Single and two-mode mechanical squeezing of an optically levitated nanodiamond via dressed-state coherence

Nonclassical states of macroscopic objects are promising for ultrasensitive metrology as well as testing quantum mechanics. In this work, we investigate dissipative mechanical quantum state engineering in an optically levitated nanodiamond. First, we study single-mode mechanical squeezed states by magnetically coupling the mechanical motion to a dressed three-level system provided by a Nitrogen-vacancy center in the nanoparticle. Quantum coherence between the dressed levels is created via microwave fields to induce a two-phonon transition, which results in mechanical squeezing. Remarkably, we find that in ultrahigh vacuum quantum squeezing is achievable at room temperature with feedback cooling. For moderate vacuum, quantum squeezing is possible with cryogenic temperature. Second, we present a setup for two mechanical modes coupled to the dressed three levels, which results in two-mode squeezing analogous to the mechanism of the single-mode case. In contrast to previous works, our study provides a deterministic method for engineering macroscopic squeezed states without the requirement for a cavity.

Recently, levitated nanoparticles with internal degrees of freedom, such as the nitrogen-vacancy (NV) center with a single spin, have been studied theoretically to test quantum wavefunction collapse models [23,24] and quantum gravity [25] in vacuum. More recently, optical levitation of nanodiamonds in low vacuum has been demonstrated experimentally [26,27], paving the way for preparing quantum states of mechanically oscillating levitated nanoparticles.
In the present work, the nanoparticle mechanical motion is coupled to the single NV center spin via a magnetic field gradient, without requiring a cavity [23,24]. Distinct from the works cited above [62,63], our method does not require a measurement-based technique, but instead relies on a microwave field-induced spinstate coherence for generating steady-state mechanical squeezing in both the single-mode and two-mode cases. By applying two microwave fields coupling the 0ñ | and 1  ñ | states of the NV center ground-state triplet [57], a dressed three-level system is created to induce a two-phonon transition in the mechanical oscillator, an interesting effect which has not been studied before, to the best of our knowledge, in spin-optomechanical systems. Our work thus promotes the subarea of levitated nanomechanics with a new method of creating macroscopic nonclassical states [23][24][25].
To arrive at our results, we employ a master equation approach to describe the mechanical motion, by tracing out the spin degree of freedom in the Born-Markov approximation. This approach is enabled by applying optically induced dissipation [65] to the spin triplet states leading to relaxation rates much stronger than the spin-mechanical coupling. For the single-mode case, we find remarkably that quantum squeezing is achievable at room temperature with experimentally achievable ultrahigh vacuum and feedback cooling techniques [13]. For moderate vacuum, quantum squeezing is possible with precooled phonon occupation number. For the two-mode case, we propose a setup such that both modes are coupled to the dressed states in exactly the same way as for the single-mode case. The analytical results for both the single-mode and the twomode squeezing are equivalent to each other. We also present numerical results in a wide range of parameters for single-mode squeezing, which are applicable to the two-mode case.
The analysis presented using an optically levitated nanodiamond is quite general, therefore the proposal can also be extended to related systems, such as, nanodiamonds using magneto-gravitational traps [66] or Paul traps [67,68], which avoid optical scattering, or a single NV center coupled to a cantilever [57].
2. Single NV center coupled to one mechanical mode 2.1. The model We consider a single NV center nanodiamond optically trapped in vacuum and executing harmonic center of mass motion along all three directions in space, as shown in figure 1 with the gradient B 0 is applied to couple the mechanical motion and the electron spin of the NV center. The magnetic field is assumed to be lying in the z-y plane of the spin axes and making an angle j with the z axis. The Hamiltonian of the system is Figure 1. The configuration considered in section 2. The green circle denotes an optically levitated nanodiamond oscillating in a harmonic potential (black curve) along the x coordinate. A magnetic field is also applied along the same direction. The arrow on the circle denotes the direction of the spin S z axis corresponding to an NV center contained in the nanodiamond. We note that the spin axes are not aligned with the coordinate axes in general. As shown in the figure, S S , y z and B x all lie on the same plane. Not shown is S x , which is perpendicular to B x and points out of the plane of the paper.
where the coupling constants g ge sin and the dressed states are   a  b  c   sin  0  cos  ,  ,  cos  0  sin  ,  6 q q , and tan 2 2 2 The eigenvalues of the dressed states are The dressed states of equation (6) are shown in figure 2(b), along with the oscillator phonons of energy m w which couple to the NV center via the terms in the second line of equation (5). The effective detunings 1 D and 2 D will be derived later in the text.
We note that in our model the coupling field 1 W provides external control of the hybridization and eigenfrequencies of the single spin levels. In the eigenbasis of H NV , the mechanical motion couples to two transitions of the eigenstates, which is promising for creating mechanical squeezing because of the implied twophonon transition. A related scheme has been considered for coherent three-level atoms coupled to a cavity field via a two-photon transition for quantum noise quenching and optical field squeezing [69]. Finally, we note that the orientation of the magnetic field gradient only adds a phase to the mechanical-spin coupling in the eigenbasis.

