Using a hybrid system (Cooper pair box plus nanomechanical resonator) in the presence of Kerr nonlinearities and losses to control the entropy of the subsystems

Optics Communications ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Invited Paper

Using a hybrid system (Cooper pair box plus nanomechanical resonator) in the presence of Kerr nonlinearities and losses to control the entropy of the subsystems C. Valverde a,b,c,n, A.N. Castro b, B. Baseia d,e a

Câmpus Henrique Santillo, Universidade Estadual de Goiás, 75.132-903 Anápolis, GO, Brazil Universidade Paulista (UNIP), 74.845-090 Goiânia, GO, Brazil c Escola Superior Associada de Goiânia (ESUP), 74.840-090 Goiânia, GO, Brazil d Instituto de Física, Universidade Federal de Goiás, 74.690-900 Goiânia, GO, Brazil e Departamento de Física, Universidade Federal da Paraíba, 58.051-970 João Pessoa, PB, Brazil b

ar t ic l e i nf o

a b s t r a c t

Article history: Received 14 November 2015 Received in revised form 12 January 2016 Accepted 15 January 2016

We consider the Jaynes–Cummings model to describe the interaction of a Cooper pair box and a nanomechanical resonator in the presence of a Kerr medium and losses. The evolution of the entropy of both subsystems and the Cooper pair box population inversion were calculated numerically. It was found that population inversion and entropy increase when the frequency of the nanoresonator is time-dependent, even in the presence of losses; the effect is very sensitive to detuning and disappears in resonant regime. We also compare effects of the losses on each subsystem. & 2016 Published by Elsevier B.V.

Keywords: Jaynes–Cummings model Cooper pair box Nanomechanical resonator Kerr medium Losses

1. Introduction Quantum information processing with hybrid systems has attracted much attention over the past years [1,2]. These systems combine advantages of atoms, spins, and solid-state devices with applications in quantum information and quantum computation [3,4]. They also benefit from compatible advantages of the individual subsystems and provide potential opportunity to overcome obstacles to quantum state engineering [5,6]. An early and important example, extensively explored in quantum optics, is the atom-field coupled system. Inspired by several results and the limitations of optical cavities, many researchers in the field first substituted studies performed in the optical domain for others performed in the microwave domain of superconducting cavities coupled to Rydberg atoms, and then going to the quantum electrodynamics circuit coupled to nanoresonators: this is implemented through a system composed by a Cooper pair box (CPB) that interacts with a nanomechanical resonator (NR), constituting an important example of a hybrid system. The CPB associated to an n Corresponding author at: Câmpus Henrique Santillo, Universidade Estadual de Goiás, 75.132-903 Anápolis, GO, Brazil. E-mail address: [email protected] (C. Valverde).

NR substitutes the atom in the early atom-field system and this allows us to extend the generation of nonclassical states of the light field to the microwave domain ( ∼ 1 GHz) with wide applications in nanocircuits. It has been explored in several works, e.g. photon blockade [7]; phonon blockade [8]; Bell inequality violations [9]; Landau–Zener transitions [10]; atomic physics and quantum optics [11]; mechanisms for photon generation from a quantum vacuum [12]; quantum simulation [13] and quantum circuit combined with electron spins of nitrogen-vacancy centers [14,15]. Among the various works dealing with this hybrid system, e.g., [16–18], only a few of them treat the situation where one of the frequencies [19,20] or the amplitude [21] varies with time. In the CPB-NR system these variations affect its dynamic properties, e.g., by changing the transition rates of excitations in the subsystems [22]. We can also mention the coupling rate determination in optomechanical systems via the optical mass detection technique in terms of coupled NR system [23,24]; a recent comprehensive review on optical mass sensors is given in [25]. The transmission-line shunted plasma oscillation qubit, in short termed as transmon [26–30], is based on the CPB system and relevant knowledge has been acquired on transmon-based quantum computing [31,32]. In this report we will employ the Jaynes–Cummings model to

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treat a CPB-NR system in the presence of losses; a nonlinear Kerr medium is also added to include influences of nonlinearities. The addition of Kerr nonlinearity in this system has been applied in nanomechanical resonators, e.g. to obtain high precision measurements [33,34]; to study the influence of nonlinearity on the system stability [35]. We investigate the time evolution of the CPB population inversion, I (t ), and the statistical properties of both subsystems. We consider the NR initially in a coherent state and the CPB in its excited state. The influence caused by the time-dependent NR frequency on the properties of both subsystems is also studied. We consider the combined effects of nonlinearity and losses on the dynamics of the CPB population inversion and on the NR entropy, the latter being used as a measure of entanglement (here meaning mixed state). We compare the results obtained in the resonant case (ωNR = ωCPB ) with those obtained for small detunings between the CPB and NR. The influences of losses from the NR and from the CPB upon the mentioned properties are also compared.

