Abstract

We show that a cavity optomechanical system formed by a mechanical resonator simultaneously coupled to two modes of an optical cavity can be used for the implementation of a deterministic quantum phase gate between optical qubits associated with the two intracavity modes. The scheme is realizable for sufficiently strong single-photon optomechanical coupling in the resolved sideband regime, and is robust against cavity losses.

© 2015 Optical Society of America

1. Introduction

Simple quantum information protocols and quantum gates have been recently implemented with high fidelity in trapped ions and in circuit cavity QED (see e.g., Ref. [1] for a review). In an efficient quantum network, the information elaborated by a solid state processor at a node should be then robustly encoded in single-photon qubits for long-distance communication and distribution of quantum information. The possibility to implement high-fidelity two-qubit gates between single-photons would greatly facilitate such quantum information routing; an example is provided by perfect Bell-state discrimination for quantum teleportation and entanglement swapping [2], which could be implemented deterministically if a quantum phase gate (QPG) between single photon qubits would be available [3]. It is well known that to obtain such a QPG one needs nonlinearities implicit in post-processing measurements or a nonlinear system [4]. A particularly convenient QPG is obtained when the conditional phase shift between the two photonic qubit is equal to π, because in this latter case the QPG is equivalent, up to local unitary transformations, to a C-NOT gate [2, 5]. Using all-optical solutions this is not easy to achieve because to process the information one needs strong photon-photon interaction. In fact, to implement quantum information with photons, a nonlinear interaction is needed either to build a two-photon gate operation [6] (but the commonly achievable χ3 susceptibility factor is typically too small), or the nonlinearity implicit at the detection stage in linear optics quantum computation [7].

Then, one has to think differently as for example using electromagnetically induced transparency (EIT) [8, 9]. Indeed the possibility of reducing the speed of light in atomic media with Λ-type levels was proposed and experimentally obtained [10]. Although the two weak fields wave functions traveling with slow group velocity can have a very high nonlinear coupling when they propagate in an atomic media with Λ-like level configurations [11, 12], we need, however, to obtain the same result at the single photon level and this is extremely hard to achieve or even impossible with Λ-like configurations. A full quantum analysis has shown that a large nonlinear cross-phase shift is achievable using an atomic M level structure [1315], but it was also shown that a trade-off between the size of the conditional phase shift and the fidelity of the gate exists. This can be avoided in the transient regime, which is however experimentally challenging. More recently, important results have been achieved by exploiting two different solutions able to provide the required effective nonlinearities: i) the strong dispersive coupling of light to strongly interacting atoms in highly excited Rydberg states [1621]; ii) a fiber-integrated cavity QED system employing a whispering gallery mode resonator strongly coupled to a single Rubidium atom [22].

Nevertheless, in recent years it has been proposed [23], and then experimentally realized [2427], that EIT-like effects could be obtained also within cavity optomechanical systems. Furthermore, it is well known that the ponderomotive action of light, together with the backaction of the mechanical oscillator interacting with it, is responsible for an effective optical Kerr nonlinearity [28], which in turn, may give rise to interesting quantum phenomena, such as squeezing of the cavity output light, as predicted a couple of decades ago [29, 30] and recently experimentally achieved [3133].

In this paper we will show that a QPG for simple photonic qubits with a conditional phase shift equal to π is achievable by employing a cavity optomechanical system with sufficiently large single-photon optomechanical coupling, and relatively high mechanical Q-factor Qm. Refs. [34, 35] first suggested that multi-mode optomechanical systems in the single-photon strong coupling regime could be exploited for quantum information processing with photons and phonons, and a first example of optomechanical implementation of a QPG has been recently provided in Ref. [36]. Here we further develop these ideas, by proposing a much simpler scheme, which requires the control of only two cavity modes and of a single mechanical resonator, rather than four optical cavity modes and two mechanical resonators as in Ref. [36]. Recent progress in the realization of strongly coupled nano-optomechanical systems [3739] suggests that the QPG scheme proposed here could be implemented in the near future.

2. The model

We consider an optomechanical system consisting of a mechanical resonator interacting with two optical modes, which is described by the following Hamiltonian

H^=h¯ω1a^1a^1+h¯ω2a^2a^2+h¯ωmb^b^+h¯(g1a^1a^1+g2a^2a^2)(b^+b^),
where, âi() and a^i(b^) are the annihilation and creation operators for the optical cavity (mechanical) modes, with frequency ωi/2π and ωm/2π respectively, and with [a^i,a^i]=[b^,b^]=1; gi = (i/dx)xzpf is the i-th single-photon optomechanical coupling rate, with xzpf=h¯/2mωm the spatial size of the zero-point fluctuation of the mechanical oscillator.

We focus on the simplest choice for an optical qubit, the state space spanned by the lowest Fock states of an optical mode, |0〉 and |1〉; to be more specific we want to implement a QPG between the optical qubits associated with two optical cavity modes of the optomechanical system under study. The generic initial (pure) state of the two optical qubits is given by

|ψin=α00|01|02+α01|01|12+α10|11|02+α11|11|12,
corresponding in general to an entangled state of the two modes with up to two photons. A simple proof-of-principle demonstration could be achieved by restricting to factorized input states of the two cavity modes, which could be provided by two weak laser pulses driving the two selected cavity modes at frequencies ωL1 and ωL2, similar to the preliminary experimental demonstration of a QPG given in Ref. [6]. In this case the two cavity modes are prepared in a product of two coherent states with amplitudes αd1 and αd2, |ψ(0)〉 = |αd11|αd22 ≃ [|0〉1 +αd1|1〉1][|0〉2 +αd2|1〉2], where the latter expression is valid for |αd1|, |αd2| ≪ 1. For input laser powers Pj, cavity detunings Δj = ωjωLj, and decay rates κj, j = 1, 2, the amplitudes are given by αdj=2Pjκj/[h¯ωLj(κj2+Δj2)].

It is convenient to move to a frame rotating at the corresponding driving laser frequency for each cavity mode, providing therefore the phase reference for each optical qubit; this is equivalent to move to the interaction picture with respect to the free optical Hamiltonian H0=h¯ωL1a^1a^1+h¯ωL2a^2a^2, in which the system Hamiltonian becomes

H^=h¯Δ1n^1+h¯Δ2n^2+h¯ωmb^b^+h¯ωmf^n^(b^+b^),
where we have used the cavity mode photon number operators n^j=a^ja^j, and we have defined = [g11 + g22]/ωm.

