1 and i=2 get entangled, but level i=3 remains almost disconnected. In phase III, λ≥3/2, level i=3 gets entangled too. Interlevel entanglement grows with λattaining the maximum value of 0.84 at the limiting point (α0(∞), β0(∞)) =(1,1)for N=50. This behavior of the interlevel entropy for the 3CAT variational state can be also appreciated by looking at the stationary (magenta) curve in Fig. 2with relation to the isentropic curves. Concerning atom entanglement, Fig. 8shows a better agreement between variational and numerical results, showing a rise of entanglement when the coupling strength λgrows across the three phases, attaining values close to the large Nmaximum values L a 1=1 and L a 2=5/6 at the limiting point (α0(∞), β0(∞)) →(1,1). The entanglement growth is more abrupt between phases I and II than between phases II and III. We see that both, level and atom entanglement measures capture differences between the three phases, even for finite N, and therefore they can be considered as precursors of the corresponding QPT. The main features of the inter-atom entanglement entropy for the 3CAT variational state are also captured by the trajectory of the stationary (magenta) curve in Figs. 3and 4through the isentropic curves. 123 Entanglement and U(D)-spin squeezing in symmetric multi-quDit… Page 23 of 27 304 Fig. 7 Level entanglement linear entropies L  i(for levels i=1,2 and 3) of the ground state of the threelevel atom LMG model Hamiltonian (56), for N=50 atoms, as a function of the control parameter λ (in units). Critical points, at which a QPT takes place, are marked with vertical grid lines, whereas the horizontal grid line labels the asymptotic value L  i→1−2/√πN≃0.84 of the entropies. We compare exact results, obtained from numerical diagonalization of the Hamiltonian, with variational (analytical) results obtained from a parity symmetry restoration (in terms of Schrödinger cats) of mean-field results Fig. 8 One-qutrit L a 1and two-qutrit L a 2entanglement linear entropies of the ground state of the three-level atom LMG model Hamiltonian (56) as a function of the control parameter λ(in units). Critical points, at which a QPT takes place, are marked with vertical grid lines, whereas horizontal grid lines label the asymptotic values L a 1→1and L a 2→5/6 of the entropies. We compare numerical with variational results for N=50 atoms In Fig. 9, we represent the D=3 spin total squeezing parameter ξ2 D(55)ofthe variational and numerical ground states for N=50 atoms, as a function of the control parameter λ(in units). The results reveal a clear growth of squeezing at the critical points, the change being more abrupt at these points for the variational (parity-adapted mean field) than for the numerical ground state. Note that the variational ground state only shows squeezing (ξ2 D<1) at the critical points, whereas the numerical ground state exhibits squeezing for any λ= 0. Looking at the stationary (magenta) curve of Fig. 5, we appreciate that it practically lies in regions of no squeezing (in red color) except near the critical points, where squeezing suddenly increases (yellow color regions). 123 304 Page 24 of 27 M. Calixto et al. Fig. 9 D-spin total squeezing parameter ξ2 D(55) of the ground state of the three-level atom LMG model Hamiltonian (56) as a function of the control parameter λ(in units). Critical points, at which a QPT takes place, are marked with vertical grid lines. We compare numerical with variational results for N=50 atoms 7 Conclusions and outlook We have extended the concept of pairwise entanglement and spin squeezing for symmetric multi-qubits (namely identical two-level atoms) to general symmetric multi-quDits (namely identical D-level atoms). For it, we have firstly computed expectation values of U(D)spin operators Sij in general symmetric multi-quDit states like: U(D)-spin coherent states, their adaptation to parity (Schrödinger DCAT states), and an extension of NOON states to Dlevels (NODON states). The reduced density matrices to one and two quDits extracted at random from a symmetric multi-quDit state exhibit atom entanglement for DCAT states, but not for U(D)-spin coherent states. We have used entanglement to characterize quantum phase transitions of LMG D-level atom models (we have restricted to D=3 for simplicity), where DCAT states (as an adaptation to parity of mean-field spin coherent states) turn out to be a reasonable good variational approximation to the exact (numerical) ground state. We have also proposed an extension of standard SU(2)-spin squeezing to SU(D)-spin operators, which recovers D=2 as a particular case. We have evaluated SU(3)-spin squeezing of the ground state of the LMG three-level atom model, as a function of the control parameter λ, and we have seen that squeezing grows in the neighborhood of critical points λc, therefore serving as a marker of the corresponding quantum phase transition. A deeper study and discussion of squeezing in these models requires a phase-space approach in terms of a coherent (Bargmann) representation of states, such as the Husimi and Wigner functions, and it will be the subject of the future work. Acknowledgements We thank the support of the Spanish MICINN through the Project PGC2018-097831B-I00 and Junta de Andalucía through the Projects SOMM17/6105/UGR, UHU-1262561 and FQM-381. JG also thanks MICINN for financial support from FIS2017-84440-C2-2-P. AM thanks the Spanish MIU for the FPU19/06376 predoctoral fellowship. We all thank Octavio Castaños for his valuable collaboration in the early stages of this work. Funding Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. 123 Entanglement and U(D)-spin squeezing in symmetric multi-quDit… Page 25 of 27 304 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. 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