† where a^ (a^ ) is the creation (annihilation) operator for the NR excitations, corresponding to the frequency ω and mass m; EJ0 and Ec are respectively the energy of each Josephson junction and the charge energy of a single electron; C1 and CJ0 stand for the input capacitance and the capacitance of each Josephson tunnel, reπ spectively; Φ0 = e is the quantum flux and Ng = C1Vg /2e is the charge number in the input with the input voltage Vg and ωc is the CPB frequency. We have used the Pauli matrices to describe our system operators, where the states 0 and 1 represent the number of extra Cooper pairs in the superconducting island. So we have, σ^z = 1〉〈1 − 0〉〈0 and Ec = e2/C1 + 4C 0J . The magnetic flux can be written as the sum of two terms,

(2)

Φe = Φb + Bℓx,

where the term Φb is the induced flux, corresponding to the equilibrium position of the NR and the second term describes the contribution due to the NR vibration; B represents the magnetic field created in the loop and ℓ is the length of the NR. We write the † displacement x^ as x^ = x (a^ + a^ ), where x0 represents the NR 0

amplitude oscillation. Substituting Eq. (2) in Eq. (1) and controlling the flux Φb we can adjust cos (πΦb/Φ0) = 0 to obtain

2. The Hamiltonian system The present procedure is inspired by the work of Ref. [33] where the authors use a CPB coupled to two resonant NRs to generate pairs of superposition coherent states 180° out of phase with each other. A superconductor CPB charge qubit is adjusted to the input voltage Vg of the system through a capacitor with an input capacitance C1. Observing the configuration shown in Fig. 1 we see three loops: a small loop on the left, another on the right, and a great loop in the center. The control of the external parameters of the system can be implemented via the input voltage Vg and the three external fluxes ΦL , Φr and Φe. The control of these parameters allows us to make the coupling between the CPB and the NR. In this way one can induce small neighboring loops. The great loop contains the NR and its effective area in the center of the apparatus changes as the NR oscillates, which creates an external flux Φe (t ) that provides the coupling between the CPB and the NR. We will consider ℏ = 1 and assume as identical the four Josephson junctions, with the same Josephson energy EJ0, the same being assumed for the external fluxes ΦL and Φr, i.e., with the same magnitude, but opposite sign: ΦL = − Φr = Φx . In this way we can write the Hamiltonian describing the entire system as

⎛ πΦx ⎞ ⎛ πΦ ⎞ ⎛ 1⎞ † ^ H = ωa^ a^ + 4Ec ⎜ Ng − ⎟ σ^z − 4E 0J cos ⎜ ⎟ cos ⎜ e ⎟ σ^x, ⎝ ⎠ ⎝ Φ0 ⎠ ⎝ Φ0 ⎠ 2

(1)

⎛ πBℓx^ ⎞ ⎛ πΦx ⎞ ⎛ 1⎞ † ^ ⎟⎟ σ^x H = ωa^ a^ + 4Ec ⎜ Ng − ⎟ σ^z − 4E 0J cos ⎜ ⎟ sin ⎜⎜ ⎝ ⎝ Φ0 ⎠ 2⎠ ⎝ Φ0 ⎠

and making the approximation πBℓx/Φ0 ⪡1 we find the Hamilto† † ^ 1 nian in the form, H = ωa^ a^ + 2 ωc σ^z + λ 0 (a^ + a^ ) σ^x , where the constant coupling λ0 and the effective energy ωc ( ℏ = 1) are given by λ 0 = − 4E 0J cos (πΦx/Φ0)(πBℓx0/Φ0), ωc = 8Ec (Ng − 1/2). In the rotating wave approximation the above Hamiltonian results in the form, † † ^ 1 Heff = ωa^ a^ + 2 ωc σ^z + λ 0 (σ^+a^ + a^ σ^−).

