Xiao Xiao(肖驍),Ming-Yue Zhao(趙明月),Shu-Min Yu(于書敏),and Yi-Bo Gao(高一波),?

1Department of Physics,Renmin University of China,Beijing 100872,China

2College of Applied Sciences,Beijing University of Technology,Beijing 100124,China

1 Introduction

Actually,any quantum system interacting with the environment(the bath)can not be isolated from the environment completely.[1]In quantum information and quantum computation,[2]the decay process of quantum system induced by quantum fluctuations of the bath is very important for the qubit.In quantum optics,[3]the Jaynes–Cummings model has been one of the most important models,[4]which describes the light-matter interaction of a two-level atom and a single mode of the quantized electromagnetic field.[5]Among these light-matter interaction issues,[6]the revivals and collapses of the atomic population inversion(also named Rabi oscillation)has been studied in the literatures.[7?9]

Decay of Rabi oscillation has also been used as a tool to characterize the decoherence in superconducting qubits(charge qubit,phase qubit,and flux qubit).[10?11]Recently,in circuit QED system,[12]the researchers have performed spectroscopic measurements of a superconducting qubit dispersively coupled to a nonlinear resonator driven by a pump microwave field.[13]Also in nanomechanical QED system,the integration of Josephson junction qubit and nanomechanical resonators are attracting considerable attentions.[14?17]The dynamics of all these qubit-resonator systems could be described by the Jaynes–Cummings Hamiltonian.

When intrinsic nonlinearity of nanomechanical resonator[18]is considered in the coupled qubit-resonator system,superconducting qubit can be used to probe quantum fluctuations of nonlinear resonator.[19]And the nonlinearity can be used to create nonclassical states in mechanical systems[20?21]and selectively address the nanomechanical qubit transitions in quantum information processing.[22]

In previous studies,[23]master equation approach has been used to deal with the issues in open quantum system.In this paper,considering the influence of the environment on this nanomechanical QED system,we can use microscopic master equation approach[24]to solve time evolution of the density operator for the qubit-resonator system and study the temporal Behavior of Rabi Oscillation.

The paper is structured as follows.In Sec.2,a nonlinear Jaynes–Cummings model[25]is used to describe the dynamics of the coupled qubit-nanomechanical resonator system.In Sec.3,using microscopic master equation approach,we solve time evolution of density operator for the qubit-resonator system.The probability on excited state of the qubit is calculated to show the temporal process of Rabi oscillation.Finally,the results are summarized.

2 The Qubit-Resonator System

In nanomechanical QED system,we can use a Jaynes–Cummings type Hamiltonian to describe the dynamics of the qubit-resonator system consisting of a charge qubit and a nanomechanical resonator system,

Considering the nonlinearity of nanomechanical resonator,the Hamiltonian for this qubit-resonator system writes[26]

Here the rotating-wave approximation(ωq= ωa= ω)and~=1 is adopted.Corresponding to charge qubit and nanomechanical resonator,the lowering(raising)operator σ?(σ+)and the annihilation(creation)operator a(a?)satisfy the commutation relation[σ?,σ+]= σzand[a,a?]=1.The Hamiltonian in Eq.(2)describes the dynamics of a nonlinear Jaynes–Cummings model,[25]and a quartic potential x4[18]gives linear part χa?a and nonlinear part χ(a?a)2which leads to the phonon-phonon interaction in nanomechanical QED systems.The g is the coupling constant and the χ is the nonlinearity parameter(χ ? g).

Solving the Hamiltonian HS,we get the ground state|E0i=|00i with energy E0= ?ω/2 and excited state doublets

with energy

Some parameters are defined,i.e.,?=and θ=arcsin(g/?).

With the loss of nanomechanical resonator,the total Hamiltonian

consists of three parts,i.e.,the system part HS,the interaction part

and the bath part

where bjand b?jare bosonic annihilation and creation operators for the bath oscillators with mode frequency ωj(j=1,2,...),the ξjis the coupling strength between nanomechanical resonator and bath.

In this paper,we adopt microscopic master equation approach[24]to solve time evolution of density operator(ρ)for the qubit-resonator system,our master equation is

where L is a time-independent linear superoperator.

Using the microscopic master equation approach in Ref.[24],we obtain the eigen-equations

The{ρk}is a set of eigenoperators due to the superoperator L with the eigenvalue{λk}for the index k.

Given initial state of the qubit-resonator system,the initial reduced density operator ρ(0)is expanded in terms of ρk,

where the Cks are time-independent coefficients.The results in Ref.[24]tell us that time evolution of reduced density operator ρ will be

Now only one excitation is interested,our truncated basis{|E1+i,|E1?i,|E0i}consists of the three lowest eigenstates due to the Hamiltonian HS.

Now we can rewrite the master equation in Eq.(3),

Here the non-unitary parts of dissipative dynamics are described by L+ρ and L?ρ,

The superoperator L±describe the transitions between the higher excited state|E±i and the ground state|E0i induced by the environment.The corresponding decay rate γ1+(γ1?)describes the transition from the excited state|E1+i(|E1?i)to the ground state|E0i,these transitions are induced by the interaction between the system and the environment.