3. Hamiltonian dynamics

In order to have a physical description of how the effective optical nonlinearity provided by the optomechanical interaction allows to implement the QPG, we first study the ideal case with no optical and mechanical losses, in which the dynamics is determined solely by the Hamiltonian of Eq. (3), i.e., by the unitary operator Û(t) = eiĤt/h̄. In such a case the dynamics can be exactly solved: in fact, profiting from the fact that both photon number operators j are conserved, and moving to a photon-number-dependent displaced frame for the mechanical resonator, one can rewrite the unitary evolution operator in a form in which the optical and mechanical evolution operators are conveniently factorized. In fact, acting with the photon number conserving mechanical displacement operator

D^(f^n^)=exp[(b^b^)f^n^],
which separates the Hamiltonian according to
D^(f^n^)H^D^(f^n^)=H^opt+H^b,
where
H^opt=h¯Δ1n^1+h¯Δ2n^2h¯ωmf^n^2=h¯Δ1n^1+h¯Δ2n^2h¯(g1n^1+g2n^2)2ωm,
H^b=h¯ωmb^b^,
one gets
U^(t)=U^opt(t)D^(f^n^)U^b(t)D^(f^n^),
where Ûopt(t) = eoptt/h̄, and Ûb(t) = ebt/h̄. Therefore the dynamics of the two optical modes is mostly determined by the effective unitary operator Ûopt(t), possessing either self-Kerr terms n^j2 and the cross-Kerr term −2h̄g1g2tn̂12/ωm; the other factor D^(f^n^)U^b(t)D^(f^n^) however also affects the optical mode dynamics since it entangles them with the mechanical resonator, by correlating the resonator position with the photon numbers.

The photon number conserving dynamics allows to stay within the logical space described above, i.e., the one spanned by optical Fock states with no more than one photon (and this will remain true even when we will include optical losses). In this case one can always fix the two detunings in order to eliminate completely the effect of the self-Kerr terms. In fact, within this subspace, n^j2=n^j, and therefore, taking Δj=gj2/ωm, j = 1, 2, one has the effective unitary operator Ûopt(t) = exp [2ig1g2tn̂12/ωm], yielding a nonlinear conditional phase shift ϕnl(t) = 2g1g2t/ωm only when each cavity mode has one photon, i.e.,

|01|02|01|02;|01|12|01|12;|11|02|11|02;|11|12ei2g1g2tωm|11|12.
Therefore we expect to get a conditional phase shift equal to π when the interaction time t is equal to
tπ=πωm2g1g2.
We now evaluate the exact Hamiltonian evolution in order to see to what extent the interaction with the mechanical resonator affects and modifies the ideal QPG dynamics defined by Eqs. (10) and (11). The natural choice for the initial state is the factorized state ρ^(0)=|ψinψ|ρ^bth, where |ψin is the generic initial state of Eq. (2), and ρ^bth is the thermal equilibrium state of the mechanical resonator, with mean thermal phonons. By renumbering |0〉1|0〉2 → |0〉, |1〉1|0〉2 → |1〉, |0〉1|1〉2 → |2〉, |1〉1|1〉2 → |3〉, we can write the state of the whole system at time t as
ρ^(t)=U^(t)|ψinψ|ρ^bthU^(t)=k,l=03αkαl*U^opt(t)|kl|U^opt(t)D^(fk)U^b(t)D^(fk)ρ^bthD^(fl)U^b(t)D^(fl),
where
D^(fk)=exp[fk(b^b^)]
is now a displacement operator acting only on the mechanical resonator degree of freedom, with a c-number displacement fk, with f0 = 0, f1 = g1/ωm, f2 = g2/ωm, f3 = (g1 + g2)/ωm. We are interested in the state of the two optical qubits only, and therefore we have to trace over the mechanical resonator. Using the explicit expression of Ûopt(t) (with the choice of detuning specified above), and performing the trace, the reduced state of the optical modes reads
ρ^opt(t)=k,l=03ck,l(t)αkαl*exp[i2g1g2tωm(δk,3δl,3)]|kl|,
where δk,l is the Kronecker delta, while ck,l(t) is the factor describing the decoherence caused by the interaction with the mechanical resonator and whose explicit expression is given by (see the appendix for its derivation)
ck,l(t)=exp[(fkfl)2(1cosωmt)(2n¯+1)+i(fl2fk2)sinωmt].
We quantify the QPG performance with the fidelity relative to the ideal pure target state corresponding to a π conditional phase shift, that is
|ψtgt=α00|01|02+α01|01|12+α10|11|02+eiπα11|11|12=k=03αkeiπδk,3|k.
The corresponding fidelity F(t) can be written as
F(t)ψtgt|ρ^opt(t)|ψtgt=k,l=03ck,l(t)|αk|2|αl|2exp[i(2g1g2tωmπ)(δk,3δl,3)].
Actually, the QPG performance can be characterized by the gate fidelity, which is the average of the above quantity over all possible input states of the two qubits [41]. Since the averages are given by |αk|4¯=1/8, ∀k, and |αk|2|αl|2¯=1/24, ∀kl [41], using the explicit expression for ck,l(t) of Eq. (15) implying in particular that ck,k(t) = 1 ∀k and that ck,l(t) = cl,k(t)*, and the explicit values of the c-numbers fk, we get
(t)=ψtgt|ρ^opt(t)|ψtgt¯=12+112{exp[g12ωm2(1cosωmt)(2n¯+1)][cos(g12ωm2sinωmt)+cos(g12+2g1g2ωm2sinωmt+π2g1g2tωm)]+exp[g22ωm2(1cosωmt)(2n¯+1)][cos(g22ωm2sinωmt)+cos(g22+2g1g2ωm2sinωmt+π2g1g2tωm)]+exp[(g1g2ωm)2(1cosωmt)(2n¯+1)]cos(g22g12ωm2sinωmt)+exp[(g1+g2ωm)2(1cosωmt)(2n¯+1)]cos[(g1+g2ωm)2sinωmt+π2g1g2tωm]}.
From Eqs. (17) one can see that the gate fidelity achieves the ideal value of unity when two conditions are satisfied: i) the conditional phase shift is equal to π, (or more generally to an odd multiple of π), 2g1g2t/ωm = (2m + 1)π (integer m); ii) ck,l(t) = 1, ∀k, l. Eq. (18) shows that the latter conditions are achieved for generic nonzero couplings g1 and g2 only after every mechanical oscillation period, i.e., when ωmt = 2, p = 1, 2,.... The QPG will be minimally affected by losses for the shortest interaction time tπ of Eq. (11), and therefore, the ideal conditions for a unit-fidelity QPG with a conditional phase shift equal to π are
2g1g2tπωm=πωmtπ=2πg1g2=ωm24.
Therefore, when optical and mechanical losses are negligible, one can realize an ideal QPG with a simple optomechanical setup by fixing the interaction time between the mechanical resonator and the cavity modes according to Eq. (11), and provided that the single-photon optomechanical couplings can be tuned to the strong coupling condition of Eq. (19). The interaction time can be controlled in tunable optomechanical systems in which the interaction can be turned on and off, as it could be done for example in the optomechanical setup of Ref. [40], where a vibrating nanobeam is coupled to the evanescent field of a whispering gallery mode of a microdisk. It is also relevant to stress that under these conditions the QPG is practically insensitive to the effect of thermal noise acting on the resonator, because at the exact gate duration tπ the fidelity becomes completely independent upon the mean thermal phonon number . Eq. (18) shows that instead the gate fidelity significantly drops for increasing as soon as ttπ.

