1. Introduction

The ability to utilize focused laser beams for the precise control, manipulation, and confinement of a single particle or more particles under vacuum conditions has emerged as a powerful tool for experimental exploring fundamental questions in physics. As an example let us mention thermodynamic processes taking place in the underdamped regime at the microscale and nanoscale [1–3] and the emergence of non-conservative instabilities in thermally excited motion of a single particle [4] and more optically coupled particles [5–9]. Very vivid activities are focused on testing the macroscopic limits of quantum optomechanics [10,11] enabled by recent experimental achievements in cooling of the center-of-mass motion of an optically levitated nanoparticle, serving as a model of macroscopic quantum harmonic oscillator close to the lowest energy states. The successful cooling strategies (for one or more-dimensional cooling) use either the active force feedback acting upon the levitated nanoparticle and proportional to the detected motion of the levitated nanoparticle [12–21] or an anti-Stokes light scattering from an optically levitated nanoparticle into a cavity mode [22–24].

Although the above-mentioned remarkable achievements have been reached with bulk laboratory systems, the technologies leading to the miniaturization of various experimental platforms have matured [25]. This includes also the optical microcavities that have been frequently used for nanoparticle position detection [26–28], characterization [29], and trapping [30–34]. The combination of small mode volume and high-quality factor of the microcavity ensures a strong interaction between the confined optical field and a nanoparticle placed into it. A special branch of microcavities represents a one-dimensional photonic crystal microcavity (PhC) consisting of a single row of holes in a silica photonic waveguide. If the PhC is properly designed and manufactured, its quality factor, defined as the ratio of the resonance wavelength to the spectral full width at half maximum of the cavity transmissivity resonance profile $Q = \lambda /\Delta \lambda$, can exceed $10^5$ [35]. A systematic approach for such designs was reported in [36] and a cavity with $Q = 10^9$ was proposed.

In this paper, we followed the recommended design for one-dimensional PhC with tapered waveguide [36] and analyzed its properties for optical confinement of a nanoparticle. We calculated optical forces acting upon a nanoparticle and optomechanical coupling regimes for various displacements from the center of the hole and frequency detunings of the mode frequency from the resonant frequency of the empty cavity.

2. Geometry and optical properties of an empty photonic crystal microcavity

The geometry of the PhC is drawn in Fig. 1(a)) and follows the deterministic method proposed by Quan and Loncar [36]. The whole concept is based on the state-of-the-art microfabrication methods [37] using silicon-on-insulator (i.e., SiO$_2$) wafer technology. The PhC structure is made of silicon of thickness $t=220$ nm, it is undercut, surrounded by a vacuum, and held on its edges on an SiO$_2$ layer. The PhC structure contains $2 i_{\mathrm {max}}-1$ holes of the same radius $r= 150$ nm that are equally separated with their centers by $a=520$ nm. The width $w(i)$ of the PhC structure around $i^\mathrm {th}$ hole decreases from the central width $w_{\mathrm {center}}=700$ nm to the edge ones $w_{\mathrm {end}}=500$ nm following the quadratic dependence $w(i) = w_{\mathrm {center}} - (i-1)^2(w_{\mathrm {center}} - w_{\mathrm {end}})/(i_{\mathrm {max}-1})^2$.

 

Fig. 1. Geometry of the analyzed photonic crystal microcavity (PhC) and its optical properties without a nanoparticle. a) Schematics of the analyzed PhC and the system of coordinates used in the analyses. The total number of holes is equal to $2 i_\mathrm {max}-1=13$, where $i_\mathrm {max}=7$ corresponds to the number of holes from the middle one to the edge one. The width of the silicon part of the PhC decreases quadratically from the middle hole, having index $i=1$, following $w(i) = w_{\mathrm {center}} - (i-1)^2(w_{\mathrm {center}} - w_{\mathrm {end}})/(i_\mathrm {max}-1)^2$. b) and c) present slices $xy$ and $xz$ of the spatial distribution of the relative field intensity $E^2_\mathrm {r}= |{\bf E}_\mathrm {c}(x,y,z)|^2/\mathrm {max} \{ |{\bf E}_\mathrm {c}(x,y,z)|^2\}$. d) The modal volume $V_m$, defined by Eq. (1), is calculated for the PhCs with the increasing number of holes $i_\mathrm {max}$. e) Dependence of the quality of the empty cavity (i.e., without a nanoparticle) $Q = \lambda /\Delta \lambda$ calculated from the cavity spectral properties for an increasing number of holes $i_\mathrm {max}$ (similar to Fig. 2(j)).

