Staircase in magnetization and entanglement entropy of spin squeezed condensates
SStaircase in magnetization and entanglement entropy of spin squeezed condensates
H. M. Bharath, M. S. Chapman and C. A. R. S´a de Melo
School of Physics, Georgia Institute of Technology (Dated: November 9, 2018)Staircases in response functions are associated with physically observable quantities that responddiscretely to continuous tuning of a control parameter. A well-known example is the quantiza-tion of the Hall conductivity in two dimensional electron gases at high magnetic fields. Here, weshow that such a staircase response also appears in the magnetization of spin-1 atomic ensemblesevolving under several spin-squeezing Hamiltonians. We discuss three examples, two mesoscopicand one macroscopic, where the system’s magnetization vector responds discretely to continuoustuning of the applied magnetic field or the atom density, thus producing a magnetization staircase.The examples that we consider are directly related to Hamiltonians that have been implementedexperimentally in the context of spin and spin-nematic squeezing. Thus, our results can be readilyput to experimental test in spin-1 ferromagnetic Rb and anti-ferromagnetic Na condensates.
In the integer quantum Hall effect, the Hall conductiv-ity changes discretely to continuous tuning of the mag-netic field [1, 2]. In general, when a system respondsdiscretely to a continuous change of a control parame-ter, a staircase structure appears in its response func-tion, which is a distinctive characteristic of quantization.Such phenomenon is significant on two counts. First, onecan stabilize the system on a step of the staircase, thatis, the flat region between two discrete jumps. Second,these stable states are potentially topological and maycarry topological invariants of the system’s phase space.The quantum Hall effect has been observed in fermionictwo-dimensional (2D) electron gases [3, 4].Bosonic analogues of quantum Hall states have beenpredicted to exist in rotating, weakly interacting Bose-Einstein condensates (BEC)[5–11]. A spinless, non-inteacting, rotating BEC in a harmonic trap is charac-terized by Landau levels, similar to a 2D electron gasin a magnetic field [5]. For a rotating BEC, the trapfrequency plays the role of the effective magnetic fieldand the corresponding lowest Landau level is degener-ate in the angular momentum about the axis of rotation.This means that there are multiple angular momentumeigenstates within the lowest Landau level, thus, a weakinteraction in the system may select one of these angularmomentum eigenstates as the ground state of the systemdepending on the ratio of the interaction strength andthe cyclotron frequency [5, 6]. Thus, the system’s angu-lar momentum responds discretely to continuous tuningof the effective magnetic field, in analogy with the quan-tum Hall effect. Recently, such phenomena has been pre-dicted even in a spin-1 BEC [7] and a pseudo spin-1/2BEC[10].For the bosonic examples discussed above, the interac-tion plays a pivotal role in the emergent angular momen-tum staircase as a function of the effective magnetic field.Two other quantum phenomena that also arise from in-teractions are squeezing and many body entanglement.Spin squeezed states have been prepared in bosonic sys-tems [12–19] and used to enhance the precision in a mea- surement, for example, of the applied magnetic field.They are characterized by noise in the transverse spincomponent that is lower than any classical state and aregenerally prepared with the help of an interaction termin the Hamiltonian. Two of the most common modes ofpreparing squeezed states, one-axis twisting and two-axiscounter twisting, involve interactions [20].In this letter, we show three examples of spin-squeezingHamiltonians, realizable in spin-1/2 and spin-1 BECs,that are characterized by a staircase response in themagnetization. First we show this for one-axis twist-ing Hamiltonian. Second, we demonstrate that an in-teracting ferromagnetic spin-1 BEC, where spin-nematicsqueezing has been demonstrated [16], also displaysa staircase. Third, we consider an interacting anti-ferromagnetic spin-1 BEC, where a staircase is also ob-tained in the direction of the magnetization. The firsttwo examples are mesoscopic, while the third is a macro-scopic phenomenon. We also propose experiments to ob-serve these effects.
Staircase in one-axis twisting.
