Cooling Flexural Modes of a Mechanical Oscillator by Magnetic Trapped Bose-Einstein Condensate Atoms
CCooling Flexural Modes of a Mechanical Oscillator by Magnetic TrappedBose-Einstein Condensate Atoms
Donghong Xu , , and Fei Xue , ∗ Anhui Province Key Laboratory of Condensed Matter Physics at Extreme Conditions,High Magnetic Field Laboratory, Chinese Academy of Sciences,Hefei 230031, Anhui, People’s Republic of China University of Science and Technology of China, Hefei 230026, People’s Republic of China Collaborative Innovation Center of Advanced Microstructures,Nanjing University, Nanjing 210093, People’s Republic of China (Dated: November 14, 2018)We theoretically study cooling of flexural modes of a mechanical oscillator by Bose-Einstein Con-densate (BEC) atoms (Rb87) trapped in a magnetic trap. The mechanical oscillator with a tinymagnet attached on one of its free ends produces an oscillating magnetic field. When its oscillatingfrequency matches certain hyperfine Zeeman energy of Rb87 atoms, the trapped BEC atoms arecoupled out of the magnetic trap by the mechanical oscillator, flying away from the trap with stolenenergy from the mechanical oscillator. Thus the mode temperature of the mechanical oscillator isreduced. The mode temperature of the steady state of the mechanical oscillator, measured by themean steady-state phonon number in the flexural mode of the mechanical oscillator, is analyzed. Itis found that ground state (phonon number less than 1) may be accessible with optimal parametersof the hybrid system of a mechanical oscillator and trapped BEC atoms.
I. INTRODUCTION
Mechanical oscillators play an important role in ap-plications of ultra-high precise measurements and sens-ing of displacements [1], masses [2] and nuclear spin po-larization fluctuations [3, 4]. Mechanical oscillators arealso expected to play an important role in exploring var-ious meso-scopic quantum phenomena. They have beenused in study of fundamental quantum physics, such asgenerating entangled states and deterministic entangle-ment [5, 6]. Mechanical oscillators could serve as a quan-tum data bus in quantum computations [7, 8]. Reducingthermal vibrations of mechanical oscillators is importantin their practical applications and fundamental physicsstudies of micro and nano-mechanical oscillators. There-fore, great efforts are done in cooling down the thermalvibrations of mechanical oscillator to a temperature ofquantum regime both in theoretical and experimental re-search. On the experimental side, although passive cool-ing of a mechanical oscillator down to quantum groundstate has been demonstrated in recent experiment by us-ing a microwave-frequency mechanical oscillator of giga-hertz frequency (about 6 GHz), using standard dilutionrefrigerator (20mK) [9]. For mechanical oscillators withoscillating frequencies at MHz, the passive cooling by di-lution refrigerator is hard. Significant effort has beendevoted to developing active cooling methods in hybridsystem, such as cooling mechanical oscillator by opticalfield enhanced by a cavity [10], by single electron transis-tor [11], or by coupling it to a superconduction LC circuit[12]. Proposals suggest that the ground state cooling ofmechanical oscillator is possible [13–15]. Recently, the ∗ xuef@hmfl.ac.cn hybrid system cooling evokes some attentions as an in-triguing cooling proposal for mechanical oscillators, suchas atom-assisted cooling in opto-mechanics system. Spe-cially, the atom based hybrid system was proposed fewyears ago [16, 17]. Recently, by applying the Landau-Zener theory, a cooling scheme in such hybrid system, isproposed [18].Coupling between mechanical oscillators and alkaliatoms Rb in gas state via Hyperfine Zeeman splittingof these atoms at room temperature was experimentallydemonstrated [19]. A few years ago, interactions betweena mechanical oscillator and Bose-Einstein-Condensation(BEC) atoms via surface force of the mechanical