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We calculated the electromagnetic field distribution in the PhC using a three-dimensional finite-difference-time domain (3D-FDTD) method implemented in Meep open source software [38]. The simulation domain consists of the PhC structure shown in Fig. 1(a)), the vacuum of minimal thickness 1 $\mu$m filling the space between the structure, and the planar perfectly matched layers serving as the boundary of the simulation domain. We used the same computational grid of 20 nm in all performed simulations.

As the input parameters, we took refractive indices of silicon (3.46) and silica (1.44) corresponding to the wavelength 1550 nm. It corresponds to the center of the spectrum of the Gaussian light pulse covering wavelengths 1450-1650 nm. The beam propagated through the PhC structure along the positive direction of the $x$ axis and its electric and magnetic field components were determined at each computational voxel. The spectral transmissivity of the PhC structure was determined from the squared magnitude of the electric field intensity $|{\bf E}|^2$ evaluated at the right part of the PhC as detector (blue line) in Fig. 1(a)).

Figures 1(b))-c) reveal the spatial distribution of $|{\bf E}_\mathrm {c}|^2$ for the resonant wavelength $\lambda _\mathrm {c0} = 1554.68$ nm, corresponding to resonant cavity frequency $\omega _\mathrm {c0}/2\pi = 192.8337$ THz of the empty PhC. The highest value of $|{\bf E}_\mathrm {c}|^2$ can be found in the middle hole of the PhC structure. The volume of the mode over the whole simulation domain is expressed as [32].

Figure 1(e)) illustrates the exponential increase of the cavity quality factor $Q$ by two orders of magnitude as the number of holes ($i_\mathrm {max}$) increases. Because of the Fourier uncertainty principle [38], the simulation need to run for a time inversely related to the frequency resolution we would like to obtain, it means that for high $Q$ its convergence is longer. In the subsequent numerical analysis, we considered only the case $i_\mathrm {max}=7$, i.e., the case illustrated in Fig. 1(a)), which is good compromise between high $Q$ and relatively short simulation time. Should anyone desire to compute the case with higher $Q$ we made the source code publicly available [40]. However, in order to calculate a higher $Q$ cavity case, one may alternatively to use some frequency based method, such as finite element method.

3. Optical confinement of a nanoparticle in the PhC

We considered a nanoparticle placed in the center of the middle hole. We calculated the spatial electromagnetic field distribution in the whole simulation domain over the whole spectral width defined by exciation light pulse that is Gaussian distributed in both time and frequency domain and propagates from the input port. Consequently, we shifted the nanoparticle to new positions, as Figs. 2(a))-c) illustrate, and recalculated the electromagnetic field distribution for each of them. The field distributions were used to calculate the spectral transmissivity of the cavity for various nanoparticle positions, as well as an optical force acting upon the nanoparticle [41]. To get the optical force, we employed the Meep built-in function to calculate the Maxwell stress tensor and integrated it over the cylindrical surface surrounding the particle. The final optical force is normalized to the input laser power.

