Detection of weak forces based on noise-activated switching in bistable optomechanical systems
Detection of weak forces based on noise-activated switching in bistable optomechanical systems
Samuel Aldana,Christoph Bruder,and Andreas Nunnenkamp
Department of Physics,University of Basel,Klingelbergstrasse 82,CH-4056Basel,Switzerland
(Dated:September 30,2014)We propose to use cavity optomechanical systems in the regime of optical bistability for the detection of weak harmonic forces.Due to the optomechanical coupling an external force on the mechanical oscillator modulates the resonance frequency of the cavity and consequently the switching rates between the two bistable branches.A large difference in the cavity output ?elds then leads to a strongly ampli?ed homodyne signal.We determine the switching rates as a function of the cavity detuning from extensive numerical simulations of the stochastic master equation as appropriate for continuous homodyne detection.We develop a two-state rate equation model that quantitatively describes the slow switching dynamics.This model is solved analytically in the presence of a weak harmonic force to obtain approximate expressions for the power gain and signal-to-noise ratio that we then compare to force detection with an optomechanical system in the linear regime.
PACS numbers:42.65.Pc,42.50.Lc,42.50.Wk,07.10.Cm
I.INTRODUCTION
The ?eld of cavity optomechanics is historically closely re-lated to the problem of force sensing in the context of gravita-tional wave detection [1–4],and the fundamental limit of force sensitivity can be traced back to the quantum-mechanical na-ture of the detector,the so-called standard quantum limit [5].Most optomechanical devices to date operate in the regime where the radiation pressure is suf?ciently weak on the single-photon level so the coupling between phonons and photons can be linearized.Examples for exciting progress in this area include the observation of ground-state cooling [6–8],pon-deromotive squeezing [9–11],radiation-pressure shot-noise [12,13],and mechanical zero-point motion via sideband ther-mometry [14–17]as well as the demonstration of displace-ment detection close to the standard quantum limit [18–21].Advances in fabricating optomechanical devices promise increasingly large coupling strengths [22]making nonlinear quantum effects [23,24]a possible reality in the near future.It is thus of great interest to study how the intrinsically non-linear radiation pressure can be exploited in novel devices.In this paper we propose sensitive force detection exploiting optical bistability in an optomechanical system [25–27].The optomechanical system we consider consists of a laser-driven optical cavity whose resonance frequency is modulated by the displacement of a mechanical oscillator [28–30].Under cer-tain conditions the system exhibits an optical bistability,i.e.it has two classically stable states with potentially largely differ-ent cavity ?elds.Shot-noise ?uctuations in the coherent drive of the cavity will cause transitions between the two branches whose switching rates can depend strongly on cavity detun-ing.A weak periodic forcing of the mechanical resonator will modulate the cavity detuning and thus the switching rates al-lowing the detection of weak forces in the cavity spectrum.Note that exploiting periodic modulation of switching rates in bistable systems to detect small coherent signals has also been discussed in the context of stochastic resonance [31–33]and Josephson bifurcation ampli?ers [34–36].
In the following we calculate numerically the switching dy-namics in the single-photon strong-coupling regime and zero temperature limit using a stochastic quantum master equation.
We obtain the switching rates and their dependence on the de-tuning from the residence time distribution.We then develop a two-state rate equation model allowing us to write the output spectral density of the amplitude quadrature as the sum of a low-frequency noise background and a signal peak caused by the weak harmonic force.The homodyne signal amplitude de-pends linearly on the force amplitude and on the difference be-tween the cavity output ?elds.Bistable optomechanical sys-tems can thus be used as linear ampli?ers whose bandwidth is the switching rate and which have a potentially large gain for low-frequency signals.
The remainder of the present paper is organized as follows.In Sec.II we introduce the model for an optomechanical sys-tem (OMS)with an additional external force driving the me-chanical oscillator and present the stochastic master equation describing the system state conditioned on continuous homo-dyne detection.In Sec.III we investigate numerically noise-induced switching in a bistable OMS.We obtain time traces of the homodyne photocurrent,the residence time distribu-tions,and the switching rates as a function of cavity detuning.In Sec.IV we describe the slow switching dynamics and the in?uence of a harmonic force within a two-state rate equa-tion model with periodically modulated switching rates.In Sec.V we ?nd expressions for the noise spectral density and the signal amplitude of the homodyne photocurrent,based on the two-state rate equation model,and compare them to quan-tum trajectory results.Finally,we compare the power gain and signal-to-noise ratio of force detection with a bistable OMS to those achievable with an OMS in the linear regime.