With respect to the system Hamiltonian HSin Eq.(2),the eigenoperators ρks are obtained,

The corresponding eigenvalues λks(for k=1,2,3,...,9)are

and

3 Rabi Oscillation

In traditional cavity QED theory,[5]the Rabi oscillation means that there exists energy exchange of one photon between a two-level atom and a single mode quantized field in cavity.Considering the nonlinearity of nanomechanical resonator,we study the decay process of Rabi oscillation in the nonlinear Jaynes–Cummings model described by the Hamiltonian in Eq.(2).

Given the initial state of the qubit-resonator system|ψ(0)i=|ei? |0i,it means that the qubit is in excited state|ei and the resonator is in vacuum state|0i.Thus,the initial reduced density operator reads

Expanding ρ(0)into some eigenoperators ρks,we obtain a set of coefficients Cks,

According to Eq.(6),the time evolution of density operator for the qubit-resonator system is calculated as

Here the probability of the qubit in the excited(upper)state|ei is

It characterizes the temporal behavior of Rabi oscillation in the qubit-resonator system,decay process of Rabi oscillation owns the periodic structure of time oscillating.

The nanomechanical resonator is assumed to be an ideal resonator(χ=0),and ignoring the difference of decay rates(γ1+= γ1?= γ),then the probability Pe(t)becomes

It describes the well known Rabi oscillation in Jaynes-Cummings model.[4]Comparing the results in Eq.(8)with Eq.(9),we find that nonlinearity parameter χ and decay rates γ1+6= γ1?modify the periodic structure of time oscillating in Rabi oscillation.

To further clarify the dependence of the probability Pe(t)on nonlinearity parameter and decay rates clearly,some figures are plotted with parameters ω=1.0,g=0.1.[14,22,27]Here,we take the frequency ω as the unit for all these parameters.

In Fig.1,the probability Pe(t)versus time t is plotted with parameters χ =0 and γ1+= γ1?=0.004.Figure 1 shows the well known Rabi oscillation,it verifies the results in Eq.(9).

In Fig.2,the probability Pe(t)versus time t is plotted with parameters χ =0.04,γ1+=0.007 and γ1?=0.001.Figure 2 shows that nonlinearity parameter χ and decay rates(γ1+6= γ1?)modify the periodic structure of time oscillating in Rabi oscillation,which is obviously different from Fig.1.

Fig.1 Rabi oscillation in Jaynes–Cummings model.Some parameters are χ =0 and γ1+= γ1? = γ =0.004.

Fig.2 The probability Pe(t)vs.the time t,where the excite states(|1+i and|1?i)own the different decay rate(γ1? =0.001 and γ1+=0.007)and χ =0.04.

Based on those results in Eqs.(8)–(9),we find that nonlinearity parameter χ slows down the time-oscillating period of Rabi oscillation T=(2?)?1.In the following,we will study how nonlinearity parameter χ and decay rates(γ1+6= γ1?)affect the temporal behavior in Rabi oscillation solely.

Firstly,assuming the same decay rates γ1?= γ1+= γ and nonlinearity parameter χ 6=0,the probability in Eq.(8)becomes

The dependence of the probability Pe(t)on nonlinearity parameter χ is plotted in Fig.3.When cos2(?t)=0,the minimum of the probability decays exponentially,which is different from the well known Rabi oscillation in Fig.1.

Fig.3 The probability Pe(t)vs.the time t,where the excite states(|1+i and|1?i)own the same decay rate(γ1+= γ1? = γ =0.004)and nonlinearity parameter χ=0.04.

Fig.4 The probability Pe(t)vs.the time t,the excite states(|1+i and|1?i)of the qubit-resonator system own the different decay rate(γ1? =0.001 and γ1+=0.007)and nonlinearity parameter χ=0.

Secondly,assuming no nonlinearity χ=0 and different decay rates(γ1+6= γ1?),the probability in Eq.(8)becomes

The dependence of the probability Pe(t)on different decay rates is plotted in Fig.4.When cos2(gt)=0,the minimum of the probability

shows that the difference of decay rates between γ1?and γ1+does not affect the short-time behavior and long-time behavior of Rabi oscillation obviously,which is seen in Fig.4.

Analytically,we can study the short-time behavior of Rabi oscillation,then the probability Pe(t)becomes

Ignoring the nonlinearity of nanomechanical resonator or the difference of decay rates,i.e.,χ =0 or γ1?= γ1+,the probability Pe(t)becomes

According to the results in Eqs.(12)and(13),we find that both nonlinearity parameter χ and the difference of decay rates(γ1+6= γ1?)affect dominate the short-time behavior of Rabi oscillation jointly.Also,these two factors speed up the decay of Rabi oscillation in short-time limit.

4 Conclusions

In summary,we have studied the dynamics of the nanomechanical QED system consisting of a charge qubit and a nanomechanical resonator.The temporal behavior of Rabi oscillations is analytically studied while the intrinsic nonlinearity of nanomechanical resonator is considered.With the loss of nanomechanical resonator,microscopic master equation approach is used to calculate the excitedstate probability of charge qubit in this nonlinear Jaynes-Cummings model.These results show that nonlinearity parameter and decay rates can a ff ect time-oscillating and decaying of Rabi oscillation solely or jointly.

Acknowledgments

We thank the discussions with Professor Peng Zhang and Dr.Ming Hua.

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