 

Fig. 2. Spatial field distribution in the photonic crystal microcavity without and with a nanoparticle placed in the cavity center. a)-c) Illustration where the nanoparticle (the color circles) is placed (within the middle hole of the PhC) in individual planar cross-sections. The nanoparticle’s motion is over $\pm$ 50 and $\pm$ 400 nm in the $x-y$ plane and along the $z$ axis, respectively. d)-f) The spatial distribution of the relative field intensity $E_\mathrm {r}^2$ in $x-y$, $x-z$, and $y-z$ planes for the empty cavity at its resonant frequency $\omega _\mathrm {c0}$. g)-i) The spatial distribution of $E_\mathrm {rp}^2 =|{\bf E}_\mathrm {p}(x,y,z)|^2/\mathrm {max} \{ |{\bf E}_\mathrm {c}(x,y,z)|^2 \}$ at the same resonant frequency $\omega _\mathrm {c0}$ if a nanoparticle (the white circle) of diameter 100 nm is placed in the center of the middle hole. j) Transmission spectra of the empty cavity (blue) and for a nanoparticle of diameter 100 nm and 150 nm placed to the center of the middle hole.

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Figures 2(d))-f) show in detail the distribution of the relative field intensity $E_\mathrm {r}^2$ of the empty cavity in the middle hole over three perpendicular planar cross-sections passing through the center of the middle hole. The relative field intensity $E^2_\mathrm {r}$ is weaker in the hole center and increases towards the hole edges for the empty cavity (i.e. without a nanoparticle) at its resonance frequency $\omega _\mathrm {c0}$. This situation corresponds to the maximum of the blue transmissivity curve in Fig. 2(j)) at the zero frequency detuning $\delta \omega =0$. However, if a silica nanoparticle is placed in the center of the middle hole, see Fig. 2(g))-i), the cavity resonance frequency $\omega _\mathrm {cp}(0,0,0)$ is red-shifted with respect to the resonance frequency of the empty cavity $\omega _\mathrm {c0}$ by $\delta \omega _\mathrm {p0} = \omega _\mathrm {cp}(0,0,0)-\omega _\mathrm {c0}$. We numerically analyzed the cavity spectral properties for nanoparticles of two sizes appropriate for experimental levitation $d_1=100$ nm and $d_2=150$ nm in diameter and different frequency shifts $\delta \omega _\mathrm {p0}$ were obtained for each nanoparticle placed in the center of the middle hole: $\delta \omega ^\mathrm {100\,nm}_{p0}=-45.8$ GHz and $\delta \omega ^\mathrm {100\,nm}_{p0} = -154.2$ GHz.

Figures 2(g))-i) compare the distribution of the relative field intensity $E_\mathrm {rp}^2 =|{\bf E}_\mathrm {p}|^2 / \mathrm {max} \{ |{\bf E}_\mathrm {c}|^2 \}$ with a particle in the center of the hole calculated at the resonant frequency of the empty cavity. Since the resonant cavity frequency with a nanoparticle shifts when compared to the resonant frequency of the empty cavity, and the line-width of the cavity resonance is relatively narrow, the field amplitude decreased by an order of magnitude. Moreover, the spatial field distribution significantly changes in the cavity, local maxima, referred also as photonic "nano-jets" [42], appear along $y$-axis in Fig. 2(g)) and i).

Figures 3(a))-c) show the magnitudes of the optical forces calculated for $\delta \omega _\mathrm {p0}$, i.e., the cavity is in resonance with a nanoparticle placed at the center of the middle hole: $\delta \omega ^\mathrm {100\,nm}_{p0}/2\pi = - 45.8$ GHz and $\delta \omega ^\mathrm {150\,nm}_{p0}= - 154.2$ GHz, for nanoparticle of diameter 100 nm and 150 nm, respectively, assuming laser power 1 mW. The smaller particle (blue curves in Fig. 3(a)-c) is stably trapped in the hole center in all 3 directions. The maximal trapping force is the weakest in the $x$ direction but has almost the same trapping stiffnesses in $y$ and $z$ directions. The optical forces acting on the bigger particle (red curve) in $x$ and $z$ direction are stronger compared to the smaller one, as could be expected from Rayleigh approximation for this case. Due to the stronger gradient of the optical field close to the microcavity edge the bigger nanoparticle is pulled out of the microcavity center for relatively small particle displacement $\pm$18 nm along $y$ the direction. This effect limits both the trapping stability as well as trap stiffness for bigger nanoparticles. On the other hand, the PhC trapping stiffness when compared to the stiffness of high NA optical tweezers is approximately 5 order of magnitude bigger for all three dimensions assuming the same trapping laser power.

