Assuming that the atom is initially prepared in the state cos(θ/2) |e⟩ + sin(θ/2) |g⟩ and the ideal cavity field is prepared in the Fock state |n⟩, the whole system is initially prepared in the state |ψ(0)⟩ = (cos(θ/2) |e⟩ + sin (θ/2) |g⟩) ⊗ |n⟩, via the atom–photon interaction inside the cavity, and the state vector at time t is 6 Using the Schrödinger equation i∂ψ(t)/∂t = H₃ψ(t) and the initial conditions of the system, the quantum state coefficient of the system is obtained as with 7 Considering the atom and cavity field as a bipartite system, in the basis (|+,n + 1⟩,| +,n⟩, |−,n + 1 ⟩, |−, n⟩), the density matrix of the system can be obtained.

Tracing over the degrees of the freedom of the cavity, the reduced density matrix of the atom can be obtained as 8

Then by the basic vector transformation In the basis (|e⟩, |g⟩ ), the reduced density matrix element of the atom can be obtained as follows: 9

In order to quantify the atomic coherence described by the reduced density matrix (9), we adopt the l₁ norm of coherence 10

It can easily be seen that the atomic coherence described by the reduced density matrix (9) is the absolute value of the non-diagonal element of the density matrix.

According to Eqs. (9) and (10), it is easy to calculate the atomic coherence 11

The linear entropy of the two-level atom can measure the entanglement of the atom and the cavity field^[37] 12

Now we consider the more realistic case of a leaky cavity in the system. The reduced density operator ρ (t) of the atom and the leaky cavity is described by the standard master equation: where H = gcos²(θ/2)(e^iΛ′tS₊a + e^−iΛ′tS₋a⁺) and Γ is the photon escape rate of the cavity.

Suppose that the initial state of the system is 13 by the basic vector transformation equation (13) is turned into 14 |−0⟩ does not evolve with time, and by the quantum trajectory approach,^[38,39] |+0⟩ has two possible quantum states |ϕ_no (t)⟩ = (c₁(t)|+0⟩ + c₂(t)|−1⟩) and |ϕ_yes(t)⟩ = c₃(t)|−0⟩ : |ϕ_no (t)⟩ indicates that no photon jump from the cavity to the environment occurs between time t₀ and t, and |ϕ_yes (t)⟩ indicates the fact that one photon jump appears.

The density operator is obtained as 15 and |c₃ (t)|² = 1 − |c₁(t)² − |c₂(t)|² can easily be obtained from Eq. (15).

To cope with this quantum feature it is assumed that there are no jumps up to time t. Then the “jumpless” state |ϕ_no (t)⟩ obeys the nonunitary Schrödinger equation 16 with .

By Eq. (16), with the initial conditions c₁(0) = 1, c₂(0) = 0, the solutions of c₁(t) and c₂(t) are found, respectively, to 17 with .

It is worth noting that the evolutionary process will display the Markovian behavior with the weak coupling regime g < 0.5Γ, and the non-Markovian dynamics occurs, accompanied by an oscillatory reversible decay with the strong coupling regime g > 0.5Γ.

The density operator of the system at time t, in the basis (|+1⟩, |+0⟩, |−1⟩, |−0⟩), can be obtained as 18 with

Using the same method as the above, the reduced density matrix elements of the atom are 19 The reduced density matrix element of the cavity in the basis (|0⟩, |1⟩) is 20 The l₁ norm of coherence for the atom, 21 and for the cavity, 22

By using the same method as the above, the linear entropy of the two-level atom can be obtained as follows: 23

We should point out that the quantum system used to measure the coherence is a single qubit system and an X-type quantum system, for this kind of system, l₁ norm coherence and other methods, such as trace distance measurement^[15] and the robustness of coherence,^[40] are the same. The conclusions are independent of the choice of measurement.

In Fig. 1, with the increase of the driving field strength, the value of the atomic coherence becomes larger, and the atom–field entanglement is smaller. That is to say, the atomic coherence can be protected by adjusting the driving field strength.

Fig. 1.

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	Fig. 1. Plots of atomic coherence (solid line) and atom–field entanglement (dot line) versus rescaled time gt with ω = 2, ω0 = 1, ωc = 1, n = 1, (a) Ω = 1, (b) Ω = 2.

In Fig. 2, the atomic coherence and the atom–field entanglement evolve periodically with rescaled time gt. With the increase of the frequency detuning Δ, the oscillation amplitudes of the atomic coherence and the atom–field entanglement become larger. This phenomenon is interpreted as follows. The larger the detuning Δ, the weaker the coupling between the driving field and the atom is, and the stronger the effective coupling between the atom and the cavity field becomes, and the more the resources of these two kinds (the atomic coherence and the atom–field entanglement) are transformed, so the larger their oscillation amplitudes become.

Fig. 2.

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	Fig. 2. Curves of atomic coherence (solid) and atom–field entanglement (doted) versus rescaled time gt with ω = 2, Ω = 1, n = 1, (a) Δ = 2, (b) Δ = 3.

