Enhancement of gravity estimation in modulated optomechanics

Journal Pre-proof Enhancement of gravity estimation in modulated optomechanics Ling-Juan Feng, Gong-Wei Lin, Yue-Ping Niu, Shang-Qing Gong

PII: DOI: Reference:

S0030-4018(19)31180-0 https://doi.org/10.1016/j.optcom.2019.125217 OPTICS 125217

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Optics Communications

Received date : 9 October 2019 Revised date : 24 December 2019 Accepted date : 28 December 2019 Please cite this article as: L.-J. Feng, G.-W. Lin, Y.-P. Niu et al., Enhancement of gravity estimation in modulated optomechanics, Optics Communications (2019), doi: https://doi.org/10.1016/j.optcom.2019.125217. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier B.V.

Journal Pre-proof *Declaration of Interest Statement

Declaration of interests ☐ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

Journal Pre-proof *Manuscript Click here to view linked References

Enhancement of gravity estimation in modulated optomechanics Ling-Juan Feng, Gong-Wei Lin,∗ Yue-Ping Niu,† and Shang-Qing Gong‡ Department of Physics, East China University of Science and Technology, Shanghai 200237, China

interaction than that in the absence of the additional Coulomb interaction [35].

INTRODUCTION

II.

MODEL AND SOLUTIONS

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Gravimeters [1, 2] have many potential applications in fundamental physics, geodesy, geophysics, and quantum metrology. With the rapid progress of laser cooling and trapping techniques [3, 4], cold atom (or Bose-Einstein condensates) gravimeters [5–12] provide a new platform for performing the precision measurement of gravitational acceleration. Since then, theoretical and experimental works have been devoted to the ultrasensitive measurement [13–19]. In recent years, the optomechanical interaction between optical and mechanical modes has attracted extensive attention [20], since it can be applied to investigate the quantum precision measurements [21–35]. In particular, kim [34] and Bose [35] respectively have shown that the gravitational acceleration can be estimated with the use of optomechanical couplings. However, the gravity measurements in the optomechanical system using a strong optomechanical coupling have not yet been fully explored. In this paper, we propose a quantum estimation scheme to measure the gravitational acceleration by using the strong optomechanical coupling in the modulated optomechanical system. Different from the previous works [34, 35], the optomechanical interaction can be enhanced to the strong-coupling regime by the parametric amplification process. Specifically, like the case of modulation of mechanical spring constant [36], this process in the squeezed mechanical mode can be obtained by the additional Coulomb interaction. Then, we produce the unitary evolution operator at one mechanical period, decoupled completely from the mechanical oscillator, acting only on the cavity field. We finally apply estimation tools from the Fisher Information (FI) about homodyne detection and Quantum Fisher information (QFI) [37, 38] to analyze the optimal accuracy of gravitational acceleration with currently available parameters [35]. It is found that the sensitivity for measuring the gravitational acceleration could be improved with more than one order of magnitude in the presence of the additional Coulomb

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of

We present an efficient method for quantum estimation of gravitational acceleration in a modulated optomechanical system. We use the squeezed mechanical mode, which could potentially enhance the optomechanical interaction between cavity field and mechanical oscillation into the strong-coupling regime. We show that the strong optomechanical coupling can significantly improve the measurement sensitivity of gravitational acceleration. The proposed scheme can provide a platform for further investigations on precision measurement of gravitational acceleration.

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∗

† ‡

[email protected] [email protected] [email protected]

Fixed mirror

L

x

Charged mechanical oscillator

+Q1

r0 -Q2

Charged body

FIG. 1. (Color online) Schematic representation of the cavity optoelectromechanical system.