(4)

Next, we consider a more general scenario, substituting ω → ω (t ) = ω0 + f (t ), λ 0 → λ (t ) = λ 0 (1 + f (t ) /ω0 )1/2, where ω0 is the natural frequency of the NR, f (t ) is due to an external agent acting upon the NR. Thus we have the NR frequency ω (t ) and the detuning f (t ) = ω (t ) − ωc , which turns the Hamiltonian time-dependent. Here we assume that ω0 = ωc ⪢f (t ) and this leads the coupling to λ (t ) ≃ λ 0 . This result allows us to neglect the coupling variation with time. In addition, we consider the presence of the term κ to stand for the time-dependent loss affecting the CPB, the term δ being the same for the NR, and χ (t ) = χ0 + εf (t ) is the response time of the Kerr medium [36–38]. This extended and somewhat realistic scenario requires the substitution of the Ha^ by another ^ , given by miltonian H / eff ^ = ω (t ) a^ †a^ + 1 ω σ^ + λ (t ) a^σ^ + a^ †σ^ + χ (t ) a^ † 2a^ 2 − i κ 1〉〈1 − i δa^ †a^ . / + − c z 2 2 2

(

Fig. 1. Model for the CPB-NR coupling.

(3)

)

(5)

In Eq. (5), the first (second) term describes the NR (CPB) subsystem, the third term represents the CPB-NR coupling, the fourth term stands for the time dependence of the Kerr medium, and the fifth and the sixth terms, κ and δ, were mentioned above. The operators σ± = σx ± iσy leads to σ^+ = |1〉〈0|, σ^− = |0〉〈1|; κ is the CPB decay coefficient from the excited state 1 to the fundamental state |0〉, δ is the same for the NR and λ (t ) is the time-dependent coupling between the CPB and NR. Since the response time of the Kerr medium is assumed so fast, the medium follows the NR adiabatically and the third-order nonlinear susceptibility can be modulated by the NR frequency ω (t ). Actually, the inclusion of loss in the system makes the treatment somewhat more realistic since dissipation is ubiquitous in the real world; as a consequence Eq. (5) corresponds to the evolution of a system described by a nonHermitian Hamiltonian, widely used in the literature [39–43]. In a recent work studying quantum entropy and other properties using

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the Buck–Sukumar model, including loss as done here, and also using a non-Hermitian Hamiltonian [44], the results are similar to those found in Ref. [45] using the master equation at zero temperature via a Hermitian Hamiltonian for the same Buck–Sukumar model. The wave function that describes our system can be written as,

+ C0, n ( t ) 0, n ⎤⎦,

n

(6)

0.6

where C1, n (t ) and C0, n (t ) are respectively the probability amplitudes of the states 1, n and 0, n , which means the CPB in its excited state 1 or ground state 0 with elementary excitations n in the NR. As mentioned before, at t ¼0 the system is decoupled, with the CPB initially in its excited state 1 and the NR in a coherent state α represented by,

0.4

0.2

0.0 0

∞

| α〉 =

∑ Fn |n〉. n= 0

(a)

0.8

I(t)

∞

∑ ⎡⎣ C1, n ( t ) 1, n

|Ψ ( t ) 〉 =

3

25

50

75

100

125

150

175

λ0t

(7)

The total wave function in the initial state can be written as ∞ |Ψ (0)〉 = ∑n Fn |1, n〉, with the initial amplitudes C0, n (0) = 0 and

0.8

∑n = 0 C1, n (0) 2 = 1. The Schrödinger equation for the present system, described by non-Hermitian and time-dependent Hamiltonian in Eq. (5) is,

(b)

∞

^ = − i / |Ψ ( t ) .

(8)

We obtain the following set of coupled equations of motion for the amplitude probabilities C1, n (t ) and C0, n + 1 (t ):

∂C1, n ( t ) ∂t

0.6

I(t)

d |Ψ ( t ) dt

0.4

0.2

⎛ ⎞ ω 1 = ⎜ − inω (t ) − i c − iχ (t )(n2 − n) − (κ + nδ ) ⎟ C1, n ( t ) ⎝ ⎠ 2 2 − iλ (t ) (n + 1) C0, n + 1 ( t ),

(9)

0.0 0

∂C 0, n + 1 ( t ) ∂t

25

50

75

100

125

150

175

200

λ0t

⎞ ⎛ ω 1 = ⎜ − i (n + 1) ω (t ) + i c − iχ (t )(n2 + n) − (n + 1) δ⎟ ⎠ ⎝ 2 2 × C 0, n + 1 ( t ) − iλ (t ) (n + 1) C1, n ( t ),

200

(10)

1.0

(c)

whose solutions allow us to obtain the entropy of the NR subsystem and the population inversion of the CPB. 0.8