First, we consider apseudo spin-1/2 BEC under the one-axis twisting Hamil-tonian, H = χS z , where S z is the total spin operator inthe z -direction and χ represents the strength of two bodyinteractions in the system [20]. By applying a magneticfield p in the z -direction, we obtain a staircase struc-ture in the ground state magnetization of the Hamilto-nian, H = χS z − pS z . We use units where (cid:126) = 1, S z is dimensionless, χ and p are frequencies. The eigen-states of S z are also eigenstates of this Hamiltonian.The energy of the eigenstate with a magnetization m is E m = χm − pm , for m = − N , − N + 1 , · · · N , where N is the number of atoms in the condensate. By minimiz-ing the energy, we obtain the ground state magnetization m gs = [ p χ ], where [ x ] represents the integer closest to x .Here, pχ plays the role of the control parameter to whichthe magnetization responds discretely. The initial step ofthe magnetization staircase occurs when pχ < − N , withmagnetization m gs = − N , while the final step occurswhen pχ > N , with magnetization m gs = N . In between, a r X i v : . [ c ond - m a t . qu a n t - g a s ] A p r FIG. 1.
Staircase in the one-axis twisting Hamiltonian: (a) Shows the ground state magnetization as a function of thestrength of the applied field p for constant interaction χ in the one-axis twisting Hamiltonian H = χS z − pS z , with (blue curve)and without (black curve) the perturbation (cid:15)S x . (b) Shows the corresponding ground state and the first excited state energiesaround the level crossings between m = − m = 0, as well as m = 0 and m = +1. In the absence of the perturbation,there are true level crossings, but when the perturbation is added, gaps open and thereby smooth the staircase. The term (cid:15)S x is also responsible for changing the system’s magnetization, which is otherwise conserved. (c) Shows the entanglement entropyof the local ground state as a function of the control parameter p/χ . The black curve shows the entanglement without theperturbation a written in Eq.1, while the blue curve shows the entanglement with the perturbation for (cid:15)χ = 0 . the m gs responds discretely to continuous variation of p as shown in Fig. 1(a).Every step in this staircase is a distinct quantum stateand every jump corresponds to a level crossing. Theeigenenergies in the vicinity of a level crossing are shownin Fig. 1(b). Notice that this is a true level crossing,even when the system size is small, that is, it is not anavoided level crossing. Consequently, in order to observethis effect, one has to facilitate each jump in the staircaseby opening up a gap at the level crossing. This can bedone by adding a weak field (cid:15) in the x -direction leadingto the Hamiltonian H = χS z − pS z − (cid:15)S x , where (cid:15) is alsoin units of Hz. The resulting energy gaps for crossingsbetween states with m gs = − m gs = 0, as well as m gs = 0 and m gs = +1 are shown in Fig. 1(b). Theterm (cid:15)S x also smoothes out the staircase in Fig. 1(a) andis responsible for changing the system’s magnetization,which is otherwise conserved.The quantum states in this magnetization staircase arerelated to the familiar Dicke ladder [21], where transitionsbetween neighboring total angular momentum states ofatoms can occur coherently leading to superradiance.An experiment where the control parameter pχ is slowlyswept from − N to N would induce a transfer of the atompopulation between the spin states, one atom at a time.Furthermore, this is also a way of deterministically pro-ducing all the Dicke states in this ladder, most of whichare highly entangled [22, 23]. In an experiment, the sys-tem can be initialized at m = − N or m = N , where itis completely unentangled. As the control parameter pχ is tuned, the magnetization m increases in integer steps and the corresponding entanglement entropy also stepsup, peaking at m = 0, see Fig. 1(c). The entanglemententropy for magnetization m , in terms of the magnetiza-tion per atom, µ = mN , is given by [24] E = − (cid:18) − µ (cid:19) log (cid:18) − µ (cid:19) − (cid:18)
12 + µ (cid:19) log (cid:18)
12 + µ (cid:19) . (1)The perturbation (cid:15)S x , that was added to maintain adia-baticity at the level crossing, also perturbs the entangle-ment entropy, as shown in Fig. 1(c). The large dips in theentanglement entropy that appear at the level crossingsare characteristic of a singular perturbation on the de-generate ground state space. Indeed, at the level crossingbetween magnetizations m and m + 1, the unperturbedground state is a two dimensional space spanned by theeigenstates {| m (cid:105) , | m + 1 (cid:105)} of S z with eigenvalues m and m + 1, respectively. The perturbation breaks this degen-eracy and picks one state from this space as the groundstate. For instance, with an (cid:15)S x perturbation, the groundstate is | m (cid:105)−| m +1 (cid:105)√ , independent of (cid:15) . This state has alower entanglement entropy than | m (cid:105) and | m + 1 (cid:105) , andit corresponds to the dip in the blue curve in Fig. 1(c).Thus, when (cid:15) →
0, the blue curve approaches the blackcurve at every point, excluding the level crossings.There are several experimental systems where spinsqueezing has been demonstrated using the one-axistwisting Hamiltonian including trapped ion systems [12,17], Bose-Einstein condensates [25], double well [13] andcavity systems [14, 26–28]. Any of these realizations canbe used to observe this effect. In an Rb condensate,this Hamiltonian is realized using the hyperfine levels | F = 1 , m = 0 (cid:105) and | F = 2 , m = − (cid:105) as the pseudospin-1/2 states. The squeezing term S z can be producedusing a Feshbach resonance [25] and the linear term, S z can be generated using microwave dressing. Experimentswith N = 300 atoms and N χ ∼
20 Hz have been demon-strated with a detection noise of ∼ χS z lies in introduc-ing convexity into the energy functional. The energy, E m = χ ( m − m pχ ) is a convex function in the discretevariable m and the control parameter pχ contributes alinear term in this function. The minima of a convexfunction can be shifted by adding a linear term, howeverthese shifts are discontinuous since the variable is dis-crete. This is the primary characteristic of the groundstate energy of Hamiltonian which results in a staircasephenomena. Next, we use this observation to identify astaircase in the magnetization of a ferromagnetic spin-1BEC, as a second example. Staircase in a ferromagnetic spin-1 BEC.