Driving-induced dissipation
The electron spin in the NV center is notable for its long coherence time (on the order of 1 ms) even at room temperature [65,70,71]. In order to induce fast dissipation in the spin system, which is necessary for generating steady state mechanical squeezing, we apply two optical fields with the same Rabi frequency p W driving the ground-state spin levels to the excited states E 1 ñ | and E 2 ñ | via spin-conserving transitions [72], which de-excite to states 1  ñ | with a decay rate 1 g , and to the state 0ñ | with an effective decay rate 0 g , as shown in figure 3. By considering spin-mechanical couplings g g , s c ( )and microwave fields 0 W ( , 1 W ) much weaker than the optical Rabi frequency p W , we find the nonzero steady-state density matrix elements in the dressed eigenbasis of ). As can be seen from equations (7)-(10), dissipative driving can be used to control the the populations and coherences for the NV dressed states. The mechanical oscillator interacts with the NV spin due to the presence of the magnetic field. In the steady-state, therefore, the mechanical frequency is shifted by the NV spin, while mechanical motion can be engineered via the mechanicalspin interaction through the driving-induced dissipation. We substantiate these statements below.

The reduced master equation of the mechanical oscillator
In the interaction picture, we can write equation ( -, and we have used the rotating-wave approximation. The approximation is valid when , , , . We now trace out the spin degree of freedom to obtain the reduced master equation for the mechanical oscillator density matrix m r , which is our system of interest i.e.  where Tr s denotes the trace over the spin degree of freedom. We note that the steady-state density matrix elements ba r and bc r , due to the fast dissipation and strong Rabi frequencies, are zero to the lowest order. The first-order perturbation of these quantities are given by the spin-mechanical interaction H I in appendix B. By substituting for bc r and ba r in equation (12), we obtain (also shown in figure 2 (a)), and excited states E 1 ñ | and E 2 ñ | . The optical fields with Rabi frequency p W pump the population from 1  ñ | to E 1 ñ | and E 2 ñ | , which decay to 0ñ | and 1  ñ | with effective decay rates 0 g and 1 g , respectively.
corresponds to the standard Lindblad operator, and m g is the effective decay rate of the mechanical oscillator and n 1 e 1 ) is the effective mean phonon number due to both the surrounding gas and the trapping beam [12]. The mechanical fluctuations due to the optical pump p W are negligible as shown in the following discussion. In equation (13), the terms proportional to A -(A + ) describe the dissipation-induced cooling (heating) due to coupling of the mechanical motion to the transitions from cñ The terms proportional to δ are the mechanical frequency shifts due to the mechanical-spin interaction. The terms proportional to S j or S j * denote the mechanical squeezing via a twophonon transition using the single NV spin. The explicit expressions of A -, A + , δ, and S j are derived in the appendix B.