 

Fig. 3. Dependencies of components of an optical force $\mathbf { F}(x,y,z,\delta \omega )$ acting upon a nanoparticle at various positions and for various frequency detunings $\delta \omega =\omega _\mathrm {p}(x,y,z)-\omega _\mathrm {c0}$ of the beam. a) $F_x(x,0,0,\delta \omega _\mathrm {p0})$, b) $F_y(0,y,0,\delta \omega _{p0})$, c) $F_z(0,0,z,\delta \omega _\mathrm {p0})$ correspond to particular force components, particle positions and the beam detuning $\delta \omega _\mathrm {p0} = \omega _\mathrm {cp}(0,0,0)-\omega _\mathrm {c0}$ giving maximal cavity transmission if a nanoparticle is placed at the center of the middle hole, i.e., $(x, y, z)=(0, 0, 0)$ see e.g., Fig. 5(c)) and f). Blue and red curves correspond to the nanoparticle diameter 100 nm and 150 nm, respectively. d)-f) The optical potential in units of $k_BT$ corresponding to each of the force curves in a)-c). g)-l) Colormaps showing the full force profiles for various nanoparticle positions in $x$, $y$, and $z$ directions and for spectral shifts covering red ($\delta \omega <0$) and blue ($\delta \omega >0$) frequency detunings for particle diameter 100 nm d)-f) and 150 nm g)-i). The white curves denote the region of zero force corresponding to the stable or unstable region. The blue and red full lines highlight the frequency shift leading to the maximal cavity transmission. Vertical black lines in c), g), i) depict the outer edges of the hole in the $z$ direction.

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Furthermore, for both investigated nanoparticle sizes we observed that the force $F_x(x,0,0,\delta \omega _\mathrm {p0})$ increases faster than linearly as the particle gets further away from the trap center. It corresponds to an increase of the trap stiffness with the particle displacement, $x$ and this behavior is known as the hardening spring stiffness of the Duffing nonlinearity. Such type of hardening stiffness nonlinearity is rather rare because in the majority of typical optical traps (e.g., a focused laser beam, standing wave) the opposite softening stiffness is observed.

This behavior of force profiles can be also seen in the potential profiles $U$ that are depicted in Fig. 3(d)-f). Each curve corresponds to the potential obtained by numerical intergation of the corresponding curve in Fig. 3(a-c)). The stiffening potential in $x$ axis as well as unstable trapping of 150 nm diameter particle in $y$ axis can be clearly seen here.

Figures 3(g))-l) show the force profile for a broad range of cavity detunings $\delta \omega$ for particle diameter 100 nm (Fig. 3(g)-i) and diameter 150 nm (Fig. 3(j)-l) calculated for all three axes. We assumed constant input laser power of 1 mW for both particle sizes and all frequency shifts. The frequency shifts corresponding to the resonance wavelength with a nanoparticle in the middle of the hole are marked by blue and red lines for smaller and bigger nanoparticles, respectively.

The corresponding force components in $x$ and $y$ directions are symmetric with respect to the frequency detuning from the resonance condition at $\delta \omega _\mathrm {p0}$. However, the asymmetry appears in the $z$ direction where the magnitude of $F_z$ becomes significantly stronger for blue detuning. Moreover, as the nanoparticle is displaced from $z=0$, the frequency detuning providing maximal $F_z$ shifts linearly with $z$ up to blue-detuned $\delta \omega$ values, where the nanoparticle confinement along the other axes $x$ and $y$ is not possible anymore. For such blue-detuned $\delta \omega$ the hardening stiffness nonlinearity along $z$ axis appears and one observes the force behavior similar to the self-induced back action (SIBA) [32]. It means, the force changes slowly for a nanoparticle placed near the hole center but rapidly increases as the nanoparticle approaches the hole edges leading to an almost rectangular potential well, see [32].