We consider the cavity optoelectromechanical system depicted in Fig. 1 , where the optical cavity mode vertically couples to the charged mechanical oscillator via radiation pressure force. This mechanical oscillator which can be free to move under the influence of the gravitational field, and the charged body are coupled by the Coulomb force. The Hamiltonian describing this system is given by H = H0 + Hom + Hco + ∆U, where [26–36, 39, 40] 1 p2 , H0 = ~ωc a† a + mω 2 x2 + 2 2m Hom = −~χ0 a† ax, Hco = −

ke Q1 Q2 , |r0 − x|

(1)

Journal Pre-proof 2

− ke Qr01 Q2

x r0

x2 r02

0 Htot = ~ωc a† a+~ω 0 b†s bs −~ χ0 a† a + S 0 + G01


2

/ω 0 . (4)

0 From the above Hamiltonian Htot , we can define the corresponding time evolution operator 2

a)ωc t −i(b†s bs )ω 0 t i(χ0 a† a+S 0 +G01 ) /ω 0 t

of

†

U (t) = e−i(a

e

e

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tonian Hco = [41, 42]. Then the 1+ + total Hamiltonian H can be rewritten as
H 0 = ~ωc a† a + ~ (ω − 2G2 ) b† b − ~G2 b†2 + b2
− ~(χa† a + S + G1 ) b† + b , (2)

where x (p) are related to the annihilation (creation) operator of the mechanical oscillator mode b (b† ) by

1/2 1/2 x = [~/ (2mω)] b† + b and p = i [~mω/2] b† − b , e Q1 Q2 G2 = k2mωr is the effective mechanical coupling con3 0 p stant, χp= ωc /L ~/ (2mω) is the coupling strength, p ke Q1 Q2 S = mg 1/ (2~mω) and G1 = 1/ (2~mω). In r02 Eq.(2), we have neglected the zero-point energy from the second term and the constant term from the Coulomb interaction. In order to improve precision measurement of the gravitational acceleration, we make use of parametric amplification in the mechanical oscillator mode [36]. In terms of a squeezing transformation b = cosh(r)bs + sinh(r)b†s , with preferred squeezed mechanical mode bs and squeezing parameter r = (1/4) ln[ω/(ω − 4G2 )], H 0 is transformed to

Htot = ~ωc a† a + ~ω 0 b†s bs − ~ χ0 a† a + S 0 + G01 b†s + bs , (3) where the second and third terms in Eq.(2) have been diagonalized by the squeezing transformation, and are simplified to a mechanical oscillator with the transformed mechanical frequency ω 0 = (ω − 4G2 ) exp(2r), S 0 = S exp(r) and G01 = G1 exp(r). χ0 = χ exp(r) describes the optomechanical radiation-pressure strength, which could be significantly enhanced by adjusting properly the mechanical coupling G2 . It can lead to the realization of the strong-coupling regime in the presence of the Coulomb interaction. Then, applying the polaron transformation

.

(5)

Then, multiplying on the left by U and on the right by U † and using

† 0 † 0 e−i(bs bs )ω t [ χ0 a† a + S 0 + G01 b†s − bs ]ei(bs bs )ω t
0 0 = χ0 a† a + S 0 + G01 (b†s e−iω t − bs eiω t ), (6) we get

˜

˜

2

0

0

˜ S+G1 ) (ω t−sin ω t) U (t) = ei(χn+ ˜ G ˜ 1 )(b† η−bs η ∗ )] −i(b† bs )ω 0 t ˜ S+ s s , × e[(χn+ e

re-

Here the first term H0 is the free Hamiltonian with a (a† ) and x (p) being the annihilation (creation) operator of the cavity mode with frequency ωc and the position (momentum) operator of the mechanical oscillator mode with frequency ω and mass m, respectively. The second term Hom describes the optomechanical interaction between the cavity field and the mechanical resonator by assuming no retardation effects [39, 40]. χ0 = ωc /L is the optomechanical coupling strength with L being the cavity length. The third term Hco denotes the Coulomb interaction between the charged mechanical oscillator and the charged body. ke is the electrostatic constant. Q1 is the positive charge on the charged mechanical oscillator, and -Q2 is the negative charge on the charged body. r0 represents the equilibrium separation between the charged mechanical oscillator and the charged body. The final term presents the gravitational potential energy with g being the local gravitational acceleration. Considering x  r0 , with the series expansion of the second order x/r0 , we obtain h the Coulomb i interaction Hamil-

0 Htot = U † Htot U [43–46], where the translation operator

U = exp[ χ0 a† a + S 0 + G01 /ω 0 b†s − bs ], and the BakerCampbell-Hausdorff formula [47], we obtain

pro

∆U = −mgx.

