# Vacuum Rabi Splitting in Nanomechanical QED System with Nonlinear Resonator

###### Abstract

Considering the intrinsic nonlinearity in a nanomechanical resonator coupled to a charge qubit, vacuum Rabi splitting effect is studied in a nanomechanical QED (qubit-resonator) system. A driven nonlinear Jaynes-Cummings model describes the dynamics of this qubit-resonator system. Using quantum regression theorem and master equation approach, we have calculated the two-time correlation spectrum analytically. In the weak driving limit, these analytical results clarify the influence of the driving strength and nonlinearity parameter on the correlation spectrum. Also, numerical calculations confirm these analytical results.

###### pacs:

85.85.+j, 85.25.Cp## I introduction

In quantum optics Scully (1997) and quantum information, DiVincenzo (2000) the well known Jaynes-Cummings model, Jaynes (1963) one of the most important models, describes the light-matter interaction between a two-level quantum system (qubit) and a boson (resonator). Generally in cavity QED system, Raimond (2001) nanomechanical QED system Xue (2007) and circuit QED system, Blais (2004)vacuum Rabi splitting effect has been used to characterize the coupling strength between a qubit and a resonator.

Recently, a nanomechanical resonator with frequency of the order of 1 GHz approaches the quantum regime, Roukes (2003, 2009) it is getting closer to test the basic principles of quantum mechanics. When a superconducting qubit Makhlin (2001); Nori (2005) is coupled to a nanomechanical resonator, Cleland (2002) we can study quantum optical properties, such as, quantum decoherence, Gao (2013); Schwab (2002) Rabi oscillation, Gao (2016) vacuum Rabi splitting, Gao (2009) classical-quantum transition Sun (2006) and phonon blockade. LiuYX (2010)

Increasing the amplitude of driving, the nonlinearity response of nanomechanical resonator Roukes (2013) is not negligible which can be used to detect the classical-quantum transition. Itamar (2007) When intrinsic nonlinearity of nanomechanical resonator Hartmann (2014) is considered in the qubit-resonator system, we can use superconducting qubit to probe quantum fluctuations of nonlinear resonator. Blais (2013) Recently, the nonlinearity can be exploited to generate nonclassical states in mechanical systems Paris (2015); Hartmann (2015) and selectively address the nanomechanical qubit transitions in quantum information processing. Hartmann (2013)

In this paper, the dissipative dynamics of the qubit-resonator system is solved by master equation approach and quantum regression theorem. Carmichael (1999) Comparing with previous results, Gao (2009) the influence of the nonlinearity of nanomechanical resonator on vacuum Rabi splitting effect is studied analytically and numerically.

The paper is organized as follows. In Sec. II, using a driven nonlinear Jaynes-Cummings model, we describe the dynamics of a qubit-resonator system consisting of a superconducting qubit and a nonlinear nanomechanical resonator. In Sec. III, the two-time correlation spectrum is calculated analytically. In Sec. IV, vacuum Rabi splitting effect is studied. Also numerical simulations confirm our analytical results. Finally, our conclusions are summarized.

## Ii Model

In nanomechanical QED system, consisting a superconducting qubit and a nanomechanical resonator, we can use a driven Jaynes-Cummings model to describe the dynamics of this qubit-resonator system, Gao (2009)

(1) | |||||

Here is a driven Jaynes-Cummings type Hamiltonian. The lowering (raising) operator () and the annihilation (creation) operator () are defined to describe the qubit with frequency and the resonator with frequency respectively. The commutation relations, and , are satisfied. The last term in Eq. (1) is a classical drive with driving constant and driving frequency . The denotes the interaction strength between the qubit and the resonator.

Considering the nonlinearity of nanomechanical resonator, nanomechanical resonator is not assumed to be an ideal resonator again. Moving into a frame with rotating frequency , the total Hamiltonian for this qubit-resonator system writes Gao (2009)

(2) | |||||

which describes the dynamics of a driven nonlinear Jaynes-Cummings model. The linear part and nonlinear part come from a quartic potential . Hartmann (2014) Here the nonlinearity parameter is small, .

Some parameters in the Hamiltonian Eq. (2) are

(3) |

The detuning between the frequencies of the resonator and the qubit is

(4) |

## Iii two-time correlation spectrum

In previous results, Gao (2009) the induced electromotive force between two ends of nanomechanical resonator is

(5) |

To characterize the vacuum Rabi splitting in this nanomechanical QED system, the two-time correlation spectrum for the is

(6) |

Based on those results in Eqs. (5,6), the two-time correlation function writes

(7) | |||||

Considering the interaction with the environment, we use the master equation to describe dissipative dynamics of the qubit-resonator system, Carmichael (1999)

(8) | |||||

where the density operator describes the time evolution of the qubit-resonator system. The latter two terms in Eq. (8) describe the decay process of the resonator and the qubit respectively. The parameters and denote the decay rates of the resonator and the qubit. Here the Markov approximation is satisfied.

