Squeezing of light field in a dissipative Jaynes-Cummings model
SSqueezing of light field in a dissipative Jaynes-Cummings model ∗ Hong-Mei Zou † , Mao-Fa Fang Key Laboratory of Low-dimensional Quantum Structures and Quantum Control of Ministry of Education,College of Physics and Information Science, Hunan Normal University, Changsha, 410081, China(Received XX; revised manuscript received X XX)Based on the time-convolutionless master-equation approach, we investigate squeezing of lightfield in a dissipative Jaynes-Cummings model. The results show that squeezing light can be gen-erated when the atom transits to a ground state from an excited state, and then a collapse-revivalphenomenon will occur in the squeezing of light field due to atom-cavity coupling. Enhancing theatom-cavity coupling can increase the frequency of the collapse-revival of squeezing. The strongerthe non-Markovian effect is, the more obvious the collapse-revival phenomenon is. The oscillatoryfrequency of the squeezing is dependents on the resonant frequency of the atom-cavity.
Keywords: squeezing of light field, Jaynes-Cummings model, non-Markovian envi-ronment
PACS:
Squeezed light, as one of quantum properties of light field, has been widely investigated sinceit was observed in a groundbreaking experiment of an atomic vapor of Sodium atoms[1]. In thepast 30 years, many significant progresses have been acquired in experimental and theoreticalresearches on squeezed light so that squeezed light has a wide range of applications in quantuminformation processing and quantum metrology[2, 3, 4].The basic idea of squeezing can be understood by considering the quantum harmonic os-cillator. For a vacuum state, the variance of the position and momentum observables equals ¯ h , so we say that a state of a single harmonic oscillator exhibits squeezing if the variance ofthe position or momentum observables is below ¯ h . In order to obey Heisenberg’s uncertaintyrelation[5, 6], both position and momentum observables cannot be simultaneously squeezed. Inthe quantum optics theory, the Hilbert space associated with a mode of the electromagneticfield is isomorphic to that of the mechanical harmonic oscillator, thus the position and momen-tum observables in the harmonic oscillator correspond to the electric field magnitudes measuredat specific phases, that is, the position and momentum operators of the mechanical harmonicoscillator can be expressed by the creation and annihilation operators of a quantum field, i.e.ˆ x = (cid:113) ¯ h ω (ˆ a + ˆ a † ), ˆ p = − i (cid:113) ¯ hω (ˆ a − ˆ a † ) and [ˆ x, ˆ p ] = i ¯ h . Accordingly, the field measurements inan electromagnetic wave are affected by quantum uncertainties. For the coherent and vacuumstates, this uncertainty is ( (cid:52) x ) · ( (cid:52) p ) = ¯ h , called the standard quantum limit[7, 8, 9], † Corresponding author. E-mail:[email protected], tel:13807314064 a r X i v : . [ qu a n t - ph ] N ov here ( (cid:52) X ) = (cid:104) X (cid:105) − (cid:104) X (cid:105) ( X = x, p ). But squeezed states of light field exhibit uncertaintiesbelow the standard quantum limit[10]. This means that, when we turn on the squeezed light,we see less noise than no light at all. Hence, squeezed light is an interesting physical reality andhas a variety of applications, such as precision measurements of distances, gravitational wavedetectors, universal quantum computing, dense coding and quantum key distribution, and soon. Though the original definition of squeezed states refers to the squeezing of the quadra-ture amplitudes, squeezing of other quantities has also been studied including photon numbersqueezing and two-mode squeezing[11, 12]. And, there exist many physical processes that canbe employed to prepare single- and two-mode states of the electromagnetic field. For examples,single-mode squeezing lights and two-mode squeezing lights can be generated via parametricdown-conversion[13, 14, 15], in atomic ensembles[1, 16, 17] and in optical fibers[18, 19].Study of squeezed light is a hot issue in quantum optics. Recently, a lot of new worksare reported in the research area of squeezed light. The authors in Ref.