Furthermore, in the $x$ direction we observe that for 100 nm diameter particle and strong blue detunings $\delta \omega >30$ GHz the original trapping location in the center of the PhC central hole becomes unstable and two new stable positions are formed. Even though these strongly blue detuned forces are weak, one may stabilize the particle in the central hole by increasing the trapping power by a factor $\sim 65 \times$. This is only slightly more than the ratio of the cavity transmission for on resonance and strong blue detuned case, i.e., the peak intensity of the electromagnetic field would be approximately the same as in the case of resonant detuning. In such a case, a double-well having a shallow barrier of $\sim 0.15\ k_BT$ will be created. It demonstrates that a smart interplay between the frequency detuning and incident power could modify the profile of the trapping potential of the microcavity. A similar effect, however, does not appear in any of the two other coordinate axes. To visualize this effect, the positions of zero forces are marked by the white curves in all maps in Fig. 3. One can see that the separation of the trapping sites can be tuned by the light frequency, with the traps being more separated for bigger blue detuning. However, another set of unstable positions appears for bigger shifts $\delta \omega > 100$ GHz (white curves in the top corners of Fig. 3(d)), and for shifts above $\delta \omega > 500$ GHz (data not showed) the stable traps completely disappear and, thus the studied PhC cannot stably trap the particle at all. Similar behavior in $x$ direction appears even for the particle diameters for even more strongly blue detuned frequency $\delta \omega > 90$ GHz.

4. Stochastic simulation of the nanoparticle motion in the PhC

In order to predict the behavior of a nanoparticle trapped in the PhC we performed a set of computer simulations of the particle motion. The evolution of the particle trajectory is calculated based on the discretized Langevin equation [43]. We assumed the ambient pressure of 1 mBar where one expect reasonably fast thermalization of the simulation initial state, however the oscillatory behavior and non-linear effects would be dominate in the consequent data analysis over the effects of gas molecule collisions. For both particle sizes (i.e., force profile) we simulated corresponding trajectories in $xyz$ of length 5 s each with a timestep of 10 ns. Such long trajectories at moderate pressure guarantee that the simulated process is fully thermalized with its environment as the characteristic damping time $\tau =1/\Gamma \simeq 0.14$ ms and the trajectory lenght is 36 000$\tau$. In order to prevent collisions of a nanoparticle with the PhC wall and to keep the nanoparticle confined close to the center, we took the shallowest direction in the trapping potential, i.e. $x$ axis for 100 nm and $y$ axis for 150 nm diameter particle. Moreover, we would like to keep the light power on minimal levels that would prevent the damage of the PhC waveguide structure. Therefore, we place limit on maximal particle displacement from the equilibrium position, i.e. 50 and 17.5 nm for 100 and 150 nm diameter particle in $x$ and $y$ direction, respectively. The smaller particle limit was selected on the edge of the computed force range and for the bigger particle the limit is close to the point where force $F_y$ becomes negative. Furthermore, we assumed that the particle would never reach the trap distance corresponding to 7$\sigma _{\mathrm {thrm}}$ (probability lower than $10^{-11}$), where $\sigma _{\mathrm {thrm}}$ is the standard deviation of the particle in the linear harmonic trap. As a next step we calculate the linear stiffness on the trap center using force profiles shown in Fig. 3 and we scale this stiffness by changing excitation laser power so that the predicted standard deviation of the particle in the optical traps is 7.1 nm or 2.5 nm. Assuming the trapping on PhC resonance the required power is 0.38 mW and 0.70 mW for the smaller or bigger nanoparticle, respectively. This approach may be applied even for any excitation frequency, however, due to smaller trap stiffness the power requirements increase. Such a laser power is at the same level reported previously for trapping using photonic cavity in water [30] or for detection of 1.5 nm small gold nanoparticles in air using nanobeam cavity [44]. However, a thermo-optomechanical pulsation was reported in vacuum at order of magnitude lower power for silicon nitride nanobeam cavity [45]. However, it is our goal here present an initial parameter ranges that would allow optical levitation of a particle in PhC structure. I.e. thanks to the cavity Q factor the excitation laser creates a potential well of sufficient depth to localize the nanoparticle far from the silicon walls.