4. Dissipative dynamics

Let us now consider the realistic situation in order to see to what extent optical losses, mechanical damping, and thermal noise affect this ideal gate behavior. In that case the evolution is no more analytically tractable, and we will consider the numerical solution of the master equation for the density matrix of the optomechanical system under study.

Introducing the cavity modes decay κi (i = 1, 2), the mechanical damping γm = ωm/Qm, and the mean thermal phonon number associated with the reservoir of the mechanical resonator, , the master equation in the usual Born-Markov approximation can be written as [42]

ddtρ^(t)=1ih¯[H^,ρ^(t)]+κ12(2a^1ρ^(t)a^1a^1a^1ρ^(t)ρ^(t)a^1a^1)+κ22(2a^2ρ^(t)a^2a^2a^2ρ^(t)ρ^(t)a^2a^2)+γm2(n¯+1)(2b^ρ^(t)b^b^b^ρ^(t)ρ^(t)b^b^)+γm2n¯(2b^ρ^(t)b^b^b^ρ^(t)ρ^(t)b^b^),
where Ĥ is the Hamiltonian in Eq. (3).

In Fig. 1 we compare the behavior of the gate fidelity (t) in the absence of damping and losses of Eq. (18) versus the dimensionless interaction time ωmt, either at = 0 (red dot-dashed curve), and at = 10 (full black curve) with the corresponding curves in the presence of optical and mechanical damping processes. These latter curves (the dotted blue line corresponds to = 0, while the green dashed line to = 10) are obtained from the numerical solution of the master equation Eq. (20) in the case κ1 = κ2 = 10−2 ωm, Qm = 106, while we have fixed the couplings according to the ideal strong coupling condition of Eq. (19) g1 = g2 = ωm/2.

 figure: Fig. 1

Fig. 1 Numerical solution of the master equation Eq. (20) for the gate fidelity (t) versus the dimensionless interaction time ωmt. We compare four different cases: i) zero damping and losses γm = κ1 = κ2 = 0 and = 10 (full black line); ii) zero damping and losses and = 0 (red dashed-dotted line); iii) with damping and losses (κ1 = κ2 = 10−2ωm, Qm = 106,) and = 10 (green dashed line); iv) with damping and losses (κ1 = κ2 = 10−2 ωm, Qm = 106,) and = 0 (blue dotted line). The numerical solutions for the zero damping and loss case are indistinguishable from the analytical expression of Eq. (18) either at = 0 and at = 10. In all cases we have fixed the couplings according to the ideal strong coupling condition of Eq. (19), g1 = g2 = ωm/2.

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We see, as expected, that in the absence of optical and mechanical losses, (t) = 1 exactly at the interaction time tπ of Eq. (11), regardless the value of the temperature of the mechanical reservoir. At different times the gate performance is strongly affected by thermal noise; what is relevant is that in the presence of realistic values of mechanical damping and of optical loss rates, this scenario is still maintained, with a limited decrease of the gate fidelity.

5. Conclusions

We have proposed a simple optomechanical setup which is able to implement an ideal QPG with a conditional phase shift equal to π between two optical qubits associated with the lowest Fock states (zero and one photon) of an optical cavity mode. The scheme is minimal because it employs only two modes of a high-finesse optical cavity and a single mechanical resonator coupled to them. The scheme is robust in the presence of realistic values of optical and mechanical losses and, if the interaction time is appropriately fixed, it is almost completely insensitive to the thermal noise acting on the mechanical resonator. The most stringent and challenging condition is the required strong optomechanical coupling condition, given by Eq. (19), in which the single-photon optomechanical coupling must be of the order of the mechanical resonance frequency. Such a condition has not been achieved yet in current solid-state nanomechanical setups, for which record values corresponds to gjm ∼ 10−3 [3739]. On the contrary, such a strong coupling situation is normally achieved in ultracold atom realizations of cavity optomechanics, where the mechanical resonator corresponds to the collective motion of an ensemble of trapped ultracold atoms; for example one has g/ωm ∼ 0.3 in Ref. [31]. The limitation in these latter systems is represented by cavity losses, because in this case one is typically far from the resolved sideband regime κ/ωm ≪ 1 which is required here in order that cavity losses do not alter significantly the effective cross-Kerr nonlinear interaction mediated by the resonator, responsible for the QPG dynamics. Therefore the present proposal could be implemented in experimental optomechanical platforms able to combine a significantly large single photon coupling g/ωm ∼ 0.5 with a resolved sideband operation condition κ/ωm ≪ 1.

6. Appendix

We now derive the explicit expression for the decoherence coefficients ck,l(t) of Eq. (15). From Eqs. (12)(14) and using the cyclic property of the trace, one has

ck,l(t)=Trb[D^(fl)U^b(t)D^(fl)D^(fk)U^b(t)D^(fk)ρ^bth],
which is a thermal average of a combination of displacement operators. We notice then that the factor U^b(t)D^(fl)D^(fk)U^b(t) within the trace is just the Heisenberg time evolution for a time t of a displacement operator of a free mechanical resonator, so that
U^b(t)D^(fl)D^(fk)U^b(t)=exp[(b^eiωmtb^eiωmt)(flfk)].
Inserting this solution within Eq. (21), and using the property of the displacement operator (α)(β) = (α + β)exp[iIm(αβ*)], one gets
ck,l(t)=Trb{D^(flfk)(eiωmt1)ρ^bth}exp[i(fl2fk2)sinωmt].
Performing the thermal average of the final displacement operator, according to
exp[αb^α*b^]th=exp[|α|2(n¯+12)],
we finally get Eq. (15).

