Abstract

We have found an incorrect formula to estimate the root mean square (rms) amplitude of the molecular vibration in the molecular optomechanics systems reported by J. Liu et al. [Photon. Res. 5, 450–456 (2017) [CrossRef]  ]. In contrast with common optomechanical systems, the equilibrium position for molecular optomechanics systems used in the letter cannot be achieved by the equipartition theorem. Here, we achieve the effective temperature of molecules and then provide a corrected formula for the estimation of the rms amplitude of molecular motion. Using defined effective temperature into our new formula, we show that the minimal measurable force is (F1.7×1014N), which it is 1 order bigger than the result of the main paper, and it is also in accordance with numerical calculation through the Lindblad master equation.

© 2020 Chinese Laser Press

The model introduced in Ref. [1] by Jian Liu et al. consists of two adjacent molecules placed in the hot spot of a plasmonic cavity, which directly interact through dipole–dipole interaction and are also driven by two color laser beams (i.e., a strong pump and a weak probe laser), as shown in Fig. 1. The optomechanical theory has been applied to investigate the coupling behavior of molecules so that the full Hamiltonian in a rotating frame is then given by

H^=δpuc^c^+i=1,2ωia^ia^ii=1,2gic^c^(a^i+a^i)λ(a^1a^2+a^1a^2)+iΩpu(c^c^)+iΩpr(c^eiδtc^eiδt),
where c^,a^i are annihilation operators for optical and the vibrational mode of molecule (i), λ is the dipole–dipole coupling constant, δpu=ωcωpu is pump cavity detuning, gi is optomechanical coupling constant, and Ωpu,Ωpr are the laser pumping and probing rate, respectively. Using the Lindblad master equation, the dynamical behavior of open whole systems can investigate as follows [2]:
dρ^dt=1i[H^,ρ^]+Lc^+La^1+La^2,
where
La^i=γi2(n¯i+1)Da^i{ρ^}+γin¯i2Da^i{ρ^},i=1,2,Lc^=κ2Dc^{ρ^},
where DA^{ρ^} is defined as the Lindblad super operator,
DA^{ρ^}=2A^ρ^A^A^A^ρ^ρ^A^A^,A=c,a1,a2,
and κ, γ1,γ2 are the damping of the plasmon and the first and second molecule, respectively.

 

Fig. 1. Schematic of the model and relevant parameters.

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Although two molecules are coupled to a common heat bath at temperature Tbath (authors had considered room temperature, i.e., Tbath=300K), optomechanical interaction between molecules and dipole–dipole can modify the temperature of molecules. Hence, we define the effective mean phonon number as

nieff=a^ia^i=tr(a^ia^iρss),
where ρss is the steady-state density operator. As a direct result, the effective temperature of each molecule yields [3]
Tieff=ωiKBlog(1+1nieff),i=1,2.
In the symmetric case, when ω1=ω2 and g1=g2, the effective temperature of both molecules is the same, i.e., T1eff=T2eff (note that two molecules are coupled to a common heat bath). However, for the case λ=0.2THz, we have plotted the effective temperature of molecules as a function of optomechanical coupling constant with two different heat baths (Fig. 2; all parameters are considered similar to the main paper). The effective temperature of molecule GBT becomes T1eff=T2eff320K, when the molecules are coupled to a room temperature heat bath. One can also analytically derive the effective temperature of molecules through adiabatic elimination of plasmon mode, in which an effective Hamiltonian for the two molecules is derived and also two mutually coupled molecules become coupled to an effective common thermal bath [3,4].

 

Fig. 2. Effective temperature of molecules as a function of optomechanical coupling constants (notice that all parameters are considered similar to the main paper, i.e., κ2π=33THz, ωm2π=32.2THz, g/2π=70GHz, λ=0.2THz, ΩPu2=0.22eV2, δ=0, and we also ignore the probe laser Ωpr=0 because it is weak and far detuned from plasmonic cavity center frequency and hence does not play a role in heat transferring).

