Cooling a mechanical resonator by quantum interference in a triple quantum dot
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J a n Cooling a mechanical resonator by quantum interference in a triple quantum dot
Shi-Hua Ouyang,
1, 2
Chi-Hang Lam, and J. Q. You Department of Applied Physics, Hong Kong Polytechnic University, Hung Hom, Hong Kong, China Department of Physics and Surface Physics Laboratory (National Key Laboratory), Fudan University, Shanghai 200433, China (Dated: November 8, 2018)We propose an approach to cool a mechanical resonator (MR) via quantum interference in a triplequantum dot (TQD) capacitively coupled to the MR. The TQD connected to three electrodes isan electronic analog of a three-level atom in Λ configuration. The electrons can tunnel from theleft electrode into one of the two dots with lower-energy states, but can only tunnel out from thehigher-energy state at the third dot to the right electrode. When the two lower-energy states aretuned to be degenerate, an electron in the TQD can be trapped in a superposition of the degeneratestates called the dark state. This effect is caused by the destructive quantum interference betweentunneling from the two lower-energy states to the higher-energy state. Under this condition, anelectron in the dark state readily absorbs an energy quantum from the MR. Repeating this process,the MR can be cooled to its ground state. Moreover, we propose a scheme for verifying the coolingresult by measuring the current spectrum of a charge detector adjacent to a double quantum dotcoupled to the MR.
PACS numbers: 03.65.Ta, 85.35.Be, 42.50.Gy
I. INTRODUCTION
Mechanical resonators (MRs) with a high resonant fre-quency and a small mass have wide applications and areattracting considerable recent attentions [1, 2]. Tech-nically, these MRs can be used as ultrasensitive sen-sors in high-precision displacement measurements [3], de-tection of gravitational waves [4] or mass detection [5].Also, quantized MRs can be useful in quantum infor-mation processing. Indeed, quantized motion of buck-ling nanoscale bars has been proposed for qubit imple-mentation [6, 7] and also for creating quantum entangle-ment [8–10]. However, for all these applications, a basicprerequirement is that the dynamics of the MRs mustapproach the quantum regime.Quantum behaviors of a MR are usually suppressed bythe coupling to its environment. One way to approachthe quantum regime is to increase its resonant frequencyso that an energy quantum of the MR becomes largerthan the thermal energy. Recently, MRs based on metal-lic beams [11] and carbon nanotubes [12] have been de-veloped, which have resonance frequencies of several hun-dred megahertz. However, for a MR with a frequency of200 MHz, a temperature lower than 10 mK (below thepresent dilution refrigerator temperature) is required tomaintain the MR at the quantum regime. To attain thequantum regime, one needs to cool the MR further viacoupling to an optical or an on-chip electronic system.Numerous experiments on cooling a single MR via radia-tion pressure or dynamical backaction have been reported(see, e.g., [13–21]). Theoretically, cooling by coupling toa Cooper pair box [22] or to a three-level flux qubit [23]via periodic resonant coupling have also been proposed.In these schemes, a strong resonant coupling between theMR and the qubit is required to cool the MR to its groundstate.
A. Sideband cooling of a MR
In the weak coupling regime, a conventional methodfor cooling the MR is the sideband cooling approach (see,e.g., Ref. [24–31]). In this case, a MR is coupled to a two-level system (TLS) in which the two states can be elec-tronic states in quantum dots [24–26], photonic statesin a cavity [27–30], or charge states in superconduct-ing qubits [32]. In order to achieve ground-state coolingin the sideband cooling approach, the resolved-sidebandcooling condition ω m ≫ Γ (with ω m denoting the os-cillating frequency of the MR and Γ the decay rate ofthe TLS) must be followed in order to selectively drivethe lowest sideband of the TLS. Then, the excitation ofthe TLS from the ground state | g i to the excited state | e i and the subsequent decay from this excited state tothe ground state will, on average, decrease an energyquantum in the MR. This process can be described by | g, n i → | e, n − i → | g, n − i , where n denotes to thestate with n phonons. However, the frequency of a typ-ical MR is about 100 MHz [11, 12]. It is in general ofthe same order of the decay rate of the two-level system.This indicates that the resolved-sideband cooling condi-tion is not easy to fulfill. Violating the condition meansthat the processes of a carrier transition and a subsequentsideband transition (i.e., | g, n i → | e, n i → | g, n + 1 i ) willoccur. This will heat up the MR instead and suppressany ground-state cooling [29]. B. Cooling atomic motion via quantuminterference in a three-level atom
