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]
† γ † HD =a ^ a^ + ωA^Jz + √ 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 √ coupling constant, for N , takes the value γc = ωA/2 [Hepp, Lieb 1973]. → ∞
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−|α| /2 αν 2j 1/2 √ j+m |αi⊗|ζi = j ζ |νi ⊗ |j, mi . 2 ∞ ν! j + m 1 + |ζ| 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) ≡ hα| ⊗ hζ|HD|αi ⊗ |ζi 1 = p2 + q2 − j ω cos θ + 2pjγ q sin θ cos φ . 2 A
We define 1 α = √ (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 ωA > 0,
θc = 0, qc = 0, pc = 0,
|γ| < γc ; q 2 p 4 θc = arccos(γc/γ) , qc = −2 j γ 1 − (γc/γ) cos φc,
pc = 0, φc = 0, π,
|γ| > γc . √ γc = ωA/2 . From the expressions for the critical values in the superradiant regime, where θc 6= 0, one gets
qc √ sin θc √ = − ωA √ 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 ^ |α, ζi± = N± 2 1 ± exp(iπΛ) |α, ζi = N± |αi ⊗ |ζi ± | − αi ⊗ | − ζi .
where Λ^ = j + N^ ph + ^Jz and the normalization
v ,u N! u 1 − |ζ|2 N = 1 t2 1 ± exp (−2 |α|2) . ± 1 + |ζ|2
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 − ωA (cos θ) ± 2 1 ± e±(p2+q2)(cos θ)∓N
√ ±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 Γ 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 |ψki = ck(λ, ν)|νi|j, λ − j − νi . ∞ 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 Γ -6.0 Γ 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 ωA = 1 and N = 20 atoms.
q q , Θ , Θ N N
1.0 1.0
0.5 0.5
Γ Γ 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
Θ Γ = 0.545 Γ = 0.550 Γ = 0.560 1.5 1.5 1.5 ΘΘ Θ
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 ωA = 1.
The values {Emin, qmin, θmin} are from left to right, {−10.1887, −0.935972, 0.29446}, {−10.1963, −0.964931, 0.30329} y {−10.2137, −1.05681, 0.331208}, 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\ Γ Γ 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 Γ Γ 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 Γ 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
FHkΓ+1, kΓL
1.00000
0.99995
0.99990
0.99985
0.99980 Γ 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 ωA = 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 −2|α|2 2 2 2j2 SL = 1 − N± 1 ± e 1 + (1 − |ζ| ]) 2 + 1 ∓ e−2|α| 2 1 + (1 − |ζ|2)2j2
(b) The von Neumann entropy
(±) (±) (±) (±) (±) SVN = −λ+ log λ+ − λ− log λ− where (±) −2|α|2 2 2j λ+ = N± 1 ± e 1 + (1 − |ζ| ])
(±) −2|α|2 2 2j λ− = N± 1 ∓ e 1 − (1 − |ζ| ])
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
Γ 0.2 0.4 0.6 0.8 1.0
For N = 20 atoms and ωA = 1.
SVN
0.6
0.5
0.4
0.3
0.2
0.1 Γ 0.2 0.4 0.6 0.8 1.0
2012– Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Comparison SAS vs Exact of linear and VN entropies
SL " 0.6 SL " 0.6 0.5 0.5 Γ 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 Γ 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0
For N = 10 atoms and ωA = 1.
q S VN" 1.0
0.8
0.6
0.4
0.2
Γ 0.5 1.0 1.5
2012– Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Linear Entropy for several number of atoms
SLq + 0.8
0.6
0.4
0.2
0.0 Γ 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60
We show the linear entropy as a function of γ for the following number of atoms:
NA = 40, 80, 160, and 320.
2012– Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Matter Squeezing Coefficients and Mandel parameter
The matter squeezing coefficients are defined as follows
√ ∆J⊥ ξ = 2 q , |h~Ji|
while for the photon part one has the parameter ∆N Q = ph − 1 . hNphi For the CS one can prove that ξ = 1 and a Poisson statistics that is Q = 0
2012– Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Matter and field statistics for SAS vs Exact
j$5, Ωa $1 j$5, Ωa $1 coh Ξ# coh Ξ# 3.0 ! " 3.0 ! " 2.5 2.5 2.0 2.0 Γ 1.5 Γ 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8 1.5 1.0
0.5 1.0
Matter squeezing parameters ξx (dashed) and ξy (dotted) for the even and odd SAS compared with the corresponding exact quantum results for N = 10 atoms
and ωA = 1.
2012– Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Universal invariant
The equivalent quantum operators to the phase space variables are q and θ are q q − 2ha^†a^i , 2 θ − arccos − h^J i . → N z
→
2 YNph] - q N N ArcCosX -J j \ Θ z 0.2 0.4 0.6 0.8 1.0 1.2 0.2 0.4 0.6 0.8 1.0 1.2 -0.2 -0.2
-0.4 -0.4
-0.6 -0.6 -0.8 -0.8 -1.0 -1.0 -1.2 -1.2 -1.4 -1.4
All the plots fall in the same curve, the results are presented for N = 10 atoms and
ωA = 1. However, it does not change for larger values of the number of atoms. 2012– Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Conclusions
The symmetry adapted states constitute an excellent approximation to the ground state and first excited state of the Dicke Model. We show that the CS approximation gives rise to significant differences with respect to the SAS case. The overlap or the fidelity of the variational SAS with respect to the exact solutions are close to one except in a small vicinity of the phase transition. For this reason, the fidelity is a very good tool to detect the position of the quantum phase transitions. We prove that most of the observables detect the quantum phase transition, the linear and von Neumann entropies also present singular values in that critical point. Besides we have shown that for a finite number of atoms the quantum phase transitions as a function of γ changes from its thermodynamical value. We prove that the curve
r „ « 2 † 2 ha^ a^i vs. arccos − h^Jzi , N N
is a universal invariant for the Dicke model.
2012– Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012 Thank you
2012– Beauty in Physics: Theory and Experiment Matter-Field Entanglement Cocoyoc, 2012