In this case, the number operator is defined to characterize the total number of excitations of the qubit-resonator system. The operator satisfies

where is the state of the qubit-resonator system, the index () denotes the state of the qubit (resonator) and satisfies () for ().

In this paper, the weak driving limit is adopted, When time approaches infinite, , the qubit-resonator system will decay into the vacuum state . The weak driving will induce the transitions from the vacuum state to the other excited states ( and ). Under the lowest order perturbation theory, we need to consider only one-phonon excitation in the qubit-resonator system,

(9) |

The Hilbert space for the reduced density matrix in Eq. (8) reduces into a smaller subspace with a truncated basis

(10) |

Thus, in this truncated basis, the corresponding density matrix elements satisfying the master equation in Eq. (8) are

(11) |

In the long-time limit, the system stays in a steady state, we can take and . Here can see that () scales as the order of and scales as the order of . Keeping terms of the and dropping the higher terms of the , we get

(12) |

Applying Laplace transformation, we have solved

(13) |

Here the coefficients (, and ) are dependent of time, i.e.,

(14) |

(15) |

and

for .

From the above results in Eq. (12), we can calculate the single-time function

(16) |

where and the index means the steady state. Through some simple calculations, we obtain the steady solution of the density matrix element ,

(17) |

Using the quantum regression theorem Gao (2009); Carmichael (1999); Xiao (1999), we can obtain the two-time correlation function

(18) |

The other two-time functions can be obtained as

(19) |

(20) |

and

(21) |

Based on the above results in Eqs. (18,19,20,21), the two-time correlation function for the induced electromotive force is obtained,

(22) |

In the limit of weak driving, we can neglect the terms (, , , and ) which are proportional to the in Eq. (22).

Using the formula,

the correlation spectrum in Eq. (6) is calculated as

(23) |

Where some parameters are defined,

for .

## Iv vacuum rabi splitting

Generally in vacuum Rabi splitting, the splitting frequency provides the information of the coupling between the qubit and the resonator. As seen in Eq. (23), nonlinearity parameter modifies the decay rate () and central frequency () of two peaks in the spectrum . And in the limit of weak driving, the does not affect the spectrum , it means that we can use this driven nonlinear Jaynes-Cummings model to characterize vacuum Rabi splitting effect very well. The corresponding splitting frequency between two peaks in the spectrum is

(24) |

which is independent of the driving strength and determined by the couple strength , the detuning , decay rate () and nonlinearity parameter .

Assuming nanomechanical resonator as an ideal resonator, , the spectrum in Eq. (23) will be same as the previous results. Gao (2009) When the resonant condition () and the strong-coupling limit () are adopted, we obtain the well-known splitting frequency Raimond (2001)

(25) |

To further clarify the dependence of correlation spectrum on nonlinearity parameter and the driving strength more clearly, the resonant condition is adopted, . The other parameters are and GHz is taken as the unit for all these parameters. Gao (2009); Hartmann (2013); Schwab (2002)

Numerical calculations by QuTiP python (2013) are illustrated with some plots in the following. In Fig. 1, we choose the values of and , the increasing of the number of total excitations tells that the value of is suitable in the other plots of the spectrum .

Figure 2 shows, in the limit of weak driving (), there are two peaks in the spectrum and the increasing of the driving strength does not affect the spectrum obviously, which confirms the analytical calculations in Eq. (23). Then we take the value of in the following plots.

As shown in Fig. 3, the increasing of nonlinearity parameter leads to the shift of central frequency of two peaks dramatically and does not change the distance between the centers of two peaks obviously because nonlinearity parameter is much smaller than the coupling , we can see it in Eq. (24).

Figure 2 and Figure 3 demonstrate the dependence of the spectrum on the driving strength and nonlinearity parameter . These numerical results agree with those analytical results in Eq. (23), both of them maintain that the weak driving strength does not change the splitting frequency , nonlinearity parameter changes the heights of two peaks ( and ) and the shifts of central frequency ( and ) obviously.

## V conclusions

In this paper, vacuum Rabi splitting effect is studied to provided the information of the coupling . Considering the intrinsic nonlinearity in nanomechanical resonator, a driven nonlinear Jaynes-Cummings model is used to describe the dynamics of the qubit-resonator system. Using quantum regression theorem, the dissipative dynamics of the qubit-resonator system is solved by master equation approach. Here, the two-time correlation spectrum is analytically calculated to clarify the dependence of correlation spectrum on the driving strength and nonlinearity parameter . Because of small nonlinearity in nanomechanical resonator, we find that nonlinearity parameter leads to the shifts of central frequency ( and ) and does not change the splitting frequency obviously. In Fig. 2 and Fig. 3, numerical results plotted by QuTiP agree with the analytical results in Eq. (23).