For laser cooling of atoms, an alternative approach [33]based on quantum interference in the internal degreesof freedom of the atoms without the need to follow theresolved-sideband cooling condition has been proposed.In this approach, an additional state is coupled to the ex-cited state | i of the TLS to form a Λ-shaped three-levelsystem [see Fig. 1(b)]. The two lower-energy states in thisthree-level system are tuned to be degenerate. Due to dis-sipation of the excited state, the atom will eventually ar-rive at a particular superposition of the two lower-energystates which is orthogonal to the excited state. This phe-nomenon results from the destructive quantum interfer-ence between the two transitions | i → | i and | i → | i and the superposition state is called the dark state [34].When atomic motion is also considered, the carrier tran-sition ( | , n i → | , n i ) and thus the heating process of theatomic motion in the sideband cooling scheme is hencesuppressed. It was shown that atomic motion can becooled to its ground state in the non-resolved sidebandregime [33]. C. Cooling a MR via quantum interference in atriple quantum dot
In the present work, we propose a new scheme to coola MR via capacitive coupling to a triple quantum dot(TQD) schematically displayed in Fig. 1(a). We considerthe strong Coulomb-blockade regime so that at most oneelectron is allowed to present at one time in the TQD.The TQD acts as a three-level system in Λ configura-tion, in which the two dot states | i and | i (i.e., thesingle-electron orbital states in dots 1 and 2) are cou-pled to a third (excited) state | i via two tunnel barri-ers [see Fig. 1(a)]. Here, the degrees of freedom of theMR is analogous to the motional degrees of freedom ofthe atoms discussed above and the TQD is an electronicanalog of a three-level atom. We will show that by prop-erly tuning the gate voltages, one can degenerate the twolower-energy states and obtain a dark state in the TQD.By capacitively coupling to the TQD, the MR can becooled to its ground state, in full analogy to the slowingdown of the atoms via quantum interference. Compar-ing with the cooling of atoms [33], our approach has thefollowing potential advantages: (i) Our cooling system iscompletely electronic and can be conveniently fabricatedon a chip. (ii) Simply by adjusting the gate voltages, itis easy to achieve the two degenerate lower-energy statesrequired for realizing destructive quantum interference inthe TQD. Moreover, in contrast to the cooling of a MRby coupling it to a superconducting qubit [35], the decayrate Γ of the higher-energy state of the TQD, which isequal to the rate of electrons tunneling from the TQD tothe electrode, is tunable in our case by varying the gatevoltage.Moreover, we also propose a method to verify whetherthe MR is successfully cooled by coupling it to a doublequantum dot (DQD). This DQD can be reduced fromthe TQD by applying appropriate gate voltages. Whenthe DQD and the MR is tuned into a strongly disper-sive regime in which the transition frequency differencebetween the two subsystems is much larger than the FIG. 1: (color online) Schematic diagram of a TQD system.Dots 1 and 2 are both tunnel-coupled to dot 3 (with interdotcoupling strengths Ω and Ω , respectively) while they areonly capacitively coupled to each other. A MR is capacitivelycoupled to dots 1 and 3 of the TQD. (b) A three-level systemin Λ configuration. The energy detunings between the twolower-energy states and the third excited state are respectively∆ and ∆ . The coupling strength between the state | i ( | i )and the state | i is Ω (Ω ). coupling strength between them, the coupling betweenthe MR and the DQD only yields a phonon-number-dependent Stark shift to the transition frequency of theDQD. This Stark shift corresponds to the shift of theresonant peak in the current spectrum of a charge detec-tor. Thus, by measuring the shift of the resonant peak,one can readout the phonon-number state and examinewhether the MR is successfully cooled or not.This paper is organized as follows. Section II intro-duces a microscopic model for the coupled MR-TQDsystem. We show that the TQD is an electronic ana-log of a three-level atom driven by two electromagneticfields. Also, we show how the TQD evolves into the darkstate. In Sec. III, we derive a master equation to describethe quantum dynamics of the coupled MR-TQD system.With this master equation, we further derive in Sec. IVthe master equation of the MR by eliminating the TQDdegrees of freedom. Moreover, we calculate the steady-state average phonon occupancy of the MR and showthat the MR can indeed be cooled to its ground state byusing the quantum interference in the TQD. In Sec. V,we propose a method to verify if the MR is successfullycooled by measuring the full-frequency current spectrumof a charge detector. Section VI summarizes our results.In the appendix, we give a detailed derivation of the mas-ter equation for the reduced density matrix of the MR. II. A MECHANICAL RESONATOR COUPLEDTO A TRIPLE QUANTUM DOTA. Model
