CLASSIFICATION OF OPTOMECHANICAL INTERACTION AND THE DISCOVERY OF COHERENT COUPLING
2.2 Classification of optomechanical couplings
In order to classify an optomechanical system without any ambiguity, we formu- late the following steps:
1. Write the total Hamiltonian of the optomechanical system including all the optical and mechanical degrees of freedom and the coupling among them in any convenient basis.
2. Transform the Hamiltonian to the canonical form shown by Eq. (2.6), where Ω(x) andΓ(x) are diagonal and ˆa(x) is the set of cavity eigenmodes. Envi- ronmental modes ˆbcan be chosen correspondingly.
3. Classify the type of optomechanical interaction by thex-dependence feature in ˆa(x),Ω(x), andΓ(x).
We can follow these steps to illustrate the classification of the optomechanical couplings into three types mentioned in the Introduction. We consider a specific example with two cavity modes ˆ𝑎1,𝑎ˆ2 and one mechanical degree of freedom 𝑥, and expand the𝑥-dependence up to a linear order in𝑥, where𝑔1,2and𝑔𝛾1,𝛾2are the expansion coefficients of diagonal terms inΩ(𝑥)andΓ(𝑥) matrices:
1. Dispersive coupling, where the eigenfrequencies depend on the mechanical oscillation: Ω(x), the example reads:
Ω(𝑥) =
𝜔1−𝑔1𝑥 0 0 𝜔2−𝑔2𝑥
!
. (2.8)
2. Dissipative coupling, where the rates of coupling to the external modes depend on the mechanical oscillation: Γ(x), the example reads:
Γ(𝑥) =
p2𝛾1+𝑔𝛾1𝑥 0
0 p
2𝛾2+𝑔𝛾2𝑥
!
. (2.9)
3. Coherent coupling, where the eigenmodes depend on the mechanical os- cillation: ˆa(x), and the derivative of any optical mode ˆ𝑎𝑖 with respect to any mechanical displacement 𝑥𝑗 includes only the other optical modes (see Eq. (2.7)):
𝜕𝑎ˆ𝑖(x)
𝜕 𝑥𝑗 x=0
=Õ
𝑙≠𝑖
𝜕 𝑓𝑖𝑙(x)
𝜕 𝑥𝑗 x=0
ˆ
𝑎𝑙(0). (2.10)
Coherent<latexit sha1_base64="/B5lTSchtZwowtf7beuCcxtPubw=">AAAB83icbVDLSgMxFM3UV62vqks3wSK4KjMi6LLYjcsK9gGdoWTSO21oHkOSEcrQ33DjQhG3/ow7/8a0nYW2HggczrmHe3PilDNjff/bK21sbm3vlHcre/sHh0fV45OOUZmm0KaKK92LiQHOJLQtsxx6qQYiYg7deNKc+90n0IYp+WinKUSCjCRLGCXWSWGoBW6qMWiQdlCt+XV/AbxOgoLUUIHWoPoVDhXNhMtSTozpB35qo5xoyyiHWSXMDKSETsgI+o5KIsBE+eLmGb5wyhAnSrsnLV6ovxM5EcZMRewmBbFjs+rNxf+8fmaT2yhnMs0sSLpclGQcW4XnBeAh00AtnzpCqGbuVkzHRBNqXU0VV0Kw+uV10rmqB44/XNcad0UdZXSGztElCtANaqB71EJtRFGKntErevMy78V79z6WoyWvyJyiP/A+fwDNr5GF</latexit><latexit sha1_base64="/B5lTSchtZwowtf7beuCcxtPubw=">AAAB83icbVDLSgMxFM3UV62vqks3wSK4KjMi6LLYjcsK9gGdoWTSO21oHkOSEcrQ33DjQhG3/ow7/8a0nYW2HggczrmHe3PilDNjff/bK21sbm3vlHcre/sHh0fV45OOUZmm0KaKK92LiQHOJLQtsxx6qQYiYg7deNKc+90n0IYp+WinKUSCjCRLGCXWSWGoBW6qMWiQdlCt+XV/AbxOgoLUUIHWoPoVDhXNhMtSTozpB35qo5xoyyiHWSXMDKSETsgI+o5KIsBE+eLmGb5wyhAnSrsnLV6ovxM5EcZMRewmBbFjs+rNxf+8fmaT2yhnMs0sSLpclGQcW4XnBeAh00AtnzpCqGbuVkzHRBNqXU0VV0Kw+uV10rmqB44/XNcad0UdZXSGztElCtANaqB71EJtRFGKntErevMy78V79z6WoyWvyJyiP/A+fwDNr5GF</latexit><latexit sha1_base64="/B5lTSchtZwowtf7beuCcxtPubw=">AAAB83icbVDLSgMxFM3UV62vqks3wSK4KjMi6LLYjcsK9gGdoWTSO21oHkOSEcrQ33DjQhG3/ow7/8a0nYW2HggczrmHe3PilDNjff/bK21sbm3vlHcre/sHh0fV45OOUZmm0KaKK92LiQHOJLQtsxx6qQYiYg7deNKc+90n0IYp+WinKUSCjCRLGCXWSWGoBW6qMWiQdlCt+XV/AbxOgoLUUIHWoPoVDhXNhMtSTozpB35qo5xoyyiHWSXMDKSETsgI+o5KIsBE+eLmGb5wyhAnSrsnLV6ovxM5EcZMRewmBbFjs+rNxf+8fmaT2yhnMs0sSLpclGQcW4XnBeAh00AtnzpCqGbuVkzHRBNqXU0VV0Kw+uV10rmqB44/XNcad0UdZXSGztElCtANaqB71EJtRFGKntErevMy78V79z6WoyWvyJyiP/A+fwDNr5GF</latexit><latexit