[20] investigated thespatial distribution of quantum fluctuations in a squeezed vacuum field. The authors in Ref.[21]discussed influence of virtual photon process on the generation of squeezed light from atoms inan optical cavity. The authors in Ref.[22] considered in detail a system of two interferometersaimed to the detection of extremely faint phase-fluctuations. The authors in Ref.[23] researchedtheoretically the squeezing spectrum and second-order correlation function of the output lightfor an optomechanical system, and so on. In this present work, we investigate squeezing oflight field in a dissipative Jaynes-Cummings model and analyze influences of initial atomicstates, atom-cavity couplings, non-Markovian effects and resonant frequencies on the squeezing.The results show that a collapse-revival phenomenon will occur in the squeezing of light field.Enhancing the atom-cavity coupling can increase the frequency of the collapse-revival. Thestronger the non-Markovian effect is, the more obvious the collapse-revival phenomenon is.The oscillatory frequency of squeezing is dependents on the resonant frequency of the atom-cavity.The outline of the paper is the following. In Section 2, we introduce a dissipative Jaynes-Cummings model. In Section 3, we review the concept of squeezing of light field. Results andDiscussions are given in Section 4. Finally, we sum up the report in Section 5. We consider a composite system of a two-level atom interacting with a cavity which iscoupled to a bosonic environment[24, 25]. The Hamiltonian is written as (¯ h = 1) H = H JC + H r + H I (1)here H JC = 12 ω σ z + ω a † a + Ω( aσ + + a † σ − ) ,H r = (cid:88) k ω k b † k b k , I = ( a † + a ) (cid:88) k g k ( b † k + b k ) , (2)where σ + = | e (cid:105)(cid:104) g | and σ − = | g (cid:105)(cid:104) e | are the raising and lowering operators of the qubit, and σ z = | e (cid:105)(cid:104) e | − | g (cid:105)(cid:104) g | is a Pauli operator for the qubit with transition frequency ω [26]. a † and a are the creation and annihilation operators of the cavity field. b † k and b k are the creation andannihilation operators of the environment. Ω is the coupling between the atom-cavity and g k is the coupling between the cavity-environment.Let us suppose that there is one initial excitation in the atom-cavity system and the en-vironment is at zero temperature. Neglecting the atomic spontaneous emission and the Lambshifts, in the dressed-state basis {| E (cid:105) , | E − (cid:105) , | E (cid:105)} , using the second order of the time con-volutionless(TCL) expansion[27], the non-Markovian master equation for the density operatorΛ( t ) is˙Λ( t ) = − i [ H JC , Λ( t )] + γ ( ω + Ω , t )( 12 | E (cid:105)(cid:104) E | Λ( t ) | E (cid:105)(cid:104) E | − {| E (cid:105)(cid:104) E | , Λ( t ) } )+ γ ( ω − Ω , t )( 12 | E (cid:105)(cid:104) E − | Λ( t ) | E − (cid:105)(cid:104) E | − {| E − (cid:105)(cid:104) E − | , Λ( t ) } ) , (3)where | E ± (cid:105) = √ ( | g (cid:105) ± | e (cid:105) ) are the eigenstates of H JC with energy ω ± Ω and | E (cid:105) = | g (cid:105) is the ground state with energy − ω . The timedependent decay rates for | E − (cid:105) and | E (cid:105) are γ ( ω − Ω , t ) and γ ( ω + Ω , t ) respectively.We take a Lorentzian spectral density of the environment, i.e. J ( ω ) = 12 π γ λ ( ω − ω ) + λ , (4)where γ is related to the relaxation time scale τ S by τ S = γ − and λ defines the spectral width ofthe coupling which is connected to the reservoir correlation