Steady-state entanglement generation for non-degenerate qubits

The gist

We propose a scheme to dissipatively produce steady-state entanglement in a two-qubit system, via an interaction with a bosonic mode.

System Model and Hamiltonian

Let us consider a system of two qubits, with different resonance frequencies ω(1)e and ω(2)e, which are coupled to the same bosonic mode with frequency ωm, as illustrated in Fig. 1(a) In the Schrödinger picture, the system Hamiltonian reads Hˆ = ω(1)e σˆ(1)ee + ω(2)e σˆ(2)ee + ωmaˆ†â + għ â σˆ(1)+σˆ(2)+ h.c.i, (1) Here, the bosonic mode is symmetrically detuned from each of the qubits, so that ω(1)e = ωm − ∆ and ω(2)e = ωm + ∆ In analogy with the typical three-level Λ systems, we consider that Ei and Ψ−i play the roles of the two ground states, while Ψ+i is the excited state, as depicted in Fig. 1(b) According to Eq. (6), the transitions Ei ↔ Ψ+i and Ψ−i ↔ Ψ+i have effective coupling strengths √2g and ∆, respectively The Hamiltonian in Eq. (6) has a dark eigenstate (an eigenstate without a Ψ+i component) given by Di = −∆p∆2 + 2g2/√2gEi + √2g/p∆2 + 2g2/√2gΨ−i.(7) Just as in other three-level systems [43], when the condition g ∆ is fulfilled, the dark state D transforms to Ψ−i, which is our maximally entangled target state

Steady-State Entanglement Production

To describe the main mechanism responsible for the generation of entanglement, we restrict ourselves, for the moment, to the single-excitation subspace In this subspace, Ei = Gi ⊗ 1i and Ψ±i = Φ±i ⊗ 0i, where Gi = g, gi and Φ±i = (e, gi ±g, ei) / √2 In analogy with the typical three-level Λ systems, we consider that Ei and Ψ−i play the roles of the two ground states, while Ψ+i is the excited state The Hamiltonian in Eq. (6) has a dark eigenstate (an eigenstate without a Ψ+i component) given by Di = −∆p∆2 + 2g2/√2gEi + √2g/p∆2 + 2g2/√2gΨ−i.(7) A significant difference between our system and a typical three-level Λ-system is the fact that neither Ei or Ψ−i are actually ground states Because of this, the system keeps spontaneously decaying to the true zero energy ground state Gi⊗0i For this reason, the analogy becomes more accurate as the mode dissipation rate κ and the spontaneous decay rate Γ become negligible (κ, Γ g, ∆) One way to circumvent the decay and maintain the system in the one-excitation subspace is to keep injecting energy into the mode We consider this energy injection as an additional term Hˆpump = ε ↠+ â (8) In our simulations, ε was kept constant at ε = κ To characterize the steady-state entanglement between the qubits, we choose the monotone quantifier concurrence [45, 46] We numerically simulate the full system dynamics, taking into account the decoherence and also higher excited states As shown in Fig. 2(a), we obtain a strong entanglement within a large region of parameters ∆ and g Moreover, we see that satisfying the EIT condition g ∆ is a necessary but not sufficient condition to reach our maximally entangled target state

Entanglement Dynamics and Optimization

We have chosen four sets of parameters to illustrate the system dynamics: A shows the situation where g ∆ and the population is lead to the respective dark state, which has a Gi character [see Eq. (7)], and no entanglement is observed in B, we have g = ∆ which leads to weakly entangled dark state given by a mixture of g, gi and Φ−i In C we have an interesting situation where the EIT condition g ∆ is fulfilled, the dark state a Φ−i character, but the detuning is so small that it takes too long to populate Φ−i, and since the system keeps decaying to the true ground state Gi ⊗ 0i, the entanglement is affected D shows a near-optimal situation, with g ∆ and ∆ Γ, driving approximately all the population to the dark state Φ−i In Fig. 2(b) we show the behavior of the concurrence as a function of the detuning for a fixed value of g = κ We observe that the concurrence decreases significantly when we move away from the optimal point ∆ ≈ 0.1g This finding is in accordance with what was previously discussed for the parameters sets A and C This optimal ratio between g and ∆ is influenced by the system’s decay rate Γ A reduced value of Γ allows us to achieve entanglement for smaller detunings Focusing on the entanglement dynamics, we show in Fig. 2(c) the concurrence over time for the cases A, B, C and D We also include a visual representation of the time evolved partial density matrix, where the bosonic mode has been traced out, showing that we achieve the maximally entangled target state Φ−i for the near-optimal set of parameters D

Adiabatic Population Transfer

Inspired by the schemes to counteract decoherence developed for multilevel atoms, we propose a STIRAP like process to efficiently populate the entangled state Ψ−i To this end, we consider a trapped ion implementation, where two ions are confined in a harmonic potential and coupled to the same motional mode, as depicted in Fig. 3(a) To perform the adiabatic population transfer, the system must be initially prepared in the state Ei, which consists of the two ions in the ground state and the motional state with one excitation The initial state Ei corresponds to the dark state when ∆ g By reversing this condition to g ∆ the dark state adiabatically transforms to Ψ−i By manipulating the coupling strength g and the detuning ∆, which can be accomplished by changing the power of a laser resonant to the red sideband transition and by magnetic field gradient varying in time, respectively, we can control the adiabaticity throughout the process so that all of the population is transferred coherently to the final maximally entangled state Ψ−i, with a negligible population in the state Ψ+i (see Appendix A)<ref:2205.

1. The AI system can generate maximally entangled steady states in two-qubit systems by engineering dissipation to combat decoherence, as shown by the proposal to dissipatively produce steady-state entanglement in a two-qubit system, via an interaction with a bosonic mode. This allows for the creation of long-lived quantum correlations suitable for robust quantum communication protocols.

2. The AI can implement an adiabatic population transfer scheme using STIRAP-like processes to coherently drive the system to the final maximally entangled state Ψ−i, with a negligible population in the state Ψ+i. This capability enables high-fidelity state preparation essential for quantum computation routines or sensing applications.

3. The AI can optimize coupling parameters by leveraging symmetry, as the paper shows that the symmetry in the energy shifts between the emitters with respect to the bosonic mode actually allows us to achieve a stronger entanglement and "This interaction with the quantized mode provides a coupling regime strong enough so that the time scales of the effective interactions are much faster than the qubit relaxation, leading to higher degrees of entanglement." This leads to more efficient hardware design for generating high-entanglement resources.

4. The AI can dynamically adjust system parameters during state preparation by employing time-dependent control functions, such as ∆ (t) = ∆max squared [1 - tanh λ(t − t0)] and a time-varying coupling g (t) = gmax squared [1 + tanh λ(t − t0)], to achieve near-perfect coherent population transfer, even when simultaneous control is not possible.