sha1_base64="/B5lTSchtZwowtf7beuCcxtPubw=">AAAB83icbVDLSgMxFM3UV62vqks3wSK4KjMi6LLYjcsK9gGdoWTSO21oHkOSEcrQ33DjQhG3/ow7/8a0nYW2HggczrmHe3PilDNjff/bK21sbm3vlHcre/sHh0fV45OOUZmm0KaKK92LiQHOJLQtsxx6qQYiYg7deNKc+90n0IYp+WinKUSCjCRLGCXWSWGoBW6qMWiQdlCt+XV/AbxOgoLUUIHWoPoVDhXNhMtSTozpB35qo5xoyyiHWSXMDKSETsgI+o5KIsBE+eLmGb5wyhAnSrsnLV6ovxM5EcZMRewmBbFjs+rNxf+8fmaT2yhnMs0sSLpclGQcW4XnBeAh00AtnzpCqGbuVkzHRBNqXU0VV0Kw+uV10rmqB44/XNcad0UdZXSGztElCtANaqB71EJtRFGKntErevMy78V79z6WoyWvyJyiP/A+fwDNr5GF</latexit>
!1
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Dispersive
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!2 ↵2x
!2
!2
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!3+↵3x
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@
1 C C C C A
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Dispersive
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Figure 2.1: The comparison between coherent coupling and dispersive coupling:
theΩ(x) matrix under the basis of ˆa(0). Each color block represents a frequency degenerate subspace, which can have one or more modes. Different blocks have different frequencies, e.g.,𝜔1 ≠ 𝜔2 ≠ 𝜔3. The influence of mechanical oscillation shows up asx-dependent perturbation, which will either directly add on to diagonal terms as dispersive coupling, e.g., the𝛼3𝑥term, or will show up in off-diagonal terms to couple different modes. The off-diagonal coupling within one color block, e.g., the𝛼2𝑥 term, will open the degeneracy and also cause dispersive coupling. While the coupling between blocks, e.g., the𝛼12𝑥term, won’t change the eigenfrequencies 𝜔1,2and will cause coherent coupling. See main text for detailed discussion.
The simple example can be taken from Eq. (2.5) where the original modes ˆ
𝑎1,2become mixed by𝑥(see Eq. (2.4)) and the corresponding eigenfrequency matrix:
Ω00(𝑥)=
𝜔1 0 0 𝜔2
!
, (2.11)
doesn’t depend on𝑥: 𝑑Ω00(𝑥)/𝑑𝑥 =0.
Following our classification strategy, one can clearly distinguish among the different types of interactions, even in cases where several couplings coexist.
In addition to the definition above, a physical picture of coherent coupling also helps to understand this new concept. The difference between dispersive coupling and coherent coupling can sometimes be not obvious: both of them can be expressed as coupling of optical modes by mechanical motion under some specific cavity basis, as showed the example in Eq. (2.1). The distinction between them is illustrated in Fig. 2.1: when mechanical motion couples different optical modes, these modes are either frequency-degenerate or have different frequencies. When the unperturbed modes are frequency-degenerate, the coupling via the mechanical motion breaks the degeneracy and leads to newx-dependent eigenfrequencies, which are the sign