The device layout of a MR coupled to a TQD is shownin Fig. 1(a). The TQD is connected to three electrodesvia tunneling barriers. In the TQD, dots 1 and 2 are onlycapacitively coupled to each other and electrons cannottunnel directly between them. Such capacitively coupleddots have already been achieved in experiments (see, e.g.,[36]). In contrast, electrons can tunnel between dots 1and 3 as well as between dots 2 and 3. Here we focuson the strong Coulomb-blockade regime, so that at mosta single electron is allowed in the TQD. Thus, only fourelectronic states need to be considered in the TQD, i.e.,the vacuum state | i , and states | i , | i and | i corre-sponding to a single electron in the respective dot. TheMR is capacitively coupled to dots 1 and 3 and this isschematically shown in Fig. 1(a).The total Hamiltonian of the whole system reads H total = H + H int + H T + H ep . (1)The unperturbed Hamiltonian H is defined by H = H leads + H TQD + H R + H ph , (2)where terms on the R.H.S. of Eq. (2) denote Hamiltoni-ans of the electrodes, the TQD, the MR and the thermalbath given by H leads = X ik E ik c † ik c ik , (3) H TQD = − ∆ a † a − ∆ a † a +(Ω a † a + Ω a † a + H . c . ) , (4) H R = ω m b † b, (5) H ph = X q ω q b † q b q . (6)We have put ¯ h = 1 and the energy of the state | i ischosen as the zero-energy point. c † ik ( c ik ) is the creation(annihilation) operator of an electron with momentum k in the i th electrode ( i = 1 , a † i creates anelectron in the i th dot. The phonon operators b † and b respectively create and annihilate an excitation of fre-quency ω m in the MR. In Eq. (6), the thermal bath ismodeled as a bosonic bath with b † q ( b q ) being the creation(annihilation) operator at freqency ω q .The electromechanical coupling between the MR anddots 1 and 3 of the TQD is given by H int = − g ( a † a − a † a )( b † + b ) , (7)with a coupling strength g = ηω m . For a typical elec-tromechanical coupling, η ∼ . H T = X ik (Ω ik a † i c ik + H . c . ) , (8)where Ω ik characterizes the coupling strength betweenthe i th dot and the associated electrode via tunnelingbarrier. Moreover, the coupling of the MR to the outsidethermal bath is characterized by H ep = X q Ω q ( b † q b + H . c . ) , (9)with a coupling strength Ω q . B. Analogy between TQD and Λ -type three-levelatom in two driving electromagnetical fields We now show that in the absence of the MR, our TQDsystem is analogous to a typical Λ-type three-level atomin the presence of two classical electromagnetical fields.This field-driven three-level system is often used in quan-tum optics for producing a dark state (see, e.g., [34]).The Hamiltonian of the field-driven Λ-type three-levelsystem can be written as H Λ = ω a † a + ω a † a + ω a † a +Ω a cos( ω a t )( a † a + a † a )+Ω b cos( ω b t )( a † a + a † a ) , (10)where ω i ( i = 1 , i th statein the three-level system. Also, ω a and ω b are the fre-quencies of the two driving fields and Ω a and Ω b are thecorresponding driving strengths. In order to eliminatethe time-dependence of the Hamiltonian in Eq. (10), wetransform the system into a rotating frame defined by U R = e iRt with R = ω a a † a + ω b a † a − ω ( a † a + a † a + a † a ) . (11)The transformed Hamiltonian is e H Λ = U − R H Λ U R + i ˙ U − R U R , (12)where the first term is evaluated as (within the rotating-wave approximation) U − R H Λ U R = ω a † a + ω a † a + ω a † a + Ω a a † a + a † a ) + Ω b a † a + a † a ) , (13)and the second term gives i ˙ U − R U R = ω a a † a + ω b a † a − ω ( a † a + a † a + a † a ) . (14)Thus, Eq. (12) reduces to e H Λ = − ∆ a † a − ∆ a † a +Ω ( a † a + a † a ) + Ω ( a † a + a † a ) , (15)where ∆ = ω − ω − ω a and ∆ = ω − ω − ω b are thefrequency detunings while Ω = Ω a / = Ω b / C. “Dark state” in the TQD
From the study of quantum optics, the existence of adark state in a Λ-type three-level atom when the lower-energy states become degenerate, i.e., ∆ = ∆ , is able tosuppress absorption or emission. Below we demonstratethat a similar dark state also exists in the TQD [38].After tracing over the degrees of freedom of the elec-trodes, the quantum dynamics of the TQD in the absenceof the MR is described by˙ ρ d = L TQD ρ d = − i [ H TQD , ρ d ] + Γ D [ a † ] ρ + Γ D [ a † ] ρ d + Γ D [ a ] ρ d , (16)where ρ d is the reduced density matrix of the TQD andΓ i ( i = 1 , i th dot. The notation D for any operator A is given by D [ A ] ρ = AρA † −
12 [ A † Aρ + ρA † A ] . (17)Considering equal energy detunings of the lower energystates | i and | i with respect to the excited state | i ,i.e., ∆ = ∆ = ∆, the eigenstates | g i , |−i and | + i ofthe TQD become | g i = β | i − α Ω (Ω | i + Ω | i ) , |−i = 1Ω (Ω | i − Ω | i ) , | + i = α | i + β Ω (Ω | i + Ω | i ) , (18)where α = cos( θ/ β = sin( θ/ θ = 2Ω / ∆, andΩ = p Ω + Ω . The corresponding eigenenergies are E g = − ∆ + φ , E − = − ∆ , E + = − ∆ − φ , (19)with φ = √ ∆ + 4Ω . For simplicity, we consider equalcouplings of the three quantum dots to the correspond-ing electrodes, i.e., Γ = Γ = Γ ≡ Γ. Based on the