Acknowledgments

This work has been supported by the European Commission (ITN-Marie Curie project cQOM and FET-Open Project iQUOEMS), and by MIUR (PRIN 2011).

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41. J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390–393 (1997). [CrossRef]  

42. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, Berlin1994). [CrossRef]  

References

  • View by:

  1. M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: An outlook,” Science 339, 1169–1174 (2013).
    [Crossref] [PubMed]
  2. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Springer, 2000).
  3. D. Vitali, M. Fortunato, and P. Tombesi, “Complete quantum teleportation with a Kerr nonlinearity,” Phys. Rev. Lett. 85, 445–448 (2000).
    [Crossref] [PubMed]
  4. D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–578 (1997).
    [Crossref]
  5. S. Lloyd, “Almost any quantum logic gate is universal,” Phys. Rev. Lett. 75, 346–349 (1995).
    [Crossref] [PubMed]
  6. Q.A. Turchette, C.J. Hood, W. Lange, H. Mabuchi, and H.J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710–4713 (1995)
    [Crossref] [PubMed]
  7. E. Knill, R. Laflamme, and G.J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001)
    [Crossref] [PubMed]
  8. E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. XXXV, 257–354 (1996).
    [Crossref]
  9. S.E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
    [Crossref]
  10. L.V. Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
    [Crossref]
  11. M. D. Lukin and A. Imamoğlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000).
    [Crossref] [PubMed]
  12. M. D. Lukin and A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature 413, 273–276 (2001).
    [Crossref] [PubMed]
  13. C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: Full quantum analysis,” Phys. Rev. A 73, 010301(R) (2006).
    [Crossref]
  14. C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Cross phase modulation in a five-level atomic medium: semi-classical theory,” Eur. Phys. J. D 40, 281–296 (2006).
    [Crossref]
  15. S. Rebic, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase gate operation based on nonlinear optics,” Phys Rev A 74, 032301 (2006).
    [Crossref]
  16. A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
    [Crossref] [PubMed]
  17. D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with Rydberg atoms,” Phys. Rev. Lett. 107, 213601 (2011).
    [Crossref] [PubMed]
  18. T. Peyronel, O. Firstenberg, Q. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
    [Crossref] [PubMed]
  19. V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
    [Crossref]
  20. O. Firstenberg, T. Peyronel, Q. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
    [Crossref] [PubMed]
  21. B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
    [Crossref] [PubMed]
  22. J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, “Nonlinear π phase shift for single fibre-guided photons interacting with a single resonator-enhanced atom,” Nat. Photonics. 8, 965–970 (2014).
    [Crossref]
  23. G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
    [Crossref]
  24. S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science, 330, 1520–1523 (2010).
    [Crossref]
  25. J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
    [Crossref] [PubMed]
  26. A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
    [Crossref] [PubMed]
  27. M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
    [Crossref]
  28. K. Hammerer, C. Genes, D. Vitali, P. Tombesi, G.J. Milburn, C. Simon, and D. Bouwmeester, “Nonclassical states of light and mechanics,” in “Cavity Optomechanics: Nano- and Micromechanical Resonators Interacting with Light,” (SpringerBerlin Heidelberg), edited by M. Aspelmeyer, T.J. Kippenberg, and F. Marquardt, eds. pag. 25–56, (2014).
    [Crossref]
  29. S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994).
    [Crossref] [PubMed]
  30. C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
    [Crossref] [PubMed]
  31. D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
    [Crossref]
  32. A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
    [Crossref] [PubMed]
  33. T. P. Purdy, P-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X3, 031012 (2013).
  34. M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Ref. Lett. 109, 063601 (2012).
    [Crossref]
  35. K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
    [Crossref] [PubMed]
  36. W.-Z. Zhang, J. Cheng, and L. Zhou, “Quantum control gate in cavity optomechanical system,” J. Phys. B 48, 015502 (2015).
    [Crossref]
  37. C. Baker, W. Hease, D.-T. Nguyen, A. Andronico, S. Ducci, G. Leo, and I. Favero, “Photoelastic coupling in gallium arsenide optomechanical disk resonators,” Opt. Express 22, 14072–14086 (2014)
    [Crossref] [PubMed]
  38. K. C. Balram, M. Davanço, J. Y. Lim, J. D. Song, and K. Srinivasan, “Moving boundary and photoelastic coupling in GaAs optomechanical resonators,” Optica 1, 414–420 (2014).
    [Crossref]
  39. S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, “Silicon optomechanical crystal resonator at millikelvin temperatures,” Phys. Rev. A 90, 011803(R) (2014).
    [Crossref]
  40. D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. J. Kippenberg, “Measurement and control of a mechanical oscillator at its thermal decoherence rate,” arXiv:1410.6191 [quant-ph].
  41. J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390–393 (1997).
    [Crossref]
  42. D. F. Walls and G. J. Milburn, Quantum Optics (Springer, Berlin1994).
    [Crossref]

2015 (1)

W.-Z. Zhang, J. Cheng, and L. Zhou, “Quantum control gate in cavity optomechanical system,” J. Phys. B 48, 015502 (2015).
[Crossref]

2014 (5)

C. Baker, W. Hease, D.-T. Nguyen, A. Andronico, S. Ducci, G. Leo, and I. Favero, “Photoelastic coupling in gallium arsenide optomechanical disk resonators,” Opt. Express 22, 14072–14086 (2014)
[Crossref] [PubMed]

K. C. Balram, M. Davanço, J. Y. Lim, J. D. Song, and K. Srinivasan, “Moving boundary and photoelastic coupling in GaAs optomechanical resonators,” Optica 1, 414–420 (2014).
[Crossref]

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, “Silicon optomechanical crystal resonator at millikelvin temperatures,” Phys. Rev. A 90, 011803(R) (2014).
[Crossref]

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref] [PubMed]

J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, “Nonlinear π phase shift for single fibre-guided photons interacting with a single resonator-enhanced atom,” Nat. Photonics. 8, 965–970 (2014).
[Crossref]

2013 (5)

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
[Crossref] [PubMed]

T. P. Purdy, P-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X3, 031012 (2013).

M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: An outlook,” Science 339, 1169–1174 (2013).
[Crossref] [PubMed]

O. Firstenberg, T. Peyronel, Q. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref] [PubMed]

2012 (5)

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
[Crossref]

T. Peyronel, O. Firstenberg, Q. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Ref. Lett. 109, 063601 (2012).
[Crossref]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

2011 (4)

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref] [PubMed]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with Rydberg atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref] [PubMed]

2010 (2)

