Frequency response of the decoherence in a Duffing oscillator and the dispersion of the Wigner function in the Fourier domain

Chinese Journal of Physics 54 (2016) 906–911

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Frequency response of the decoherence in a Duffing oscillator and the dispersion of the Wigner function in the Fourier domain Shi-Hui Zhang∗, Wu-Qing Wang Department of Mathematics and Physics, North China Electric Power University, Baoding, PR China

a r t i c l e

i n f o

Article history: Available online 14 October 2016 PACS: 03.65.Yz 05.45.-a 03.65.Sq Keywords: Decoherence Frequency response Wigner function

a b s t r a c t The decoherence measured by the linear entropy is shown to be closely related to the dispersion of the Wigner function in the Fourier domain. The latter corresponds to the oscillations of the Wigner function in the phase space and can be influenced by the external driving force in driven nonlinear systems, e.g., in a Duffing-type oscillator. In the further investigations on the Duffing-type oscillator, the growth of the entropy of the system is found to be significantly dependent on the frequency of the driving force. Pronounced response peak occurs in the driving frequency response curve of the entropy of the system. Furthermore, there is good correspondence between the frequency response curve of the entropy and that of the dispersion of the Wigner function in the Fourier domain. © 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

1. Introduction The environment-induced decoherence is a central concept in physics. It is viewed as a major obstacle for realizing the applications of quantum coherence in the fields such as quantum computer and quantum simulation. It has received much attention for many years and still of great interest [1–7]. Early studies have shown that decoherence is closely related to chaos [4–7]. Besides, it is also found that universal decoherence exists in systems with gravitational potential [8] and macroscopic quantum resonators [9]. During recent years, the experimental tests of quantum phenomena in mesoscopic and macroscopic objects provide new and novel insights into the fundamental questions in quantum physics involving decoherence [9]. In the studies on the quantum phenomena in mesoscopic and macroscopic systems, driven nonlinear oscillators are of fundamental interests [10–20]. Known to all, frequency response is one of the important features in driven nonlinear systems. It is widely used in information technologies and also can be found in many studies on mesoscopic and macroscopic systems [10–20]. Due to the coupling to the environment, the driven nonlinear systems mentioned above are often subjected to the influence of the environment. The latter tends to lead to decoherence. However, the frequency response of the decoherence in driven nonlinear systems has not been clarified. In this paper, the relations of the decoherence to the dispersion of the Wigner function in the Fourier domain and the frequency response of the decoherence are investigated in an open driving oscillator by means of the Wigner function. The decoherence measured by the linear entropy is shown to arise from the loss of the high frequency components of

∗

Corresponding author. E-mail address: [email protected] (S.-H. Zhang).

http://dx.doi.org/10.1016/j.cjph.2016.10.003 0577-9073/© 2016 The Physical Society of the Republic of China (Taiwan). Published by Elsevier B.V. All rights reserved.

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the Wigner function in the Fourier domain. Its growth rate increases with the increase of the dispersion of the Wigner function in the Fourier domain. The latter corresponds to the increase of the oscillations of the Wigner function in the phase space. It can be influenced by the driving force and shown driving frequency dependence in a periodically driven nonlinear system, e.g., in a Duffing-type oscillator. In the investigations on the Duffing-type oscillator, the average growth rate of the entropy of the system is found to significantly depend on the frequency of the external driving force. It increases with increasing the driving frequency from zero until a maximum is reached, after which it rapidly drops off to a plateau. Accordingly, a pronounced peak can be seen in the frequency response curve of the entropy of the system. Furthermore, good correspondence is found between the frequency response curves of the entropy and the dispersion of the Wigner function in the Fourier domain. 2. Decoherence and the dispersion of the Wigner function in the Fourier domain The model studied in this work is a periodically driven nonlinear system, which is of fundamental and technical interests in the studies related to decoherence and mesoscopic devices [4–6,10–13,18–22]. Its Hamiltonian can be written as

HS =

p2 + V (q, t ), 2m

(1)

where V (q, t ) = V0 (q ) + f0 q cos(ωt ). V0 (q) is the potential of the nonlinear oscillator and is responsible for the external driving force. The quantum state of the system (1) is described by the Wigner function [23]. The latter is the Wigner–Weyl transform of the density matrix and can be written as −1

W (q, p; t ) = (2π h ¯)





dyeipy/ h¯ q −





y y  ρ (t )q + 2 2



(2)

It was introduced to study the quantum corrections to classical statistical mechanics and is the quantum analogue of the classical phase-space distribution [23]. The Wigner function is of fundamental importance in many fields of quantum physics, e.g., in quantum computation [24] and in quantum optics [25]. Besides, it can be measured directly via homodyne tomography [25] and its oscillations in phase space are signature of the quantum interference [25–27]. The coupling of the system (1) to environment is modeled via the Ohmic environment in high-temperature and weakcoupling limit. In this case, the time evolution of the Wigner function of the system (1) can be described as [4,28,29]

∂t W = {HS , W }PB +

 (−1 )n ( h ¯ /2 )2n 2n+1 2n+1 ∂ V ∂ p W + D∂ p2W, ( 2 n + 1 )! q

(3)

n≥1

where {}PB is the Poisson bracket, the last term on the right hand side arises from the coupling of the system to the environment and D is the so-called diffusion coefficient. In the Wigner representation, the decoherence of the system (1) can be evaluated by the linear entropy [30,31]

S(t ) = 1 − 2π h ¯





dq

d pW 2 (q, p; t ).