II.MODEL
We consider an optomechanical system (OMS)in which the position of a mechanical oscillator modulates the resonance frequency of an optical cavity.The system consists of a me-chanical mode with resonance frequency ωm and an optical mode with frequency ωc which are coupled by the radiation-pressure interaction.The optical mode is driven by a laser with strength and frequency ωd .In a frame rotating at the
a r X i v :1409.8082v 1 [q u a n t -p h ] 29 S e p 2014
drive frequency ωd the Hamiltonian reads ( =1)
?H =??0?a ??a ?i ?a ??a ? +ωm ?b ??b ?g 0?a ??a ?b +?b ? ,(1)where ?a and ?b are bosonic annihilation operators for the op-tical and mechanical mode,?0=ωd ?ωc is the detuning
between driving and cavity frequency,and g 0is the optome-chanical coupling.We also add an external periodic force on the mechanical resonator with amplitude g 1and frequency ?
?H F =?g 1sin(?t )
?b +?b ? .(2)A complete description of the system additionally requires
the optical damping rate κ,the mechanical energy dissipation rate γm ,and the mean phonon number in thermal equilibrium n th =0corresponding to a zero-temperature reservoir.
The dissipative dynamics of the OMS undergoing contin-uous homodyne measurement of the cavity output can be de-scribed with the It?o stochastic master equation (SME)[37,38]
d ?ρc =L [?ρc ]dt +H [?ρc ]dW ,
(3)
L [?ρc ]=?i ?H +?H F ,?ρc +κD ?a [?
ρc ]+(n th +1)γm D ?b [?ρc ]+n th γm D ?b ?[?
ρc ],(4)H [?ρc ]=√κ
?a ?ρc +?ρc ?a ?? ?a +?a ? c ?ρc ,
(5)
where dρc =?ρc (t +dt )??ρc (t ), ?a +?a ? c =Tr [(?a +?a ?)?ρc ],and dW is a Wiener increment with E [dW ]=0and E [dW 2]=dt .E [?]is the ensemble average and the Lindblad terms have the usual form,D ?o [?
ρ]=?o ?ρ?o ??(?o ??o ?ρ+?ρ?o ??o )/2.The ?rst term in Eq.(3)is the Liouvillian describing the coherent evo-lution due to the Hamiltonian and the decoherence originating from the coupling to the environment.The second term called innovation describes the effect of a measurement of the ampli-tude quadrature,?X =?a +?a ?,with homodyne detection of the
cavity output ?eld.The innovation term conditions the evolu-tion of the quantum state ?ρc (t )on the homodyne photocurrent
I c (t )=
√κ
?X (t ) c +dW dt
,(6)
which is the sum of a conditioned expectation value of ?X
and a ?uctuating term originating from the shot noise of the local oscillator (here we have assumed unit detection ef?ciency).We will refer to the result for a particular noise realization of ?ρc (t )and I c (t )as a quantum trajectory.Taking the ensem-ble average of Eq.(3)we recover the unconditional quantum state ?ρ(t )=E [?ρc (t )]which is a solution to the quantum mas-ter equation
˙?ρ=L [?ρ].
(7)
In the following we calculate the evolution of the quantum state ?ρc (t )by numerically integrating Eq.(3)[39]and use the time traces of the homodyne photocurrent I c (t )to investigate the switching dynamics in the regime of optical bistability.To quantify the in?uence of the external mechanical force on the cavity output we use the time-averaged spectral density
S out II (ω)=lim t →∞
dτe iωτE [I c (t +τ)I c (t )].(8)
FIG.1.(Color online)Noise-activated switching in a bistable op-tomechanical system (OMS).(a)Homodyne photocurrent I c (t )for a
representative quantum trajectory.The OMS switches between two
bistable states that are close to the two stable solutions ˉX
±(dashed lines)of the nonlinear mean-?eld equations (MFEs)(10).From the time trace I c (t )the residence times τ±can be extracted.We show both the conditioned expectation value of the amplitude quadrature ?X
(t ) c (grey solid)and the homodyne photocurrent I c (t )after ap-plying a low-pass ?lter (black solid).(b)From a suf?ciently long tra-jectory we can obtain the probability distribution p (I c )of the ?ltered homodyne photocurrent whose double-peak structure is a signature
of the bistable behavior.(c)Stable ˉX
±(black solid)and unstable (black dashed)solutions to the MFEs (10)as a function of the bare detuning ?0.We indicate the stable states (circles)between which the system shown in (a)and (b)switches.The ?gure also shows
the steady-state expectation value ?X
ss (red solid)interpolating be-tween the bistable solutions ˉX
±.(d)A blow-up of the region marked grey in panel (c).Additionally,we plot the weighted average of the
mean-?eld solutions p ss ?ˉX ?+p ss +ˉX +(black dots)where the prob-abilities p ss ±are given by Eq.(11).The parameters are ωm /κ=5,γm /κ=1/2,g 0/κ=1/√
2, /κ=1.5,and ?0/κ=?1.45(a,b).
For ?nite,but suf?ciently long sampling times T the spectral
density can be obtained using the Wiener-Khintschin theorem
from a quantum trajectory as S out
II (ω)=|I T (ω)|2where
I T (ω)=1
√T T 0dt e iωt I c (t )(9)
is the windowed Fourier transform of the homodyne photocur-rent I c (t ).In this way we replace the ensemble average by a time average.In the following we will numerically simulate a single,suf?ciently long quantum trajectory instead of calcu-lating averages over an ensemble of quantum trajectories.