System dynamics-analytical results
To study the system dynamics of the mechanical oscillator, we derive from the reduced master equation (13) that The steady-state solutions to the above equations are given by To obtain maximum mechanical squeezing, we define the quadrature variance rotated in the phase-space plane such that and the criteria for quantum squeezing is given by figure 2(b)). Therefore, steady-state squeezing is possible since the spin-mechanical cooling dominates over the spin-mechanical heating. We obtain to the first-order the quantities: The physical origins of these terms can be seen from the above expressions in this limiting case. The cooling term is due to the absorption of phonons from the mechanical oscillator and proportional to the population in the lowest level cc r . The dressed-state spin coherence (proportional to ac r ) gives rise to squeezing. The mechanical frequency shift is due to a non-resonant two-phonon process between the states añ | and bñ | . Here we also used the condition 0 1 G » G which can be satisfied by applying a strong resonant microwave field coupling the excited spin triplet states [72,73] to suppress the dissipation from states . Therefore, the steady-state mean phonon number due to the dissipative cooling is given by where the cooling rate is given by . By using the above conditions, we obtain The quadrature variance is then given by Using equation (27), we can see that for 1 , the quadrature x 2 D ( ) can be smaller than 1/4 when d d 0 ss á ñ † , which demonstrates quantum squeezing of the mechanical motion near the ground state. To obtain quantum ground state cooling, we require the strong cooperativity condition, i.e., g n 1 (24) and (25), and strong coupling condition [57], i.e., g , ). Similar to that in [57], the quadratic frequency shift can be completely suppressed by choosing a suitable dressed-state angle θ such that 1 sin tan 0 Distinct from [57], the favorable angle θ can be satisfied with many sets of parameters Δ, 0 W and 1 W rather than one set of parameters. The reason is that we have three free parameters Δ, 0 W and 1 W (instead of two in [57]) which are constrained by two conditions bc m w w = and 1 sin 2 q . This is another advantage of having the extra control parameter 1 W . We consider numerical parameters explicitly in the next section for quantum ground state cooling and quantum squeezing. We first consider the case of no coupling ( 0 1 W = ) between the 1  ñ | NV states (see figure 2) as this coupling is not essential to the physics, and only provides fine control as shown below. We plot the numerical results for d d ss á ñ † , x 2 D ( ) , A -, and A + using the solutions equations (16) and (17). First, we observe that ground-state cooling [57] is possible with strong cooperativity, i.e. g n 1 as shown in figure 4(a). In this case the cooling processes dominate the heating. Second, we observe in figure 4(b) that the quadrature variance , which implies quantum squeezing of the one quadrature of the mechanical oscillator. We find that the region for which the quantum squeezing occurs qualitatively agrees with the region d d 1 ss á ñ  † , as discussed analytically in section 2.4. To understand the cooling and the squeezing, we plot Aand A + in figure 5. As m 0 w W varies between 0 and 2 , we see an optimal cooling limit is obtained by balancing the cooling and heating effects from the single spin. . We find quantum squeezing can be achieved when n 3 10 th 3´, which corresponds to an initial temperature ∼0.1 K. This initial temperature of the mechanical oscillator may be achieved with cryogenic techniques or by using feedback cooling [8,12,13]. Furthermore, we plot the quantity and g m w keeping other parameters constant, in figure 6(c). We see from the figure that stronger g is preferred for realizing quantum squeezing as long as the Born-Markov approximation is valid.