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Now, we provide a corrected formula for estimation of the root mean square (rms) amplitude of molecular motion and show that the equipartition theorem does not work in molecular optomechanics systems in contrast with the common optomechanical system. The rms amplitude of molecular motion is defined as follows:

xirms=x^i2=2meffωm(a^i+a^i)2,i=1,2,
where meff is considered as the effective mass of the benzene ring and the atom of sulfur or gold for molecules of thiophenol or GBT (meff3.6×1026kg) [1]. Using Eq. (5) and the commutation relation, one can easily show that
xirms=2meffωm(2nieff+1),i=1,2.
Then, the rms amplitude can also write as a function of defined effective temperature using Eq. (6),
xirms=2meffωm[21+exp(ωmKBTieff)+1],i=1,2.
After some calculation, the above equation is recast to
xirms=2meffωmcoth(ωm2KBTieff),i=1,2.
In the classical limit KBTωm, the above equation becomes the classical rms amplitude derived by equipartition theorem, in which the average energy is equally shared among all degrees of freedom in the same portion, so we have
xirms=KBTieffmeffωm2,i=1,2
(because of cothx1x for x1). But, here system is in the other extreme, i.e.,
KBTωm.
That means the energy of the resonator (ωm/2π=32.2THz, or ωm133meV) is much larger than the room temperature energy (KBTieff28meV for GBT), and the high temperature is not valid anymore. Molecular vibrations are indeed “frozen” at room temperature, so that their amplitude of motion is dominated by quantum zero point fluctuations instead of classical thermal fluctuations, which can change the main result of the letter about the minimal measurement force between molecules through a plasmonic cavity. For this point, we plot the minimal measurement force as a function of molecular frequency for three approaches, equipartition theorem (dashed line), numerically by Lindblad master equation in QUTIP package [5] (solid line), and our corrected formula (star dots) in Fig. 3, in which, for the numerical approach, we achieve the rms amplitude as follows:
xirms=2meffωmtr[(a^i+a^i)2ρss],i=1,2.
As shown in Fig. 3, the results of our corrected formula are matched with numerical Lindblad approach, which shows that the corrected formula is trustable in quantum regime. In order to clarify the equipartition theorem formula does not work as well, we plot the minimal measurement force for two different effective temperatures (cryogenic and almost room temperature for GBT), which shows a big deviation especially at high molecular frequency (ωm/2π25THz such as GBT) and cryogenic effective temperature. In this situation, coth(ωm2KBTieff)1, and rms amplitude becomes similar to zero-point motion, i.e.,
xirms=2meffωm,
which shows it is dominated by quantum zero-point fluctuations instead of a thermal environment.

 

Fig. 3. Comparison between different methods (equipartition theorem, numerical, and corrected formula) for minimal measurable force as a function of molecular frequency, with two different effective temperatures.

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In particular, we achieve the rms of molecules GBT in the main paper at effective temperature (Tieff320 K) using our correct formula [Eq. (10)],

xirms0.0027nm.
And also the distance between molecules becomes
R0.88nm.
Therefore, the minimal measurable force is given as the following relation:
FdVintdR=6e24πε0R4x1x21.7×1014N.
One can also investigate the rms amplitude of quantum molecular optomechanical systems through quantum counterparts of the energy partition theorem described by the corresponding frequency probability distributions [6,7].

Acknowledgment

I would like to express my special appreciation and thanks to my advisor Dr. Johannes Feist for assistance with developing our revised paper.

Disclosures

The authors declare no conflicts of interest.

REFERENCES

1. J. Liu and K. D. Zhu, “Coupled quantum molecular cavity optomechanics with surface plasmon enhancement,” Photon. Res. 5, 450–456 (2017). [CrossRef]  

2. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, 2002).

3. S. M. Ashrafi, R. Malekfar, A. R. Bahrampour, and J. Feist, “Optomechanical heat transfer between molecules in a nanoplasmonic cavity,” Phys. Rev. A 100, 013826 (2019). [CrossRef]  

4. A. Xuereb, A. Imparato, and A. Dantan, “Heat transport in harmonic oscillator systems with thermal baths: application to optomechanical arrays,” New J. Phys. 17, 055013 (2015). [CrossRef]  

5. J. R. Johansson, P. D. Nation, and F. Nori, “QuTiP: ‘an open-source python framework for the dynamics of open quantum systems,’” Comput. Phys. Commun. 183, 1760–1772 (2012). [CrossRef]  

6. P. Bialas, J. Spiechowicz, and J. Luczka, “Partition of energy for a dissipative quantum oscillator,” Sci. Rep. 8, 16080 (2018). [CrossRef]  

7. P. Bialas, J. Spiechowicz, and J. Luczka, “Quantum analogue of energy equipartition theorem,” J. Phys. A 52, 15LT01 (2019). [CrossRef]  