The Hamil-tonian of a ferromagnetic spin-1 BEC of Rb atoms,confined to an optical dipole trap and with an appliedmagnetic field of B z along the z -direction is [29] H = N (cid:88) i =1 (cid:18) − (cid:126) m ∇ i + V T ( r i ) (cid:19) + 4 π (cid:126) m (cid:88) i>j δ ( r i − r j ) (cid:88) F =0 , F (cid:88) m F = − F a F | F, m F (cid:105)(cid:104) F, m F | + N (cid:88) i =1 (cid:18) µ B g F B z L zi + µ B (cid:126) ∆ B z L zi (cid:19) . (2)Here, V T is the dipole trapping potential, the interac-tion between pairs of atoms is modeled by a δ functionpotential and it involves two s-wave scattering lengths, a and a , corresponding to the possible total spin of thetwo interacting atoms, both of which are in the spin-1state. In addition, the relevant Land´e g-factor is g F and L zi is the spin operator for the i -th atom. The hyper-fine splitting between the F = 1 and F = 2 levels is∆. Assuming that the trap is sufficiently tight, one canapproximate the ground state by a product of a spatialwave function common to all spin modes and a collective N -atom spin state. This is also known as the single modeapproximation (SMA). Under SMA, the spin part of theHamiltonian is H = cS + qQ zz − pS z , (3)where c < c = π (cid:126) ( a − a )3 m (cid:82) | φ ( r ) | d r , where φ ( r ) is the common spatialwave function. The total spin operator of all the atoms is S , the strength of the quadratic Zeeman term is q = µ B (cid:126) ∆ and the linear Zeeman contribution is p = µ B g F B z . The collective spin and second rank tensor operatorsare S z = (cid:80) Ni =1 L zi and Q zz = (cid:80) Ni =1 L zi , respectively.This Hamiltonian has been used to produce spin-nematicsqueezed states [16].We show that the quadratic Zeeman effect induces anenergy that is convex in the system’s magnetization andtherefore, with c and q fixed to appropriate values, wecan obtain an analogous staircase in this system. TheHamiltonian commutes with S z and therefore, it has si-multaneous eigenstates with the latter. Let us denotethese eigenstates by | n, m (cid:105) , with( cS + qQ zz ) | n, m (cid:105) = λ nm | n, m (cid:105) S z | n, m (cid:105) = m | n, m (cid:105) (4)The eigenenergy of this state is E nm = λ nm − pm . Ob-taining the ground state involves a simultaneous min-imization over n and m . We define the function E m as the minimal value of E nm over all n , correspondingto the ground state energy of the Hamiltonian for fixedmagnetization. The global ground state is obtained byminimizing E m over m . The Zeeman term p contributes FIG. 2.
Convexity of the energy:
The minimum en-ergy eigenvalue E m of the ferromagnetic Hamiltonian H = cS + qQ zz − pS z is a convex function of the magnetization m . For the purpose of this illustration, we have used N = 10.The minima of these curves correspond to the ground statemagnetization. Because p is the coefficient of a linear term in m , changing it has the effect of shifting the minimum. Thefour values of p/ | c | have their minima are different values of m , leading to a staircase response of the ground state magne-tization as p/ | c | is changed. FIG. 3.