Experimental accessiblity
Remarkably, we find, at initial room temperature environment for the mechanical oscillator, that quantum squeezing is feasible with our system for ultrahigh vacuum with feedback cooling. In ultrahigh vacuum ( 10 8 <mbar), as demonstrated recently for an optically levitated nanoparticle [13], the gas damping rate is on the order of 10 g 6 g~-Hz, which corresponds to 10 m g 12 w g~. As an example, we consider an optically levitated nanodiamond with a radius 50 nm and a mechanical oscillation frequency 2 1.0MHz m w p = along x axis in a magnetic field gradient of ∼10 5 T m −1 . Recent experiment has produced a strong magnetic field gradient of ∼10 6 T m −1 in a 23 nm position shift from a magnetic tip [74]. For a magnetic field gradient of ∼10 5 T m −1 and an initial mean phonon number 10 3 , the maximum magnetic field strength is about 10 −5 T. Hence the maximum frequency splitting on the levels 1  ñ | is on the order of 1 MHz, much smaller than the zero-magnetic field splitting 2.88 GHz between 1  ñ | and 0ñ | . Therefore, the strong magnetic field gradient does not generate a strong magnetic field to change the groundstate frequencies appreciably in our model.
To obtain an optical-induced dissipation rate 4 1.  [65]. For 8MHz p W~, the corresponding optical pump power is smaller than 1 W m [75], which has a negligible effect on the mechanical motion fluctuation due to the optical scattering [3].
To reduce the mean phonon number of the mechanical oscillator due to both the surrounding gas and the optical trapping field, feedback cooling of the nanoparticle can be employed by introducing extra mechanical damping from feedback [8,12,13]. For completeness, we briefly discuss the implementation on feedback cooling of an optically levitated nanoparticle [8,12,13]. The optical trapping beam is scattered by the nanoparticle and the scattered light carries the information of the nanoparticle position. By doing an interferometric detection of the scattered and unscattered light, the position of the nanoparticle can be measured [5]. The position information is then fed back to modulate the intensity of the trapping beam which controls the stiffness of the harmonic trap. Feedback cooling is achieved by increasing the trap stiffness when the nanoparticle moves away from the trap center and reducing it when the nanoparticle moves towards the center. The final phonon number of the nanoparticle is determined by its initial phonon number, the gas pressure, the detection efficiency, and the feedback strength [12]. We estimate that with a feedback-induced mechanical damping 10 Hz fb 3 g~, quantum squeezing is achievable at room temperature when the initial phonon occupation number is reduced to n 2 th~. These numbers correspond to a strong cooperativity g n 10 . Our prediction is within the reach of a recent experiment, where a final phonon number of 63 has been demonstrated with feedback cooling [13].
We note that using the driving field 1 W , it is possible to control the energy difference between dressed states.
Our model requires 0 ab w > for the rotating-wave approximation to be valid. We plot the value of ab w versus 0 W and 1 W in figure 6(d) and we find the condition is satisfied for the parameter regime where mechanical quantum squeezing can be engineered.
To summarize, single-mode quantum squeezed mechanical state is feasible using our model in ultrahigh vacuum, even at room temperature.
3. Single NV center coupled to two mechanical modes 3.1. The model The optically levitated nanodiamond has three harmonic oscillations independent of each other for small oscillation amplitudes, which is an excellent platform for multimode mechanical quantum state engineering. By applying magnetic field gradient in both x and y directions of the harmonic oscillations, as shown in figure 7, we can couple two mechanical modes to the single spin of the NV center nanodiamond. The magnetic field gradient We assume x and y coordinate axes of the mechanical motions are in the plane of the spin operator components S x and S z . The angles between B x and S z , and between B y and S z , are j and 2 p j -, respectively. The interaction Hamiltonian of the single spin and the mechanical motions are given by (see appendix C) is the annihilation operator in x (y) direction and g 1z , g 1x , g 2z , and g 2x are the spin-mechanical coupling strengths for the two modes in the spin orientations defined in the appendix C. The electron spin dynamics is the same as in the single mechanical mode case, where the spin is driven by two microwave fields coupling between states 0ñ | and 1  ñ | , and an effective field coupling between states 1 + ñ | and 1 -ñ | . In the eigenbasis of H NV , the total Hamiltonian is given by where we assume the two modes have the same frequency m w , and 4 j p = for simplicity such that g g sin . This configuration is possible for a nanoparticle trapped in an optical field, where the frequencies of two transverse modes can be made very close to each other [8]. The other quantities are the same as in the single mechanical mode case. The second line in the above equation describes the interaction between the electron spin S z with the two mechanical modes in the dressed-state basis, similar to that of the single-mode case if we replace d with d d 2 ) . The last term in equation (29) describes the coupling between the electron spin S x with d d = , such that g x =0. Physically, this corresponds to a magnetic field with the proper position dependence along the direction of S z . In the following, we study how to engineer two-mode mechanical squeezed states under this condition. For slightly different magnetic field gradients B 0 and B 1 , we will consider the first-order perturbation of the extra interaction term with both analytical and numerical results.
By considering the mechanical frequency m w resonant coupled to the transition from bñ | to cñ | and far detuned from the other transition of añ | to bñ | , we can write H, in the interaction-picture under the rotatingwave approximation as are the same as the single mode case. The advantage of this configuration is such that the superposed mode d d 2 can be cooled efficiently to its ground-state similar to the single mode case while squeezing process is engineered via the two-phonon transition of the superposed mode mediated by the dressed-state spin levels.

