Bound entanglement in the Jaynes–Cummings model

Abstract

We study in detail entanglement properties of the Jaynes-Cummings model assuming a two-level atom (qubit) interacting with the first N levels of an electromagnetic field mode (qudit) in a cavity. In the Jaynes-Cummings model, the number operator is the conserved quantity that allows for the exact diagonalization of the Hamiltonian and thus we study states that commute with this conserved quantity and whose structure is preserved under the Jaynes-Cummings dynamics. Contrary to the common belief, we show that there are bound entangled states that satisfy the symmetries imposed by the conservation of the number of excitations when N > 3. Furthermore we show that the Jaynes-Cummings interaction can be used to generate bound-entanglement between the atom and the mode.

Perturbative expansion of entanglement negativity using patterned matrix calculus

“…which was shown in Ref. [51] to have zero negativity while remaining entangled (these states are 'bound' entangled [52,53]). The negativity and second-order Rényi entropy are plotted in Fig.…”

“…Jaynes-Cummings model A commonly used model in quantum optics is the Jaynes-Cummings model (JCM), which characterizes a two-level atom interacting with a single quantized mode of a bosonic field. The JCM has been the subject of much theoretical and experimental work [49,50], including recent theoretical studies of its entanglement properties [39,51]. The JCM Hamiltonian is given in units of = 1 by…”

“…whereâ is the bosonic annihilation operator for the field, σ = |g e| lowers the atom from the excited state |e to the ground state |g , ω is the frequency of the bosonic mode, ∆ is the detuning between the mode and the atomic transition frequency, and g is a coupling constant [51]. The Hamiltonian conserves total excitation number 1 ⊗â †â +σ †σ ⊗ 1, restricting the dynamics to systems of size 2 × N , where N is the number of Fock states of the bosonic mode that are coupled to by the initial conditions.…”

Entanglement and nonlocality in diagonal symmetric states of N qubits

“…Notice that ρ JC ∈ B(C 2 ⊗C M ) but when expressed in the Dicke basis they can be mapped onto a diagonal symmetric states; ρ JC ∈ B((C 2 ) ⊗M ). For the particular case of M = 4, these states read [23]: The generic PPT region given by Eqs. (12).…”

“…Notice that ρ JC ∈ B(C 2 ⊗C M ) but when expressed in the Dicke basis they can be mapped onto a diagonal symmetric states; ρ JC ∈ B((C 2 ) ⊗M ). For the particular case of M = 4, these states read [23]: with a, b ∈ R ≥ 0 and a > b. Interestingly enough, this family of states fulfill E 1 = 0 and E 3 = 0 and they are extremal points in the subset of states satisfying PPT Γ 1 ≥ 0 but they do not fulfill that PPT Γ 2 > 0. Thus, they are PPT-edge states with respect to the partition ρ Γ 1 JC .…”