Enhanced sensing of millicharged particles using nonlinear effects in an optomechanical system

Jian Liu; KaDi Zhu

doi:10.1364/OE.26.002054

1. Introduction

Millicharged particles(MCPs) or epsilon charged particles are the hypothetical particles with an electric charge $e^{'} = ε e,$ which is much smaller than the elementary charge $e .$ The existence of the particles with small, unquantized electric charge can be introduced into the standard model in a variety of ways [1]. There has been interest in the possibility of a small, nonzero electric charge for the neutrino [2,3]. The possibility that dark matter carries a fractional or epsilon charge has been considered recently [4–6]. The search for the epsilon charges for the range from keV to GeV progressed through the efforts of many direct experiments and indirect observations over many years [7–11]. Now the search for stable particles with millicharges in bulk matter is still on the initial stage. Previous research using magnetic levitation electrometers or Millikan oil drop techniques in matter did not have sensitivity less than $0.1 e$ [12–16]. Recently, Moore et al. report the search for particles with $ε \geq 10^{- 5}$ using optically levitated microspheres [17]. Chen et al. propose to use a high-Q whispering-gallery-mode (WGM) optical microresonator to detect the surface and bulk charge of a dielectric nanoparticle [18]. Based the optical trapping scheme, we propose a levitated sensor to detect the epsilon charge in nano-scale with improved sensitivity of $ε \sim 10^{- 14}$ in this paper by the cavity pump-probe technology.

Cavity optomechanics, which explores the interaction between optical field and mechanical motion, has witnessed rapid advances in recent years [19]. One of the principal advantages of cavity-optomechanical systems is the built-in readout of mechanical motion via the light field transmitted through (or reflected from) the cavity. Various of schemes have been proposed for the measuring of the parametric displacement [20], mechanical Fock-state detection [21], optical feedback cooling [22], extremely sensitive force sensing [23,24] and electrical charges measurement [18,25]. Recently, the optical pump-probe technique has become a popular topic, which offers an effective method to study the light-matter interaction. The optical pump-probe technology includes a strong pump field and a weak probe field. The strong pump laser is used to stimulate the system to generate coherent optical effect, while the weak laser plays the role of probe beam. Therefore, the linear and nonlinear optical effects can be observed on the probe spectrum in the two field's control scheme.

Most recently, this pump-probe scheme has also been realized experimentally in optical cavity [26–28]. In the present paper, we apply the pump-probe fields to the cavity-nanosphere coupling system and study theoretically the optical nonlinear in the optomechanical system. The results show that a Kerr peak with ultra narrow linewidth can be achieved through the high vacuum allowed. Based on the frequency shift induced by the millicharge, the detection of signature of MCPs in bulk matter can be carried out by the transmission spectrum. We expect that the optomechanical detector could improve the search for epsilon charges via an enhanced sensitivity of $10^{8} ~ 10^{9}$ than current experiments [17].

The remainder of the paper is organized as follows. Section two describes the theoretical model. Section three gives discussions about the measurement mechanism: 1) with the charged ring, the charge of the nanosphere is mapped to its mechanical resonant frequency. 2) with optical optomechanical system, very tiny drift in mechanical frequency could be readout optically. Section four studies the constraints and limits of the measurement system, discusses the thermomechanical noise and photon shot noise. We check the frequency stability and radiation damping rate respectively. The last section contains the summary of the main results of the present work.

2. Models

As shown in Fig. 1(a), a fused silica nanoshpere with radius $a = 50 n m$ is placed inside a Fabry-Perot cavity and optically trapped at the antinodes of the optical lattice. The dielectric object is subjected to an electric field generated by a homogeneously charged ring with radius $R_{0} = 50 μ m .$ Then we radiate a strong pump field(with frequency $ω_{p}$ ) and a weak probe(with frequency $ω_{p r}$ ) field on the cavity.