time τ R by τ R = λ − . If λ > γ , therelaxation time is greater than the reservoir correlation time and the dynamical evolution of thesystem is essentially Markovian. For λ < γ , the reservoir correlation time is greater than orof the same order as the relaxation time and non-Markovian effects become relevant[28, 29, 30].When the spectrum is peaked on the frequency of the state | E − (cid:105) , i.e. ω = ω − Ω, the decayrates for the two dressed states | E ± (cid:105) are respectively expressed as[24] γ ( ω − Ω , t ) = γ (1 − e − λt )and γ ( ω + Ω , t ) = γ λ + λ { λ sin t − cos t ] e − λt } .If the initial state of the atom-cavity isΛ(0) = Λ (0) Λ (0) Λ (0)Λ (0) Λ (0) Λ (0)Λ (0) Λ (0) Λ (0) , (5)we can acquire the matrix elements at all times from Equation (3)Λ ( t ) = c Λ (0) , Λ ( t ) = c Λ (0) , Λ ( t ) = c Λ (0) , Λ ( t ) = c Λ (0) , Λ ( t ) = c Λ (0) , Λ ( t ) = c Λ (0) + c Λ (0) + c Λ (0) , (6)3ere c = e − f , c = e − i Ω t e − ( f + f ) , c = e − i ( ω +Ω) t e − f ,c = e − f , c = e − i ( ω − Ω) t e − f ,c = 1 − c , c = 1 − c , c = 1 , (7)where f = 12 (cid:90) t γ ( ω − Ω , t (cid:48) ) dt (cid:48) = 12 ( γ t + γ λ ( e − λt − f = 12 (cid:90) t γ ( ω + Ω , t (cid:48) ) dt (cid:48) = 12 γ λ + λ [ t − e − λt sin (2Ω t )4Ω + λ + ( λ − )( e − λt cos (2Ω t ) − λ (4Ω + λ ) ] . (9) To analyze the squeezing property of light field, we review two quadrature operators X and X having a phase difference π [10], i.e. X = 12 ( a + a + ) ,X = 12 i ( a − a + ) (10)where [ X , X ] = i . The Heisenberg uncertainty relationis given by( (cid:52) X ) · ( (cid:52) X ) ≥
116 (11)where ( (cid:52) X j ) = (cid:104) X j (cid:105) − (cid:104) X j (cid:105) (j=1,2). Consequently, the fluctuation in the amplitudes X j ofthe quadrature operators is said to be squeezed if X j satisfies the condition( (cid:52) X j ) <
14 (12)or F j = ( (cid:52) X j ) − < ρ f ( t ) = ρ ( t ) ρ ( t ) ρ ( t ) ρ ( t ) , (14)we can get (cid:104) X j (cid:105) = T r ( X j ρ f ( t )) and (cid:104) X j (cid:105) = T r ( X j ρ f ( t )). Then we insert (cid:104) X j (cid:105) and (cid:104) X j (cid:105) intoEquation (12) or Equation (13) and can analyze numerically the squeezing properties of thelight field. 4 Results and Discussions
We set that the initial state of the atom-cavity is | ψ (0) (cid:105) = ( cos ( θ ) | e (cid:105) + e iϕ sin ( θ ) | g (cid:105) ) A | (cid:105) f , (15)where A indicates the atom, f expresses the cavity field. θ is an amplitude parameter and ϕ isa phase parameter. We can obtain the reduced density matrix ρ f ( t ) by using Equation (6). Inthe following, we analyse the influences of the initially atomic state, the atom-cavity coupling,the resonant frequency of the atom-cavity and the non-Markovian effect on the squeezing oflight field. Figure 1. (Color online)The influence of the initially atomic state on the squeezing of lightfield in the Markovian regime( λ = 5 γ ). (a)The dynamics revolution of the squeezing factor F as functions of ϕ and γ t when θ = π ; (b)The dynamics revolution of the squeezing factor F as functions of θ and γ t when ϕ = 0. Other parameters are Ω = γ and ω = 10 γ . In Figure 1, we describe the influence of the initial atomic states on the squeezing of lightfield in the Markovian regime( λ = 5 γ ). Figure 1(a) shows the squeezing factor F as functionsof ϕ and γ t when θ = π . From Figure 1(a), we see that the F is obvious t dependent but is ϕ independent. For a certain value of ϕ , the F varies to a negative value from zero and thenoscillates when time t increases, namely, there is not the squeezing in the initial state but theatomic transition can generate squeezing light, and then a collapse-revival phenomenon willoccur in the squeezing of light field due to the atom-cavity coupling. For different ϕ , the F has similarly oscillatory behavior as time t increases. Figure 1(b) depicts the squeezing factor F as functions of θ and γ t when ϕ = 0. From Figure 1(b), we find that, the F is not onlydependent on t but also dependent on θ . For different θ , there are the diverse evolution curvesof F , for examples, there is no any squeezing phenomena when θ = π while the F is squeezedfor some certain values of θ . 