We used the trajectories achieved from the stochastic simulations of the nanoparticle motion using the nonlinear force profile corresponding to the excitation laser power discussed above and, further, we calculated power spectral densities (PSDs) of nanoparticle positions, see Fig. 4(a))-c). For each size of the nanoparticle and axis of coordinates, the PSD maximum corresponds to the oscillation frequency $\Omega _\mathrm {m}$ of the nanoparticle along a particular axis. The highest oscillation frequencies ($\Omega _{\mathrm {m}, z}/2\pi \approx 2.5-3$ MHz) appear along $z$ axis and the weakest along $x$ axis ($\Omega _{\mathrm {m}, x}/2\pi \approx 1.5$ MHz). In both cases, i.e. along $x$ and $z$ axis the oscillation frequency of the bigger particle is higher due to bigger trapping power. This is not the truth for $y$ axis where the oscillation frequency of the smaller particle is higher that of the bigger particle. This difference would even increase if the excitation power would be the same for both particles. The simple Rayleigh (dipole) approximation of optical forces calculation predicts that for these particle sizes the oscillation frequencies should be approximately the same. In our case the highly localized light intensity hotspots in the cavity modify the simplest view and lead to this behavior.

 

Fig. 4. Power spectral densities of simulated 1D particle motion in $x$ a), $y$ b), and $z$ c) directions for particle diameter 100 nm (blue curves) and 150 nm (red curves), respectively. As we explained in the main text, We assumed trapping power 0.38 mW and 0.70 mW for each particle size.

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5. Optomechanical coupling

Inserting a nanoparticle into the cavity mode induces so-called dispersion coupling [39,46] between the cavity and the mechanical oscillations of the nanoparticle. This coupling is given by a change of the effective refractive index, and depends on the ratio of particle to cavity mode volume. For bulk cavities this ratio is typically $V_\mathrm {par}/V_\mathrm {cav} \approx 10^{-9}$, but for our PhC cavity it is several orders of magnitude larger $V_\mathrm {par}/V_\mathrm {PhC} \approx 10^{-2}$. In the field of cavity optomechanics, the system cavity – nanoparticle is typically described using the following Hamiltonian [46]

Following [47] we denote the linear and quadratic single photon coupling rates as $g_0 = \frac {\partial \omega }{\partial x} x_\mathrm {zpf}$ and $g_q = \frac {1}{2} \frac {\partial ^2 \omega }{\partial x^2} x_\mathrm {zpf}^2$ and the zero-point fluctuation of the particle motion $x_\mathrm {zpf} = \sqrt {\hbar /2 m \Omega _\mathrm {m}}$ with mass $m$. Inserting the Eq. (3) into Eq. (2) we get a Hamiltonian in linearized form

Using our numerical model we characterized the magnitude of the optomechanical coupling. Figures 5 shows shift of cavity resonance in the presence of nanoparticle using transmission spectra of the cavity relatively to the empty cavity. These transmission spectra were calculated for two nanoparticle diameters 100 nm (first row) and 150 nm (second row), placed along $x$ (first column), $y$ (second column), or $z$ (third column) axis. The red curves show the frequency shift $\delta \omega$ giving maximal transmission for particle displacements along $x$, $y$, and $z$ axes. This frequency shift $\delta \omega$ we employed to calculate the linear optomechanical coupling rates $g_0$ in Fig. 5 g)-i). For the calculation of the coupling rates we considered angular frequencies of particle oscillation in each axis which we obtained by means of the stochastic motion simulation, see Section 4 and Fig. 4 for more details. Even for very high oscillatory frequencies $\sim$ 3.5 MHz estimated for 150 nm diameter particle in $z$ direction we may reach linear single photon coupling rate $g_0$ up to 1 MHz assuming particle displacement of 115 nm, i.e., to the cavity edge. This, however, is very hard to achieve experimentally. A more experimentally feasible option would be to displace the nanoparticle using electric field. The chip design allows for manufacturing electrodes at a distance closer than $1\, {\mathrm{\mu} \mathrm{m}}$. To displace the nanoparticle 5 nm from the center, the needed electric field is $6\, {\mathrm{V}/\mathrm{\mu} \mathrm{m}}$. At this displacement the linear single photon coupling rate is $g_0 \approx 100\, \rm {kHz}$. Please note that typical values $g_0$ for bulk cavities are in the order of several Hz [39,46,48,49] and for similar PhC cavity designs $g_0$ is of the order of tens of kHz [26,50].