Fig. 1 (a)Schematic diagram of the proposed setup for detecting millicharged particles by optical pump-probe technology in the cavity. Our approach involves optically trapping a fabricated nanosphere in the antinode of an optical standing wave. To eliminate the polarization force background, we use a homogeneously charged ring to produce symmetrical electric field near the nanosphere and the nanosphere should be trapped at the center of the ring. (b) Force analysis for the millicharged particles in the nanosphere.

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For small oscillation amplitudes, the levitated particle experiences a harmonic trapping potential with a trap stiffness(or the spring constant) $k = ω_{n}^{2} m,$ where $ω_{n}$ is the trapping frequency and $m$ is the mass of the nanosphere. The effective spring constant can be defined as $k^{'} = k + \nabla F,$ here $\nabla F$ represents the force gradient at the equilibrium position of the levitated nanosphere. If we consider the interaction between the electric field gradient and the single MCP, we get $\nabla F = ε e \nabla E,$ here $\nabla E$ is the electric field gradient. The resonance frequency $ω_{n}$ can be converted into ${ω^{'}}_{n} = \sqrt{(k + \nabla F) / m} .$ If the differentiation is sufficiently small, we get ${ω^{'}}_{n} ≅ ω_{n},$ and $\nabla F / 2 k ≅ ω_{n} / {ω^{'}}_{n} - 1.$ Assume $d ω = {ω^{'}}_{n} - ω_{n},$ based on the frequency shift of the resonance mode, one can obtain

\nabla F = \frac{2 k}{ω_{n}} d ω .

Here the frequency shift is dependent linearly on the force gradient via the linearized responsivity

2 k / ω_{n} .

To probe the MCPs, we electricize the ring homogeneously with an assumed linear density. $υ$ The main systematic effect limiting the measurement sensitivity in [17] is the interacting primarily between the electric field and the dipole moments of the nanospheres. In order to minimize this effect, we suggest using the homogeneously charged ring. The nanosphere can be trapped at the centre of the ring thereby the applied electric field is distributed symmetrically along the horizontal axis. Thus the resultant polarization force in the nanosphere is zero. In cylindrical coordinates, the position of the center of the nanosphere is $(0, 0, 0)$ as shown in Fig. 1(b). Now we consider a MCP appears in the nanosphere with the position $(r, ϕ, z) .$ The nanosphere will experience a force due to the electromagnetic interaction between the charged ring and the MCP. The electric field along $z$ axis at the MCP's location can be obtained by

E_{z} = \frac{q}{4 π^{2} Ε_{0}} \frac{2 z}{\sqrt{R_{0}^{2} + r^{2} + z^{2} + 2 R_{0} r} (R_{0}^{2} + r^{2} + z^{2} - 2 R_{0} r)} O,

where

R_{0}

is the radius of the ring,

q = 2 π R_{0} υ

is the total quantity of electric charge of the ring,

Ε_{0}

is the vacuum permittivity,

O = (π / 2) [1 - {(1 / 2)}^{2} U^{2} - {(1 \cdot 3 / 2 \cdot 4)}^{2} (U^{4} / 3) - \cdot \cdot \cdot]

is a total elliptical integral and the parameter

U^{2} = 4 R_{0} r / (R_{0}^{2} + r^{2} + z^{2} + 2 R_{0} r) .

Since the scale of the nanosphere is much smaller than the ring, namely

r, z ≪ R_{0},

the electric field can be obtained within approximation,

E_{z} \approx \frac{q}{4 π^{2} Ε_{0}} \frac{2 z}{R_{0}^{3}} .

Then we can obtain the force gradient

\nabla F_{z} = \frac{2 ε e q}{4 π^{2} Ε_{0} R_{0}^{3}} .

We replace Eq. (1) in the equation for

\nabla F_{z}

and considering

k = ω_{n}^{2} m,

it is easy to show that the single MPC will induce a shift as

d ω = \frac{ε e υ}{2 π Ε_{0} R_{0}^{2} ω_{n} m} .

Interestingly, we find that the shift is independent with the position of the MPC in the nanosphere in this case. That is to say, no matter where it appears, the shift is merely decided by the total electron charge

ε e .