(4)

It can be expressed in terms of the reduced density of the system S(t ) = 1 − T rρ 2 (0 ≤ S ≤ 1). Its growth indicates the decay of the off-diagonal elements of the reduced density matrix of the system and thus corresponds to the loss of the quantum coherence [30]. In other words, the decoherence increases as S increases from 0 to 1. To investigate the environment-induced decoherence, it is necessary to explore the evolution of the Wigner function of the system. The latter is described by Eq. (3). Its solution in the Heisenberg picture is

W (q, p; t ) = exp [h0 t + hE t + F (t )]W (q, p; 0 ),



n (− h2 ) 2−2n /(2n

where h0 = −( p/2m )∂x + (∂qV )∂ p + ¯ time interval δ t, Eq. (5) can be approximated as

(5)

 + 1 )! ∂x2n+1V ∂ p2n+1 , hE = D∂ p2 and F (t ) = ( f0 /ω ) sin(ωt ). For a short

W (q, p; t + δt ) = UD (δt )W0 (q, p; t + δt ) + O(δt 2 ),

(6)

where UD (δt ) = exp(hE δt ) and W0 (q, p; t + δt ) = exp[h0 δt + F (t )]W (q, p; t ). W0 (q, p; t + δt ) corresponds to the evolution in the absence of diffusion. By taking the Fourier transform with respect to p, Eq. (6) becomes

(q, p ; t + δt )  e−Dδt p2f W 0 (q, p ; t + δt ), W f f

(7)

(q, p ; t ) = dpe where W W (q, p; t ) is the Fourier transform of W(q, p; t) with respect to p. In Eq. (7), the exponential f arises from U (δ t) and is caused by the coupling to the environment. Its exponent increases with |p | decay factor before W 0 E f and thus it induces the decay of the oscillations of the Wigner function in the phase space via cutting off the high-frequency in the Fourier domain. This can lead to decoherence, since the spatial oscillations of the Wigner function components of W 0 are signature of the quantum interference and indicates the coherence of the quantum state of the system. −i p f p

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Furthermore, according to Eq. (7), the loss of the high-frequency components of the Wigner function in the Fourier domain increases with the increase of the dispersion of the Wigner function in the Fourier domain along pf . Thus, the latter can enhance the decoherence. This can be further confirmed by taking the square of the magnitude of the two sides of Eq. (7) and integrating together with x and pf , after which we obtain





dq

(q, p ; t + δt )|2  d p f |W f





dq

0 (q, p ; t + δt )|2 . d p f e−2Dδt p f |W f 2

(8)

Since Dδ t is very small, Eq. (8) can be approximately described as





dq

(q, p ; t + δt )|2  d p f |W f





0 (q, p ; t + δt )|2 d p f |W f   0 (q, p ; t + δt )|2 , − 2Dδt dq d p f p2f |W f dq

(9)





(q, p ; t )|2 = dq d pW 2 (q, p; t + δt ) according to the Parseval’s theorem [32]. Thus, Eq. (9) can be where dq d p f |W f rewritten as

δS  2D δt





dq

0 (q, p ; t + δt )|2 d p f p2f |W f

(10)





with δ S = dq d pW02 (q, p; t + δt ) − dq d pW 2 (q, p; t + δt ). δ S is the entropy growth induced by the diffusion during the time interval between t and t + δt. Thus, Eq. (10) indicates the growth rate of the entropy S(t) during the time evolution of the Wigner function. As shown by Eq. (10), for a given D, the growth rate of the entropy is S(t) determined by

Pf2 (t ) =





dq

0 (q, p ; t + δt )|2 . d p f p2f |W f

(11)

In other words, the growth of the entropy S(t) during a specified time interval [0, τ ](or in other words, the average growth rate of the entropy S(t) for a specified time interval [0, τ ]) is determined by the accumulation of Pf2 (t ) with time, i.e.