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(7)

where n = a† a is the number operator of the cavity 0 ˜1 = field, η = (1 − e−iω t ), χ ˜ = χ0 /ω 0 , S˜ = S 0 /ω 0 , G 0 0 G1 /ω , and we have omitted the free evolution of the † cavity field e−i(a a)ωc t [34, 35, 43, 44]. We assume an initial state ρ(t = 0) = |αihα| ⊗ ρTh , where |αi is the initial state of the cavity field mode and ρTh = P coherent 1 † −~βω 0 (r) is the thermal squeezed state e S(r)|kihk|S k Z of the mechanical oscillator, with the squeezing operator †2 2 r 1 S(r) = e 2 (bs −bs ) , the partition function Z = 1−e−~βω 0 and β the inverse temperature. The time evolution of the system in Eq. (7) leads to the state at time t = 2π/ω 0 given by 2

ρ(t) = e−|α|

X αn α∗m √ n!m! m,n

2

2

2

˜

˜

˜ S+G1 )(n−m)] × ei2π[χ˜ (n −m )+2χ( |nihm| X †2 2 0 r 2 †2 r 1 ⊗ e−~βω e 2 (bs −bs ) |kihk|e 2 (bs −bs ) . Z

(8)

k

From the equation, it is readily seen at one mechanical period t = 2π/ω 0 that the oscillator state returns to its initial state, and the cavity field is decoupled from the mechanical oscillator. This means that the oscillator does not impact our measurement schemes. Next, we consider in detail how to estimate the gravitational acceleration by local estimation theory [37]. Optimal estimators of the parameter λ in classical and quantum estimation theory are those saturating the CramerRao theorem [38] Var(λ) ≥

1 , M F (λ)

(9)

which establishes the ultimate lower bound on the variance Var(λ), where M is the number of measurements, and

Journal Pre-proof 3

F (λ) ≤ H (λ) ≡ T r[ρλ L2λ ] = T r[∂λ ρλ Lλ ],

(11)

|Ψ(2π/ω 0 )i = e−

|α|2 2

X αn ˜ G ˜ 1 )2 ˜ S+ √ ei2π(χn+ |ni, n! n

and the field density matrix

× ei2π[χ˜

X αn α∗m √ n!m! m,n

2

(12)

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2

ρ(2π/ω 0 ) = e−|α|

˜ G ˜ 1 )(n−m)] (n2 −m2 )+2χ( ˜ S+

2

× ei2π[χ˜

2

X αn α∗m √ n!m! m,n

˜ G ˜ 1 )(n−m)] (n2 −m2 )+2χ( ˜ S+

2

×

On the other hand, the Quantum Fisher Information (QFI) in this case of the pure state in Eq. (12) becomes   H(g) = 4 h∂g Ψ|∂g Ψi − |h∂g Ψ|Ψi|2  2 = |α|2 8π χ∂ ˜ g S˜ . (16)

Inspection of Eqs. (15) and (16) shows that when φ = 0 and π/2 the FI can coincide exactly with the QFI, and the FI and the QFI are dependent of χ ˜ and are linear in the average number of photons |α|2 . From Fig. 2, it is seen that the increase of χ ˜ can significantly enhance the estimation precision of the gravitational acceleration. The enhancement is because the mechanical parametric amplification under the additional Coulomb interaction can induce strong optomechanical coupling. As shown in Fig. 3, the optomechanical coupling could be enhanced by adjusting some parameters of the mechanical oscillator under the additional Coulomb interaction, such as the distance r0 and the charge Q1 = Q2 = Q, which can improve the measurement sensitivity.

|nihm|. (13)

