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Matter-Field Entanglement Within the Dicke Model

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Matter-Field Entanglement Within the Dicke Model Matter-Field Entanglement within the Dicke Model O. Casta~nos,R. L´opez-Pe~na,J. G. Hirsch, and E. Nahmad-Achar Instituto de Ciencias Nucleares, UNAM 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Content The Dicke Hamiltonian Energy Surface of coherent states (CS) Symmetry adapted coherent states (SAS) Comparison variational vs Exact Results Fidelity Linear and von Neumann entropies Matter squeezing coefficient Conclusions 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 The Dicke Hamiltonian The Dicke model describes a cloud of cold atoms interacting with a one-mode electromagnetic field in an optical cavity [Dicke 1954] y γ y HD =a ^ a^ + !A^Jz + p a^ +a ^ ^J+ + ^J- : N Consider only the symmetric configuration of atoms i.e., N = 2j. The model presents a quantum phase transition when the p coupling constant, for N , takes the value γc = !A=2 [Hepp, Lieb 1973]. ! 1 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Energy surface for CS The energy surface, or associated classical Hamilton function, is formed by means of the expectation value of the Hamiltonian with respect to a test state. We consider as test state the direct product of coherent states of the groups HW(1), for the electromagnetic part (Glauber 1963) and SU(2), for the atomic part (Arecchi et al 1972) , i.e., 2 +j e-jαj =2 αν 2j 1=2 p j+m jαi⊗jζi = j ζ jνi ⊗ jj; mi : 2 1 ν! j + m 1 + jζj ν=0 m=-j X X 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Energy surface for CS The energy surface is given by H(α, ζ) ≡ hαj ⊗ hζjHDjαi ⊗ jζi 1 = p2 + q2 - j ! cos θ + 2pjγ q sin θ cos φ : 2 A We define 1 α = p (q + i p) ; 2 θ ζ = tan exp (i φ) ; 2 where (q; p) correspond to the expectation values of the quadratures of the electromagnetic field and (θ, φ) determine a point on the Bloch sphere. 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Critical points of the CS Energy surface The critical points that minimize the energy surface, for !A > 0, θc = 0; qc = 0; pc = 0; jγj < γc ; q 2 p 4 θc = arccos(γc/γ) ; qc = -2 j γ 1 -(γc/γ) cos φc; pc = 0; φc = 0; π, jγj > γc : p γc = !A=2 . From the expressions for the critical values in the superradiant regime, where θc 6= 0, one gets qc p sin θc p = - !A p cos φc : N 2 cos θc 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Symmetry Adapted Coherent States The test states (CS) do not respect the symmetry of the Dicke Hamiltonian. Then it is convenient to define states adapted to the discrete symmetry present in HD, 1 ^ jα, ζi± = N± 2 1 ± exp(iπΛ) jα, ζi = N± jαi ⊗ jζi ± j - αi ⊗ j - ζi : where Λ^ = j + N^ ph + ^Jz and the normalization v ,u N! u 1 - jζj2 N = 1 t2 1 ± exp (-2 jαj2) : ± 1 + jζj2 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Energy surface for SAS The energy surface for the symmetry adapted states is given by 1 2 2 2 hHDi± = ± p + q 1 - 2 1 ± e±(p2+q2)(cos θ)∓N 2 N ±1 tan θ cos θ - !A (cos θ) ± 2 1 ± e±(p2+q2)(cos θ)∓N p ±p tan θ sin φ + q ep2+q2 sin θ cos φ (cos θ)-N + 2 N γ : p2 q2 N e + (cos θ)- ± 1 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Energy surface for SAS There are two procedures, one substitutes the analytical critical points for the CS in the energy surface for the SAS; the same procedure can be used to determine the expectation values of other observables of interest (OC et al Phys. Rev A 84 (2011)). The second one is the following: For a given number of atoms N together with a fix value of the coupling parameter γ, we determine the values qc, pc, θc, φc, where the energy surface presents a minimum. The minima always occur for pc = 0 and φc = 0; π. In this presentation I will show you results for φc = 0. The values of qc and θc have to be determine numerically. 