τ 2

−1 τ P (t )dt since τ is specified. 2 0 Pf (t )dt . The latter can be evaluated by Pf = τ 0 f As can be seen from Eq. (11), the value of Pf2 (t ) increases with the dispersion of the Fourier transform of the Wigner

function with respect to p along the pf axis. Therefore, the latter can be used as an indicator of Pf2 (t ). It can be measured by the variance of the amplitude of the Wigner function along pf in the Fourier domain, i.e.

 p2f =  p f 2  −  p f 2 =





dq

0 (x, p ; t + δt )|, d p f p2f |W f

(12)

(x, p ; t )| is an odd function for p and its integral over the full space (i.e., p ) equals to zero. Accordingly, where p f |W 0 f f f

τ −1 2 (t )dt can be used as an indicator of P 2 . Since the increase of the dispersion of the Wigner function in the 2 p f = τ  p 0 f f Fourier domain can induce the increase of Pf2 , it can enhance the growth of the entropy of the system, according to Eqs. (10) and (11). This is consistent with the conclusions drawn from Eq. (7). For a periodically driven anharmonic oscillator like (1), the presence of the external driving force can change the energy of the system and cause transitions between energy levels. Therefore, it can affect the oscillations of the Wigner function in the phase space and further influence the growth of the entropy of the system, according to the discussion above. Moreover, the influences of the driving force on the energy and energy levels of the system depend on its driving frequency. Thus, the influences of the driving force on the dispersion of the Wigner function in the Fourier domain and the entropy growth of the system can also show driving frequency dependence, and even lead to “resonance” in the frequency response of the entropy growth of the system in periodically driven nonlinear oscillators, e.g., in a Duffing-type oscillator. This can be seen from the investigations on a Duffing-type oscillator. 3. Frequency response of the decoherence in a Duffing-type system The Duffing-type oscillator is a fundamental periodically driven nonlinear system in the studies related to decoherence and mesoscopic devices [4–6,10–12,15–17]. Its Hamiltonian is

HS =

p2 1 α + mω02 q2 + q4 + f0 q cos ωt 2m 2 4

(13)

with V (q ) = mω02 q2 /2 + α q4 /4 . α characterizes the strength of the nonlinearity. Hereafter, α , f0 and ω are in units of







m2 ω03 / h ¯ , mω03 h ¯ and ω0 respectively. Besides, t, q and p are in units of ω0−1 , h ¯ /mω0 and mω0 h ¯ respectively. With Eqs. (2)–(4) and (13), the driving frequency response of the decoherence of the Duffing oscillator is investigated. Each calculation is started with a coherent state, which widely used in the theoretical and experimental works. Its Wigner

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Fig. 1. Entropy S vs time t for α = 0.1, f = 1 and D = 0.0 0 02. The Wigner function is initial centered at (q0 , p0 ) = (1, 0 ). The values of the driving frequencies are ω = 0.5 (solid line), ω = 1.3 (dashed line) and ω = 2 (dotted line).

Fig. 2. Frequency response curves of (a) S(τ ), (b) Pf2 and (c)  p2f . Parameters are the same as in Fig. 1. The steps of the driving frequency ω are 0.05 for the interval [1, 2] and 0.1 for the other intervals. The maximum values of the response peaks in (a), (b) and (c) all occurs at ω = 1.3.

function reads

(q − q0 )2 ( p − p0 )2 − , (14) π σq σ p σq 2 σ p2   h where σq = σ p = 1 (σ q and σ p are in units of ¯ /mω0 and mω0 h ¯ , respectively). The parameter values are α = 1/10, f = 1, D = 1/50 0 0 and (q0 , p0 ) = (1, 0 ) (D is in unit of mω02 h ¯ ). W (x, p; 0 ) =

1



exp −

The results for three typical driving frequencies are illustrated in Fig. 1. The values of the driving frequencies equal to 0 (solid line), 1.3 (dashed line) and 2 (dotted line). From Fig. 1, one can see that the growth of the entropy of the system with time is influenced by the driving force and significantly depends on the driving frequency. Especially, the growth of the entropy of the system for ω = 1.3 is much faster than those for ω = 0.5 and ω = 2. At t = 10 0 0, the value of the entropy S(t) for ω = 1.3 is over three times greater than that for ω = 2. In other words, the growth rate of the entropy is prominently different for the driving force with different frequency. This is consistent with the above discussions. In order to further clarify the relation of the growth of the entropy to the driving frequency, more detailed investigations are made by increasing ω from 0 to 3 with the same parameter values as in Fig. 1. The results are presented in Fig. 2(a), where S(τ ) denotes the value of S(t) at t = τ (t = 10 0 0 as shown in Fig. 1). For a specified τ , the value of S(τ ) reveals its