3. Time evolution of the population inversion The present approach also allows us to investigate the CPB dynamics in a non-perturbative approach. One way to characterize the influence of the NR upon the CPB is shown by the CPB population inversion. This parameter is defined as ∞

I (t ) =

∑

2⎤ ⎡ C (t ) 2 − C 0, n + 1 (t ) ⎦. ⎣ 1, n

n= 0

I(t)

0.6

0.4

0.2

(11)

To calculate this property, I(t), we will assume the NR frequency varying with time as ω (t ) = ω0 + f (t ). The third order nonlinear susceptibility is modulated as χ (t ) = χ0 + εf (t ) and we also assume the NR initially in a coherent state with the mean excitation number n¯ = 25 and ωc /λ 0 = ω0/λ 0 = 20k . We consider the time evolution of the population inversion for different values of the decay coefficients κ and δ.

0.0 0

3

6

9

12

15

18

λ0t Fig. 2. Time evolution of the population inversion for different values of the parameters κ (t ) and δ (t ) for: n = 25, ω0/λ 0 = ωc /λ 0 = 20k , f (t ) = 0 ; χ0 /λ 0 = 0.2; (a) κ /λ 0 = 0.0 and δ /λ 0 = 0.0 ; (b) κ /λ 0 = 0.01 and δ /λ 0 = 0.0 ; (c) κ /λ 0 = 0.0 and δ /λ 0 = 0.01.

3.1. Resonant case: f (t ) = 0 If the model is restricted to the usual case, with the NR frequency constant, Eqs. (9) and (10) can be solved numerically. In Fig. 2a, we plotted the inversion I(t) as a function of time. In the

absence of losses, the average inversion, defined by its value during the collapse, is greater than zero, close to 0.5, when we consider the value χ0 /λ 0 = 0.2. However, neglecting the presence of the Kerr medium ( χ0 /λ 0 = 0), the average inversion vanishes, as

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expected. Here the third-order nonlinear susceptibility represents the coupling of the NR and the Kerr medium. The higher the value of χ the stronger the coupling of the NR with the Kerr medium, and vice-versa. Considering the case χ0 /λ 0 = 0.2 and the system with no loss ( κ = δ = 0), we observe the collapse-revival effect (see Fig. 2a). In this figure, the horizontal line crossing the value I (t ) = 0.5 of the average inversion is due to the presence of nonlinearity, whose absence turns the average inversion null. We note in this figure the amplitude of oscillation decreasing with time, due to the presence of the Kerr medium. However, when we only consider loss in the CPB ( κ ≠ 0, δ = 0) the collapse-revival effect reappears, with subsequent oscillations that smoothly vanish (see Fig. 2b). Contrarily, when the loss only affects the NR (κ ¼0, δ ≠ 0) such oscillations rapidly vanish. This shows that the spoiling effect caused by the NR loss dominates that coming from the loss in the CPB, see Figs. 2c and b. This could be attributed to the NR being a system with an energy spectrum having infinite degrees of freedom (many oscillation modes) while the CPB is a system with an energy spectrum having only two degrees of freedom ( 0 and 1 states).

0.8

(a)

0.6

0.4

I(t)

4

0.2

0.0

-0.2

-0.4 0

15

30

45

60

75

90

λ0t

(b)

0.6

3.2. Off-resonant case: f (t ) = τ sin (ω′)

4. Time evolution of the entropy

+ + − − SNR (t ) = − {πNR (t ) ln πNR (t ) + πNR (t ) ln πNR (t )} ,

where with 〈C |S〉 =

0.2

0.0

0

15

30

45

60

75

90

λ0t 0.8

(c)

0.6

I(t)

0.4

0.2

0.0

0

Nowadays, some devices are based on quantum mechanical phenomena, and this is also true for information transmission. For example, in optical communication a polarized photon can carry information. Now, the entropy of entanglement is defined for pure states as the von Neumann entropy of one of the reduced states, e.g., the NR entropy,

± (t ) = 1 (〈C |C 〉 + 〈S|S〉 ± 1 [( C |C πNR 2 2 ∞ 2 〈S|S〉 = 〈C |C 〉 = ∑n = 0 C1, n (t ) , ∞ S|C * = ∑n = 0 C1,⁎ n + 1 (t ) C0, n + 1 (t ).