III.
NOISE-ACTIV ATED SWITCHING IN BISTABLE OMS
We investigate the dynamics of an OMS in a regime where
the mechanical resonator acts like an effective Kerr nonlinear-ity for the optical mode [27].As a consequence the system
3
can exhibit optical bistability,a phenomenon characterized by the presence of two stable mean-?eld states.In a semiclassi-cal approximation the steady-state amplitudes of the optical ˉa and mechanical modes ˉb are obtained by solving the coupled mean-?eld equations (MFEs)
0= i ?0?κ2
ˉa +ig 0ˉa ˉb +ˉb ? + ,0=? iωm +γm 2 ˉb +ig 0|ˉa |2
.
(10)An analysis of the nonlinear MFEs (10)shows that the OMS undergoes a bifurcation when the driving amplitude exceeds the threshold value bif =31/4(κ3ωm /18)1/2/g 0.As a con-sequence three solutions for ˉa exist in a certain range of nega-tive detuning ?0.The two solutions ˉa ±with the smallest and largest amplitude |ˉa |are stable and referred to as the upper and lower branches of the bistable system.
Shot-noise ?uctuations in the cavity drive will cause transi-tions between the stable branches.This effect dubbed noise-activated switching has been investigated e.g.in the case of a Kerr medium theoretically [40–44]and experimentally [45].In Fig.1(a)we show the homodyne photocurrent I c (t )for a representative quantum trajectory.We observe that the OMS switches between two bistable states characterized by two dif-ferent values of I c (t )and corresponding approximately to √κˉX ±where ˉX ±=ˉa ±+ˉa ?±.After applying a low-pass ?lter to the raw quantum trajectory data we can extract the residence times τ±from the time trace I c (t ).From a suf?-ciently long trajectory we obtain the probability distribution p (I c )for the homodyne photocurrent,shown in Fig.1(b).It features a double peak,a signature of the bistable behavior.In Fig.1(c)we show the mean-?eld amplitude quadrature,ˉX =ˉa +ˉa ?,as function of the detuning ?0obtained from the solutions to the nonlinear MFEs (10).We also calculate
the steady-state expectation value ?X
ss from the QME (7)which interpolates between the two bistable solutions ˉX
±.Figures 2(a)and 2(b)show histograms R (τ±)of residence times in the upper and lower branches,respectively,which we extracted from the quantum trajectory shown in Fig.1(a)in-cluding statistical error bars.We ?t the data with exponential distribution functions R ?t (τ±)=W ?exp(?W ?τ±)and de-termine the switching rates W ?from the upper to the lower branch and vice versa [46].In Fig.2(c)we plot the switching rates W ±as a function of cavity detuning ?0.
In steady state the probability to ?nd the OMS in the upper or lower branch,p ss ±,is related to the switching rates via
p ss ±=
W ±
W ++W ?
.
(11)
The probability p ss ±is the fraction of time spent by the system in the upper and lower branch,respectively.It can be written as T ±/(T ++T ?),where T ±is the average residence time and
is given by T ±= τ±R (τ±)dτ±=W ?1
?.If the ?uctuations in each branch ˉa ±are small compared to their phase-space separation |ˉa +?ˉa ?|,the average homodyne photocurrent I ss =E [I c (t )],or equivalently the steady-state
expectation value ?X
ss =I ss /√κ,is well approximated
by FIG.2.(Color online)Residence time distributions and switching rates .(a)Histogram R (τ+)of residence times in the upper branch extracted from the quantum trajectory in Fig.1(a)with statistical er-ror bars.The solid line is an exponential ?t R ?t (τ+)=W ?e ?W ?τ+excluding the ?rst bin.(b)Same as (a)but for the residence times in the lower branch.We determine the switching rate W +by ?tting the histogram R (τ?)with the distribution R ?t (τ?)=W +e ?W +τ?.(c)Switching rates W ±as a function of ?0.Parameters are identical to those in Fig.1and with ?0/κ=?1.45(a,b).
the weighted average of the mean-?eld solutions
I ss √κ p ss ?ˉX ?+p ss +
ˉX + .(12)
In Fig.1(d)we show a blow up of Fig.1(c)for detunings in
the bistable regime.Additionally,we also plot p ss ?ˉX ?+p ss +ˉX +
where the probabilities p ss ±are given by Eq.(11).We see that the switching dynamics of bistable OMS in this regime can be accurately captured by a two-state model.