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
[Crossref]

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science, 330, 1520–1523 (2010).
[Crossref]

2006 (3)

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: Full quantum analysis,” Phys. Rev. A 73, 010301(R) (2006).
[Crossref]

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Cross phase modulation in a five-level atomic medium: semi-classical theory,” Eur. Phys. J. D 40, 281–296 (2006).
[Crossref]

S. Rebic, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase gate operation based on nonlinear optics,” Phys Rev A 74, 032301 (2006).
[Crossref]

2001 (2)

M. D. Lukin and A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature 413, 273–276 (2001).
[Crossref] [PubMed]

E. Knill, R. Laflamme, and G.J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001)
[Crossref] [PubMed]

2000 (2)

D. Vitali, M. Fortunato, and P. Tombesi, “Complete quantum teleportation with a Kerr nonlinearity,” Phys. Rev. Lett. 85, 445–448 (2000).
[Crossref] [PubMed]

M. D. Lukin and A. Imamoğlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000).
[Crossref] [PubMed]

1999 (1)

L.V. Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

1997 (3)

S.E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–578 (1997).
[Crossref]

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390–393 (1997).
[Crossref]

1996 (1)

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. XXXV, 257–354 (1996).
[Crossref]

1995 (2)

S. Lloyd, “Almost any quantum logic gate is universal,” Phys. Rev. Lett. 75, 346–349 (1995).
[Crossref] [PubMed]

Q.A. Turchette, C.J. Hood, W. Lange, H. Mabuchi, and H.J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710–4713 (1995)
[Crossref] [PubMed]

1994 (2)

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994).
[Crossref] [PubMed]

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

Agarwal, G. S.

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
[Crossref]

Alegre, T. P. M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Allman, M. S.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

Andronico, A.

Arcizet, O.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science, 330, 1520–1523 (2010).
[Crossref]

Arimondo, E.

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. XXXV, 257–354 (1996).
[Crossref]

Aspelmeyer, M.

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, “Silicon optomechanical crystal resonator at millikelvin temperatures,” Phys. Rev. A 90, 011803(R) (2014).
[Crossref]

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
[Crossref] [PubMed]

Baker, C.

Balram, K. C.

Bawaj, M.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Behroozi, C.H.

L.V. Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

Bennett, S. D.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

Biancofiore, C.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

Bimbard, E.

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Botter, T.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
[Crossref]

Bourzeix, S.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

Bouwmeester, D.

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–578 (1997).
[Crossref]

K. Hammerer, C. Genes, D. Vitali, P. Tombesi, G.J. Milburn, C. Simon, and D. Bouwmeester, “Nonclassical states of light and mechanics,” in “Cavity Optomechanics: Nano- and Micromechanical Resonators Interacting with Light,” (SpringerBerlin Heidelberg), edited by M. Aspelmeyer, T.J. Kippenberg, and F. Marquardt, eds. pag. 25–56, (2014).
[Crossref]

Brahms, N.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
[Crossref]

Brooks, D. W. C.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
[Crossref]

Chan, J.

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Chang, D. E.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Cheng, J.

W.-Z. Zhang, J. Cheng, and L. Zhou, “Quantum control gate in cavity optomechanical system,” J. Phys. B 48, 015502 (2015).
[Crossref]

Chuang, I. L.

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Springer, 2000).

Cicak, K.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

Cirac, J. I.

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390–393 (1997).
[Crossref]

Cohen, J. D.

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, “Silicon optomechanical crystal resonator at millikelvin temperatures,” Phys. Rev. A 90, 011803(R) (2014).
[Crossref]

Davanço, M.

Deléglise, S.

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S. Rebic, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase gate operation based on nonlinear optics,” Phys Rev A 74, 032301 (2006).
[Crossref]

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: Full quantum analysis,” Phys. Rev. A 73, 010301(R) (2006).
[Crossref]

Regal, C. A.

T. P. Purdy, P-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X3, 031012 (2013).

Reynaud, S.

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

Rivière, R.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science, 330, 1520–1523 (2010).
[Crossref]

Safavi-Naeini, A. H.

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, “Silicon optomechanical crystal resonator at millikelvin temperatures,” Phys. Rev. A 90, 011803(R) (2014).
[Crossref]

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
[Crossref] [PubMed]

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Ref. Lett. 109, 063601 (2012).
[Crossref]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Scheucher, M.

J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, “Nonlinear π phase shift for single fibre-guided photons interacting with a single resonator-enhanced atom,” Nat. Photonics. 8, 965–970 (2014).
[Crossref]

Schilling, R.

D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. J. Kippenberg, “Measurement and control of a mechanical oscillator at its thermal decoherence rate,” arXiv:1410.6191 [quant-ph].

Schliesser, A.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science, 330, 1520–1523 (2010).
[Crossref]

Schoelkopf, R. J.

M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: An outlook,” Science 339, 1169–1174 (2013).
[Crossref] [PubMed]

Schreppler, S.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
[Crossref]

Sharypov, A. V.

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref] [PubMed]

Sheng, J.

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref] [PubMed]

Simmonds, R. W.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

Simon, C.

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref] [PubMed]

K. Hammerer, C. Genes, D. Vitali, P. Tombesi, G.J. Milburn, C. Simon, and D. Bouwmeester, “Nonclassical states of light and mechanics,” in “Cavity Optomechanics: Nano- and Micromechanical Resonators Interacting with Light,” (SpringerBerlin Heidelberg), edited by M. Aspelmeyer, T.J. Kippenberg, and F. Marquardt, eds. pag. 25–56, (2014).
[Crossref]

Sirois, A. J.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

Song, J. D.

Srinivasan, K.

Stamper-Kurn, D. M.

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
[Crossref]

Stannigel, K.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

Stanojevic, J.

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Sudhir, V.

D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. J. Kippenberg, “Measurement and control of a mechanical oscillator at its thermal decoherence rate,” arXiv:1410.6191 [quant-ph].

Teufel, J. D.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

Tombesi, P.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: Full quantum analysis,” Phys. Rev. A 73, 010301(R) (2006).
[Crossref]

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Cross phase modulation in a five-level atomic medium: semi-classical theory,” Eur. Phys. J. D 40, 281–296 (2006).
[Crossref]

S. Rebic, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase gate operation based on nonlinear optics,” Phys Rev A 74, 032301 (2006).
[Crossref]

D. Vitali, M. Fortunato, and P. Tombesi, “Complete quantum teleportation with a Kerr nonlinearity,” Phys. Rev. Lett. 85, 445–448 (2000).
[Crossref] [PubMed]

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994).
[Crossref] [PubMed]

K. Hammerer, C. Genes, D. Vitali, P. Tombesi, G.J. Milburn, C. Simon, and D. Bouwmeester, “Nonclassical states of light and mechanics,” in “Cavity Optomechanics: Nano- and Micromechanical Resonators Interacting with Light,” (SpringerBerlin Heidelberg), edited by M. Aspelmeyer, T.J. Kippenberg, and F. Marquardt, eds. pag. 25–56, (2014).
[Crossref]

Tualle-Brouri, R.