Staircase in the magnetization direction: (a) shows the ground state magnetization vector of an anti-ferromagneticcondensate with Hamiltonian H = cS + pS x + αQ xz , for three different values of c with N = 100. The last term in theHamiltonian induces the tilting of the magnetization vector by specific angles, depending on where the system is on thestaircase. (b) shows the tilt angle for N = 20 as a function of c/p , a staircase, but in contrast with the previous examples, thistime it is not only in the magnitude of magnetization, but also in the direction. The blue curve shows the smoothened staircaseafter adding an (cid:15)Q xx perturbation, with (cid:15) = 0 . p . The inset shows the ground state entanglement entropy as a function ofthe control parameter. In both (a) and (b), α = 0 . p . a linear term to E m leading to E m = min n { λ nm − pm } = min n { λ nm } − pm (5)We use | c | as our energy unit, and show in Fig. 2 that E m is a convex function of m . Consequently, the groundstate magnetization varies through discrete values of m ,when the control parameter p/ | c | is tuned. When q (cid:28) | c | ,the energy E m ≈ −| c | N ( N + 1) − pm is linear in m andhas a minimum at m = N . When q (cid:29) | c | and q > p , theenergy E m ≈ q | m | − pm has a minimum at m = 0. Uponvariation of q between these two extremes, E m must havea minimum between m = 0 and m = N , and must be aconvex function of m as seen in Fig. 2 .Thus, we obtain a similar staircase structure in themagnetization, when p | c | is varied adiabatically. Like theprevious example, the flat areas in the staircase corre-spond to distinct quantum states and a discrete jumpcorresponds to a level crossing, which needs to be facili-tated by opening up an energy gap. Again, this can bedone by perturbing the Hamiltonian with a weak field inthe x -direction (cid:15)S x . In typical experiments [16], | c | ∼ q ∼ | c | , indicating that the emergence of themagnetization staircase is also accessible to existing tech-niques. Similar to the previous example, the entangle-ment entropy also has a staircase structure.Both of the examples discussed so far are mesoscopic inthe sense that the values of the control parameter corre-sponding to adjacent steps are separated by ∼ N , where N is the number of atoms. Therefore, in the limit oflarge atom numbers, it is increasingly more difficult toresolve the different jumps. However, next we show that in an anti-ferromagnetic condensate, a similar staircasestructure appears as a truly macroscopic manifestation,where, the jumps are macroscopically separated. Staircase in an anti-ferromagnetic spin-1 BEC.
Weconsider a spin-1 anti-ferromagnetic BEC with an ap-plied field p in the z -direction leading to the Hamilto-nian H = cS − pS z , where c > | s, m (cid:105) with − s ≤ m ≤ s and s = 0 , , , · · · , N (assuming N is even), due to bosonicsymmetry. Here, s is the total spin of the system, thatis, S | s, m (cid:105) = s ( s + 1) | s, m (cid:105) . The eigenenergy of | s, m (cid:105) is E sm = cs ( s + 1) − pm . When p >
0, the ground statehas m = + s . In this case the energy E s = min m E sm = cs + ( c − p ) s (6)is a convex function in s . In contrast to previous ex-amples, the control parameter is the coefficient c of thequadratic term instead of the field p in the linear Zeemancontribution.The ground state value of s is the non-negative integerclosest to p − c c . When c = 0, the ground state has s = N and when c ≥ p , it has s = 0. Because s has a staircasestructure, so does the systems magnetization. The levelcrossings in this staircase occur at values of c given by c s = p s − s = 2 , , · · · , N. (7)The magnetization of the ground state is given by (cid:104) (cid:126)S (cid:105) = (0 , , s ) and develops a staircase structure when c is tuned. We show now that by adding a suitable per-turbation to the Hamiltonian, this staircase