FIG. 2: (color online) Effective tunneling processes of elec-trons through a TQD represented in the eigenstate basis.An electron can tunnel from the left electrode into the threeeigenstates | + i , |−i and | g i , with rates Γ β , Γ and Γ α , re-spectively. Note that the total tunneling rate is 2Γ becausean electron tunnels from the left electrode to the TQD viatwo tunnel barriers (each having a tunneling rate Γ). In theeigenstate | + i ( | g i ), it will tunnel out to the right electrodewith a rate Γ α (Γ β ). However, if the electron occupies thedark state |−i , no further tunneling occurs and the electronis trapped. eigenstate basis of the TQD in Eq. (18), the equationsof motion for the reduced density matrix elements of theTQD are obtained from Eq. (16) as˙ ρ = − ρ + Γ β ρ gg + Γ α ρ ++ + Γ αβ ( ρ + g + ρ g + ) , ˙ ρ gg = Γ α ρ − Γ β ρ gg − Γ2 αβ ( ρ + g + ρ g + ) , ˙ ρ −− = Γ ρ , ˙ ρ ++ = Γ β ρ − Γ α ρ ++ − Γ2 αβ ( ρ + g + ρ g + ) , ˙ ρ + g = − i ( E + − E g ) ρ + g − Γ2 ρ + g − Γ2 αβ ( ρ ++ + ρ gg ) − Γ αβρ . (20)Figure 2 schematically show effective electron tunnelingprocesses through the TQD as described by Eq. (20).Starting from an initially empty TQD, an electron cantunnel from the left electrode into any of the three eigen-states, with tunneling rates Γ β , Γ and Γ α for eigen-states | + i , |−i and | g i , respectively. An electron in theeigenstate | + i ( | g i ) then tunnels out of the TQD to theright electrode with a rate Γ α (Γ β ). However, if theelectron occupies the state |−i , no further tunneling oc-curs because, being orthogonal to | i , it is decoupled fromthe right electrode. Therefore, an electron in the TQDwill be trapped in the state |−i , which is called the darkstate in quantum optics [38]. This dark state results fromthe destructive quantum interference between the tran-sition | i → | i (i.e., the electron tunneling from state | i to state | i in the TQD system) and the transition | i → | i (i.e., the electron tunneling from state | i tostate | i ). III. EFFECTIVE HAMILTONIAN ANDMASTER EQUATION FOR THE COUPLEDMR-TQD SYSTEM
We now study the coupled MR-TQD system. Ratherthan analyzing directly the energy exchange between theMR and the TQD which involves tedious algebra, weapply a canonical transform U = e S on the whole system,where S = η ( a † a − a † a )( b − b † ) . (21)The transformed Hamiltonian is given by H = U H total U † = H leads + H ph + H ep + ω m b † b − ∆ a † a − ∆ a † a +[Ω a † a B + Ω a † a B + H . c . ] , + X k (cid:2) Ω k a † c k B + Ω k a † c k +Ω k a † c k B † + H . c . (cid:3) , (22)where we have neglected a small level shift of η ω m tothe states | i and | i and we have also defined B = exp[ − η ( b − b † )] . (23)To describe the quantum dynamics of the coupled MR-TQD system, we derive a master equation (under theBorn-Markov approximation) by tracing over the degreesof freedom of both the electrodes and the thermal bath.Up to second order in η , the master equation can bewritten as dρdt = − iω m [ b † b, ρ ] − i [ H TQD , ρ ] − i [ V ( b † − b ) , ρ ] + L T ρ + L D ρ, (24)where V = η (cid:2) ( a † a − a † a ) + Ω ( a † a − a † a ) (cid:3) , (25) L T ρ = Γ D [ a † ] ρ + Γ D [ a † ] ρ + Γ D [ a ] ρ + η Γ (cid:0) D [ a † b † ] ρ + D [ a † b ] ρ (cid:1) + η Γ (cid:0) D [ a b † ] ρ + D [ a b ] ρ (cid:1) , (26) L D ρ = γ [ n ( ω m ) + 1] D [ b ] ρ + γn ( ω m ) D [ b † ] ρ. (27)Here, the Liouvillian operator L T ρ accounts for the dissi-pation due to the electrodes and L D ρ represents the dis-sipation at the MR induced by the thermal bath. Also, γ denotes the decay rate of excitations in the MR inducedby the thermal bath and n ( ω m ) is the average boson num-ber at frequency ω m in the thermal bath. IV. GROUND-STATE COOLING OF THE MRA. Master equation for the reduced density matrixof the MR
In the limit γ ≪ g ≪ ω m , the TQD is weakly coupledto the MR and can be regarded as part of the environ-ment experienced by the MR. The degrees of freedom ofthe TQD can then be adiabatically eliminated [27, 39]and the master equation for the reduced density matrix µ of the MR is given by (see Appendix)˙ µ = − i ( ω m + δ m )[ b † b, ρ ] + 12 { γ [ n ( ω m ) + 1] + A − ( ω m ) }× [2 bµb † − ( b † bµ + µb † b )]+ 12 [ γn ( ω m ) + A + ( ω m )][2 b † µb − ( bb † µ + µbb † )] , (28)where δ m is the driving-induced shift of the MR fre-quency. In Eq. (28), the additional terms A + and A − areinduced by the coupling with the TQD. With this mas-ter equation, one obtains the equation of motion for thephonon-number-probability distribution, p n = h n | µ | n i ,of the MR: dp n dt = (cid:8) γ [ n ( ω m ) + 1] + A − (cid:9) [( n + 1) p n +1 − np n ]+[ γn ( ω m ) + A + ][ np n − − ( n + 1) p n ] , (29)Moreover, the equation of motion for the average phononnumber, h n i = P n np n , in the MR can be obtained fromEq. (29) as d h n i dt = − ( γ + W ) h n i + γn ( ω m ) + A + , (30)where W = A − − A + . In order to cool the MR, one needs W > A − > A + ). B. Steady-state solution