We first consider the Fisher Information (FI) related to homodyne detection at estimating the gravitational acceleration, corresponding to the eigenvalue equation x ˆφ |xφ i = √xφ |xφ i [48], where x ˆφ = [a exp (−iφ) + a† exp (iφ)]/ 2 is the quadrature operator with φ being the real phase, and xφ and |xφ i are eigenvalue and eigenstate. When φ = 0 and π/2, x ˆ0 and x ˆπ/2 denote the position and momentum operators for the cavity field. In particular, hn|xφ i = π −1/4 2−n/2 (n!)−1/2 exp (−x2φ /2)Hn (xφ ) exp (inφ) [48], where Hn is the Hermite polynomials of order n. Then, we can get the conditional probability of obtaining the outcome xφ when the parameter has the gravitational acceleration p(xφ |g) = e−|α|

(15)

re-

where Lλ is the Symmetric Logarithmic Derivative. Let us assume the initial state |Ψ(0)i = |αi, with |αi being the coherent state for cavity field, since the unitary evolution operator in Eq. (7) acts only on the coherent state of the field after a mechanical period t = 2π/ω 0 . By using evolution operator we obtain the state of cavity field

= [Im (α) cos (φ) − Re (α) sin (φ)]2  2 × 8π χ∂ ˜ g S˜ .

of

where p(x|λ) is the conditional probability of obtaining the value x in the parameter λ and ∂λ = ∂/∂λ is the derivative with respect to the parameter λ. For quantum states ρλ , the Quantum Fisher Information (QFI) can be given by

Information simplifies to  2 F (g) = − 4π χ∂ ˜ g S˜ (e−iφ α − eiφ α∗ )2

pro

F (λ) is Fisher Information (FI) and Quantum Fisher Information (QFI). The Fisher Information (FI) can be calculated by Z (∂λ p(x|λ))2 F (λ) = dx , (10) p(x|λ)

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e−xφ Hn (xφ )Hm (xφ )e−iφ(n−m) √ , π 1/2 2(n+m)/2 n!m!

(14)

and the corresponding Fisher Information F (g) =  2 R [(n−m)p(xφ |g)]2 − 4π χ∂ ˜ g S˜ dx . Through the generatp(xφ |g) P 2 ∞ n ing function e2xt−t = n=0 t Hn /n! [48], the Fisher

FIG. 2. (Color online) Fisher Information for homodyne detection F (g) (blue dots) and Quantum Fisher Information H(g) (red triangles) as a function of χ. ˜ The parameters are taken from Ref. [35] as |α|2 ≈ 106 , ωc = 1014 Hz, L = 10−5 m, ω = 103 Hz, m = 10−6 Kg, and φ = π/2.

Now we address the experiment feasibility of the proposed scheme. Recently, cavity optomechanics in the experiments has achieved some progresses, including normalmode splitting [49], precision measurement [50, 51], nonreciprocal amplification [52]. To achieve the proposed scheme, we use the parameters of the cavity and the mechanical oscillator as L = 10−5 m, ωc = 1014 Hz, ω = 103 Hz [53], and m = 10−6 kg [54]. The relevant parameters of the Coulomb interaction can be assumed to be

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of 2

ρI (2π/ω 0 ) = e−|α|

X αn α∗m 0 √ e−πκ/ω (n+m) n!m! m,n

× ei2π[χ˜ ×e

2

˜ G ˜ 1 )(n−m)] (n2 −m2 )+2χ( ˜ S+ −[1−2iχ ˜2 (n−m)/κω 0 ]2π/ω 0 κ 1−2iχ ˜2 (n−m)/κω 0

|α|2 1−e

|nihm|. (18)

Note that when κ → 0, the solution ρI (2π/ω 0 ) given by Eq.(18) reduces to the decoherence-free solution ρ(2π/ω 0 ) given by Eq.(13). Considering the strong-coupling regime, i.e., χ0  κ and using Eqs. (10) with κ/ω 0 = 0.1, we get F (g) ≈ 1.71 × 1031 m−2 s4 and ∆g ≈ 2.41 × 10−16 ms−2 . Compared with the decoherence-free case, we find that the decoherence is not severely affecting the homodyne measurement about the gravitational acceleration in the strong-coupling regime.