oscilla-tor was realized [20]. Recently, magnetic cantilever tipresonant with trapped cold atoms is presented [22]. Themagnetic trapped BEC [23] and mechanical oscillatorson atom chips are reviewed in Ref. [21]. In this work,we study the cooling of flexural modes of a mechanicaloscillator by coupling them to Bose-Einstein Condensateatoms ( Rb ). These atoms are trapped in a magnetictrap. A magnetic tip is attached on the free end of me-chanical oscillator. The tiny magnetic tip produces an os-cillating magnetic field around the center of the magnetictrap while the mechanical oscillator vibrates. This os-cillating magnetic field induces trapped atoms’ spin flip.These ‘flipped’ atoms then escape from the magnetic trapwith stolen energy from the mechanical oscillator. Theseatoms are excited to untrappable states, then freely ex-pand, get away from the trap [24, 25].The paper is organized as follows. In Sec. II, we in-troduce the hybrid system and its Hamiltonian. For me-chanical oscillator coupling to a single atom in the trap,the Hamiltonian of this hybrid system can be describedwith the Jaynes-Cummings model. For mechanical os-cillator coupling to an ensemble of cold atoms undis-tinguishably, the coupling constant between mechanical a r X i v : . [ qu a n t - ph ] O c t oscillator and the atoms is enhanced [16, 18, 26]. Thehybrid system is described by Tavis-Cummings Hamil-tonian. In Sec. III, we draw an analogy between theexcited atoms and the output of atom laser, then themaster equation for phonon number in mechanical oscil-lator’s fundamental mode is derived. The expression ofmean steady-state phonon number is obtained. In Sec.IV, by applying some practical parameters, it is foundthat the mechanical oscillator could be cooled down toits ground state. II. HYBRID SYSTEM OF A MECHANICALOSCILLATOR AND BEC ATOMS
The hybrid system is illustrated in Fig. 1(a). In thiswork, the atoms are trapped in a magnetic trap. Thecenter of trap is at a distance d from the mechanicaloscillator along the y-axis. A single-domain magnetic tipcreates a magnetic field with gradient G m while the me-chanical oscillator vibrates. The mechanical oscillator os-cillates along the y-axis: β ( t ) = β cos( ω m t + ϕ m ). Here, β is the amplitude of vibrations, ϕ m is the phase and ω m is the frequency of mechanical oscillator. We choose theorientation of the Cobalt magnet tip so that B ac m ( t ) is per-pendicular to the static magnetic field B around centerof the trap. Because of the shape anisotropy of tip, mag-netic moment µ m is spontaneously oriented along its longaxis, as shown in Fig. 1(a), in yellow color. Let the staticmagnetic properly set along the z-axis: B = B e z with B is the amplitude of the applied static magnetic field.Approximating the magnetic tip as magnetic dipole, wehave the field gradient which is similar to [16, 18, 27]: G m = 3 µ | µ m | / (4 πd ) around center of the trap. µ is the permeability of vacuum. Assuming the oscillatingamplitude β (cid:28) d thus the mechanic motion of the me-chanical oscillator induces an oscillating magnetic field inthe x-direction as illustrated in Fig. 1(a): B acm ( t ) = G m β ( t ) e x . (1)For BEC [Fig. 1(a), in red color], the static magneticfield B is applied along the z-axis on a group of Rb atoms, separating their degenerate hyperfine levels: F = I + J . Here, the electronic angular momentum J = 1 / I = 3 /
2. Denoting the hyperfine levelsas | F, m F (cid:105) , with m F being the projection of F on theaxis of the applied static field B , the hyperfine Zeemanlevels depend on the amplitude B ≡ | B | of the appliedmagnetic field in a special case of Breit-Rabi formula [23]: E | F,m F (cid:105) = ( − F A hf h (cid:114) m F BB hf + ( BB hf ) . (2)In Eq. (2), we have offset the zero of energy to themean of zero field energies of the atomic spin statesF = 1 and F = 2 for convenience. The zero-field energysplitting A hf = ( E F = 2 − E F = 1 ) /h ≈ .