From Eq. (30), the steady-state average phonon num-ber in the MR is n st = γn ( ω m ) + A + γ + W , (31)where the term γn ( ω m ) in the numerator is due to thethermal bath while A + results from the scattering pro-cesses by the TQD. We assume that the MR is initiallyat equilibrium with the thermal bath, so that the initialphonon number in the MR is n ( ω m ). In order to cooldown the MR significantly, one needs a large cooling rate W ≫ γ to overcompensate for the heating effect of thethermal bath. At the end of Sec. IV, we show that thiscan be achieved using typical experimental parameters.Here we consider ∆ = ∆ ≡ ∆ so that the dark stateexists. The transition rates A ± are found to be (see Ap-pendix) A ± = 2 η Ω Ω Ω ω m Γ4[Ω − ω m ( ω m ± ∆)] + ω m Γ + η Γ ρ st00 , (32)where ρ st00 is the steady-state probability of an emptyTQD. To cool the MR, one needs A − > A + , which isfulfilled either when ∆ > < ω m , or when ∆ < > ω m . Assuming also W ≫ γ , the steady-state average phonon number in the MR is approximatelygiven by n st ≈ γn ( ω m ) W + n f . (33)Here n f ≡ A + /W which gives n f = 4[Ω − ω m ( ω m − ∆)] + ω m Γ ω m ( ω m − Ω ) . (34) C. Optimal cooling condition
It is easy to see that n f reaches the minimum n min f = ( Γ4∆ ) , (35)when the term in square brackets in the r.h.s. of Eq. (34)becomes zero, i.e., Ω = ω m ( ω m − ∆) , (36)or ω m = 12 (∆ + φ ) . (37)Therefore, by properly choosing the parameters Ω , ω m ,and ∆ so that the optimal cooling condition in Eq. (36)is fulfilled and ∆ ≫ Γ, the steady-state average phononnumber in the MR can be much smaller than unity, im-plying that ground-state cooling of the MR is possible.Moreover, the phonon number n f achievable accordingto Eqs. (34) and (35) is identical to the previous resultsfor the cooling of trapped atoms via quantum interfer-ence [33]. However, the additional advantages of a solid-state cooling system proposed here are that it can befabricated on a chip and is highly controllable. Specif-ically, all the relevant parameters (i.e., the detuning ∆,the tunneling rate Γ and the interdot coupling strengthsΩ and Ω ) can be controlled by tuning the gate volt-ages in the TQD. Thus, for a fixed frequency ω m of theMR, the optimal cooling condition in Eq. (36) can beconveniently fulfilled.The underlying physics of the optimal cooling condi-tion in Eq. (36) can be understood based on the eigen-state basis of the TQD. In the limit γ ≪ g ≪ ω m con-sidered here, the TQD arrives quickly at the dark state |−i . The coupling between the MR and the TQD will FIG. 3: (color online) Contour plot of the steady-state aver-age phonon number n st in the MR as a function of the nor-malized driving detuning ∆ /ω m and the normalized interdotcoupling Ω /ω m . The two solid curves correspond to n st = 0 . .
02. The black dasded line represents Ω = ω m ( ω m − ∆),under which the MR can be optimally cooled. We have chosenΩ = Ω = Ω / √ ω m = 2 π ×
100 MHz,Γ = ω m , Q = 10 , η = 0 .
1, and n ( ω m ) = 21. excite the TQD to the state | + i most readily when thefrequency ω m of the MR is equal to the transition fre-quency ( φ + ∆) / |−i and | + i , i.e., ω m = ( φ + ∆) /
2. This corresponds to the transition |− , n i → | + , n − i . The excited electron subsequentlytunnels to the right electrode, i.e., | + , n − i → | , n − i .This whole process extracts an energy quantum from theMR. When this cycle repeats, i.e., | , n i → |− , n i →| + , n − i → | , n − i → · · · , the MR is cooled to theground state. Here we emphasize that the resonance con-dition for exciting the TQD from the state |−i to thestate | + i via the MR is equivalent to the optimal coolingcondition in Eq. (36). An electron can also relax from thedark state |−i to the ground state | g i by releasing energyto the MR. However, this heating process of the MR isstrongly suppressed because the frequency of the MR isoff-resonant to the transition |−i → | g i in the TQD.Figure 3 displays a contour plot of the steady-state av-erage phonon number of the MR ( n st ) as a function ofthe effective interdot coupling Ω (= p Ω + Ω ) and theenergy detuning ∆. Here we choose ∆ < > ω m to make sure that W >
0. For these typical parameters,a small n st < .
05 is predicted over a wide range of valueson the Ω − ∆ plane. This implies that ground-state cool-ing of the MR should be experimentally accessible. Fur-thermore, to estimate the cooling rate W , we use typicalexperimental parameters [11, 40]: ω m = 2 π ×
100 MHz,∆ = − π ×
300 MHz, and g = 2 π ×
10 MHz. The interdotcouplings are chosen as Ω = Ω ≃ π ×
141 MHz to fulfillthe optimal cooling condition Ω = ω m ( ω m − ∆). UsingEq. (32), one arrives at a cooling rate W ≈ π × Q = 10 (see,e.g., Ref. 12), one has γ = ω m /Q = 2 π × W ≫ γ can be achieved.In this case, a MR can be cooled from, e.g., an initialtemperature T = 100 mK corresponding to n ( ω m ) = 21down to T = 0 . n st = 0 . ω m ≫ Γ must be fol-lowed for ground-state cooling of a MR. For a relativelylarge decay rate, only MR with a very high frequency(which becomes fragile in experiments) can be cooled.On the other hand, for cooling via quantum interfer-ence in the TQD proposed here, the cooling conditionsΩ = ω m ( ω m − ∆) and ∆ ≫ Γ do not require a high MRfrequency.