From Figs. 1 and 2, it is interesting to find that at the initial time, the atomic coherence has a maximum value while the atom–field entanglement is zero, and the atomic coherence decreases to zero. Meanwhile the atom–field entanglement increases and reaches a maximum value. Moreover, the more the atomic coherence increases, the more the atom–field entanglement decreases. This shows that the atomic coherence itself can be transformed into the atom–field entanglement through the atom–field interaction for this system. Furthermore, the increasing frequency detuning can strengthen the transformation.

Now, we come to study the case of a pair of separate cavities each with a single atom inside and with the initial density matrix ρ(0) = ρ_Φ(0) ⊗ |0⟩₁ |0⟩₂₁ ⟨0|₂ ⟨0|, where ρ_Φ(0) = r|Φ⟩ ⟨Φ| + [(1 − r)/4]I, |Φ⟩ = cosϕ |−+⟩ + sinϕ|+−⟩, |0⟩₁ |0⟩₂ denotes the vacuum state for two cavities, with r indicating the purity of initial states and being a real number varying from 0 to 1. According to Eqs. (6) and (7) with n = 0, the density matrix of the system at time t can be obtained as where |φ⟩_i = α₀(t)|+⟩_i|0⟩_i + β₁(t)|−⟩_i|1⟩_i (i = 1,2) denotes the i-th atom–cavity evolution, in a way of the l₁ norm coherence, we can obtain the following quantum coherence for composite subsystems, i.e., q₁ ⊗ q₂, c₁ ⊗ c₂, q₁ ⊗ c₂, q₂ ⊗ c₁ and q₁ ⊗ c₁, respectively: 24 where q_i(c_i) represents the i-th atom (cavity).

Furthermore, for the initial density matrix ρ(0) = ρ_Ψ (0) ⊗ |0⟩₁ |0⟩₂₁ ⟨0|₂ ⟨0|, with ρ_Ψ(0) = r|Ψ⟩ ⟨Ψ| + [(1 − r)/4]I, |Ψ⟩ = cos ϕ |−−⟩ + sin ϕ |++⟩, the quantum coherence for composite subsystems can also be obtained, and its results are the same as those from Eq. (24). This indicates that quantum coherence of composite subsystems has the same evolution for the two initial states.

The bipartite coherence of five partitions: q₁ ⊗ q₂, c₁ ⊗ c₂, q₁ ⊗ c₂, q₂ ⊗ c₁, q₁ ⊗ c₁ is shown in Fig. 3. In the initial period of time, C_q1q2 decreases from the original value to zero, and then increases due to atom–photon interaction. The values of C_c1c2, C_q1c2, C_q2c1, C_q1c1 firstly increase from zero to a maximum, and then decrease from the maximum to zero. The phenomena mentioned above indicate that the C_q1q2 does not disappear, but transfers to other subsystems, no matter whether they are local q₁ ⊗ c₁ or non-local c₁ ⊗ c₂, q₁ ⊗ c₂, q₂ ⊗ c₁. Furthermore, based on the analysis of the results, Q₁ (t) = C_q1q2 + C_c1c2 is conserved for any value of r as shown in Fig. 3, and the Q₁ (t) value and the initial value of C_q1q2 are equal. In other words, there is mutual conversion between C_q1q2 and C_c1c2. Besides, we find that the Q₂ (t) = C_q1q2 + C_c1c2 + 2C_q1c1 |cotϕ| − 2C_q1c2 is also conserved in the case of r = 1. Interestingly, its value is also equivalent to the initial value of C_q1q2. Furthermore, we find that is also held no matter what the value of r is. It indicates that for two identical single-body systems there is a rich phenomenon of quantum coherence, and it is very meaningful to study the theory associated with coherence swapping and coherence transfer in quantum information processing.

Fig. 3.

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	Fig. 3. Plots of quantum coherence for different partitions Cq1q2 (dash line), Cc1c2 (dot line), Cq1c2 = Cq2c1 (dash–dot line), Cq1c1 (dash–dot–dot line) versus gt with ϕ = π/6. Solid line denotes the coherence invariant of Cq1q2 + Cc1c2, panels (a) and (b) show the purity of initial states r = 1, and panels (c) and (d) display the purity of initial states r = 0.6.

To better understand the effects of the driving field strength Ω on the conservation rules, it follows from Figs. 3(a) and 3(b), or Figs. 3(c) and 3(d) that Ω only influences the relative number of the bipartite coherence of partitions, and does not change the conservation rules.

It is worth pointing out that there is also a similar conservative relation of entanglement to Q_i(t) (i = 1, 2, 3) in this system,^[41–44] such as, for the initial density matrix ρ(0) = ρ_Φ(0) ⊗ |0⟩₁ |0⟩₂¹⟨0|₂ ⟨0| with r = 1, Q_i(t) (i = 1, 2, 3) of entanglement is always true, however, for the initial density matrix ρ(0) = ρ_Ψ(0) ⊗ |0⟩₁ |0⟩₂₁ ⟨0|₂ ⟨0| with r = 1, only Q₂(t) of entanglement would be conserved in the case of |cosϕ| > |sinϕ|.