835 GHz, and F = ω L Liquid stateVapor state 𝑚 𝐹 −2 −1 F = (a) β(t) μ m Mechanical oscillator 𝑧 𝑦 B m ac (t) B Rb 𝑥 (c)(b) d Τ ω L 𝜋 G H z FIG. 1. (Color online). (a) Schematic layout for the hy-brid system. The magnetic trapped BEC atoms is in redcolor and the mechanical oscillator is in green color. The tinyCobalt magnetic tip (in yellow color) attached on the free endof the mechanical oscillator produces an oscillating magneticfield around center of the trap while the mechanical oscilla-tor vibrates. The single-domain magnetic tip creates a field B acm ( t ) and couples to atomic spin of atoms. (b) Hyperfinestructure level of Rb in the static magnetic field B . Thetrappable states are in blue color and the untrappable statesare in red color. The hyperfine level | , − (cid:105) are coupled outto the mechanical oscillator when the Larmor frequency ω L is tuned to the oscillation frequency of mechanical oscillator ω m , and then, atoms are excited from | , − (cid:105) to | , (cid:105) with atransition rate Γ. (c) Field dependence of Larmor frequency ω L / π = ( E | , − (cid:105) − E | , (cid:105) ) /h between state | , − (cid:105) (liquidstate) and | , (cid:105) (vapor state). B hf ≡ A hf h/ (2 µ B ) ≈ h is the Planck con-stant, µ B is the Bohr magneton.We consider a trap potential: V T (x , y , z) =( m/ ω x + ω y + ω z ) [28]. Here, m is the atomicmass of Rb , ω x = ω y is transversal trapping frequency, ω z is the longitudinal trapping frequency. For such a puremagnetic trap, the states | , (cid:105) , | , (cid:105) , | , (cid:105) and | , − (cid:105) are attracted to the local minimum of the magnetic trap,while the states | , − (cid:105) , | , − (cid:105) , | , (cid:105) and | , (cid:105) are freeout of the magnetic trap. These eight hyperfine spinlevels of trappable (in blue color) and untrappable (inred color) and the process of atoms’ transition for Rb induced by the mechanical oscillator are shown in Fig.1(b).Because the other two trappable states | , (cid:105) and | , (cid:105) tend to change into untrappable states by two-body col-lision among atoms [23] and the collision loss is lower inF = 1 than F = 2 [16], the cold atoms in the trap are pre-pared in state | , − (cid:105) initially. We choose | , − (cid:105) ≡ | (cid:105) ,named as liquid state. We choose | , (cid:105) ≡ | (cid:105) , named asvapor state. Jaynes-Cummings Model.
We assume that all BECstate atoms are prepared in the liquid state | (cid:105) . Eachatom changing from the liquid state | (cid:105) to the vapor state | (cid:105) is expected to taken energy from the mechanical oscil-lator. The energy transfer rate depends on atoms’ “tran-sition rate”. The trapped BEC atoms in the liquid stateare assumed to be in the Thomas-Fermi (TF)regime. For B (cid:28) B hf , which is reliable in most BEC experiments[23], the Larmor frequency between the liquid state andthe vapor state can be written as: ω L = µ B | g F | B (cid:126) , (3)to the first order of B , which is plotted in Fig. 1(c).Here, g F is the Land´ e g-factor of atomic spin state F . As g I (cid:28) g J , g F can be expressed by: g F = g J [ F ( F + 1) + J ( J + 1) − I ( I + 1)] / [2 F ( F + 1)][29]. g J and g I are theLand´ e g-factors for total angular momentum of electronand nuclear spin, respectively. Therefore, the Larmorfrequency ω L can be tuned by adjusting the amplitude ofstatic field B , hence the detuning δ = ω m − ω L can alsobe tuned.The oscillating magnetic field B acm (t) interacts with theliquid state and vapour state via the Zeeman Hamilto-nian: H z = − µ · B acm ( t ) with µ = − µ B g F F [16]. In thecase of that the direction of field gradient at the centerof the trap is along x-axis, as shown in Fig. 1(a), theHamiltonian H z reads: H z = µ B g F F x G m β ( t ) . (4)We replace the mechanical oscillator’s displacement β ( t ) by the operators: β ( t ) → a qm ( a + a † ) with a qm ≡ (cid:112) (cid:126) / m eff ω m being the rms amplitude of the mechani-cal oscillator’s quantum zero-point fluctuation, m eff be-ing the effective mass of the oscillator, and a ( a † ) beingthe bosonic annihilation (creation) operator for phononsin the flexural mechanical mode.The two states of liquid state | (cid:105) and vapor state | (cid:105) can be separated from the other six hyperfine sub-levelsby the quadratic Zeeman effect [16, 18, 30]. Therefore,we could define a pseudo spin with the liquid state andvapour state as its eigenstate of σ z . with σ z is the zcomponent of Pauli matrix. With √ g F F x replacing by σ x and applying the rotating-wave approximation in Eq.(4), the interaction Hamiltonian H I of the mechanicaloscillator interacting with the two-level system can be written as [16, 18, 31]: H I = 12 (cid:126) g ( a † σ − + aσ + ) . (5)Here, g = µ B G m a qm / ( √ (cid:126) ) is the single-atom-single-phonon coupling constant and σ ± = ( σ x ± iσ y ). Thus,we have the Hamiltonian for the hybrid system of the me-chanical oscillator and a single Rb atom in the Jaynes-Cummings form: H JC = (cid:126) ω m a † a + 12 (cid:126) ω L σ z + 12 (cid:126) g ( a † σ − + aσ + ) . (6) Tavis-Cummings Model.
For Rb BEC atoms, the av-erage distance between them is much smaller than theirtypical wave length in the trap potential [32], many Rb are interacting with the mechanical oscillator simulta-neously. Assuming that N Rb atoms identically in-teract with the mechanical oscillator, the hybrid sys-tem is better described by the Tavis-Cummings Model.Thus, we define the following collective spin operators as[18, 33, 34]: S z = (cid:80) Ni =1 σ i z , S ± = √ N (cid:80) Ni =1 σ i ± , where i isthe ith atom and the sums run over the N atoms. TheseBEC state atoms are all identical particles that we donot distinguish them from each other. In view of that, weneglect the space distribution of these magnetic trappedcold atoms and assume the energy level spacing betweenthe liquid state | (cid:105) and the vapor state | (cid:105) are all equalto (cid:126) ω L [16], so that the Hamiltonian of these N two-levelsystems reads: (cid:126) ω L S z /
2. The interaction between me-chanical oscillator and this ensemble of pseudo spins canbe described as: g √ N / a † S − + aS + ) . The couplingstrength is enhanced [26]: g √ N → g N , with g N be-ing the collective coupling constant. The hybrid system,with N two-level atoms identically coupling to mechani-cal oscillator, is thus described by the Tavis-CummingsHamiltonian [16, 33]: H T C = (cid:126) ω m a † a + 12 (cid:126) ω L S z + 12 (cid:126) g N ( a † S − + aS + ) . (7)The Tavis-Cummings Hamiltonian describes the identi-cal coupling between the BEC state atoms and mechan-ical oscillator, the coupling constant is collectively en-hanced. In the following, we calculate the mean steady-state phonon number for the mechanical oscillator whenit interacts with an ensemble of Rb atoms. III. MEAN STEADY-STATE PHONON FORMECHANICAL OSCILLATOR