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Energy spectra E -0.3 -0.4 -0.5 -0.6 g 0.2 0.4 0.6 0.8 1.0 1.2 -0.7 -0.8 Energy spectra of HD, for N = 10 atoms. The size of the space is fixed to get convergence in the values of the energy spectra. 0.5 0.2 0.4 0.6 0.8 1.0 1.2 -0.5 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Quantum states The basis states can be written λ j ki = ck(λ, ν)jνijj; λ - j - νi : 1 λ=0 X ν=maxX(0,λ-2j) 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 E E -4.8 -4.6 -5.0 -4.8 -5.2 -5.0 -5.2 -5.4 -5.4 -5.6 -5.6 -5.8 -5.8 -6.0 g -6.0 g 0.40 0.45 0.50 0.55 0.60 0.65 0.40 0.45 0.50 0.55 0.60 0.65 At the left the ground state energy of exact (gray), SAS (red), CS (green), while at the right the first excited energies of exact (brown), SAS (blue) as functions of γ, for !A = 1 and N = 20 atoms. q q , q , q N N 1.0 1.0 0.5 0.5 g g 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 -0.5 -0.5 -1.0 -1.0 The phase transitions can be seen clearly in the expressions for q and θ. 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Contours Energy plots for SAS q g = 0.545 g = 0.550 g = 0.560 1.5 1.5 1.5 qq q 1.0 1.0 1.0 0.5 0.5 0.5 q 0.0 0.0 0.0 -6 -4 -2 0 2 -6 -4 -2 0 2 -6 -4 -2 0 2 We take N = 20 atoms and !A = 1. The values fEmin; qmin; θming are from left to right, f-10:1887; -0:935972; 0:29446g, f-10:1963; -0:964931; 0:30329g y f-10:2137; -1:05681; 0:331208g, respectively. The plot shows why there is a jump in the minimum critical points. 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Comparison Variational vs Exact Results XJz\ XJz\ g g 0.2 0.4 0.6 0.8 1.0 1.2 0.2 0.4 0.6 0.8 1.0 1.2 -2 -2 -4 -4 -6 -6 -8 -8 -10 -10 Quantum, SAS, and CS results for hJzi. XJz\ XJz\ -7.0 -7.0 -7.5 -7.5 -8.0 -8.0 -8.5 -8.5 -9.0 -9.0 -9.5 -9.5 -10.0 -10.0 g g 0.50 0.52 0.54 0.56 0.58 0.60 0.50 0.52 0.54 0.56 0.58 0.60 Comparison between the variational CS, SAS and EXACT results for the even case. The comparison between SAS and EXACT results for the odd case. 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Fidelity F F 1.0 1.00 0.95 0.9 0.90 0.8 0.85 0.80 0.7 0.75 Γ g 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.45 0.50 0.55 0.60 0.65 The fidelity is a measure to establish how similar is a probability distribution to another one. When the system is pure this reduces to the overlap between the corresponding states. At the left, the fidelity between the exact and symmetry adapted states using the critical points associated to the CS while at the right using the critical points of SAS. 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Quantum phase transitions FHkg+1, kgL 1.00000 0.99995 0.99990 0.99985 0.99980 g 0.0 0.2 0.4 0.6 0.8 1.0 Determination of the value of γ by means of the fidelity for N = 20 atoms and !A = 1, we use a step of ∆γ = 0:001. The mimimum value of the fidelity is given by (γ, F(γ + ∆γ, γ)) = (0:567; 0:999807). 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Linear and VN Entropies A pure bipartite state is entangled if it can not be written as a tensorial product of the two parts. To measure it, one uses: 2 (a) The linear entropy SL = 1 - Tr(ρM), and it takes the form (±) 4 -2jαj2 2 2 2j2 SL = 1 - N± 1 ± e 1 + (1 - jζj ]) 2 + 1 ∓ e-2jαj 2 1 + (1 - jζj2)2j2 (b) The von Neumann entropy (±) (±) (±) (±) (±) SVN = -λ+ log λ+ - λ- log λ- where (±) -2jαj2 2 2j λ+ = N± 1 ± e 1 + (1 - jζj ]) (±) -2jαj2 2 2j λ- = N± 1 ∓ e 1 -(1 - jζj ]) 2012{ Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Linear and VN entropies for SAS SL 0.4 0.3 0.2 0.1 g 0.2 0.4 0.6 0.8 1.0 For N = 20 atoms and !A = 1.
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