In what follows, we propose a pump-probe scheme to readout the very tiny drift in mechanical frequency optically.

Then let’s denote the cavity mode annihilation (creation) operator by $σ (σ^{+})$ with the commutation relation $[σ, σ^{+}] = 1.$ The bosonic annihilation (creation) operators of the nanosphere oscillator are represented by $s$ and $s^{+}$ with $[s, s^{+}] = 1.$ Thus the Hamiltonian of the whole system in a rotating frame can be expressed as follows [29–31]

H = ℏ Δ_{c} σ^{+} σ + ℏ ω_{n} s^{+} s - ℏ g σ^{+} σ (s^{+} + s) - i ℏ Ω_{p} (σ - σ^{+}) - i ℏ Ω_{p r} (σ e^{i Δ_{p r} t} - σ^{+} e^{i Δ_{p r} t}),

where

g

is the optomechanical coupling rate,

Δ_{c} = ω_{c} - ω_{p}

is pump-cavity detuning. We also introduce the Rabi frequency of the pump field and the probe field inside the cavity

Ω_{p} = \sqrt{2 P_{p} κ / ℏ ω_{p}}

and

Ω_{p r} = \sqrt{2 P_{p r} κ / ℏ ω_{p r}},

where

P_{p}

is the pump power,

P_{p r}

is the power of the probe field, respectively. We then define the operator

n = s^{+} + s

and the detuning of the probe and pump field

Δ_{p r} = ω_{p r} - ω_{p} .

The temporal evolutions of

σ

and

n

can be obtained by solving the quantum Langevin equations with the damping terms

\frac{d σ}{d t} = - (i Δ_{c} + κ) σ + i g n σ + Ω_{p} + Ω_{p r} e^{- i Δ_{p r} t} + \sqrt{κ} {\hat{a}}_{i n},

\frac{d^{2} n}{d t^{2}} + γ_{n} \frac{d n}{d t} + ω_{n}^{2} n = 2 ω_{n} g σ^{+} σ + \hat{ξ} (t),

where

κ

is the total energy damping rate of the cavity,

γ_{n}

is the damping rates of the vibrational mode of the nanosphere. In Eq. (7),

{\hat{a}}_{i n}

is the

δ

-correlated Langevin noise operator of the cavity-field quantum vacuum fluctuation, which has zero mean,

〈 {\hat{a}}_{i n} (t) {\hat{a}}_{i n}^{+} (t^{'}) 〉 = δ (t - t^{'}),

〈 {\hat{a}}_{i n}^{+} (t) {\hat{a}}_{i n} (t^{'}) 〉 = 0.

The motion of nanosphere are affected by thermal bath of Brownian and non-Markovian stochastic process,

\hat{ξ} (t)

is the Brownian noise force with zero mean value [32,33]. Following standard methods from quantum optics, we derive the steady-state solutions to Eqs. (7) and (8) by setting all the time derivatives to zero,

σ_{0} = \frac{Ω_{p}}{(i Δ_{c} + κ) - i g n_{0}},

n_{0} = \frac{2 g {| σ_{0} |}^{2}}{ω_{n}} .

We define the parameter

ω_{0} = {| σ_{0} |}^{2},

hence we can obtain

Ω_{p}^{2} = ω_{0} [κ^{2} + {(Δ_{c} - \frac{2 g^{2} ω_{0}}{ω_{n}})}^{2}] .

One can always rewrite each Heisenberg operator as the sum of its steady-state mean value and a small fluctuation with zero mean value as follows

σ = σ_{0} + δ σ,

n = n_{0} + δ n .

Inserting these equations into the Langevin equations, one can safely neglect the nonlinear terms such as

δ σ δ n .