IV .TWO-STATE MODEL WITH SLOWLY AND PERIODICALLY MODULATED SWITCHING RATES
The in?uence of the periodic force (2)on the switching dy-namics can be described with a two-state rate equation model
˙p ±(t )=±W +(t )p ?(t )?W ?(t )p +(t )
=?W (t )p ±(t )+W ±(t )
(13)
where p ±(t )is the probability for the system to be in the vicin-ity of the branch ˉa ±satisfying p ++p ?=1,W ±(t )are the
time-dependent switching rates,and W (t )=W +(t )+W ?(t ).For a mechanical forcing that is slow on the time scale of
intra-branch ?uctuations,i.e.? κ,ωm ,the in?uence of ?H
F can be reduced to an adiabatic change of the resonator equi-librium position that is given by 2(g 1/ωm )sin(?t )in units of its zero-point amplitude.This leads to a slow variation of the cavity detuning ?0+2(g 0g 1/ωm )sin(?t )and will only affect the long-time dynamics of the optical mode,i.e.the switching
4 behavior,by modulating the switching rates
W±(t)=W0±+W1±sin(?t).(14)
Here,W0±denote the switching rates in absence of the external
force g1=0and,assuming that for a weak force the switching
rates depend linearly on the detuning,we have
W1±=
2g0g1
m ?W0±
.(15)
The steady-state solution to the rate equation(13)for peri-odic switching rates W±(t)with period T?=2π/?is itself periodic and given by[47]
p±(t)=
1
1?e?W T?
T?
dt W±(t?t )
×e?W t exp
?
t
t?t
δW(t )dt
(16)
with W= T?
W(t)dt/T?andδW(t)=W(t)?W.For
the transitions rates W±(t)in Eq.(14),W=W0++W0?and δW(t)=(W1++W1?)sin(?t).Expanding the exponential in Eq.(16)and neglecting higher harmonics,we obtain in the limit|W1++W1?| ?the long-time solution
p±(t) W0±
W
±
W1+W0??W1?W0+
W
W2+?2
sin(?t?φ)(17)
whereφ=arctan
?/W
.The?rst term in Eq.(17)corre-
sponds to p ss±,the steady-state probability to?nd the system in the upper or lower branch in absence of the external force. The second term is a slow periodic modulation of these prob-abilities and we will use them to characterize the in?uence of an external force on the homodyne photocurrent I c(t).
V.DETECTION OF WEAK PERIODIC FORCES WITH A BISTABLE OPTOMECHANICAL SYSTEM
We will now analyze our force detection scheme by exam-ining the output spectral density of the homodyne photocur-
rent S out
II (ω).In brief,the spectral density is the sum of two
contributions,a noise background and a signal contribution,
S out II(ω)=S noise
II (ω)+S signal
II
(ω).(18)
The noise background S noise
II (ω)quanti?es the power per unit
bandwidth of the noise interfering with detection at frequency ω.As we will show,in our detection scheme,the main contri-
bution to S noise
II (ω)at low frequencies originates from the in-
coherent switching of I c(t)between the two stable branches.
A weak harmonic force with frequency?produces a coher-ent modulation of the homodyne photocurrent with amplitude I(?)and thus contributes a delta peak to the spectral density
S signal II (ω)=
π
2
I(?)2[δ(ω??)+δ(ω+?)].(19)
For a?nite sampling time T one expects the signal peak height
to be S signal
II
(?)=πI(?)2/(2?ω)where?ω=2π/T is the
?nite frequency resolution of the spectral density.
We will use two quantities to quantify the ampli?cation and
the sensitivity of our proposed detector scheme.The?rst one
is the ratio I(?)/g1which relates the modulation amplitude of
the homodyne photocurrent I(?)(output signal amplitude)to
the forcing amplitude g1(input signal amplitude).This ratio
characterizes ampli?cation with a dimensionless power gain
G(?)=κ
I(?)
g1
2
(20)
expressing the ratio of the signal output power∝I(?)2to the
signal input power∝g21.To quantify the sensitivity of our
scheme we will use the signal-to-noise ratio(SNR)de?ned as
SNR=
1
?ω
?+?ω/2
???ω/2
S out II(ω)dω
S noise
II
(?)
.(21)
For a suf?ciently long sampling time T,the noise background
S noise
II
(ω)is approximately constant over the frequency win-
dow?ω=2π/T.Thus,SNR=S signal
II
(?)/S noise
II
(?)+1,
i.e.the SNR depends only on the ratio of the output signal and
the noise background power at the signal frequency?.
Our two-state rate equation model allows us to?nd approx-
imate expressions for the noise spectral density S noise
II
and sig-
nal amplitude I(?).We will compare these analytical results
to quantum trajectory simulations https://www.sodocs.net/doc/2411859397.html,ing the gain G(?)
and SNR to characterize our detection scheme we will be able
to compare its performance to force detection with an OMS
in the linear regime.We will derive analytical expressions for
the modulation amplitude I lin(?),the power gain G lin,and the
noise background S noise
II,lin
.We then express S noise
II,lin
as a func-
tion of the power gain G lin and the OMS parametersωm,κ,
andγm so we can compare the sensitivity of the two different
schemes,bistable OMS and linear OMS,at?xed power gain.
A.Two-state approximation for the output spectral density
Describing the switching dynamics within the two-state rate
equation model allows us to?nd analytic expressions for the
low-frequency part of the output spectral density S out
II
(ω).As
stated above,Eq.(18),S out
II
(ω)can be separated into a noise
background S noise
II
(ω)and the signal part S signal
II
(ω).