I(t)

0.4

When the system is non resonant, with τ /λ 0 = 10 and ω′/λ 0 = 1, and assuming loss only in the CPB subsystem ( κ ≠ 0, δ = 0), we observe a periodical behavior of the CPB population inversion, with amplitude decay over time – see Fig. 3a; in this case the collapse-revival effect of population inversion is not observed. However, this behavior is modified when the loss comes from the NR, with the extinction of oscillations in the CPB population inversion (not shown in figures). These effects are due to detuning. Comparing Fig. 3a with Fig. 3b and assuming loss only in the CPB, we note that the maximum and minimum amplitudes of the population inversion decrease when the frequency of detuning ω′ increases by 20 times. This is due to the approximation of the external frequency ω′ to the frequencies of both subsystems. Considering ω′ increasing and loss affecting only the CPB – see Fig. 3b, the system continues displaying population inversion and oscillations, the latter exhibiting small periods, an effect not shown when the loss comes only from the NR – see Fig. 3c. As shown in Figs. 3b and c the variation of the average amplitude diminishes as the frequency ω′ increases. We close this section by pointing out that the irregular behavior of the inversion is due to the dependence of the Rabi frequencies on the photon number and its distribution. Also, we have found that the larger the losses the more quickly the oscillation amplitude is suppressed and also more quickly the population inversion reaches its maximum value and its asymptotic value 0 – see Fig. 3c.

3

6

9

12

15

18

λ0t Fig. 3. Time evolution of the population inversion for different values of the parameters κ (t ) and δ (t ) for: n = 25, ω0/λ 0 = ωc /λ 0 = 20k , ε/λ 0 = 0.001; τ /λ 0 = 10 ; f (t ) = τ sin (ω′t ) ; χ0 /λ0 = 0.2; (a) κ /λ 0 = 0.01; δ /λ 0 = 0.0 and ω′/λ 0 = 1; (b) κ /λ 0 = 0.01; δ /λ 0 = 0.0 and ω′/λ 0 = 20; (c) κ /λ 0 = 0.0 ; δ /λ 0 = 0.01 and ω′/λ 0 = 20 .

(12) − S|S )2 + 4 C |S ∑n = 0 C0, n + 1 (t ) 2, ∞

2 ]1/2 )

and

In our calculations using Eq. (12) we will assume the NR frequency varying in the form ω (t ) = ω0 + f (t ). The third order nonlinear susceptibility is modulated in the form χ (t ) = χ0 + εf (t ), and we also assume the NR initially in a coherent state with the

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4.1. Resonant case: f (t ) = 0

0.6

We will use the values of parameters of Fig. 4a. In an ideal system the parameters κ and δ are null, as assumed in Fig. 4a: in this figure the maximum of the NR entropy is close to ln 2. After the start of the interaction the NR entropy gradually tends to its minimum, then returns to its maximum and remains oscillating regularly due to the sequence of energy exchanges between the systems. First, assuming that a small value of loss only affects the CPB, say κ /λ 0 = 0.01, the NR assumed ideal (δ = 0) for the case χ0 /λ 0 = 0.2, the maximum value of the entropy shows no significant change for small times – see Fig. 4b whereas this maximum increases for larger times (between 25 and 75) – see Figs. 4a and b. For λ 0 t > 75 the amplitude of the entropy oscillations goes to zero, while holding their periodicity and both subsystems going to their vacuum state, null entropy. If instead we consider the small loss only affecting the NR, say δ /λ 0 = 0.01, then the entropy oscillations vanishes rapidly (cf. Fig. 4c), which shows the entropy being more sensitive to the loss coming from the NR than that coming from the CPB. For larger values of the parameters δ and κ, the entropy movies rapidly to zero, as expected, due to the passage of both subsystems to their respective ground states.

SNR(t)

0.5 0.4 0.3 0.2 0.1

(a)

0.0 0

25

50

75

100

λ0t 0.7 0.6

SNR(t)

0.5

4.2. Off-resonant case: f (t ) = τ sin (ω′)

0.4 0.3 0.2 0.1

(b)

0.0 0

25

50

75

100

λ0t 0.5

0.4

SNR(t)

5

0.3

0.2

0.1

(c)

0.0 0

4

8

12

16

λ0t Fig. 4. Time evolution of the entropy for different values of the parameters κ (t ) and δ (t ) for: n = 25, ωc /λ 0 = ω0/λ 0 = 20k , χ0 /λ0 = 0.2, f (t ) = 0 . (a) κ /λ 0 = 0, 0λ 0 and δ /λ 0 = 0.0 ; (b) κ /λ 0 = 0.01 and δ /λ 0 = 0.0 ; (c) κ /λ 0 = 0.0 and δ /λ 0 = 0.01.

mean number of photon n¯ = 25 and ωc /λ 0 = ω0/λ 0 = 20k . Next we consider the time evolution of the entropy for different values of the decay coefficients κ and δ, for the resonant and off-resonant cases.