τ average growth rate during the time interval from 0 to τ , since the latter (i.e., τ −1 0 S˙ (t )dt ) equals to S(τ )/τ . In Fig. 2(a), the value of S(τ ) has a marked driving-frequency dependence. It increases with increasing ω from 0 until a maximum is reached at ω = 1.3. After reaching the maximum, it rapidly decreases. Accordingly, a pronounced peak occurs in the frequency response curve of S(τ ). Near the response peak, the entropy growth and thus the decoherence is significantly increased by the driving force. Known to all, quantum coherence is an essential ingredient in quantum-enhanced technologies. For instance, it is crucial for the generation of entanglement. Therefore, the enhancement of the decoherence by the driving force should receive additional consideration in driven nonlinear systems like the Duffing-type oscillator considered here. As discussed above, the influence of the driving force on the decoherence can be ascribed to that of the driving force on the dispersion of the Wigner function in the Fourier domain  p2f . Specifically, the growth of the entropy with time

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(x, p ; t ) with respect to p for ω = 0.5, ω = 1.3 and ω = 2 with Fig. 3. Contour plots of the amplitude of the Fourier transform of the Wigner function W f t = 100; (b) as in (a) but with t = 10 0 0. The values of the driving frequency are presented in the upper right corner of each panel.

depends on Pf2 , which can be approximately evaluated by  p2f . For comparison purpose, the frequency response curves of Pf2 and  p2f are presented in Fig. 2(b) and (c) respectively. Comparisons between Fig. 2(a), (b) and (c) show the good correspondence between the driving frequency response curves of S(τ ), Pf2 and  p2f . This is consistent with the discussion above and can also be confirmed by the contour plots of the amplitude of the Fourier transform of the Wigner function with respect to p, which are displayed in Fig. 3. For comparison, the contour plots in Fig. 3(a) are obtained with t = 100, while those in Fig. 3(b) are obtained with t = 10 0 0. The values of the driving frequencies are shown in the upper right corner of each panel. As can be seen from Fig. 3(a), the Wigner function of the system at t = 100 for ω = 1.3 has larger dispersion in the Fourier domain along the pf direction than those for ω = 0.5 and ω = 2. Accordingly, the value of Pf2 and  p2f for ω = 1.3 are both larger than those for ω = 0.5 and ω = 2, as can be seen from Fig. 2(b) and (c). The increase of Pf2 can enhance the growth of the entropy of

the system, as expressed by Eqs. (10) and (11). Thus, the value of the linear entropy is larger for ω = 1.3 than for ω = 0.5 and ω = 2 in Fig. 2(a). Besides, the enhancement of the growth of the entropy corresponds to that of the loss of the highfrequency components of the Wigner function, as shown by Eq. (7). This can be confirmed by the contour plots displayed in Fig. 3(b). In Fig. 3(b), the loss of high-frequency components of the Wigner function in the Fourier domain along pf at t = 10 0 0 is greater for ω = 1.3 than for ω = 0.5 and ω = 2. Especially, as displayed in Fig. 3(b), the Wigner function for ω = 1.3 has lost most of the high-frequency components along the pf direction in the Fourier domain and become a narrow trip at t = 10 0 0. That is, the spatial oscillation of the Wigner function for ω = 1.3 at t = 10 0 0 has been almost totally washed out. Accordingly, in Fig. 2(a), the value of the linear entropy for ω = 1.3 at t = 10 0 0 is very close to its upper limit. This reveals the relations between the growth of the entropy and the decay of the spatial oscillations of the Wigner function. Furthermore, the spatial oscillations of the Wigner function, which arise from the quantum interference, can also be viewed as the signature of the nonclassicality [33,34]. Therefore, the frequency response of the linear entropy also reveals the frequency response of the loss of the nonclassicality. The latter may emerge in the quantum evolutions of driven nonlinear systems, due to the influences of the driving force and the perturbation from the environment. This could affect the quantum behaviors and should be taken into account in the studies on quantum driven nonlinear systems.

4. Conclusions Entropy growth is found to increase with the dispersion of the Wigner function in the Fourier domain. The latter can be influenced by the external driving force in a driven nonlinear system. This can lead to the driving frequency dependence of the entropy growth. In the further investigations on a Duffing-type oscillator, the growth of the entropy of the system is shown to significantly depend on the driving frequency and a pronounced response peak occurs in the frequency response of the average growth rate of the entropy of the system. This sheds some lights on the roles of the frequency response in quantum behavior of the driven nonlinear oscillators, which are fundamental systems used in the experimental tests of quantum phenomena in mesoscopic and macroscopic objects. For instance, it is necessary to devote some attention to the influence of the driving frequency response of the decoherence on the quantum coherent states in driven mesoscopic sys-

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tems, where important dynamic behaviors such as resonance are often achieved by changing the frequency of the external driving. Besides, the frequency response curve of the entropy of the Duffing-type oscillator is very similar to the nonlinear resonance of the classical Duffing oscillator, which has been observed experimentally in nano-resonator very recently [10]. Their relationship need to be studied in the near future and will be discussed in forthcoming works. This work is support by the Fundamental Research Funds for the Central Universities in China (No. 2014MS168). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]

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