 

Fig. 5. Colormaps of the PhC waveguide spectral transmission with a nanoparticle of diameter 100 (a-c) or 150 nm (d-f) placed in the middle hole and displaced from its center along $x$ a), d), $y$ b), e), or $z$ c), f) directions. The frequency shift $\delta \omega$ of the illumination frequency is taken from the empty cavity resonant frequency $\omega _{c0}/2\pi = 192.8$ THz Red curves denote the frequency shift giving the maximal transmission for the corresponding particle displacement from the hole center. Color coding range is the same in all panels (a-f) and corresponds to colorbar next to panel f). g)-i) Linear single photon coupling rates of smaller (blue) and larger (red) nanoparticle displaced from the center of the middle hole in the $x,y$, and $z$ direction, respectively.

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In the $x$ and $y$ directions the linear coupling is approximately 15 and 2 times smaller than in the $z$ directions. The value of quadratic single photon coupling rate $g_q$, calculated from the second derivative of the resonance shift due to the particle motion, is very weak with value in the order of several Hz in all three axes and thus the linear coupling rate dominates.

As we have shown in the previous sections the nanoparticle may be stable trapped in the cavity center and in combination with the relatively strong dispersive coupling the so called SIBA effect may be induced [32,33,50]. To cool the nanoparticle we need it to interact witch its velocity [9]. For cooling with SIBA this is achieved when the phase delay between the nanoparticle motion and the cavity buildup or decay is $\pi /2$ and thus the decay rate $\kappa$ of the cavity is similar to the nanoparticle oscillation frequency. For $\Omega _{m,z}/2\pi \approx \kappa /2\pi \approx 3\, \rm {MHz}$ we would need a cavity with quality $Q \approx 10^7$.

6. Conclusions

We proposed and theoretically analyzed a promising design of a vacuum optical trap inside a micro-cavity designed in the one-dimensional (1D) silicon photonic crystal waveguide with a quadratically modulated width. The numerical results predict a cavity quality factor of $Q\sim 8\times 10^5$ and an effective mode volume of approximately 0.16 $\mu \mathrm {m^3}$. If the nanoparticle is placed into the central hole of the PhC, strong optical forces arise and confine the nanoparticle in all three dimensions, however, the nanoparticle size is limited by the size of the cavity and strong optical forces present close to the cavity surface. Appropriate frequency detuning of the cavity mode and incident power enables modification of the spatial shape of the confinement potential. The optomechanical interaction between nanoparticle and cavity was analyzed showing that the linear optomechanical coupling is dominant and with value several orders of magnitude larger than value typical for bulk cavity. The strong optomechanical interaction of nanoparticles and micro-cavity may induce the SIBA [33] effect and may be used also for cooling the motion of levitated nanoparticles. At the time of writing this paper the quality factors of PhC cavities are constrained to $Q \approx 10^6$ due to manufacturing limitations, but in theoretical calculations, it is possible to achieve very high cavity quality factors of up to $Q_c \approx 10^{9}$ [36], where the strong single photon coupling regime at the ambient pressure $10^{-6}$ mbar can be achieved. This allows for a coherent exchange between the cavity field and the oscillating nanoparticle to occur within the decay times, which is necessary for true quantum control of optomechanical systems [48,51,52] and which is very hardly accessible using common cavities based on bulk mirrors.