5 igure 2. (Color online)The influence of the atom-cavity coupling Ω on the squeezing factor F (black solid line) and the atomic excited population P e (blue dotted-dashed line) versus γ t inthe Markovian regime( λ = 5 γ ). (a)Ω = γ ; (b)Ω = 2 γ . Other parameters are θ = π , ϕ = 0and ω = 10 γ . The red dashed lines indicate the envelope lines of the F -oscillations. In Figure 2, we plot the influence of the atom-cavity coupling Ω on the squeezing factor F in the Markovian regime( λ = 5 γ ). From Figure 2, we can see that the F is obviouslydependent on the atom-cavity coupling. Figure 2(a) exhibits a quick oscillation of F as time t increases due to the interaction between the atom with the dissipative cavity. The squeezingfirst increases to F = − .
13 at γ t = π/ F = 0 at γ t = 0 and then collapses tozero at γ t = π . Afterwards, the squeezing is revived to F = − .
03 at γ t = 3 π/
2, which ismuch smaller than F = − .
13. In a longer time scale, the oscillation of F is modulated bythe coupling parameter Ω(see the envelope line expressed by the red dashed line in the Figure2(a)). The collapse-revival of F -oscillation is obvious over γ t ∈ [0 ,
8] and the periods is π/ Ω.And this collapse-revival phenomenon is always accompanied by the decay and repopulationof the excited atom. The light field reaches its maximal squeezing when the P e reduces tozero from 0 .
25. Subsequently, the F -oscillation collapses to zero when the P e again risesfrom zero. Finally, this collapse-revival of F -oscillations will disappear due to the dissipationof the cavities coupling with the Markovian environments. Comparing Figure 2(a) and (b),we can know that their squeezing dynamics is similar for different Ω. The difference is inthe collapse-revival frequency of F -oscillation and in the maximal squeezing obtained. Thecollapse-revival frequency of F -oscillation in the latter case is twice the former. The maximalsqueezing obtained is obvious larger than the former. Hence, strengthening Ω can increase thecollapse-revival frequency of F -oscillation and the maximal squeezing.The physical interpretations of the above results are as follows. Squeezing light can be6enerated when the atom transits to a ground state from an excited state. And, due to theinteraction between the atom and its cavity, the photons can be exchanged between the atomand its cavity so that the collapse-revival phenomenon of F -oscillation can occur in a shorttime. On the other hand, due to the coupling of the cavity with its bosonic environment, thephotons exchanged to the cavity will continuously reduce thus the collapse-revival phenomenonof F -oscillation disappears in a long time, as shown in Figure 2(a)-(b). Figure 3. (Color online)The influence of the non-Markovian effect on the squeezing factor F (black solid line) and the atomic excited population P e (blue dotted-dashed line) versus γ t when Ω = γ . (a) λ = 3 γ ; (b) λ = 0 . γ ; (c) λ = 0 . γ . Other parameters are θ = π , ϕ = 0and ω = 10 γ . The red dashed lines indicate the envelope lines of the F -oscillations. Figure 3 exhibits the influence of the non-Markovian effect on the squeezing factor F whenΩ = γ . From Figure 3, we can find that, for the same Ω, the collapse-revival frequencies of F -oscillations are same whether the Markovian or the non-Markovian regimes. The influenceof the non-Markovian effect on the F is embodied in the maximal value of squeezing. In theMarkovian regime( λ = 3 γ ), the squeezing increases to F = − .