Since the driving fields are weak, we will identify all operators with their expectation values, and drop the quantum and thermal noise terms. Then the linearized Langevin equations can be written as

〈 δ \dot{σ} 〉 = - κ 〈 δ σ 〉 + i g (n_{0} 〈 δ σ 〉 + σ_{0} 〈 δ n 〉) + Ω_{p} + Ω_{p r} e^{- i Δ_{p r} t},

〈 δ \ddot{n} 〉 + γ_{n} 〈 δ \dot{n} 〉 + ω_{n}^{2} 〈 δ n 〉 = 2 ω_{n} g σ_{0}^{2} .

To solve these equations, we make the ansatz as follows

〈 δ σ 〉 = σ_{+} e^{- i Δ_{p r} t} + σ_{-} e^{i Δ_{p r} t},

〈 δ n 〉 = n_{+} e^{- i Δ_{p r} t} + n_{-} e^{i Δ_{p r} t} .

Upon substituting these equations into Eqs. (14) and (15), and upon working to the lowest order in

Ω_{p}

but to all orders in

Ω_{p r},

we obtain in the steady state

σ_{-} = \frac{G}{(P + κ^{2})} \frac{Ω_{p r} Ω_{p}^{2}}{B (Y^{2} - P^{2}) + 2 P G ω_{0}} .

Here

σ_{-}

is a parameter in analogy with the Kerr nonlinear optical susceptibility with

Y = - i Δ_{c} + κ,

P = i Δ_{c} - i g n_{0},

B = - Δ_{p r}^{2} - i γ_{n} Δ_{p r} + ω_{n}^{2},

G = 2 i g^{2} ω_{n} .

Using the input-output relation

σ_{o u t} (t) = σ_{i n} (t) - \sqrt{2 κ} σ (t),

the amplitude of the Kerr nonlinear can be expressed as

σ_{o u t^{-}} = - \sqrt{2 κ} σ_{-} .

The transmission intensity of the output field is then given by [26]

{| T_{o u t^{-}} |}^{2} = 2 κ {| \frac{σ_{o u t^{-}}}{Ω_{p r}} |}^{2} .

The full width at half maximum (FWHM) of the transmission peak can be defined as

F W H M = 2 | δ_{p r} |,

with

{| T_{o u t^{-}} (δ_{p r}) |}^{2} = (1 / 2) {[{| T_{o u t^{-}} |}^{2}]}_{\max},

where the detuning

δ_{p r} = Δ_{p r} - ω_{n} .

3. Measurement

For numerical work, we use a fused silica nanosphere with the density of $ρ = 2.2 g / c m^{3}$ and the dielectric constant $Ε = 2.$ The trapping frequency $ω_{n}$ depends on the trapping beam intensity, it can be modulated by the intracavity power [29]

ω_{n} = \frac{1}{2 π} {(\frac{6 K^{2} I_{0}}{ρ c} Re \frac{Ε - 1}{Ε + 2})}^{1 / 2},

here

c

is the speed of light,

K = 2 π / λ

is the wave numbers of the trapping beams with wavelength

λ = 1 μ m .

In what follows, we choose the intensity

I_{0} = 10^{- 2} W / μ m^{2},

hence one can obtain

ω_{n} = 150 k H z .

The quality factor one would get for a levitated nanosphere is solely determined by the air molecule impacts. Random collisions with residual air molecules provide the damping

γ_{n} = ω_{n} / Q

and thus, the quality factor due to the gas dissipation can be defined as [29,34]

Q = \frac{m ω_{n} v}{p A} = \frac{a ρ ω_{n} v}{3 p} .

Here

v = \sqrt{k_{B} T / m_{g a s}}

is the thermal velocity of the gas molecules,

p

the gas pressure and

A

the surface area of the resonators. For the nanosphere under the pressure

p = 10^{- 10} T o r r,

we get

Q = 1.3 \times 10^{11}

in the room temperature(

T = 300 K

). This value is considered to be within reach of moderate experimental improvements [29,35]. The optomechanical coupling between the cavity field and the nanosphere can be described by the coupling rate [29]

g = \frac{3 V_{s}}{4 V_{c}} \frac{Ε - 1}{Ε + 2} ω_{c},

where

V_{s} = (4 / 3) π a^{3}

and

V_{c} = (π / 4) L w^{2}

are the nanosphere and the optical mode volumes, respectively. To estimate the coupling, we consider the use of optical microcavities operating at

1064 n m,

corresponding to

ω_{c} = 280 T H z,

the length

L = 1 m m

[36], and mode waist

w = 50 μ m .