In absence of the external force incoherent switching causes
autocorrelations of the homodyne photocurrent to decay ex-
ponentially on a time scale W?1.We?nd the autocorrelation
function(up to an irrelevant constant I2ss)is given by
E[I c(t+τ)I c(t)]=e?W|τ|κp ss+p ss?(ˉX+?ˉX?)2+δ(τ).
(22)
The second term stems from the shot noise of the local oscilla-
tor.The?rst term is proportional to the steady-state variance
Var(?X)ss= ?X2 ss? ?X 2ss p ss+p ss?(ˉX+?ˉX?)2.Calcu-
lating Var(?X)ss from the QME(7),we?nd that this two-state
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0.3
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0200
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(c )FIG.3.(Color online)Detection of weak force with a bistable OMS .
(a)Spectral density for the homodyne photocurrent S out
II (ω)in pres-ence of a weak external force on the mechanical oscillator (black).The spectral density features a noise background and a signal peak.
At small frequencies the noise background S noise
II
(ω)(red)can be ap-proximated by a Lorentzian of width W at zero frequency,Eq.(23).
(b)and (c)Signal peak height S signal II
(?)=S out II (?)?S noise
II (?)as a function of forcing amplitude g 1(b)and forcing frequency ?(c).Black squares are quantum trajectory simulations with statistical er-ror bars.Black lines are analytical results based on the two-state rate equation model,Eq.(27),as discussed in the main text.The param-eters are the same as in Fig.1but for ?0/κ=?1.4.The weak external mechanical force has a frequency ?/κ=0.1(a,b)and an amplitude g 1/κ=0.2(a,c).The spectral density for each pair of parameters (?,g 1)is obtained from an average over hundred spectra with a frequency resolution ?ω=10?3κ.
approximation overestimates the variance in the presence of appreciable intra-branch ?uctuations around mean-?eld solu-tions.In fact,the noise background is smaller and more accu-rately given by
S noise II (ω)=2κVar (:?X
:)ss W W 2
+ω2
+1,(23)
where Var (:?X
:)ss =Var (?X )ss ?1is the normally-ordered variance of the amplitude quadrature,the colon denoting nor-mal ordering of the optical creation and annihilation opera-tors.Equation (23)satis?es the constraint that the total power of the homodyne photocurrent minus the shot-noise contribu-tion must satisfy [37], [S noise
II (ω)?1]dω2π=κVar (:?X :)ss .
The noise spectrum consists of a shot noise contribution and a Lorentzian centered at zero frequency with a half width at half maximum given by W .
Equation (17)allows us to ?nd an approximate expression
for the signal part S signal
II to the output spectral density.In the long-time limit a periodic time-dependence of the probabil-ity p ±(t )yields a periodically modulated average homodyne photocurrent
E [I c (t )]=√κ
p +(t )ˉX
++p ?(t )ˉX ? =I ss +I (?)sin(?t ?φ)
(24)
with the modulation amplitude in two-state approximation
I (?)=√κ ˉX +?ˉX ?
W 1+W 0??W 1?W 0+
W W 2
+?2
.(25)
The relationship between the average steady-state homodyne
photocurrent I ss =√κ ?
X ss ,the probabilities p ss ±,and the
transitions rates W i
±given by Eqs.(11),(12),and (15)provide a direct interpretation of I (?).The zero-frequency expression I (0)=(2g 1g 0/ωm )(?I ss /??0)is the linear response of I ss to a change in the detuning ?0.The prefactor 2g 1/ωm is the zero-frequency response of the mechanical oscillator,i.e.the change in the mechanical equilibrium position (in units of its zero-point amplitude)caused by a static force with amplitude g 1.This displacement leads to a change of the cavity detuning ?0by g 0(2g 1/ωm ).Relaxation of a bistable OMS at rate W causes an attenuation of this response at ?nite frequencies ?,
I (?)=
2g 0g 1ωm √κ? ?X
ss ??0W W 2
+?2
.