Analytical results of the two-mode mechanical oscillator
At the steady-state of the spin states, we can trace out the spin degree of freedom to obtain the reduced master equation for the two-mode mechanical oscillator similar to equation ( where the coefficients are given in equations (B.5)-(B.9), and m m j g g = is the decay rate, assumed to the same for both mechanical modes. The terms, such as the cooling, the heating, and the squeezing, in the reduced master equation for two mechanical modes are similar to those of the singe-mode case. We are interested in the steadystate properties of the two-mode system and we find at the steady-state the relevant mean values are . To obtain the maximum degree of two-mode squeezing, we choose 1 q and 2 q such that the two-mode quadrature variance is given by 1 . 34 áD ñ = áD ñ ( ) All the discussions about squeezing a single-mode mechanical oscillator apply to the two-mode case exactly under the assumption that the interaction between the spin component S x and the two mechanical modes are negligible. This assumption we made in the two-mode case is valid in the rotating-wave approximation. We also verified that d d 0 for the parameter regime of interest for the requirement of steady-state of the two modes.
For nonzero g , x ab bc w w  , we see from equation (29) that the last term results in small frequency shifts of levels a b c , , ñ ñ ñ | | | . These frequency shifts does not change much the steady-state population ij r in the dressedstate basis since g g , . However, the frequency shifts result in frequency uncertainty in bc w , which may affect the resonant coupling between levels bñ | , cñ | and b b 2 . To the first-order perturbation, the frequency uncertainty is evaluated to be |under the rotating-wave approximation.

Numerical results of the two-mode mechanical oscillator
In this section, we present numerical results of the two mechanical modes coupled to the single NV center electron spin. In our model, the two modes couple to S z for mechanical cooling and squeezing, and to S x which leads to frequency shifts in the dressed states. Ideally, we find the proper magnetic field gradients which eliminates the coupling to S x . Numerically, we plot the results of the steady-state system variables for both the ideal case and non-ideal case for comparison.
In figure 8, we plot the final phonon number of the superposed mode b b 1 2 + and the two-mode quadrature variance u 2 D ( ) for g x =0 and g g 50 for a range of parameters, which means the superposed mode b b 1 2 + is indeed cooled close to its quantum ground state. We also find that the two-mode quadrature variance can be smaller than 1/4 meaning two-mode quantum squeezing, and the two-mode quadrature variance equals to the single-mode quadrature variance for the same parameters by comparing figure 4 (b) with figure 8 (b). Second, when g g 50 x z = , we find both the final phonon number of the superposed mode and the two-mode quadrature variance are slightly reduced due to the frequency shifts.
In figure 9, we plot the range of u 1 4 for different values of m 0 w W and m 1 w W at g x =0 and g g 50 x z = . By comparing the two figures, we find the range of quantum squeezing is slightly reduced due to the mechanical coupling to S x . Therefore, the two-mode mechanical squeezed states can be engineered with our model even if in the presence of imperfections, such as coupling to S x .

Discussion
In summary, we presented a method for engineering two-mode mechanical squeezed states under similar conditions required for the single-mode case by using the proper magnetic field gradients. Therefore, the discussions on the experimental accessibility in the single-mode case are applicable to the two-mode case here, such as room temperature squeezing with feedback cooling. The mechanical coupling to S x due to the imperfect magnetic field gradients results in small dressed-state frequency uncertainty and weak mechanical squeezing reduction. As an application, the two-mode mechanical squeezed states are useful for sensitive phase measurement beyond the standard quantum limit in an interferometric setup [49,77].

Conclusion
In this paper, we have investigated quantum state engineering of an optically levitated nanodiamond coupled to a single NV center ground-state electron spin. We considered quantum state engineering of both single-mode and two-mode mechanical motions. Both analytical and numerical results have been obtained to show that single-mode squeezed states of the mechanical oscillator is feasible with the state-of-art experiments even at room temperature. We have shown that our scheme for single-mode squeezing can be readily extended to the case of two-mode squeezing, which is of interest for precision measurements.
In conclusion, we presented an experimentally realizable method for engineering both single-mode and two-mode mechanical squeezed states in an optically levitated nanodiamond via dressed-state coherence. Our work advances macroscopic quantum state engineering in cavity-free systems, and paves the way for sensitive metrology with squeezed mechanical states.   ( ) is the zero-point fluctuation in x y ( ) direction, and d d 1 2 ( ) is the annihilation operator in x y ( ) direction.