Then the optomechanical coupling rate can be obtained as

g = 1.4 \times 10^{4} H z .

In order to remove the net charge of levitated sensor in trapping, a fiber-coupled xenon flash lamp is used to illuminate the electrode surfaces near the nanosphere [17]. To determine the original frequency, we hit the cavity by the pump-probe pulses and measure the absorption of the probe beam without the applied electric field. According to Eqs. (11) and (18), we depict the Kerr nonlinear susceptibility $σ_{-}$ as a function of probe-pump detuning $Δ_{p r}$ in Fig. 2(a) with $Δ_{c} = 0.$ Two sharp Kerr peaks are located at the original frequencies of the nanosphere resonator on the spectrum. Such a phenomenon is attributed to the destructive quantum interference effect between the nanosphere mode and the beat of the two optical fields via the cavity. If the beat frequency of two lasers $Δ_{p r}$ is close to the resonance frequency of nanosphere, the mechanical mode starts to oscillate coherently, which results in Stokes-like $(Δ_{p r} = ω_{n})$ and anti-Stokes-like $(Δ_{p r} = - ω_{n})$ scattering of light from the strong intracavity field.

Fig. 2 (a)The plot of absorption spectrum as a function of probe-pump detuning. The unit of the Stokes scattering absorption intensity is an arbitrary unit(a.u.) which is a relative unit of measurement to show the ratio of amount intensity to a predetermined reference measurement. (b)Nonlinear optical spectrum of the probe field as a function of probe-pump detuning before (black solid line) and after (red dashed line and blue dash line) the action between millicharge and electric field gradient. The signals of the MCP with $ε = 10^{- 14}$ and $ε = 5 \times 10^{- 14}$ can be well recognized in the spectrum. Other parameters used are $Ω_{p} = 3 M H z,$ $Δ_{c} = 0,$ and $Q = 1.3 \times 10^{11}$

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Then we illustrate the responses of Kerr nonlinear to external field by electricize the ring homogeneously with linear density $υ = 10^{- 4} C / m .$ We depict Fig. 2(b) to show the transmission spectrum of the probe laser ${| T_{o u t^{-}} |}^{2}$ as a function of the probe detuning $δ_{p r} .$ As a MPC of $ε = 10^{- 14}$ appears in the matter, we use the black curve to show the Kerr peak in the absence of external field. Then we use the red dash curve to show the shift of $d ω = 6.4 \times 10^{- 6} H z$ can be obtained on the probe spectrum due to the interaction between the charged ring and MPC. The dash blue sharp peak shows that the enhanced peak will exhibit a larger frequency shift on the transmission spectrum as we consider the MPC of $ε = 5 \times 10^{- 14}$ in the matter. The distance of shift $d ω$ as a function of $ε$ follows a linear relationship, $d ω = 6.4 \times 10^{8} ε .$ This may provide a direct method to measure the millicharges in this coupled system. The ability to improve the resolution depends on the FWHM of the transmission peak which can be estimated as $F W H M \approx ω_{n} / Q .$ From Eq. (21) we can see that the Q-factor is mainly limited by the pressure of the residual gas $p .$ We can get $F W H M \approx 3 \times 10^{- 6} H z$ from the condition of $p = 10^{- 10} T o r r .$

In the scheme we use a nanosphere with the radius $a = 50 n m,$ the mass $m = 1.2 \times 10^{- 18} k g,$ the trapping frequency $ω_{n} = 150 k H z,$ and a charged ring with the radius $R_{0} = 50 μ m,$ linear density $υ = 10^{- 4} C / m .$ Then we can calculate from Eq. (5) that the millicharges can be measured down to a precision of $ε_{\min} \approx 4.7 \times 10^{- 15}$ in this coupling system.