V. A SCHEME FOR VERIFYING THECOOLING OF THE MRA. Quantum dynamics of coupled MR-DQDsystem in the presence of a charge detector
To verify whether the MR is cooled or not, we pro-pose a scheme in which the MR is coupled to a two-levelsystem realized by a double quantum dot (DQD). Thestate of the DQD is in turn measured by a nearby chargedetector in the form of, e.g., a quantum point contact(QPC). This setup is schematically shown in Fig. 4. Ex-perimentally, the cooling and the verification setups areall fabricated on the same chip with shared components.Dots 1 and 3 in the TQD from the above cooling setupcan make up the DQD after a gate voltage is applied todecouple dot 2 from dot 3. Also, the QPC should bedecoupled from the rest of the system during cooling byapplying a high gate voltage.The Hamiltonian of the system is given by H = H R + H DQD + H QPC + H int + H det . (38)The Hamiltonian H R of the MR and the coupling H int be-tween the MR and the DQD are already given in Eqs. (5)and (7). Here H DQD , H QPC and H det are respectively theHamiltonians of the DQD, the QPC and the coupling be-tween them and are given by H DQD = − ∆2 σ z + Ω σ x ,H QPC = X kq ω Sk c † Sk c Sk + ω Dq c † Dq c Dq ,H det = X kq ( T − χσ z )( c † Sk c Dq + H . c . ) , (39)where σ z = a † a − a † a and σ x = a † a + a † a are thePauli matrices. Also, c ik ( c † ik ) is the annihilation (cre-ation) operator for an electron with momentum k in ei-ther the source ( i = S ) or the drain ( i = D ) of the QPC. T is the transition amplitude of an isolated QPC and χ is the variation of the transition amplitude caused by theDQD. For simplicity, we assume that the detunning ∆ of FIG. 4: (color online) Schematic diagram of a MR capaci-tively coupled to a DQD which is under measurement by anearby QPC. The energy detuning between the two dot statesin the DQD is ∆ and the interdot coupling strength betweenthem is Ω . the DQD is zero. The DQD has the eigenstates | g i = √
22 ( | i − | i ) , | e i = √
22 ( | i + | i ) , (40)with | g i ( | e i ) being the ground (excited) state. Rewritingthe Hamiltonian in Eq. (38) on the eigenstate basis ofthe DQD, we have H = ω m b † b + Ω ρ z − gρ x ( b † + b ) + H QPC + X kq ( T − χρ x )( c † Sk c Dq + H . c . ) , (41)where ρ z = | e ih e | − | g ih g | and ρ x = | e ih g | + | g ih e | are thePauli matrices.We consider the coupled MR-DQD system in thestrong dispersive regime where the coupling strength ismuch smaller than the difference between the transitionfrequency 2Ω of the DQD and that of the MR, i.e., g ≪ δ = 2Ω − ω m . This regime was previously consid-ered to study whether the vibration of a MR coupled to asuperconducting circuit is classical or quantum mechani-cal [41]. In this regime, the phonon in the MR is only vir-tually exchanged between the DQD and the MR. Thus,the coupling of the DQD to the MR does not change theoccupation probability of the electron in the DQD, butonly results in phonon-number-dependent Stark shifts onenergy levels of the DQD. Moreover, the Stark shifts canbe detected by measuring the full-frequency current spec-trum of the QPC.Applying both a rotating-wave approximation and acanonical transformation U ′ = e s ′ with s ′ = η ( ρ − b † − ρ + b ) , η = g/δ, (42)to the Hamiltonian H , one obtains [up to O ( η )] H ≈ ω m b † b + 2Ω + g (2 b † b + 1) /δ ρ z + g δ ( | g ih g | + | e ih e | ) + H QPC + X kq ( T − χρ x )( c † Sk c Dq + H . c . ) . (43)From Eq. (43), after taking the trace over the degrees offreedom of the QPC, one obtains the following masterequation for the reduced density matrix elements of thecoupled MR-DQD system [42]:˙ ρ gn,gn = − γ + ρ gn,gn + ( γ − + γ d ) ρ en,en , ˙ ρ en,en = γ + ρ gn,gn − ( γ − + γ d ) ρ en,en , ˙ ρ gn,en = − iδ n ρ gn,en − ( γ + γ d ρ gn,en , (44)where δ n = 2Ω + g (2 n + 1) /δ and γ = 2 πg s g d χ eV d with g s ( g d ) being the density of states for electrons inthe source (drain) of the QPC and V d the bias voltageacross the QPC. Here γ ± = γ (1 ∓ λ n ), with λ n = δ n /eV d ,are the QPC-induced excitation and relaxation rates be-tween the ground state and the excited state of the DQD.Also, γ d is the relaxation rate resulting from the couplingof the DQD to the thermal bath. Since the dissipationrate of the MR is much smaller than that of the DQD,dissipation of the MR is neglected. In Eq. (44), the re-duced density matrix element ρ in,in ( i = g, e ) gives theoccupation probability of the state | i, n i of the coupledMR-DQD system ,while ρ in,jn ( i = j ) describes the co-herence between the states | i, n i and | j, n i . The equa-tions of motion for other elements, e.g., ρ in,jn ′ ( n = n ′ ),which are decoupled from those considered here, are notshown. Using Eq. (44) and the normalization condition p n = ρ gn,gn + ρ en,en , one finds ρ gn,gn ( t ) = ( γ − + γ d ) p n γ − (cid:2) ( γ − + γ d ) p n γ − ρ gn,gn (0) (cid:3) e − γ t ,ρ en,en ( t ) = γ + γ p n − (cid:2) γ + γ p n − ρ en,en (0) (cid:3) e − γ t ,ρ gn,en ( t ) = ρ gn,en (0) e − i ( δ n − γ ) t , (45)where γ = γ + γ d / p n is the probability that theMR is at state | n i . B. Current spectrum of the charge detector
The dc current through the QPC is related to the elec-tron occupation probability in the DQD and is givenby [43] I ( t ) = eDρ + eD ′ ρ = e D + D ′ ) + e D ′ − D ) h σ z i , (46) where D = 2 πg s g d ( T − χ ) V d ,D ′ = 2 πg s g d ( T + χ ) V d , (47)are the respective rates of electron tunneling through theQPC when dot 3 is respectively occupied or empty [43].Therefore, one can define the current operator as I ( t ) = I + I σ z ( t ) = I + I ρ x ( t ) , (48)with I , = e ( D ± D ′ ) /
2. According to the Wiener-Khintchine theorem, the power spectrum of the currentthrough the QPC is [34] S ( ω ) = Re ∞ Z e iωτ dτ [ h I ( t ) I ( t + τ ) i − h I ( t + τ ih I ( t ) i ] . (49)Substituting Eqs. (45) and (48) into Eq. (49), we get S ( ω ) /S = 1 + 2 γ γ γ + γ X n p n (1 − κp n ) γ γ + ( δ n − ω ) − γ γ γ + γ X n p n (1 + κp n ) γ γ + ( δ n + ω ) , (50)where γ = 2 πg s g d T V d , κ = ( γ + − γ − − γ d ) / γ ,and S = 2 eI is the current-noise background. FromEq. (50), one sees that the current spectrum of the QPCconsists of peaks at resonance points ω = ± δ n . Thesepeaks have width γ and heights increasing with the prob-ability p n . For instance, the peaks at the resonance point δ n = 2Ω + g (2 n + 1) δ , (51)is shifted by g (2 n + 1) /δ from 2Ω . Thus, from thispeak shift in the current spectrum, one can readout thephonon-number state of the MR.Figure 4 plots the current spectrum of the QPC withtwo different coupling strengths between the MR andthe DQD. Results for three cases in which the MR isrespectively in its ground state ( n = 0 . ≪ n = 1 .