14 at γ t = π/ F = 0at γ t = 0 and then collapses to zero at γ t = π . Afterwards, the squeezing is revived to F = − .
04 at γ t = 3 π/
2, which is much smaller than F = − .
14. That is to say, the revival7f F -oscillation is very close to zero due to the cavity dissipation, as shown in Figure 3(a).For Figure 3(b), when λ = 0 . γ , due to the memory and feedback effect of the non-Markovianenvironment, the squeezing increases to F = − .
21 at γ t = π/ F = 0 at γ t = 0 andthen collapses to zero at γ t = π . And the squeezing is revived to F = − .
15 at γ t = 3 π/ λ = 0 . γ , the maximum squeezingcan reaches F = − .
24 in a short time and the collapse-revival phenomenon of F -oscillationis very remarkable. The F will tend to a stably periodic oscillation between − .
052 to 0 . λ is, the stronger thenon-Markovian effect is, the more remarkable the collapse-revival phenomenon of F -oscillationis. Figure 4. (Color online)The influence of the atomic Bohr frequency ω on the squeezing factor F (black solid line) and the atomic excited population P e (blue dotted-dashed line) versus γ t when Ω = γ . (a) ω = 5 γ ; (b) ω = 10 γ . Other parameters are θ = π , ϕ = 0 and λ = 0 . γ .The red dashed lines indicate the envelope lines of the F -oscillations. The physical explanation is that the photons dissipated to the environment can be partlyreturned to the cavity field due to the memory and feedback effect of the non-Markovianenvironment. And the stronger the the non-Markovian effect is, the more the photons returnedis. Therefor, in the Markovian or the weak non-Markovian regimes, the F will decay to zeroin a long time(see in Figure 3(a)-(b)). If the non-Markovian effect is very strong, the photonsdissipated to the environment can be effectively fed back to the cavity so that the F -oscillationcan be well revived after they decrease to zero(see Figure 3(c)).Figure 4 displays the influence of the resonant frequency ω of the atom-cavity on thesqueezing in the non-Markovian regime( λ = 0 . γ ) and with Ω = γ . From Figure 4, we know8hat the frequency of the F -oscillations is obvious dependent on the resonant frequency ω ,which arise from the resonant coupling of the atoms with its cavity. Comparing Figure 4(a)and (b), we find that the frequency of the F -oscillations in the latter case is twice the formerbecause the resonant frequency in the latter case is twice the former. Thus, the more the valueof ω is, the more violent the Rabi oscillating of the atom-cavity is. In conclusion, we investigate the squeezing of light field in the dissipative Jaynes-Cummingsmodel by the time-convolutionless master-equation approach. We discuss in detail the influ-ences of the initially atomic state, the atom-cavity coupling, the non-Markovian effect and theresonant frequency on the squeezing when | ψ (0) (cid:105) = ( cos ( θ ) | e (cid:105) + e iϕ sin ( θ ) | g (cid:105) ) A | (cid:105) f . The resultsshow that, the atomic transition can generate squeezing light, and the collapse-revival of the F -oscillations can occur whether the Markovian or the non-Markovian regimes. The squeezingfactor F is obvious θ dependent but is ϕ independent for different the initial atomic state andthe maximal squeezing can be obtained when θ = π . The frequency of the F -oscillations isobvious dependent on the resonant frequency ω , the more the value of ω is, the bigger theoscillating frequency of the squeezing is. Strengthening the atom-cavity coupling can increasethe collapse-revival frequency of F -oscillation and the maximum squeezing obtained. Thecollapse-revival phenomenon of F -oscillation lies on the non-Markovian effect, the smaller thevalue of λ is, the stronger the non-Markovian effect is, the more remarkable the collapse-revivalphenomenon of F -oscillation is. These results may offer interesting perspectives for futureapplications of open quantum systems in quantum optical, microwave cavity QED implemen-tations, quantum communication and quantum information processing. Funding
This work was supported by the Science and Technology Plan of Hunan Province, China(Grantno. 2010FJ3148) and the National Natural Science Foundation of China (Grant No.11374096).
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