4. Constraints and limits

M. R. Vanner et al. propose a scheme to realize state preparation and state purification of a mechanical resonator using short optical pulses as discussed in [37]. During a pulsed interaction of time scale $κ^{- 1} ≪ ω_{n}^{- 1}$ the mechanical position is approximately constant, therefore this scheme can be used as an alternative to “cooling via damping” for mechanical state purification. In our shcmem, the pump pulses can also be used for realizing high-purity states of the mechanical resonator. According to [37], the purity can be improved by two pulses with separation $t_{p} = π / 2 ω_{n},$ where the initial uncertainty in the momentum becomes the uncertainty in position. Considering the coupling to a thermal bath and starting with an initial thermal state $\bar{n}$ , the effective occupation of the final state after two pulses is

{\bar{n}}_{e f f} ≅ \frac{1}{2} (\sqrt{1 + \frac{1}{χ^{4}} + \frac{π \bar{n}}{Q χ^{2}}} - 1) .

Here the position measurement strength

χ

is an important parameter in this work as it quantifies the scaling of the mechanical position information onto the light field. The resulting optimal measurement strength is given by

χ = 2 \sqrt{5} g \sqrt{N_{p}} / κ .

We consider the use of pulses of mean photon number

N_{p} = 10^{8},

the optical microcavity has an amplitude decay rate

κ = 0.5 G H z

and the optomechanical coupling

g = 1.4 \times 10^{4} H z .

For

Q = 1.3 \times 10^{11},

one can obtain

{\bar{n}}_{e f f} = 0.09,

corresponding to the effective temperature of

T_{e f f} \sim 10^{- 7} K,

indicating that ideal oscillators can be essentially decoupled from their thermal environment. Levitation under good vacuum conditions can lead to extremely low mechanical damping rates, such an approach should also facilitate the emergence of quantum behavior even in room-temperature environments.

The ultimate sensitive limits for nanomechanical resonators operating in vacuum are imposed by a number of fundamental physical noise processes. In our scheme, the thermal noise of the mechanical motion is the dominant noise source. In order to get the fundamental limits, we need to consider the minimum measurable frequency shift $d ω_{\min}$ that can be resolved in a realistic noise system. For the case of $Q ≫ 1$ [38],

d ω_{\min} \approx {[\frac{k_{B} T_{e f f} ω_{n} Δ f}{E_{c} Q}]}^{1 / 2} .

Here, the measurement bandwidth

Δ f = 1 / 2 π τ,

which is dependent upon the measurement averaging time

τ .

E_{c} = m ω_{n}^{2} {〈 x_{c} 〉}^{2}

represents the maximum drive energy.

〈 x_{c} 〉

is the maximum rms level which is still consistent with producing a predominantly linear response. For a Gaussian field distribution, the nonlinear coefficients are given by

ζ = - 2 / w^{2}

[35],where

w = 50 μ m

is the beam waist radius. For small displacements

| x_{c} | ≪ {| ζ |}^{- 1 / 2} = 3.5 \times 10^{- 5} m,

the nonlinearity is negligible. In our considerations,

x_{c}

is taken to be 3 orders of magnitude smaller, we choose

x_{c} \approx 10^{- 8} m .

Considering the quality factor

Q = 1.3 \times 10^{11},

T_{e f f} \approx 0.1 μ K,

we can achieve that

d ω_{\min} \approx 2.7 \times 10^{- 6} H z < F W H M

for the measurement averaging time

τ = 1 / 2 π Δ f \approx 0.01 s .

The result shows that the thermomechanical noise can be controlled at a level comparable to the claimed sensitivity at a short measuring time.