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

Turchette, Q.A.

Q.A. Turchette, C.J. Hood, W. Lange, H. Mabuchi, and H.J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710–4713 (1995)
[Crossref] [PubMed]

Vitali, D.

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: Full quantum analysis,” Phys. Rev. A 73, 010301(R) (2006).
[Crossref]

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Cross phase modulation in a five-level atomic medium: semi-classical theory,” Eur. Phys. J. D 40, 281–296 (2006).
[Crossref]

S. Rebic, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase gate operation based on nonlinear optics,” Phys Rev A 74, 032301 (2006).
[Crossref]

D. Vitali, M. Fortunato, and P. Tombesi, “Complete quantum teleportation with a Kerr nonlinearity,” Phys. Rev. Lett. 85, 445–448 (2000).
[Crossref] [PubMed]

K. Hammerer, C. Genes, D. Vitali, P. Tombesi, G.J. Milburn, C. Simon, and D. Bouwmeester, “Nonclassical states of light and mechanics,” in “Cavity Optomechanics: Nano- and Micromechanical Resonators Interacting with Light,” (SpringerBerlin Heidelberg), edited by M. Aspelmeyer, T.J. Kippenberg, and F. Marquardt, eds. pag. 25–56, (2014).
[Crossref]

Volz, J.

J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, “Nonlinear π phase shift for single fibre-guided photons interacting with a single resonator-enhanced atom,” Nat. Photonics. 8, 965–970 (2014).
[Crossref]

Vuletic, V.

O. Firstenberg, T. Peyronel, Q. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref] [PubMed]

T. Peyronel, O. Firstenberg, Q. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

Walls, D. F.

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, Berlin1994).
[Crossref]

Weinfurter, H.

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–578 (1997).
[Crossref]

Weis, S.

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science, 330, 1520–1523 (2010).
[Crossref]

Whittaker, J. D.

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

Wilson, D. J.

D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. J. Kippenberg, “Measurement and control of a mechanical oscillator at its thermal decoherence rate,” arXiv:1410.6191 [quant-ph].

Winger, M.

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

Xiao, M.

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref] [PubMed]

Yu, P-L.

T. P. Purdy, P-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X3, 031012 (2013).

Zeilinger, A.

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–578 (1997).
[Crossref]

Zhang, W.-Z.

W.-Z. Zhang, J. Cheng, and L. Zhou, “Quantum control gate in cavity optomechanical system,” J. Phys. B 48, 015502 (2015).
[Crossref]

Zhou, L.

W.-Z. Zhang, J. Cheng, and L. Zhou, “Quantum control gate in cavity optomechanical system,” J. Phys. B 48, 015502 (2015).
[Crossref]

Zoller, P.

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390–393 (1997).
[Crossref]

Eur. Phys. J. D (1)

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Cross phase modulation in a five-level atomic medium: semi-classical theory,” Eur. Phys. J. D 40, 281–296 (2006).
[Crossref]

J. Phys. B (1)

W.-Z. Zhang, J. Cheng, and L. Zhou, “Quantum control gate in cavity optomechanical system,” J. Phys. B 48, 015502 (2015).
[Crossref]

Nat. Photonics. (1)

J. Volz, M. Scheucher, C. Junge, and A. Rauschenbeutel, “Nonlinear π phase shift for single fibre-guided photons interacting with a single resonator-enhanced atom,” Nat. Photonics. 8, 965–970 (2014).
[Crossref]

Nature (10)

D. W. C. Brooks, T. Botter, S. Schreppler, T. P. Purdy, N. Brahms, and D. M. Stamper-Kurn, “Non-classical light generated by quantum noise driven cavity optomechanics,” Nature,  448, 476–480 (2012).
[Crossref]

A. H. Safavi-Naeini, S. Gröblacher, J. T. Hill, J. Chan, M. Aspelmeyer, and O Painter, “Squeezed light from a silicon micromechanical resonator,” Nature 500, 185–189 (2013).
[Crossref] [PubMed]

T. Peyronel, O. Firstenberg, Q. Liang, S. Hofferberth, A. V. Gorshkov, T. Pohl, M. D. Lukin, and V. Vuletic, “Quantum nonlinear optics with single photons enabled by strongly interacting atoms,” Nature 488, 57–60 (2012).
[Crossref] [PubMed]

O. Firstenberg, T. Peyronel, Q. Liang, A. V. Gorshkov, M. D. Lukin, and V. Vuletic, “Attractive photons in a quantum nonlinear medium,” Nature 502, 71–75 (2013).
[Crossref] [PubMed]

J. D. Teufel, D. Li, M. S. Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, and R. W. Simmonds, “Circuit cavity electromechanics in the strong-coupling regime,” Nature 471, 204–208 (2011).
[Crossref] [PubMed]

A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O Painter, “Electromagnetically induced transparency and slow light with optomechanics,” Nature 472, 69–73 (2011).
[Crossref] [PubMed]

L.V. Hau, S.E. Harris, Z. Dutton, and C.H. Behroozi, “Light speed reduction to 17 metres per second in an ultracold atomic gas,” Nature 397, 594–598 (1999).
[Crossref]

M. D. Lukin and A. Imamoğlu, “Controlling photons using electromagnetically induced transparency,” Nature 413, 273–276 (2001).
[Crossref] [PubMed]

D. Bouwmeester, J.-W. Pan, K. Mattle, M. Eibl, H. Weinfurter, and A. Zeilinger, “Experimental quantum teleportation,” Nature 390, 575–578 (1997).
[Crossref]

E. Knill, R. Laflamme, and G.J. Milburn, “A scheme for efficient quantum computation with linear optics,” Nature 409, 46–52 (2001)
[Crossref] [PubMed]

Opt. Express (1)

Optica (1)

Phys Rev A (1)

S. Rebic, C. Ottaviani, G. Di Giuseppe, D. Vitali, and P. Tombesi, “Assessment of a quantum phase gate operation based on nonlinear optics,” Phys Rev A 74, 032301 (2006).
[Crossref]

Phys. Ref. Lett. (1)

M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, “Enhanced quantum nonlinearities in a two-mode optomechanical system,” Phys. Ref. Lett. 109, 063601 (2012).
[Crossref]

Phys. Rev. (1)

T. P. Purdy, P-L. Yu, R. W. Peterson, N. S. Kampel, and C. A. Regal, “Strong optomechanical squeezing of light,” Phys. Rev. X3, 031012 (2013).