(26)
As stated in Eq.(19),the signal contributes a delta peak to the spectral density since the autocorrelation function of the ho-modyne photocurrent is dominated by periodic modulation in
the limit τ W ?1
,and hence factorizes,E [I c (t +τ)I c (t )]=E [I c (t +τ)]E [I c (t )].For a ?nite frequency resolution ?ω,
S signal
II (?)=πκ2?ω 2g 1g 0ωm ? ?X ss ??0 2
W 2
W 2+?2.(27)In Fig.3(a)we plot the spectral density for the homodyne
photocurrent S out
II (ω)in the presence of a weak external force.An average over hundred spectra is shown.The spectral den-sity features a low-frequency Lorentzian noise background whose frequency dependence agrees very well with our two-state approximation S noise
II (ω),Eq.(23).The height of the sig-nal peak relative to the noise level,S signal II (?)=S out II (?)?S noise
II (?),is obtained for a range of forcing amplitudes g 1and forcing frequencies ?.Comparing these quantum trajectory
simulations to Eq.(27),we ?nd that S signal
II (?)exhibits the correct quadratic dependence on the forcing amplitude g 1and Lorentzian dependence on the forcing frequency ?.The mod-ulation amplitude I (?)is about 20%smaller than expected.We suspect that this quantitive disagreement is due to the large amplitude of intra-branch ?uctuations reaching a considerable fraction of the inter-branch separation and the fact that the lin-ear approximation to the modulation of switching rates (15)is only satis?ed for the smaller values of g 1in Fig.3.The expected power gain of a bistable OMS is
G (?)= 2g 0κωm ? ?X ss ??0 2
W
2
W 2+?2.(28)We notice that ampli?cation occurs over a bandwidth given
by the switching rate W .As can be seen in Fig.1(c),the
slope ? ?X
ss /??0in the center of the bistable region is ap-proximately proportional to the difference between the two
6 mean-?eld solutionsˉX+?ˉX?.As a consequence,a large
difference in the cavity output?elds leads to a strongly am-
pli?ed homodyne signal.If the cavity is driven further away
from bifurcation,the slope increases,but the switching rate W
decreases.Thus,the gain can be made larger at the expense of
reducing the bandwidth.For low signal frequency,? W,
for which the shot-noise contribution to the noise background
S noise
II
is negligible,the SNR is independent of?,
SNR πW
?ω
g1g0
ωm
2
? ?X ss/??0
2
Var(:?X:)ss
+1,(29)
with Var(:?X:)ss and(? ?X ss/??0)2obtained from Eq.(7). These two quantities have a similar dependence on the detun-ing?0and reach their maximum at an optimal value of?0in the center of the bistable region.As a consequence,both the SNR and the gain G are maximal.
Figure4shows the dimensionless signal output power I(?)2/κ(a,b)and SNR(c,d)as a function of the signal input power(g1/κ)2(a,c)and signal frequency?(b,d).We com-pare results from quantum trajectory simulations and from our two-state rate equation model.In panel(a)we see that the bistable OMS exihibits nearly constant power gain for small forcing amplitudes g1.In panel(b)we observe that its detec-tion bandwidth is in good agreement with predictions of the two-state model and given by the switching rate W.As ex-pected,the SNR is approximately constant over the detection bandwidth as can be seen in panel(c).
B.Force detection with an OMS in the linear regime
In the linear regime the dissipative dynamics of an OMS, including the noise and signal spectral densities of its out-put?eld quadratures,can be obtained exactly from the input-output formalism[48,49].The linear regime is character-ized by a small optomechanical coupling rate,g0 κ,ωm, and a cavity driven to a coherent state with large amplitude |ˉa| 1.Under these conditions,the radiation-pressure in-teraction can be approximated by a bilinear interaction,with an enhanced coupling rate g=g0|ˉa|,between the resonator position,?b+?b?,and the amplitude quadrature,?a+?a?.The static shift of the resonator position results in an effective cav-ity detuning?=?0+g0(ˉb+ˉb?).A displacement of the mechanical resonator imprints a phase shift on the output light ?eld,which is best probed by driving the cavity on resonance,?=0,and by measuring the phase quadrature at the output [30].
Analogous to Eq.(26)we?nd an expression for the am-
plitude modulation I lin and the spectral density S signal
II,lin of the
phase quadrature in homodyne detection due to the force
I lin(?)=
G lin(?)
g1
√
κ
,
S signal II,lin (ω)=
π
2
I2lin(?)[δ(ω??)+δ(ω+?)].
(30)
Here,the equivalent power gain at frequencyωfor an OMS
in
FIG.4.(Color online)Power gain and signal-to-noise ratio(SNR).
Signal output power(a,b)and SNR(c,d)as function of the signal input power(a,c)and signal frequency(b,d).The expected values of I(?)2and the SNR according to the two-state model discussed in the main text(black line)are compared to quantum trajectory re-sults shown in Fig.3(black squares).Grey lines are a?t to the data indicating that the power gain G(?)and the SNR have the correct dependence on the signal input power and signal frequency.The ob-served power gain has a value about40%smaller than expected.In panel(b),the dotted blue line indicates the result for the largest pos-
sible power gain of an OMS operating in the linear regime G(max)
lin
, Eq.(33).In panels(c)and(d),the dashed red line indicates the SNR for an OMS in the linear regime operating at the same power gain (extracted from the quantum trajectory results)and obtained from Eq.(36).The parameters are identical to Fig.3,with an external forcing frequency?/κ=0.1(a,c)and amplitude g1/κ=0.2(b,d).
the linear regime reads
G lin(ω)=|2gκχc(ω)[χm(ω)?χ?m(ω)]|2,(31)
withχc(ω)=(κ/2?iω)?1the cavity susceptibility and χm(ω)=[γm/2+i(ωm?ω)]?1the mechanical susceptibil-ity.The zero-frequency response can be written as I lin(0)= (2g0g1/ωm)[??(
√
κˉI)]?=0,i.e.the product of a shift of the cavity detuning caused by a static force with amplitude g1and the derivative with respect to?of the average homodyne pho-tocurrent,
√
κˉI,whereˉI=?i(ˉa?ˉa?)is the mean-?eld value of the optical phase quadrature andˉa= /(κ/2?i?).At low frequency,ω κ,ωm,the power gain is approximately con-stant,
G lin(ω)=
2g0κ
m
?ˉI
?=0
2
,(32)
which is analogous to Eq.(28).