The magnetic field produced by mechanical oscillatorinduces the liquid state been coupled to the vapor statewith the Rabi frequency Ω R [18]:Ω R = µ B G m a qm < ( a + a † ) > √ (cid:126) , (8) Liquid state –Trapped ω L ∝ B Vapor state –Untrapped Ω R ∝ B mac FIG. 2. (Color online). The atoms in the liquid state aretransferred into the vapor state under the action of the mag-netic field B acm ( t ) produced by the vibrating mechanical os-cillator in half a Rabi cycle. When the atoms are excited tothe vapor state, they get away from the trap with the energyfrom mechanical oscillator. with < a qm ( a + a † ) > being thermal fluctuation of me-chanical oscillator. The untrapped motional states in theatoms has a continuum energy width which is given by µ c . This continuum energy width is BEC’s chemical po-tential [16, 32]. We here note the rate of BEC state atomschanged from liquid state to the vapor state as Γ. In thefollowing, we will call Γ as atoms’ transition rate and thetransition rate Γ is expected to be derived according toRef. [16, 25]. Neglecting the gravity, the transition rateof atoms from the liquid state | (cid:105) to the vapor state | (cid:105) reads: Γ( δ ) = ζ ( δ )Ω . (9)Here, ζ ( δ ) ≡ π (cid:126) / µ c [ (cid:112) (cid:126) δ/µ c − ( (cid:112) (cid:126) δ/µ c ) ], is relatedwith the chemical potential µ c and the detuning δ .The transition rate is negligible outside the resonanceshell and strongly enhanced within the shell [25], andthe mechanical oscillator resonance with the main axes r i = R i (cid:112) (cid:126) δ/µ c , where R i is the Thomas-Fermi radiiof the BEC. When the atoms in resonance region areexcited into the vapor state, other condensate outsidethe resonance shell will move into the resonance area,replacing the leaving ones [25].In our case, the mechanical oscillator loses its energy,and induces the atoms’ spin flips while it interacts withthe BEC atoms. The atoms become untrapped and flyaway. The atoms in the resonance shell are excited,changed from the liquid state to the vapor state. Notethe interacting time between the mechanical oscillatorand the atoms be τ . When atoms are excited, theyfreely expand and fly away from the trap with the stolenenergy from mechanical oscillator. Those atoms flyingaway from the trap play an important role in our coolingmethod. The interaction time can be approximated byhalf a Rabi cycle time: τ = π Ω R . (10) Following the master equation approach for the quan-tum laser theory in Ref. [35], we derived the meanssteady-state phonon number of the mechanical oscillator.We assume that at time t = 0, the mechanical oscillatorand magnetic trapped BEC atoms do not interact eachother, and define ρ ( t ) as the reduced density operator ofthe mechanical oscillator at time t . When the interactionis switched on at time t , the atom is excited to the vaporstate from a liquid state atom. ρ ( t + τ ) is obtained bythe super-operator M [35, 36]: ρ ( t + τ ) = M ( τ ) ρ ( t ) , (11)in which: M ( τ ) ρ ( t ) = Tr BEC [ e − iHIτ (cid:126) ρ ( t ) ⊗ | (cid:105)(cid:104) | e iHIτ (cid:126) ] . (12)Here, H I = (cid:126) g ( aσ + + H.c ) being the Jaynes-Cummings Hamiltonian in Eq. (6) at resonance in theinteraction picture, and τ is the interaction time betweenthe oscillator and the atoms. Tr BEC denotes tracing overthe variables of the two-level system of this BEC stateatoms.We assume that the BEC atoms are identically coupledto the mechanical oscillator. When k atoms are excited: ρ ( t ) = M k ( τ ) ρ (0) . (13)Here, k = Γ t is the number of excited atoms. With thepresence of the dissipation mechanical oscillator noted as L [ ρ ], differentiating Eq. (13), we get the evolution ofthe reduced density operators ρ : dρdt = Γ[ln M ( τ )] ρ + L [ ρ ] . (14)Since a single BEC atom takes little energy from