structure canbe transferred to the direction of the magnetization.Let us perturb the Hamiltonian by Q xz , which is aquadratic variable given by Q xz = (cid:80) i { L xi , L zi } for asingle atom. The Hamiltonian becomes H = cS − pS z + αQ xz . Within a given step in the staircase, p s +1 < c < p s − , we use first order perturbation theory to obtain theground state | ψ s (cid:105) = | s, s (cid:105) + αp q s | s, s − (cid:105) (8)from the unperturbed ground state | s, s (cid:105) . Here, q s = (cid:104) s, s | Q xz | s, s − (cid:105) = √ s (cid:16) N +32 s +3 (cid:17) is the relevant matrixelement [24]. In this case, the magnetization (cid:104) (cid:126)S (cid:105) = s ˆ z + αp √ sq s ˆ x (9)is tilted away from the z -axis with a polar angle given by θ s = arctan (cid:32) α √ sq s ps (cid:33) . (10)This angle has a staircase structure with c as the controlparameter as shown in Fig. 3. Similar to the previous ex-amples, the flat regions of the staircase are distinct quan-tum states and the associated level crossings need to befacilitated by the opening of a gap created by a pertur-bation of the type (cid:15)Q xx , (here, Q xx = (cid:80) Ni =1 L xi ) that in-troduces an overlap between states | s, s (cid:105) and | s ± , s ± (cid:105) .Good candidates to observe this effect experimentally are Na condensates. Typically, c ∼ N = 10 atoms. The steps in Fig. 3,corresponding to s = 1 , ,
3, are separated by a few hertzon the c axis and they are independent of the numberof atoms. Therefore, this effect is macroscopic and alsoobservable within the existing experimental systems.To conclude, we have described three examples whereatomic ensembles described by different spin-squeezingHamiltonians display a staircase structure in their mag-netizations as a response to the external tuning of acontinuous control parameter. This phenomena canbe observed in spin-1 ferromagnetic Rb and anti-ferromagnetic Na condensates, using existing experi-mental techniques.
Acknowledgments : We thank Matthew Boguslawski,Maryrose Barrios and Lin Xin for fruitful discussions.HMB and MSC would like to acknowledge support fromthe National Science Foundation, grant no. NSF PHYS-1506294. C. A. R. SdM acknowledges the support ofthe Galileo Galilei Institute for Theoretical Physics viaa Simons Fellowship, and the Aspen Center for Physicsvia NSF Grant No. PHY1607611. [1] D. J. Thouless, M. Kohmoto, M. P. Nightingale, andM. den Nijs, Phys. Rev. Lett. , 405 (1982).[2] P. Streda, Journal of Physics C: Solid State Physics ,L717 (1982).[3] K. von Klitzing, Rev. Mod. Phys. , 519 (1986).[4] H. L. Stormer, D. C. Tsui, and A. C. Gossard, Rev.Mod. Phys. , S298 (1999).[5] N. K. Wilkin and J. M. F. Gunn, Phys. Rev. Lett. , 6(2000).[6] N. R. Cooper, N. K. Wilkin, and J. M. F. Gunn, Phys.Rev. Lett. , 120405 (2001).[7] J. W. Reijnders, F. J. M. van Lankvelt, K. Schoutens,and N. Read, Phys. Rev. Lett. , 120401 (2002).[8] N. Cooper, Advances in Physics , 539 (2008).[9] S. Viefers, Journal of Physics: Condensed Matter ,123202 (2008).[10] S. Furukawa and M. Ueda, Phys. Rev. Lett. , 090401(2013).[11] T. Graß, B. Juli´a-D´ıaz, and M. Lewenstein, Phys. Rev.A , 013623 (2014).[12] V. Meyer, M. A. Rowe, D. Kielpinski, C. A. Sackett,W. M. Itano, C. Monroe, and D. J. Wineland, Phys.Rev. Lett. , 5870 (2001).[13] Y. P. Huang and M. G. Moore, Phys. Rev. Lett. ,250406 (2008).[14] I. D. Leroux, M. H. Schleier-Smith, and V. Vuleti´c, Phys.Rev. Lett. , 073602 (2010).[15] C. Gross, T. Zibold, E. Nicklas, J. Est`eve, and M. K.Oberthaler, Nature , 1165 (2010).[16] C. D. Hamley, C. S. Gerving, T. M. Hoang, E. M. Book-jans, and M. S. Chapman, Nature Physics , 305 (2012).[17] J. G. Bohnet, B. C. Sawyer, J. W. Britton, M. L. Wall,A. M. Rey, M. Foss-Feig, and J. J. Bollinger, Science , 1297 (2016).[18] K. C. Cox, G. P. Greve, J. M. Weiner, and J. K. Thomp-son, Phys. Rev. Lett. , 093602 (2016).