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TABLE I. Comparison gravimetry sensitivities √ between the √ ∆g in ms−2 and the Hz-noise ∆g/ Hz in ms−2 Hz−1/2 for different kinds of Platforms. √ Platform ∆g ∆g/ Hz time Atom Interferometry [10] 10−9e 10−8e 100si Atom-Chip Fountain Gravimeter [11] 10−10e 10−9e 100si −9e −7e FG5-X Absolute Gravimeter [14] 10 10 6.25hi −5e −7e Optomechanical Accelerometer [15] 10 10 10−3 si Magnetomechanics [19] 10−7p 10−9p 10−4 sc Optomechanics [35] 10−15p 10−16p 10−3 sc a Optomechanics 10−16p 10−18p 10−3 sc a e p Our scheme Experiments Theoretical predictions i Integration time c Cycle time or Oscillation frequency

riod of the oscillator [35, 43]. Hence, the Lindblad master equation for the optical frequencies in the interaction picture can be described as   ∂ρ(t) i 1 † 1 † † = − [HI , ρ(t)]+κ aρ(t)a − a aρ(t) − ρ(t)a a , ∂t ~ 2 2 (17)
0 † 0 0 2 0 where HI = −~ χ a a + S + G1 /ω , κ is the photon decay rate. For the initial state ρ(0) = |αi hα| , the field density matrix over one oscillation period can be found by calculating Eq. (17) [55]. It reads

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FIG. 3. (Color online) Fisher Information for homodyne detection F (g) as a function of the distance r0 and the charge Q1 = Q2 = Q. The parameters are taken from Ref. [35] as |α|2 ≈ 106 , ωc = 1014 Hz, L = 10−5 m, ω = 103 Hz, m = 10−6 Kg, and φ = π/2.

Q1 = Q2 = 2.4 nC and r0 = 4.8pmm. Then, we obtain the coupling strength χ = ωc /L ~/ (2mω) ≈ 2.3 × 103 Hz and the effective mechanical coupling constant G2 = ke Q1 Q2 ≈ 234 Hz. Using Eq. (15), we have the Fisher 2mωr 3 0

III.

CONCLUSION

In summary, we have investigated the use of the strongcoupling optomechanics to improve estimation of gravitational acceleration. In particular, the parametric amplification of the squeezed mechanical mode is generated by introducing the Coulomb interaction, which can effectively enhance the optomechanical interaction into the strong-coupling regime. Using the parameters with previous work, we show that the optomechanical interactions in the presence of the Coulomb interaction may provide a promising way for measurement of gravitational acceleration with high accuracy.

Information F (g) ≈ 6.02 ×p 1031 m−2 s4 and the corresponding sensitivity ∆g = 1/ F (g) ≈ 1.29 × 10−16 ms−2 with |α|2 ≈ 106 and φ = π/2. Compared with previous works in Table I, the sensitivity on the measure of the gravitational acceleration could be significantly improved in the strong-coupling regime. Note that even if using the same parameters of the cavity and the mechanical oscillator with previous work based on the optomechanical interaction [35], the sensitivity for measuring the gravitational acceleration could be improved with more than one order of magnitude, due to the effective enhancement of the optomechanical coupling by the additional Coulomb interaction. Finally, we briefly investigate the influence of decoherence for photon leakage from the cavity on the gravity measurement. This is because in our scheme the phonon decoherence could be negligible during a oscillation pe-

This work was supported by the National Natural Sciences Foundation of China (Grants Nos.11674094, 11774089, and 11874146) and Shanghai Natural Science Fund Project (Grant Nos.17ZR1442700, 18DZ2252400, and 18ZR1410500).

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IV.

ACKNOWLEDGEMENTS

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G.-W.L. came up with the original idea of the the oretical model; L.-J.F. performed the calculations and the simulations; Y.-P.N. and S.-Q.G. guided the research; G.-W.L. and L.-J.F. discussed the results and wrote the paper; All authors helped revise it.