themechanical oscillator, Eq. (14) can be written as [35, 37]: dρdt ≈ Γ[ M ( τ ) − ρ + L [ ρ ] . (15)The operator L in the above equation is attributedto the dissipation of mechanical oscillator and is de-fined as: L [ ρ ] = κ (n th + 1)[2 aρa † − a † aρ − ρa † a ] + κ n th [2 a † ρa − aa † ρ − ρaa † ] with κ is the energy de-cay rate and n th is the mechanical oscillator’s averagephonon number at temperature T m before cooling andn th = 1 / [exp( (cid:126) ω m /k B T m ) − < n > s = Tr[ a † aρ s ] of the mechanical oscilla-tor in the steady state ρ s from above master equation: d Tr[ a † aρ s ] dt =ΓTr( a † a [ M ( τ ) − ρ s )+ κ th + 1)Tr(2 a † aaρ s a † − a † aa † aρ s − a † aρ s a † a )+ κ th Tr(2 a † aa † ρ s a − a † aaa † ρ s − a † aρ s aa † )=0 . (16)Then we get: < n > s = n th − Γ κ Tr[ a † a (1 − M ( τ )) ρ s ] . (17)Here, Tr( a † a (1 − M ( τ )) ρ s )Γ /κ is the decreased phononnumber after a Rb atom is excited from the liquid stateto the vapor state. Noticing thatTr( a † a (1 − M ( τ )) ρ s ) > , (18)can be proved with the definition of the super-operator ˆ M ( τ ). We thus conclude that the temperature of themechanical oscillator is always cooler than that of its ini-tial state when the steady state is reached. By furthercalculation we can get that:Tr( a † a (1 − M ( τ )) ρ s ) = < n > s − Tr[ a † a M ( τ ) ρ s ] . (19)Substituting Eq. (12) into Eq. (19), and the sec-ond term of the right side of Eq. (19) can be ex-panded as: Tr[ a † a M ( τ ) ρ s ] = Tr[ a † a ( (cid:104) | e − iH I τ/ (cid:126) ρ s ⊗| (cid:105)(cid:104) | e iH I τ/ (cid:126) | (cid:105) + (cid:104) | e − iH I τ/ (cid:126) ρ s ⊗| (cid:105)(cid:104) | e iH I τ/ (cid:126) | (cid:105) )] . Here,we proceed by noting that for small time τ , with g τ / < { a † a [1 − M ( τ )] ρ s } = Tr[ a † aρ s ] − Tr[ a † a M ( τ ) ρ s ] ≈ g τ < n > s . (20)Substituting Eq. (20) into Eq. (17), the mean steady-state phonon number after cooling is given by: < n > s = n th g τ κ . (21)When there are N BEC atoms in the magnetic trap whichare identically coupled to the mechanical oscillator duringthe cooling process, the coupling constant is collectivelyenhanced [10, 16]: g → g √ N = g N . Therefore, for Ncold atoms coupling the mechanical oscillator, the meansteady-state phonon number is given by: < n > s = n th g N τ κ . (22)Combining g N , g , κ = Q m /ω m , and Eq. (10) into Eq.(22) we have: < n > s = n th πµ B (cid:126) ) N ζ ( δ ) ( G m a qm ) m ω m . (23)Equation (23) gives the mean steady-state phonon num-ber after cooling of the mechanical oscillator, with Q m be-ing the quality factor of the mechanical oscillator. In thefollowing we will discuss the mean steady-state phononnumber after cooling with some practical parameters. IV. DISCUSSION AND CONCLUSION
We now check the cooling limit with some practical pa-rameters. Let us consider a mechanical oscillator realizedby a silicon cantilever: ω m / π = 1 . m = 1 × and effective mass m eff = 10 − kg, so that the zero point fluctuation a qm = 2 . × − m. If the initial temperature of this mechanical oscilla-tor before cooling is 50 mK, the amplitude of the thermalfluctuation of mechanical oscillator is about 1 . × − m. Therefore, the assumption of g τ / < g = 8 Hz which can be realized by adjusting themagnetic tip [16, 18]. The energy decay rate for thismechanical oscillator is: κ = ω m / Q m = 2 π ×
10 Hz. The chemical potential µ c ∝ ( N ω x ω y ω z ) / [28]. Forthe frequencies of the trap: ω x / π = ω y / π = 250Hz and ω z / π = 19 Hz with the number of atomsN = 5 × [25], we have µ c / (cid:126) ≈ π × .