The low-frequency power gain,Eq.(32),can as well be ex-pressed as G lin(ω)=(8g0/ωm)2ˉn,and is proportional to the average cavity occupation on resonance,ˉn=|ˉa|2=4( /κ)2.
An OMS can only operate in the linear regime below bifurca-tion, < bif,that is for a cavity occupation below the critical value n bif=2κωm/(3
√
3g20).As a consequence,the power gain cannot be made arbitrarily large and the maximal gain has the universal value
G(max)
lin
(ω)
128
3
√
3
κ
ωm
.(33)
7 The spectral density of the noise interfering with the de-
tection of a force signal far from the mechanical resonance,
|ω?ωm| γm,referred back to the input signal is[30,49]
S noise
II,lin
(ω) G lin(ω)=
1
G lin(ω)
+G lin(ω)
ω2m?ω2
2
16κ2ω2m
+
n th+
1
2
γm
κ
ω2+ω2m
2ω2m
.
(34)
Equation(34)expresses the total measurement noise as?uc-tuations in the forcing amplitude and has three contributions. The?rst term is the imprecision noise due to the shot noise of the local oscillator.The second term is the back-action noise or radiation-pressure shot noise.The last term originates from thermal and quantum?uctuations of the resonator position.
At each frequencyω,there is an optimal gain G(opt)
lin (ω)=
2κ|χm(ω)?χ?m(?ω)|for which the measurement noise is minimal and the SNR maximal.In the limit of small frequen-cies,the optimal gain is then
G(opt) lin
4κ
ωm
.(35)
The low-frequency noise level for the optimal gain and a me-chanical resonator coupled to a zero-temperature bath(n th=
0),S noise
II,lin 2+γm/ωm,is minimal.This is commonly re-
ferred to as the standard quantum limit(SQL)of force(or position)detection.At the SQL the back-action noise and the imprecision noise are both equal to the shot-noise term.
https://www.sodocs.net/doc/2411859397.html,parison of bistable and linear detection
An OMS in the regime of optical bistability exhibits a power gain G much larger than the gain G lin of a linear OMS.The low-frequency expressions for the power gain of a bistable or linear OMS,Eqs.(28)and(32),depend on the co-ef?cients(? ?X ss/??0)2and(?ˉI/??)2,respectively.These coef?cients characterize the response of the steady-state value of the optical amplitude and phase quadratures,respectively, to a change in the detuning.The second coef?cient is propor-tional to the average cavity occupation,which is limited by ˉn lin (?)at which the SQL applies,and can even be larger than G(max) lin (?),i.e.the maximal gain for a linear OMS below bifurcation. Figure4(b)shows the dimensionless signal output power I(?)2/κas a function of the signal frequency?obtained from quantum trajectory simulations and from the two-state model. In addition,we indicate the results corresponding to a linear OMS operating at its maximal power gain G(max) lin .Note that G(?)>G(max) lin (?)within the detection bandwidth,i.e.for signal frequencies? W. As a consequence of the large gain G G(opt) lin ,the mea- surement noise S noise II unavoidably exceeds the SQL value that applies to an OMS in the linear regime,S noise II,lin 2+γm/ωm. Thus,instead of comparing the sensitivity of our scheme to a linear OMS operating at the SQL,we compare it to the sen- sitivity of a linear OMS with identical gain.The SNR of a linear OMS can be expressed as a function of its power gain G lin to compare it to results of quantum trajectory simulations. From Eqs.(30)and(34),we obtain,for small signal frequen- cies? κ,ωm and n th=0, SNR= πg21 2?ωκ 1 G lin(?) +G lin(?) ω2m 16κ2 + γm 4κ ?1 .(36) In Fig.4,we plot the SNR of a bistable OMS as function of the signal input power(g1/κ)2(c)and signal frequency? (d).In addition,we plot the SNR of a linear OMS with identi- cal parametersωm,γm,andκand operating at the same gain G lin(?)