0) or thermalized with an averagephonon number n st = 1 . g /δ < γ , and hence the measuredspectrum shows an ensemble. In this case, the phonon-number state of the MR cannot be measured. In thestrong limit (2 g /δ > γ ), however, the ensemble can beindividually resolved [Fig. 4(b)], which allows us to de-tect the phonon number and also to verify the cooling FIG. 5: (color online) Power spectrum of the current throughthe QPC when the phonon number in the MR are respec-tively n = 0 .
01 (black solid line), n = 1 (red dashed line),or given by the thermal distribution (blue dotted line), i.e., p n = n n st / (1 + n st ) n +1 with n st = 1. The coupling strengthbetween the MR and the DQD is g = 0 . ω m (a) and g = 0 . ω m (b). The other parameters are ω m = 100 MHz,Ω = 2 ω m , γ = 0 . ω m , γ = 0 . γ , and γ d = 2 γ . result of the MR. Indeed, a relatively strong coupling be-tween a MR and a quantum dot has been recently demon-strated [12]. The strong dispersive regime is thus achiev-able and one can apply the proposed coupled MR-DQDsystem to verify the cooling of the MR via measuringthe current spectrum of a nearby charge detector (e.g.,QPC). VI. DISCUSSION AND CONCLUSION
Our proposal on ground state cooling of the MR re-quires that the TQD is able to evolve into the dark state.However, the dephasing of the TQD due to coupling toother degrees of freedom in the environment can projectthe TQD into one of the three localized states | i , | i , and | i and drive the system away from the dark state [38].However, the dephasing between the two localized states | i and | i depends on the coupling strength and the en-ergy detunning between dots 1 and 2 [44]. Here in oursystem no direct coupling exists between the two local-ized states | i and | i , and thus the dephasing almost hasnegligible effects on the cooling efficiency of the MR.In summary, we have studied the cooling of a MR bycapacitive coupling to a TQD. We show that when thetwo lower-energy localized states become degenerate, theTQD will be trapped in a dark state which is decou-pled from the excited state in the absence of the MR.With the MR in resonance with the transition betweenthe dark state and the excited eigenstate in the TQD,we have shown that the MR can be cooled to its groundstate in the non-resolved sideband cooling regime . More-over, we have proposed a coupled MR-DQD system in thestrong dispersive regime for verifying the cooling result of the MR. In this regime, the coupling between the MRand the DQD induces a MR-phonon-number dependentshift of the transition frequency of the DQD. Thus thephonon-number state which characterizes the cooling re-sult of the MR can be detected by measuring the shifts ofthe resonance peaks in the current spectrum of a nearbycharge detector. Acknowledgments
This work is supported by the National Basic Re-search Program of China Grant Nos. 2009CB929300 and2006CB921205, the National Natural Science Foundationof China Grant Nos. 10534060 and 10625416, and theResearch Grant Council of Hong Kong SAR project No.500908.