###### Acknowledgements.

We thank the discussions with Professor P. Zhang.## References

- Scully (1997) M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge University Press, Cambridge, 1997).
- DiVincenzo (2000) D. DiVincenzo, Fortschr. Phys. 48 (2000) 771.
- Jaynes (1963) E. T. Jaynes and F. W. Cummings, Proc. IEEE 51 (1963) 89.
- Raimond (2001) J. M. Raimond, M. Brune, and S. Haroche, Rev. Mod. Phys. 73 (2001) 565.
- Xue (2007) F. Xue, Y. D. Wang, C. P. Sun, H. Okamoto, H. Yamaguchi, and K. Semba, New J. Phy. 9 (2007) 35.
- Blais (2004) A. Blais, R. S. Huang, A. Wallraff, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 69 (2004) 062320.
- Roukes (2003) X. M. H. Huang, C. A. Zorman, M. Mehregany, and M. L. Roukes, Nature (London) 421 (2003) 496.
- Roukes (2009) M. D. LaHaye, J. Suh, P. M. Echternach, K. C. Schwab, and M. L. Roukes, Nature (London) 459 (2009) 960.
- Makhlin (2001) Y. Makhlin, G. Schoen, and A. Shnirman, Rev. Mod. Phys. 73 (2001) 357.
- Nori (2005) J. Q. You and F. Nori, Phys. Today 58 (11) (2005) 42.
- Cleland (2002) A. N. Cleland, Foundations of Nanomechanics: From Solid-State Theory to Device Applications (Springer-Verlag, Berlin, 2002).
- Schwab (2002) A. D. Armour, M. P. Blencowe, K. C. Schwab, Phys. Rev. Lett. 88 (2002) 148301.
- Gao (2013) C. Cheng, Y. B. Gao, Commun. Theor. Phys. 60 (2013) 531.
- Gao (2016) X. Xiao, M. Y. Zhao, S. M. Yu, and Y. B. Gao, Commun. Theor. Phys. 65 (2016) 273.
- Gao (2009) Y. B. Gao, S. Yang, Y. X. Liu, C. P. Sun, and F. Nori, arxiv: 0902.2512.
- Sun (2006) L. F. Wei, Y. X. Liu, C. P. Sun, and F. Nori, Phys. Rev. Lett. 97 (2006) 237201.
- LiuYX (2010) Y. X. Liu, A. Miranowicz, Y. B. Gao, J. Bajer, C. P. Sun, and F. Nori, Phys. Rev. A 82 (2010) 032101.
- Roukes (2013) L. G. Villanueva, R. B. Karabalin, M. H. Matheny, D. Chi, J. E. Sader, and M. L. Roukes, Phys. Rev. B 87 (2013) 024304.
- Itamar (2007) I. Katz, A. Retzker, R. Straub, and R. Lifshitz, Phys. Rev. Lett. 99 (2007) 040404. V. Peano and M. Thorwart, Phys. Rev. B 70 (2004) 235401.
- Hartmann (2014) S. Rips, I. WilsonRae, and M. J. Hartmann, Phys. Rev. A 89 (2014) 013854.
- Blais (2013) F. R. Ong, M. Boissonneault, F. Mallet, A. C. Doherty, A. Blais, D. Vion, D. Esteve, and P. Bertet, Phys. Rev. Lett. 110 (2013) 047001.
- Hartmann (2015) M. Abdi, M. Pernpeintner, R. Gross, H. Huebl, and M. J. Hartmann, Phys. Rev. Lett. 114 (2015) 173602.
- Paris (2015) B. Teklu, A. Ferraro, M. Paternostro, and M. G. A. Paris, EPJ Quantum Technology 2 (2015) 16.
- Hartmann (2013) S. Rips and M. J. Hartmann, Phys. Rev. Lett. 110 (2013) 120503.
- Carmichael (1999) H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker-Planck Equations (Springer, Berlin, 1999).
- Xiao (1999) R. J. Brecha, P. R. Rice, and M. Xiao, Phys. Rev. A 59 (1999) 2392.
- Schwab (2002) A. D. Armour, M. P. Blencowe, and K. C. Schwab, Phys. Rev. Lett. 88 (2002) 148301.
- python (2013) J. R. Johansson, P. D. Nation, and F. Nori, Comp. Phys. Comm. 184 (2013) 1234.