[19] O. Hosten, N. J. Engelsen, R. Krishnakumar, and M. A.Kasevich, Nature , 505 (2016).[20] M. Kitagawa and M. Ueda, Phys. Rev. A , 5138 (1993).[21] R. H. Dicke, Phys. Rev. , 99 (1954).[22] W. Chen, J. Hu, Y. Duan, B. Braverman, H. Zhang, andV. Vuleti´c, Phys. Rev. Lett. , 250502 (2015).[23] R. McConnell, H. Zhang, S. ´Cuk, J. Hu, M. H. Schleier-Smith, and V. Vuleti´c, Phys. Rev. A , 063802 (2013).[24] See Supplementary Information.[25] H. Strobel, W. Muessel, D. Linnemann, T. Zibold, D. B.Hume, L. Pezz`e, A. Smerzi, and M. K. Oberthaler, Sci-ence , 424 (2014).[26] A. S. Sørensen and K. Mølmer, Phys. Rev. A , 022314(2002).[27] J. Borregaard, E. J. Davis, G. S. Bentsen, M. H. Schleier-Smith, and A. S. Sørensen, New Journal of Physics ,093021 (2017).[28] S. J. Masson, M. D. Barrett, and S. Parkins, Phys. Rev.Lett. , 213601 (2017).[29] T.-L. Ho, Phys. Rev. Lett. , 742 (1998).[30] E. J. Mueller, T.-L. Ho, M. Ueda, and G. Baym, Phys.Rev. A , 033612 (2006).[31] J. Wiemer and F. Zhou, Phys. Rev. B , 115110 (2004).[32] A. T. Black, E. Gomez, L. D. Turner, S. Jung, and P. D.Lett, Phys. Rev. Lett. , 070403 (2007). Supplementary Information
In this supplementary material, we present details ofthe derivation of a few important expressions used in themain text. In section I, we show how the entanglemententropy of many-body systems can be evaluated, and weillustrate our method by using the specific Hamiltoniansdescirbed in the main text. In section II, we derive theexpression for matrix element q s = (cid:104) s, s − | Q xz | s, s (cid:105) re-sponsible for the staircase structure in the direction ofthe spin vector. ENTANGLEMENT ENTROPY
A Bosonic many-body state has a unique single atomreduced density matrix, due to its symmetry. Therefore,a convenient measure of many-body entanglement of suchstates is the Von Neumann entropy of the single atom re-duced density matrix. The following simple observationhelps us determine the single atom reduced density ma-trix ρ corresponding to a many-body pure state | ψ (cid:105) andevaluate its entanglement entropy: If ˆ o is a single atomobservable operator and ˆ O = (cid:80) Ni =1 ˆ o i is the correspond-ing many-body observable, thenTr( ρ ˆ o ) = 1 N (cid:104) ψ | ˆ O | ψ (cid:105) . (11)For instance, the spin operator along the x -axis for asingle, spin-1/2 atom is L x = σ x in units of (cid:126) , where σ x is the Pauli matrix. While the many-body spin operatoralong the x -axis is simply given by S x = (cid:80) Ni =1 σ xi .The symmetry of the many-body state | ψ (cid:105) ensures thatTr( ρσ x ) = N (cid:104) ψ | S x | ψ (cid:105) . The reduced density matrix ρ can be reconstructed using the expectation values of afew different observables.We illustrate this idea with an example. Let us con-sider the ground state of the one-axis twisting Hamilto-nian H = χS z − pS z , where S z is the many-body spinoperator along the z -axis, as described in the main text.The ground state | ψ (cid:105) of this Hamiltonian is an eigenstateof S z with a magnetization given by m = (cid:104) p χ (cid:105) , the inte-ger closest to p χ . It follows from Eq. 11 that the singleatom reduced density matrix ρ of | ψ (cid:105) has spin expecta-tion values Tr( σ x ρ ) = 2 N (cid:104) ψ | S x | ψ (cid:105) = 0Tr( σ y ρ ) = 2 N (cid:104) ψ | S y | ψ (cid:105) = 0Tr( σ z ρ ) = 2 N (cid:104) ψ | S z | ψ (cid:105) = 2 mN . (12)Using the spin expectation values given above, we mayreconstruct the density matrix ρ as ρ = 12 (cid:18) + 2 mN σ z (cid:19) = (cid:18) + mN − mN (cid:19) . (13) The Von Neumann entropy of this state is E = − Tr[ ρ log( ρ )]. The explicit expression for E is given inthe main text as Eq. [1] and shown as the black curve inFig. 1(c).Next, we consider the perturbed Hamiltonian H = χS z − pS z − (cid:15)S x . In the absence of the (cid:15)S x perturba-tion, a level crossing occurs at p χ = m + 1 /