Let us now consider the variation in the detuning parameter, where τ and ω′ are parameters that modulate the NR frequency. Our discussion is limited to the condition τ ⪡ωc , ω0 and also assuming that ω′ is small to avoid interaction of the CPB with other modes of the NR. We have chosen various values τ of amplitude modulations to verify the entanglement properties of the CPB and NR. We also use various values ω′ of frequency modulation to see its influence upon the CPB-NR entanglement – see Figs. 5. Comparing Fig. 4a with Fig. 5a, we note that the entropy loses its periodical oscillations as time passes and this behavior becomes more pronounced if the frequency of detuning ω′ increases by 20 times – see Fig. 5b; we can also note diminution in the amplitudes of minima entropy. Both effects are due to detuning. Now, comparing the entropy in Fig. 4c with that in Fig. 5c and keeping the same parameters, we note the entropy going to zero in both cases; Fig. 5c displays an interesting effect: even in the presence of losses, the maximum value of the entropy grows. It reaches the value ln 2 and then it oscillates downward. This effect comes from the sinusoidal modulation of the NR frequency. The positions of maximum and minimum of entropy are due to the entanglement of the NR and CPB, the latter being destroyed by the presence of losses. When time evolves the NR and CPB lose and recover coherence periodically as classical entanglement and disentanglement occur. However, the coherence recovered by the CPB is not the same as before, and after enough long time the coherence in both subsystems vanishes. If the losses increase sufficiently the maximum values of entropy of both subsystems go to zero: hence, both subsystems that have initiated as (distinct) pure states, terminate in their (pure) vacuum states.

5. Conclusion In the present work we have considered the interaction of a CPB and an NR in the presence of a Kerr medium and losses affecting a subsystem. Concerning the influences of losses affecting the entropy of both subsystems and the CPB population inversion we observe the dominant role played by the NR upon that played by the CPB. The dissipation causes deterioration of the CPB excited

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6

0.7 0.6

SNR(t)

0.5 0.4 0.3 0.2 0.1

(a)

0.0 0

25

50

75

λ0t

dissipation (see Figs. 4c and 5c ) the same occurring for the CPB population inversion – see Figs. 2c and 3c. The larger the intensity of the NR the weaker the quantum entanglement between the NR and CPB and greater the degree of maximum mixing in the NR state. Concerning the entropy, this result is very important for information transmission: the transmission of maximum information through a quantum channel is exactly the von Neumann entropy [46,47]. These effects are very sensitive to detuning f(t), since they occur even for very small values of f(t), close to 0.04% compared with ω0 and ωc – see Figs. 3 and 2 for the population inversion and Figs. 5 and 4 for the entropy. The results suggest that it is possible to perform a dynamic control of certain properties of this system, via convenient manipulation of parameters. We hope that these results might shed light on this scenario, furnishing new insights for researchers in this area.

Acknowledgments We thank CNPq and FAPEG for their partial support.

0.7 0.6

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SNR(t)

0.5

[1] [2] [3] [4] [5] [6] [7] [8]

0.4 0.3 0.2 0.1

(b)

0.0 0

25

50

75

λ0t 0.6

[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26]

0.5

SNR(t)

[9]

0.4

0.3

0.2

[27]

0.1

[28]

(c)

0.0 0

4

8

12

[29]

16

λ0t Fig. 5. Time evolution of the entropy for different values of the parameters κ (t ) and δ (t ) for: n = 25, ωc /λ 0 = ω0/λ 0 = 20k , χ0 /λ0 = 0.2, τ /λ 0 = 10 , and ε/λ 0 = 0.001. (a) κ /λ 0 = 0.01 , δ /λ 0 = 0.0, and ω′/λ 0 = 1; (b) κ /λ 0 = 0.01, δ /λ 0 = 0.0 , and ω′/λ 0 = 20; (c) κ /λ 0 = 0.0 , δ /λ 0 = 0.01, and ω′/λ 0 = 20 .

level whereas convenient modulations of parameters favor the control of certain properties of the system. It was also shown that the certain choice of the time-dependent frequency makes the maximum value of the entropy higher, even in the presence of

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