References

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  1. J. Liu and K. D. Zhu, “Coupled quantum molecular cavity optomechanics with surface plasmon enhancement,” Photon. Res. 5, 450–456 (2017).
    [Crossref]
  2. H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, 2002).
  3. S. M. Ashrafi, R. Malekfar, A. R. Bahrampour, and J. Feist, “Optomechanical heat transfer between molecules in a nanoplasmonic cavity,” Phys. Rev. A 100, 013826 (2019).
    [Crossref]
  4. A. Xuereb, A. Imparato, and A. Dantan, “Heat transport in harmonic oscillator systems with thermal baths: application to optomechanical arrays,” New J. Phys. 17, 055013 (2015).
    [Crossref]
  5. J. R. Johansson, P. D. Nation, and F. Nori, “QuTiP: ‘an open-source python framework for the dynamics of open quantum systems,’” Comput. Phys. Commun. 183, 1760–1772 (2012).
    [Crossref]
  6. P. Bialas, J. Spiechowicz, and J. Luczka, “Partition of energy for a dissipative quantum oscillator,” Sci. Rep. 8, 16080 (2018).
    [Crossref]
  7. P. Bialas, J. Spiechowicz, and J. Luczka, “Quantum analogue of energy equipartition theorem,” J. Phys. A 52, 15LT01 (2019).
    [Crossref]

2019 (2)

S. M. Ashrafi, R. Malekfar, A. R. Bahrampour, and J. Feist, “Optomechanical heat transfer between molecules in a nanoplasmonic cavity,” Phys. Rev. A 100, 013826 (2019).
[Crossref]

P. Bialas, J. Spiechowicz, and J. Luczka, “Quantum analogue of energy equipartition theorem,” J. Phys. A 52, 15LT01 (2019).
[Crossref]

2018 (1)

P. Bialas, J. Spiechowicz, and J. Luczka, “Partition of energy for a dissipative quantum oscillator,” Sci. Rep. 8, 16080 (2018).
[Crossref]

2017 (1)

2015 (1)

A. Xuereb, A. Imparato, and A. Dantan, “Heat transport in harmonic oscillator systems with thermal baths: application to optomechanical arrays,” New J. Phys. 17, 055013 (2015).
[Crossref]

2012 (1)

J. R. Johansson, P. D. Nation, and F. Nori, “QuTiP: ‘an open-source python framework for the dynamics of open quantum systems,’” Comput. Phys. Commun. 183, 1760–1772 (2012).
[Crossref]

Ashrafi, S. M.

S. M. Ashrafi, R. Malekfar, A. R. Bahrampour, and J. Feist, “Optomechanical heat transfer between molecules in a nanoplasmonic cavity,” Phys. Rev. A 100, 013826 (2019).
[Crossref]

Bahrampour, A. R.

S. M. Ashrafi, R. Malekfar, A. R. Bahrampour, and J. Feist, “Optomechanical heat transfer between molecules in a nanoplasmonic cavity,” Phys. Rev. A 100, 013826 (2019).
[Crossref]

Bialas, P.

P. Bialas, J. Spiechowicz, and J. Luczka, “Quantum analogue of energy equipartition theorem,” J. Phys. A 52, 15LT01 (2019).
[Crossref]

P. Bialas, J. Spiechowicz, and J. Luczka, “Partition of energy for a dissipative quantum oscillator,” Sci. Rep. 8, 16080 (2018).
[Crossref]

Breuer, H.-P.

H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, 2002).

Dantan, A.

A. Xuereb, A. Imparato, and A. Dantan, “Heat transport in harmonic oscillator systems with thermal baths: application to optomechanical arrays,” New J. Phys. 17, 055013 (2015).
[Crossref]

Feist, J.

S. M. Ashrafi, R. Malekfar, A. R. Bahrampour, and J. Feist, “Optomechanical heat transfer between molecules in a nanoplasmonic cavity,” Phys. Rev. A 100, 013826 (2019).
[Crossref]

Imparato, A.

A. Xuereb, A. Imparato, and A. Dantan, “Heat transport in harmonic oscillator systems with thermal baths: application to optomechanical arrays,” New J. Phys. 17, 055013 (2015).
[Crossref]

Johansson, J. R.

J. R. Johansson, P. D. Nation, and F. Nori, “QuTiP: ‘an open-source python framework for the dynamics of open quantum systems,’” Comput. Phys. Commun. 183, 1760–1772 (2012).
[Crossref]

Liu, J.

Luczka, J.

P. Bialas, J. Spiechowicz, and J. Luczka, “Quantum analogue of energy equipartition theorem,” J. Phys. A 52, 15LT01 (2019).
[Crossref]

P. Bialas, J. Spiechowicz, and J. Luczka, “Partition of energy for a dissipative quantum oscillator,” Sci. Rep. 8, 16080 (2018).
[Crossref]

Malekfar, R.

S. M. Ashrafi, R. Malekfar, A. R. Bahrampour, and J. Feist, “Optomechanical heat transfer between molecules in a nanoplasmonic cavity,” Phys. Rev. A 100, 013826 (2019).
[Crossref]

Nation, P. D.