Phys. Rev. A (6)

G. S. Agarwal and S. Huang, “Electromagnetically induced transparency in mechanical effects of light,” Phys. Rev. A 81, 041803 (2010).
[Crossref]

S. Mancini and P. Tombesi, “Quantum noise reduction by radiation pressure,” Phys. Rev. A 49, 4055–4065 (1994).
[Crossref] [PubMed]

C. Fabre, M. Pinard, S. Bourzeix, A. Heidmann, E. Giacobino, and S. Reynaud, “Quantum-noise reduction using a cavity with a movable mirror,” Phys. Rev. A 49, 1337–1343 (1994).
[Crossref] [PubMed]

M. Karuza, C. Biancofiore, M. Bawaj, C. Molinelli, M. Galassi, R. Natali, P. Tombesi, G. Di Giuseppe, and D. Vitali, “Optomechanically induced transparency in a membrane-in-the-middle setup at room temperature,” Phys. Rev. A 88, 013804 (2013).
[Crossref]

C. Ottaviani, S. Rebic, D. Vitali, and P. Tombesi, “Quantum phase-gate operation based on nonlinear optics: Full quantum analysis,” Phys. Rev. A 73, 010301(R) (2006).
[Crossref]

S. M. Meenehan, J. D. Cohen, S. Gröblacher, J. T. Hill, A. H. Safavi-Naeini, M. Aspelmeyer, and O. Painter, “Silicon optomechanical crystal resonator at millikelvin temperatures,” Phys. Rev. A 90, 011803(R) (2014).
[Crossref]

Phys. Rev. Lett. (10)

J. F. Poyatos, J. I. Cirac, and P. Zoller, “Complete characterization of a quantum process: The two-bit quantum gate,” Phys. Rev. Lett. 78, 390–393 (1997).
[Crossref]

M. D. Lukin and A. Imamoğlu, “Nonlinear optics and quantum entanglement of ultraslow single photons,” Phys. Rev. Lett. 84, 1419–1422 (2000).
[Crossref] [PubMed]

A. V. Gorshkov, J. Otterbach, M. Fleischhauer, T. Pohl, and M. D. Lukin, “Photon-photon interactions via Rydberg blockade,” Phys. Rev. Lett. 107, 133602 (2011).
[Crossref] [PubMed]

D. Petrosyan, J. Otterbach, and M. Fleischhauer, “Electromagnetically induced transparency with Rydberg atoms,” Phys. Rev. Lett. 107, 213601 (2011).
[Crossref] [PubMed]

D. Vitali, M. Fortunato, and P. Tombesi, “Complete quantum teleportation with a Kerr nonlinearity,” Phys. Rev. Lett. 85, 445–448 (2000).
[Crossref] [PubMed]

S. Lloyd, “Almost any quantum logic gate is universal,” Phys. Rev. Lett. 75, 346–349 (1995).
[Crossref] [PubMed]

Q.A. Turchette, C.J. Hood, W. Lange, H. Mabuchi, and H.J. Kimble, “Measurement of conditional phase shifts for quantum logic,” Phys. Rev. Lett. 75, 4710–4713 (1995)
[Crossref] [PubMed]

B. He, A. V. Sharypov, J. Sheng, C. Simon, and M. Xiao, “Two-photon dynamics in coherent Rydberg atomic ensemble,” Phys. Rev. Lett. 112, 133606 (2014).
[Crossref] [PubMed]

V. Parigi, E. Bimbard, J. Stanojevic, A. J. Hilliard, F. Nogrette, R. Tualle-Brouri, A. Ourjoumtsev, and P. Grangier, “Observation and measurement of interaction-induced dispersive optical nonlinearities in an ensemble of cold Rydberg atoms,” Phys. Rev. Lett. 109, 233602 (2012).
[Crossref]

K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, “Optomechanical quantum information processing with photons and phonons,” Phys. Rev. Lett. 109, 013603 (2012).
[Crossref] [PubMed]

Phys. Today (1)

S.E. Harris, “Electromagnetically induced transparency,” Phys. Today 50(7), 36–42 (1997).
[Crossref]

Prog. Opt. (1)

E. Arimondo, “Coherent population trapping in laser spectroscopy,” Prog. Opt. XXXV, 257–354 (1996).
[Crossref]

Science (1)

M. H. Devoret and R. J. Schoelkopf, “Superconducting circuits for quantum information: An outlook,” Science 339, 1169–1174 (2013).
[Crossref] [PubMed]

Science, (1)

S. Weis, R. Rivière, S. Deléglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, “Optomechanically Induced Transparency,” Science, 330, 1520–1523 (2010).
[Crossref]

Other (4)

K. Hammerer, C. Genes, D. Vitali, P. Tombesi, G.J. Milburn, C. Simon, and D. Bouwmeester, “Nonclassical states of light and mechanics,” in “Cavity Optomechanics: Nano- and Micromechanical Resonators Interacting with Light,” (SpringerBerlin Heidelberg), edited by M. Aspelmeyer, T.J. Kippenberg, and F. Marquardt, eds. pag. 25–56, (2014).
[Crossref]

M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Springer, 2000).

D. F. Walls and G. J. Milburn, Quantum Optics (Springer, Berlin1994).
[Crossref]

D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi, and T. J. Kippenberg, “Measurement and control of a mechanical oscillator at its thermal decoherence rate,” arXiv:1410.6191 [quant-ph].