2. Withoutloss of generality, we may assume that m ≤ p χ ≤ m + 1.The ground state magnetization is m when p χ < m + and it is m + 1 when p χ > m + . Accordingly, it is con-venient to use δ = p χ − m − as a function of the controlparameter pχ . The range of δ is [ − / , /
2] and the mag-netization switches from m to m + 1 when δ crosses zero.The ground state, in the presence of the (cid:15)S x perturbationis a superposition of | m (cid:105) and | m + 1 (cid:105) , the eigenstates of S z with eigenvalues m and m + 1 respectively | ψ (cid:105) = u | m (cid:105) + v | m + 1 (cid:105) . (14)The coefficients u and v depend on δ and are given by u = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) − δ (cid:113) δ + C m (cid:15) χ v = (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) δ (cid:113) δ + C m (cid:15) χ . (15)Here, C m is the relevant Clebsch-Gordon coefficient, C m = (cid:112) ( N/ − m )( N/ m + 1). It is now straight-forward to compute the spin expectation valuesTr( σ x ρ ) = 2 N (cid:104) ψ | S x | ψ (cid:105) = 2 uvC m N Tr( σ y ρ ) = 2 N (cid:104) ψ | S y | ψ (cid:105) = 0Tr( σ z ρ ) = 2 N (cid:104) ψ | S z | ψ (cid:105) = 2 N (cid:0) m + v (cid:1) , (16)which can be used to obtain the reduced density matrix ρ = (cid:32) + m + v N uvC m NuvC m N − m + v N (cid:33) . (17)The off diagonal terms are the largest when δ = 0 thatis, when the control parameter pχ = 2 m + 1. Correspond-ingly, the Von Neumann entropy of the density matrixdefined in Eq. 17, has a dip at odd values of pχ , as shownin Fig. 1(c) of the main text. The same technique can beused to evaluate the reduced density matrix of a many-body spin-1 system, that appears in the second and thirdexamples considered in the main text. THE MATRIX ELEMENT q s Next, we show how the expression for the matrix ele-ment q s = (cid:104) s, s − | Q xz | s, s (cid:105) used in Eq. [8] of the maintext is derived. The space of symmetric states of N spin-1 atoms has ( N +1)( N +2)2 dimensions. A convenient basisfor this space is given by the normalized number states,defined as | N + , N , N − (cid:105) = ( a †− ) N − ( a † ) N ( a † +1 ) N + | vac (cid:105) ,where a † i is the creation operator for the i -th mode, | vac (cid:105) is the vacuum state, with no atoms, and N = N + + N + N − is the total number of atoms.An alternative basis, also of relevance in the presentcontext is given by the coupled spin states | s, m (cid:105) with − s ≤ m ≤ s and s = N, N − , · · · , s min . When N is even, the minimum value of s is s min = 0 and when N is odd, s min = 1. These states are the simultane-ous eigenstates of the total spin operators S and S z with eigenvalues given by S | s, m (cid:105) = s ( s + 1) | s, m (cid:105) and S z | s, m (cid:105) = m | s, m (cid:105) .To evaluate the matrix element of interest, q s = (cid:104) s, s − | Q xz | s, s (cid:105) , we need to determine the action of the opera-tor Q xz on the state | s, s (cid:105) or the state | s, s − (cid:105) . However,it is easier to determine the action of this operator on thenumber states Q xz | N + , N , N − (cid:105) = c | N + + 1 , N − , N − (cid:105) − c | N + , N + 1 , N − − (cid:105) + c | N + − , N + 1 , N − (cid:105) − c | N + , N − , N − + 1 (cid:105) , (18)where the coefficients are given by c = (cid:113) ( N + +1) N , c = (cid:113) ( N +1) N − , c = (cid:113) N + ( N +1)2 and c = (cid:113) N ( N − +1)2 .The result shown in Eq. 18 follows from the definitionof the many-body operator Q xz = (cid:80) Ni =1 { L xi , L zi } andthe matrix form of the single-atom operator { L xi , L zi } = 1 √ − − . (19)The matrix element (cid:104) s, s − | Q xz | s, s (cid:105) can be computedby expressing the coupled spin states | s, m (cid:105) in the num-ber state basis. Noting that S z | s, m (cid:105) = m | s, m (cid:105) and S z | N + , N , N − (cid:105) = ( N + − N − ) | N + , N , N − (cid:105) , the overlap (cid:104) s, m | N + , N , N − (cid:105) vanishes unless N + − N − = m . There-fore, we may write | s, s (cid:105) = N − s (cid:88) k =0 A k | k + s, N − k − s, k (cid:105) (20)Since | s, s − (cid:105) = √ s S − | s, s (cid:105) where S − is the loweringoperator, it suffices to determine A k in order to evaluatethe matrix element q s . The coefficients A k can be evalu-ated using the observation that the raising operator S + annihilates the state | s, s (cid:105) , that is S + | s, s (cid:105) = 0. Usingthe relation S + | N + , N , N − (cid:105) = c | N + + 1 , N − , N − (cid:105) + c | N + , N + 1 , N − − (cid:105) (21)in conjunction with S + | s, s (cid:105) = 0 gives the recursive rela-tion A k +1 = A k (cid:115) ( k + s + 1)( N − k − s )( k + 1)( N − k − s − . (22)Solving for A k for k > A k = A (cid:115) ( k + s )!( N − k − s − k !( N − k − s )!! , (23)where A is obtained from the normalization condition (cid:80) k A k = 1. It is useful to make the connection with hy-pergeometric functions here by noticing that the squaresof the coefficients A k are generated by a hypergeometricfunction, N − s (cid:88) k =0 A k A x k = F (cid:18) − ( N − s )2 , s + 1; − ( N − s − , x (cid:19) . (24)This follows from the observation that the above recur-sion relation on the coefficients A k is also the recursionbetween consecutive terms of a hypergeometric series: A k +1 = A k (cid:16) − ( N − s )2 + k (cid:17) ( s + 1 + k ) (cid:16) − ( N − s − + k (cid:17) k + 1) . (25)This gives a simple expression for the first coefficient A = 1 (cid:114) F (cid:16) − ( N − s )2 , s + 1; − ( N − s − , (cid:17) . (26)In fact, there is a closed-form expression for this particu-lar evaluation of hypergeometric functions where the firstargument is a negative integer, F ( − n, b ; c,
1) = ( c − b )( c − b + 1) · · · ( c − b + n − c ( c + 1) · · · ( c + n −
1) (27)Noting that ( N − s )2 is always an integer, Eqs. 23 and 26together give us all the A k coefficients.We are now ready to evaluate the matrix element q s ,which can also be written as q s = 1 √ s (cid:104) s, s | Q xz S − | s, s (cid:105) (28)This expression can be viewed as an overlap between thevectors, | ψ (cid:105) = Q xz | s, s (cid:105) and | ψ (cid:105) = S − | s, s (cid:105) , scaled by afactor of √ s . Besides Eq. 18, another expression that isuseful to evaluate this overlap is S − | N + , N , N − (cid:105) = c | N + − , N + 1 , N − (cid:105) + c | N + , N − , N − + 1 (cid:105) , (29)which shows the action of the lowering operator on anumber state. Starting from the expansion of the state | s, s (cid:105) in the number basis given in Eq. 20, and using theaction of Q xz on a number state given in Eq. 18 we mayexpand | ψ (cid:105) in the number basis. Similarly, | ψ (cid:105) can beexpanded in the number basis using Eq. 29. The overlap (cid:104) ψ | ψ (cid:105) can be written in terms of the coefficients A k as √ sq s = (cid:104) ψ | ψ (cid:105) = N − s (cid:88) k =0 s ( N − k − s + 12 ) A k . (30)The evaluation of this quantity requires a crucial sum (cid:80) k kA k , which is obtained by taking a derivative ofEq. 24: N − s (cid:88) k =0 k A k A = (cid:20) ddx F (cid:18) − ( N − s )2 , s + 1; − ( N − s − , x (cid:19)(cid:21) x =1 . (31)The derivative of a hypergeometric function can also be written in terms of a hypergeometric function as ddx F (cid:18) − ( N − s )2 , s + 1; − ( N − s − , x (cid:19) = C N,s F (cid:18) − ( N − s − , s + 2; − ( N − s − , x (cid:19) , (32)which is a standard relation. In this expression the co-efficient C N,s depends on total number of atoms N andthe total spin s as C N,s = ( s + 1) (cid:0) N − s (cid:1)(cid:0) N − s − (cid:1) . (33)We now use Eq. 27 to evaluate this expression at x = 1and obtain the desired sum of the series N − s (cid:88) k =0 kA k = ( N − s )( s + 1)2 s + 3 , (34)which leads to the final result for the matrix element q s = √ s (cid:18) N + 32 s + 3 (cid:19) ,,