In the process of atom–cavity interaction, the entanglement sudden death and entanglement sudden birth between atom–atom/atom–cavity will appear. However, the coherence will always exist and will not disappear. So the entanglement cannot keep the conservation law like quantum coherence. On the other hand, for different initial states, the times of entanglement sudden death and birth between atom-atom/atom–cavity are different, so the conservation of entanglement depends on the initial state.

Without the driving field, as displayed in Fig. 4(a), the atomic coherence decreases from the maximum to zero and then increase to a certain value, and its evolution is of damped decay. Meanwhile, the atom–cavity entanglement increases from zero to a certain value and then reduces to zero with atomic coherence. The above phenomenon indicates that the atomic coherence transforms into the atom–cavity entanglement and the cavity coherence, and the three resources eventually disappear as the cavity is dissipative. This is more visible in the weak coupling regime as displayed in Fig. 4(c). However, it can be seen clearly through Figs. 4(b) and 4(d) that the atomic coherence decreases slowly by increasing the driving field strength, no matter whether it is in the strong coupling regime or in the weak coupling regime. So for the realistic case of cavity loss in the system, although the atomic coherence and the atom–field entanglement are very sensitive to the cavity decay, they can be protected by adjusting the driving field strength.

Fig. 4.

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	Fig. 4. Plots of atomic coherence (solid line), atom–reservoir entanglement (dash line), reservoir coherence (dot line) versus gt with ω0 = 2, ωc = 1, ω = 2, η = π/2 in panels (a) and (b) strong coupling regime g = 10Γ, and in panels (c) and (d) weak coupling regime g = 0.1Γ.

With the increase of detuning Δ, no matter whether in the strong coupling regime or in the weak coupling regime as displayed in Fig. 5, we can see the atomic coherence decreases more, and the atom–cavity entanglement increases more. As the detuning increases, the effective coupling between the atom and the cavity increases, so we can see that the atomic coherence and the atom–cavity entanglement transform more in the case of larger detuning. In order to protect the atomic coherence, it is necessary to reduce the atom-driving field detuning Δ.

Fig. 5.

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	Fig. 5. Plots of atomic coherence (solid line), atom–reservoir entanglement (dash line), reservoir coherence (dot line) versus gt with Ω = 2, η = π/3 in panels (a) and (b) strong coupling regime g = 10Γ, and in panels (c) and (d) weak coupling regime g = 0.1Γ.

Considering the case of a leaky cavity in the system, there is a backflow of information from the environment to the system under suitable conditions, which means that the dynamics of the system is non-Markovian. The degree of non-Markovian dynamics can be characterized by the non-Markovianity, which can be defined as^[45] 25 where σ[t, ρ₁ (0), ρ₂ (0)] = dD[ρ₁(t), ρ₂(t)]/dt denotes the rate of change of the trace distance D[ρ₁(t), ρ₂(t)] expressed as 26 where (A denotes ρ₁(t) − ρ₂(t)), and the trace distance D describes the ability to distinguish between ρ₁(t) and ρ₂(t).

A Markovian evolution can never increase the trace distance, which means that the trace distance is a monotonically decreasing function for a Markovian progress and the σ[t, ρ₁ (0), ρ₂ (0)] ≤ 0. In other words, the trace distance is non-monotonic, which corresponds to a non-Markovian progress and the non-Markovianity N > 0. In order to calculate the degree of non-Markovianity, a specific pair of optimal initial states to maximize the σ[t, ρ₁ (0), ρ₂ (0)] are ρ₁(0) = |0⟩ ⟨0| and ρ₂ (0) = |1⟩ ⟨1|.^[45]

Figure 6 shows the trace distance D[ρ₁(t), ρ₂(t)] as a function of Γ and gt with the optimal initial states ρ₁(0) = |0⟩ ⟨0|, ρ₂(0) = |1⟩ ⟨1|. As shown in Fig. 6, with the increase of Γ, the trace distance D[ρ₁(t), ρ₂(t)] becomes monotonic and a Markovian process occurs. In contrast, with the decrease of Γ, the trace distance D[ρ₁(t), ρ₂(t)] turns non-monotonic and a non-Markovian progress occurs. The smaller the value of Γ, the bigger the change of the trace distance D[ρ₁(t), ρ₂(t)], so the larger the degree of non-Markovianty is. Furthermore, from Fig. 4, we can find the coherence and the entanglement decay slowly with the value of Γ decreasing. So there is a positive association between the non-Markovianty and coherence or entanglement.

Fig. 6.

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	Fig. 6. (color online) Trace distance D[ρ1(t),ρ2(t)] as a function of Γ and gt with the optimal initial states ρ1(0) = |0⟩ ⟨0|, ρ2 (0) = |1⟩ ⟨1|. The other parameters Ω = 1, ω0 = 2, ωc = 1.