88 kHz,so that ζ ( δ ) = 15 π (cid:126) × [ (cid:112) (cid:126) δ/µ c − ( (cid:112) (cid:126) δ/µ c ) ] / µ c ≈ . × − × [ (cid:112) (cid:126) δ/µ c − ( (cid:112) (cid:126) δ/µ c ) ]. With these pa-rameters, we can calculate the mean steady-state phononnumber < n > s of mechanical oscillator after cooling.The phonon number is depicted in Fig. 3 which is de-pending on detuning δ .The mean steady-state phonon number after coolingis shown in Fig. 3 with the initial temperature beforecooling is 50 mK. The phonon number depends on de-tuning δ is expressed in units of (cid:126) /µ c . Firstly, the meansteady-state phonon number decreases apparently as theincreasing of detuning δ before (cid:126) δ/µ c = 0 .
2. When (cid:126) δ/µ c is in the range of 0.2 - 0.6, the steady-state phononnumber change slowly with the change of the detuning.It means that the mechanical oscillator can be cooled FIG. 3. (Color online). Steady phonon number of mechani-cal oscillator after cooling depends on the detuning with theinitial temperature of mechanical oscillator is 50 mK. robustly which allows a wide range to adjust the detun-ing. The mean phonon number in steady state reachesthe minimum when (cid:126) δ/µ c = 1 /
3, we will call it as bestpoint in the following. The steady-state phonon num-ber of mechanical oscillator after cooling at best point is < n > s ≈ . ω m / π = 1 . m = 1 × and effective mass m eff = 10 − kg. The range of initial temperature ofthis mechanical oscillator in which the oscillator can becooled to the ground state is depicted in Fig. 4, withmean steady-state phonon number less than 1. We findthat the maximum initial temperature that can be cooledto ground state is about 40 mK if the detuning is set atthe best point. This temperature can be realized if theoscillator is placed in a dilution refrigerator.We also exploring the range of quality factor Q m andfrequency of mechanical oscillator ω m when it is at theinitial temperature 4.2 K in Fig. 5. We can find thatwhen the quality factor Q m is about 1 × , the groundstate can be reached even the initial temperature T m isat the 4.2 K, when the frequency of mechanical oscilla-tor ω m / π = 1 kHz. When the mechanical oscillatoris at the initial temperature of 4.2 K, we find that therange of Q m and ω m is shown at the top left corner inFig. 5. What is more, there is a wide range of mechani-cal oscillators which can be cooled to a temperature withonly few phonons, even the initial temperature is 4.2 K.It means that, the inequality Q m × f m > k B T m /h , themeasure of an experiment for which could reached intoquantum regime for the manipulation of coherent phonon[38], could be well satisfied, even for the bad quality os-cillators. Here, f m = ω m / π , is the frequency of the FIG. 4. (Color online). Mean steady-state phonon number ofmechanical oscillator after cooling, the initial temperature weconsidering is in the range of 0 - 4.2 K. FIG. 5. (Color online). Mean steady-state phonon numberof mechanical oscillator after cooling. The quality factor Q m is in the range of 10 − and frequency ω m / π is in therange of 10 Hz − Hz. The initial temperature for thesemechanical oscillators before cooling is T m = 4 . mechanical oscillator.We can come to the conclusion that, our coolingmethod with the BEC state cooling mechanical oscilla-tor is an effective proposal which can cool the mechanicaloscillator to the ground state at high initial temperaturerange. If the Cobalt bar is replaced by a Dysprosiumbar, which has a higher value of magnetic momentum µ m than Cobalt bar in unit mass, the value of single-atom-single-phonon coupling will be bigger. In this way,the mechanical oscillator will be colder than the case ofCobalt bar, and the ground state will be easier to bereached.In summary, we studied a cooling proposal for mechan-ical oscillator. We find that by applying this method, themechanical oscillator can be cooled to the ground state.We harbor the idea that that this cooling method canbe demonstrated in the near future by combining rapiddeveloping techniques for micro trap on atom chips andnano-mechanical oscillators. V. ACKNOWLEDGMENTS
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