Appendix A: Master equation for the reduceddensity matrix of the MR
In this appendix, we derive the master equation [Eq.(28)] for the reduced density matrix of the MR from themaster equation [Eq. (24)] of the coupled MR-TQD sys-tem by eliminating the degrees of freedom of the TQD.In general, the dissipation rate of the MR is much smallerthan the decay rate of the TQD, i.e., [ n ( ω m ) + 1] γ ≪ Γ.The TQD hence attains its steady-state quickly and itsperturbation to the MR can be regarded as part of the en-vironment [27, 39]. Up to the second order in η , Eq. (24)can be rewritten as dρdt = L ρ = [ L + L + L ] ρ, (A1)where L ρ = − iω m [ b † b, ρ ] − i [ H TQD , ρ ]+Γ D [ a † ] ρ + Γ D [ a † ] ρ + Γ D [ a ] ρ, (A2) L ρ = − i [ V ( b † − b ) , ρ ] , (A3) L ρ = η Γ ( D [ a † b † ] ρ + D [ a † b ] ρ )+ η Γ ( D [ a b † ] ρ + D [ a b ] ρ ) + L D ρ, (A4)are respectively the Liouvillians to zeroth, first, and sec-ond orders in η . At zeroth order in η , the quantum dy-namics of the whole system is described by˙ ρ ( t ) = L ρ ( t ) . (A5)The MR and the TQD are decoupled. Since the TQDis at its steady state most of the time, one has ρ ( t ) = ρ st d ⊗ Tr d { ρ } with ρ st d denoting the reduced density ma-trix of the TQD at steady-state and Tr d {· · · } the traceover the TQD’s degrees of freedom. Equation (A5) hasan infinite number of steady-state solutions. These so-lutions can be expanded in the basis of the eigenvectors ρ st d ⊗ | n ih n ′ | of the Liouville operator L with eigenval-ues λ nn ′ = − i ( n − n ′ ) ω m , i.e., L ρ nn ′ = λ nn ′ ρ nn ′ [33].0Here, | n i ( n = 0 , , , · · · ) denotes the n th state of theMR and ( n − n ′ ) ω m represents the energy difference be-tween the states | n i and | n ′ i . For η = 0, these stateswith different n are weakly coupled by the perturbativeterms L and L . To obtain the quantum dynamics ofthe MR, we project the system onto the subspace with azero eigenvalue ( n = n ′ ) of L . The projection operator P is defined by L P ρ = 0 . (A6)Noting g ≪ ω m (i.e., η ≪ P ρ [25] P ˙ ρ ( t ) = PL P ρ ( t ) + ∞ Z dτ PL e L τ L P ρ ( t ) . (A7)Substituting Eqs. (A3) and (A4) into Eq. (A7) and takingthe trace over the TQD degrees of freedom, the first termin Eq. (A7) becomes [39]Tr d {PL P ρ ( t ) } = 12 [ γn ( ω m ) + η Γ ρ st00 ] D [ b ] µ + 12 (cid:8) γ [ n ( ω m ) + 1] + η Γ ρ st00 (cid:9) D [ b † ] µ, (A8)and the second term givesTr d (cid:26) ∞ Z dτ PL e L τ L P ρ ( t ) (cid:27) = − i ( ω m + δ m )[ b † b, µ ] , +Re[ G ( iω m )] D [ b ] µ + Re[ G ( − iω m )] D [ b † ] µ, (A9)where µ = Tr d {P ρ } is the reduced density matrix of theMR and ρ st00 is the probability of an empty TQD at thesteady state. Here, we have defined δ m = Im[ G ( iω m ) + G ( − iω m )] . (A10)Thus, from Eqs. (A7), (A8) and (A9), one has˙ µ = − i ( ω m + δ m )[ b † b, µ ] + 12 [ γn ( ω m ) + A + ] D [ b ] µ + 12 (cid:8) γ [ n ( ω m ) + 1] + A − (cid:9) D [ b † ] µ, (A11) where A ± = 2 Re[ G ( ± iω m )] + η Γ ρ st00 . (A12)Eq. (A11) is simply the master equation (28) for the re-duced density matrix of the MR derived in Sec. IV. InEq. (A9), the correlation function G ( s ) is given by G ( s ) = − Tr d ∞ Z dtV (0) e L TQD t V (0) e st = − ∞ Z dτ h V ( t ) V (0) i e st = −h e V ( s ) V (0) i , (A13)where s = iω m . Also, e V ( s ) is the Laplace transform of V ( t ) and the Liouvillians L TQD is given in Eq. (16).To determine the correlation function G ( ± s ), one firstcalculate the Laplace transform e V ( s ) of the interactionterm V ( t ). For convenience, we introduce the vector op-erator ˆ σ for the TQD whose components are defined asˆ σ = | ih | , ˆ σ = | ih | , ˆ σ = | ih | , ˆ σ = | ih | , ˆ σ = | ih | , ˆ σ = | ih | , ˆ σ = | ih | , ˆ σ = | ih | , ˆ σ = | ih | , (A14)where the average value of each component is h ˆ σ i i =Tr { ˆ σ i ρ d } . Using this notation, one has V = 2 η Ω (ˆ σ − ˆ σ ) + η Ω (ˆ σ − ˆ σ ) . (A15)and thus G ( s ) = − η Ω [ S ( s ) − S ( s )] − η Ω [ S ( s ) − S ( s )] , (A16)where S i ( s ) = h e σ i ( s ) V (0) i with h e σ i ( s ) i being the Laplacetransform of h ˆ σ i ( t ) i . From Eq. (16), we find that h ˆ σ i ( t ) i obeys the equation of motion: d h ˆ σ ( t ) i dt = M h ˆ σ ( t ) i + B, (A17)where1 M = − Γ − Γ − Γ − i Ω i Ω − Γ − Γ − Γ − i Ω i Ω − Γ i Ω − i Ω i Ω − i Ω − i ∆ d − i Ω i Ω i ∆ d i Ω − i Ω − i Ω i Ω − i Ω λ i Ω − i Ω i Ω λ ∗ − i Ω i Ω − i Ω λ i Ω − i Ω i Ω λ ∗ , (A18)and B = (Γ , Γ , , , , , , , T . Here we have defined∆ d ≡ ∆ − ∆ , λ ≡ − ( i ∆ + Γ ), and λ ≡ − ( i ∆ + Γ ). From Eq. (A17), the steady-state solution of thevector h ˆ σ i is calculated as h ˆ σ st i = M − B. (A19)Applying the Laplace transform to Eq. (A17), one ob-tains s h e σ ( s ) i − h ˆ σ (0) i = M h e σ ( s ) i + Bs . (A20)Moreover, the i th component of the vector h e σ ( s ) i is given by h e σ i ( s ) i = X k =1 L ik [ h ˆ σ k (0) i + B k s ] . 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