J. R. Johansson, P. D. Nation, and F. Nori, “QuTiP: ‘an open-source python framework for the dynamics of open quantum systems,’” Comput. Phys. Commun. 183, 1760–1772 (2012).
[Crossref]

Nori, F.

J. R. Johansson, P. D. Nation, and F. Nori, “QuTiP: ‘an open-source python framework for the dynamics of open quantum systems,’” Comput. Phys. Commun. 183, 1760–1772 (2012).
[Crossref]

Petruccione, F.

H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, 2002).

Spiechowicz, J.

P. Bialas, J. Spiechowicz, and J. Luczka, “Quantum analogue of energy equipartition theorem,” J. Phys. A 52, 15LT01 (2019).
[Crossref]

P. Bialas, J. Spiechowicz, and J. Luczka, “Partition of energy for a dissipative quantum oscillator,” Sci. Rep. 8, 16080 (2018).
[Crossref]

Xuereb, A.

A. Xuereb, A. Imparato, and A. Dantan, “Heat transport in harmonic oscillator systems with thermal baths: application to optomechanical arrays,” New J. Phys. 17, 055013 (2015).
[Crossref]

Zhu, K. D.

Comput. Phys. Commun. (1)

J. R. Johansson, P. D. Nation, and F. Nori, “QuTiP: ‘an open-source python framework for the dynamics of open quantum systems,’” Comput. Phys. Commun. 183, 1760–1772 (2012).
[Crossref]

J. Phys. A (1)

P. Bialas, J. Spiechowicz, and J. Luczka, “Quantum analogue of energy equipartition theorem,” J. Phys. A 52, 15LT01 (2019).
[Crossref]

New J. Phys. (1)

A. Xuereb, A. Imparato, and A. Dantan, “Heat transport in harmonic oscillator systems with thermal baths: application to optomechanical arrays,” New J. Phys. 17, 055013 (2015).
[Crossref]

Photon. Res. (1)

Phys. Rev. A (1)

S. M. Ashrafi, R. Malekfar, A. R. Bahrampour, and J. Feist, “Optomechanical heat transfer between molecules in a nanoplasmonic cavity,” Phys. Rev. A 100, 013826 (2019).
[Crossref]

Sci. Rep. (1)

P. Bialas, J. Spiechowicz, and J. Luczka, “Partition of energy for a dissipative quantum oscillator,” Sci. Rep. 8, 16080 (2018).
[Crossref]

Other (1)

H.-P. Breuer and F. Petruccione, The Theory of Open Quantum Systems (Oxford University, 2002).

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Figures (3)

Fig. 1.
Fig. 1. Schematic of the model and relevant parameters.
Fig. 2.
Fig. 2. Effective temperature of molecules as a function of optomechanical coupling constants (notice that all parameters are considered similar to the main paper, i.e., κ2π=33THz, ωm2π=32.2THz, g/2π=70GHz, λ=0.2THz, ΩPu2=0.22eV2, δ=0, and we also ignore the probe laser Ωpr=0 because it is weak and far detuned from plasmonic cavity center frequency and hence does not play a role in heat transferring).
Fig. 3.
Fig. 3. Comparison between different methods (equipartition theorem, numerical, and corrected formula) for minimal measurable force as a function of molecular frequency, with two different effective temperatures.

Equations (17)

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H^=δpuc^c^+i=1,2ωia^ia^ii=1,2gic^c^(a^i+a^i)λ(a^1a^2+a^1a^2)+iΩpu(c^c^)+iΩpr(c^eiδtc^eiδt),
dρ^dt=1i[H^,ρ^]+Lc^+La^1+La^2,
La^i=γi2(n¯i+1)Da^i{ρ^}+γin¯i2Da^i{ρ^},i=1,2,Lc^=κ2Dc^{ρ^},
DA^{ρ^}=2A^ρ^A^A^A^ρ^ρ^A^A^,A=c,a1,a2,
nieff=a^ia^i=tr(a^ia^iρss),
Tieff=ωiKBlog(1+1nieff),i=1,2.
xirms=x^i2=2meffωm(a^i+a^i)2,i=1,2,
xirms=2meffωm(2nieff+1),i=1,2.
xirms=2meffωm[21+exp(ωmKBTieff)+1],i=1,2.
xirms=2meffωmcoth(ωm2KBTieff),i=1,2.
xirms=KBTieffmeffωm2,i=1,2
KBTωm.
xirms=2meffωmtr[(a^i+a^i)2ρss],i=1,2.
xirms=2meffωm,
xirms0.0027nm.
R0.88nm.
FdVintdR=6e24πε0R4x1x21.7×1014N.

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