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Figures (1)

Fig. 1
Fig. 1 Numerical solution of the master equation Eq. (20) for the gate fidelity (t) versus the dimensionless interaction time ωmt. We compare four different cases: i) zero damping and losses γm = κ1 = κ2 = 0 and = 10 (full black line); ii) zero damping and losses and = 0 (red dashed-dotted line); iii) with damping and losses (κ1 = κ2 = 10−2ωm, Qm = 106,) and = 10 (green dashed line); iv) with damping and losses (κ1 = κ2 = 10−2 ωm, Qm = 106,) and = 0 (blue dotted line). The numerical solutions for the zero damping and loss case are indistinguishable from the analytical expression of Eq. (18) either at = 0 and at = 10. In all cases we have fixed the couplings according to the ideal strong coupling condition of Eq. (19), g1 = g2 = ωm/2.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

H ^ = h ¯ ω 1 a ^ 1 a ^ 1 + h ¯ ω 2 a ^ 2 a ^ 2 + h ¯ ω m b ^ b ^ + h ¯ ( g 1 a ^ 1 a ^ 1 + g 2 a ^ 2 a ^ 2 ) ( b ^ + b ^ ) ,
| ψ in = α 00 | 0 1 | 0 2 + α 01 | 0 1 | 1 2 + α 10 | 1 1 | 0 2 + α 11 | 1 1 | 1 2 ,
H ^ = h ¯ Δ 1 n ^ 1 + h ¯ Δ 2 n ^ 2 + h ¯ ω m b ^ b ^ + h ¯ ω m f ^ n ^ ( b ^ + b ^ ) ,
D ^ ( f ^ n ^ ) = exp [ ( b ^ b ^ ) f ^ n ^ ] ,
D ^ ( f ^ n ^ ) H ^ D ^ ( f ^ n ^ ) = H ^ opt + H ^ b ,
H ^ opt = h ¯ Δ 1 n ^ 1 + h ¯ Δ 2 n ^ 2 h ¯ ω m f ^ n ^ 2 = h ¯ Δ 1 n ^ 1 + h ¯ Δ 2 n ^ 2 h ¯ ( g 1 n ^ 1 + g 2 n ^ 2 ) 2 ω m ,
H ^ b = h ¯ ω m b ^ b ^ ,
U ^ ( t ) = U ^ opt ( t ) D ^ ( f ^ n ^ ) U ^ b ( t ) D ^ ( f ^ n ^ ) ,
| 0 1 | 0 2 | 0 1 | 0 2 ; | 0 1 | 1 2 | 0 1 | 1 2 ; | 1 1 | 0 2 | 1 1 | 0 2 ; | 1 1 | 1 2 e i 2 g 1 g 2 t ω m | 1 1 | 1 2 .
t π = π ω m 2 g 1 g 2 .
ρ ^ ( t ) = U ^ ( t ) | ψ in ψ | ρ ^ b th U ^ ( t ) = k , l = 0 3 α k α l * U ^ opt ( t ) | k l | U ^ opt ( t ) D ^ ( f k ) U ^ b ( t ) D ^ ( f k ) ρ ^ b th D ^ ( f l ) U ^ b ( t ) D ^ ( f l ) ,
D ^ ( f k ) = exp [ f k ( b ^ b ^ ) ]
ρ ^ opt ( t ) = k , l = 0 3 c k , l ( t ) α k α l * exp [ i 2 g 1 g 2 t ω m ( δ k , 3 δ l , 3 ) ] | k l | ,
c k , l ( t ) = exp [ ( f k f l ) 2 ( 1 cos ω m t ) ( 2 n ¯ + 1 ) + i ( f l 2 f k 2 ) sin ω m t ] .
| ψ t g t = α 00 | 0 1 | 0 2 + α 01 | 0 1 | 1 2 + α 10 | 1 1 | 0 2 + e i π α 11 | 1 1 | 1 2 = k = 0 3 α k e i π δ k , 3 | k .
F ( t ) ψ t g t | ρ ^ opt ( t ) | ψ t g t = k , l = 0 3 c k , l ( t ) | α k | 2 | α l | 2 exp [ i ( 2 g 1 g 2 t ω m π ) ( δ k , 3 δ l , 3 ) ] .
( t ) = ψ t g t | ρ ^ opt ( t ) | ψ t g t ¯ = 1 2 + 1 12 { exp [ g 1 2 ω m 2 ( 1 cos ω m t ) ( 2 n ¯ + 1 ) ] [ cos ( g 1 2 ω m 2 sin ω m t ) + cos ( g 1 2 + 2 g 1 g 2 ω m 2 sin ω m t + π 2 g 1 g 2 t ω m ) ] + exp [ g 2 2 ω m 2 ( 1 cos ω m t ) ( 2 n ¯ + 1 ) ] [ cos ( g 2 2 ω m 2 sin ω m t ) + cos ( g 2 2 + 2 g 1 g 2 ω m 2 sin ω m t + π 2 g 1 g 2 t ω m ) ] + exp [ ( g 1 g 2 ω m ) 2 ( 1 cos ω m t ) ( 2 n ¯ + 1 ) ] cos ( g 2 2 g 1 2 ω m 2 sin ω m t ) + exp [ ( g 1 + g 2 ω m ) 2 ( 1 cos ω m t ) ( 2 n ¯ + 1 ) ] cos [ ( g 1 + g 2 ω m ) 2 sin ω m t + π 2 g 1 g 2 t ω m ] } .
2 g 1 g 2 t π ω m = π ω m t π = 2 π g 1 g 2 = ω m 2 4 .
d d t ρ ^ ( t ) = 1 i h ¯ [ H ^ , ρ ^ ( t ) ] + κ 1 2 ( 2 a ^ 1 ρ ^ ( t ) a ^ 1 a ^ 1 a ^ 1 ρ ^ ( t ) ρ ^ ( t ) a ^ 1 a ^ 1 ) + κ 2 2 ( 2 a ^ 2 ρ ^ ( t ) a ^ 2 a ^ 2 a ^ 2 ρ ^ ( t ) ρ ^ ( t ) a ^ 2 a ^ 2 ) + γ m 2 ( n ¯ + 1 ) ( 2 b ^ ρ ^ ( t ) b ^ b ^ b ^ ρ ^ ( t ) ρ ^ ( t ) b ^ b ^ ) + γ m 2 n ¯ ( 2 b ^ ρ ^ ( t ) b ^ b ^ b ^ ρ ^ ( t ) ρ ^ ( t ) b ^ b ^ ) ,
c k , l ( t ) = Tr b [ D ^ ( f l ) U ^ b ( t ) D ^ ( f l ) D ^ ( f k ) U ^ b ( t ) D ^ ( f k ) ρ ^ b th ] ,
U ^ b ( t ) D ^ ( f l ) D ^ ( f k ) U ^ b ( t ) = exp [ ( b ^ e i ω m t b ^ e i ω m t ) ( f l f k ) ] .
c k , l ( t ) = Tr b { D ^ ( f l f k ) ( e i ω m t 1 ) ρ ^ b th } exp [ i ( f l 2 f k 2 ) sin ω m t ] .
exp [ α b ^ α * b ^ ] th = exp [ | α | 2 ( n ¯ + 1 2 ) ] ,

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