The photon recoil noise should play a role for nano-scale sensors [39]. The scattered power is calculated as $P = σ_{s} I_{0}$ with $I_{0}$ the trapping beam intensity and $σ_{s}$ the scattering cross section. The scattering cross section $σ_{s} = {| α |}^{2} K^{4} / 6 π Ε_{0}^{2} .$ The particle polarizability $α = 4 π Ε_{0} a^{3} (n_{s}^{2} - 1) / (n_{s}^{2} + 2)$ and $n_{s}$ is the index of refraction of the nanosphere. Then we can obtain radiation damping rate $γ_{r a d} \approx P / m c {}^{2}.$ For the parameters used in our scheme, $I_{0} = 10^{- 2} W / μ m^{2},$ we find $γ_{r a d} \approx 10^{- 9} ≪ d ω_{\min} .$ The result shows that the minimal detectable shift is above the limit set by the photon recoil noise. Therefore, we would expect that the induced frequency shift can be resolvable in actual experiment by using the subwavelength silica nanospheres. The recoil heating rate can be given by $Γ_{r e c o i l} = P ω_{c} / 5 m c^{2} ω_{n} .$ Using $ω_{c} = 280 T H z$ and $ω_{n} = 150 k H z,$ we predicts a reheating rate of $Γ_{r e c o i l} = 0.24 H z .$ The time between two cooling pulses should be larger than the inverse of the reheating rate, which is defined as the time required to increment one quantum $ℏ ω_{n}$ of energy in the quantum harmonic oscillator. Since $t_{p} = π / 2 ω_{n} ≪ Γ_{r e c o i l}^{- 1},$ indicating that oscillators can be essentially decoupled from their thermal environment.

At last we conservatively calculate limits on the abundance per nucleon $N$ of charged particles with $ε$ in Fig. 3, and compare to previous limits from optically levitated nanospheres [17] and magnetic levitometer [14] experiments. The rectangle region A and B denotes the space directly excluded by the previous experiments above their single particle thresholds. The region C denotes the constraints for our scheme. As shown in Fig. 3, the method we have described provides the direct search for MCP in condensed matter with sensitivity of $ε \geq 10^{- 14}$ to single particle. Over this full range, the upper limit on the abundance per nucleon is at most $N \approx 10^{9} .$ The lines extending from each region show limits on the abundance below the single particle threshold for comparison. The yellow dash line shows the constraints form a high-Q whispering-gallery-mode (WGM) optical nanoresonator by the use of ${Cu}_{2} O$ nanoparticle [18]. The method we have described promises to be the most sensitive approach for detecting millicharged particles.

Fig. 3 Constraints on the abundance of millicharged particles per nucleon N versus the epsilon charge. The results from this work (red rectangle C) are compared to previous results from levitated nanospheres (blue rectangle B) [17], magnetic levitometer (green rectangle A) [14] experiments and WGM nanoresonator [18](dash yellow line). The lines extending from each region show the upper limits below the single particle threshold.

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5. Conclusion

In this work, we investigate the Kerr nonlinear spectrum in the optomechanical system where a nanosphere coupled to a cavity. We have proposed an optical method to detect the existence of MCPs in bulk matter via pump-probe technology. A direct scheme to determine the electric charge is also demonstrated. The result shows that a sharp peak can be obtained at the resonant frequency. Based on the frequency shift of the vibrational mode, the MCPs can be measured on the spectrum. The Kerr nonlinearity with an ultra narrow linewidth gives rise to an improved sensitivity of order $10^{- 14} e,$ which is much smaller than that in current experiments and many other theoretical expectations. In the resonant millicharges detection, the strain sensitivity of our setup is limited by the thermal motion of the sensor. We would expect that the claimed sensitivity can be still achieved in actual noise system in the ultrahigh vacuum.

Funding

National Natural Science Foundation of China (11274230, 11574206); the Basic Research Program of the Committee of Science and Technology of Shanghai (14JC1491700).

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Enhanced sensing of millicharged particles using nonlinear effects in an optomechanical system

Abstract

1. Introduction

2. Models

3. Measurement

4. Constraints and limits

5. Conclusion

Funding

References and links

Cited By

Figures (3)

Equations (24)

Optics Express