=G(?),where G(?)is extracted from quantum tra- jectory simulations.An important feature can be observed in panels(b)and(d)at signal frequencies in the detection band- width,? W.The power gain of the bistable OMS exceeds G(max) lin ,while the SNR is still comparable to what is expected for a linear OMS with equal gain.Our results therefore indi- cate that large-gain force detection with an OMS can be real- ized beyond bifurcation,while preserving a sensitivity that is comparable to an equivalent linear OMS. VI.CONCLUSION We have proposed bistable optomechanical systems as de- tectors of weak harmonic forces.An external mechanical force modulates the cavity frequency and thus the switching rates between the stable branches.A large difference in the respective optical output?elds will thus lead to a strong am- pli?cation of the weak signal.The noise-induced switching dynamics in the presence of a harmonic force is described by a two-state rate equation model with periodically modulated switching https://www.sodocs.net/doc/2411859397.html,ing this model,we have calculated the output signal and noise spectral density relevant to homodyne detection of the optical?eld and compared them to quantum trajectory simulations.Finally,we have also compared the power gain and signal-to-noise ratio of our detection scheme to those of an optomechanical system in the linear regime. 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The weather was so comfortable that I settled myself into the 高中英语~词性~句子成分~语法构成 第一章节：英语句子中的词性 1.名词：n. 名词是指事物的名称，在句子中主要作主语.宾语.表语.同位语。 2.形容词;adj. 形容词是指对名词进行修饰~限定~描述~的成份，主要作定语.表语.。形容词在汉语中是（的）.其标志是: ous. Al .ful .ive。. 3.动词：vt. 动词是指主语发出的一个动作，一般用来作谓语。 4.副词：adv. 副词是指表示动作发生的地点. 时间. 条件. 方式. 原因. 目的. 结果.伴随让步. 一般用来修饰动词. 形容词。副词在汉语中是（地）.其标志是：ly。 5.代词：pron. 代词是指用来代替名词的词，名词所能担任的作用，代词也同样.代词主要用来作主语. 宾语. 表语. 同位语。 6.介词：prep.介词是指表示动词和名次关系的词，例如：in on at of about with for to。其特征： 介词后的动词要用—ing形式。介词加代词时，代词要用宾格。例如：give up her（him）这种形式是正确的，而give up she（he）这种形式是错误的。 7.冠词：冠词是指修饰名词，表名词泛指或特指。冠词有a an the 。 8.叹词：叹词表示一种语气。例如：OH. Ya 等 9.连词：连词是指连接两个并列的成分，这两个并列的成分可以是两个词也可以是两个句子。例如：and but or so 。 10.数词：数词是指表示数量关系词，一般分为基数词和序数词 第二章节：英语句子成分 主语：动作的发出者，一般放在动词前或句首。由名词. 代词. 数词. 不定时. 动名词. 或从句充当。 谓语：指主语发出来的动作，只能由动词充当，一般紧跟在主语后面。 宾语：指动作的承受着，一般由代词. 名词. 数词. 不定时. 动名词. 或从句充当. 介词后面的成分也叫介词宾语。 定语：只对名词起限定修饰的成分，一般由形容 M A: Has the case been closed yet? B: No, the magistrate still needs to decide the outcome. magistrate n.地方行政官，地方法官，治安官 A: I am unable to read the small print in the book. B: It seems you need to magnify it. magnify vt.１．放大，扩大；２．夸大，夸张 A: That was a terrible storm. B: Indeed, but it is too early to determine the magnitude of the damage. magnitude n.１．重要性，重大；２．巨大，广大 A: A young fair maiden like you shouldn’t be single. B: That is because I am a young fair independent maiden. maiden n.少女，年轻姑娘，未婚女子 a.首次的，初次的 A: You look majestic sitting on that high chair. B: Yes, I am pretending to be the king! majestic a.雄伟的，壮丽的，庄严的，高贵的 A: Please cook me dinner now. B: Yes, your majesty, I’m at your service. majesty n.１．［Ｍ－］陛下（对帝王，王后的尊称）；２．雄伟，壮丽，庄严 A: Doctor, I traveled to Africa and I think I caught malaria. B: Did you take any medicine as a precaution? malaria n.疟疾 A: I hate you! B: Why are you so full of malice? malice n.恶意，怨恨 A: I’m afraid that the test results have come back and your lump is malignant. B: That means it’s serious, doesn’t it, doctor? malignant a.１．恶性的，致命的；２．恶意的，恶毒的 A: I’m going shopping in the mall this afternoon, want to join me? B: No, thanks, I have plans already. mall n.（由许多商店组成的）购物中心 A: That child looks very unhealthy. B: Yes, he does not have enough to eat. He is suffering from malnutrition. 意见应以事实为根据. 3 来自辞典例句 192. The bombers swooped ( down ) onthe air base. 轰炸机 突袭 空军基地. 来自辞典例句 193. He mounted their engines on a rubber base. 他把他们的发动机装在一个橡胶垫座上. 14 来自辞典例句 194. The column stands on a narrow base. 柱子竖立在狭窄的地基上. 14 来自辞典例句 195. When one stretched it, it looked like grey flakes on the carvas base. 你要是把它摊直, 看上去就象好一些灰色的粉片落在帆布底子上. 18 来自辞典例句 196. Economic growth and human well - being depend on the natural resource base that supports all living systems. 经济增长和人类的福利依赖于支持所有生命系统的自然资源. 12 1 来自辞典例句 197. The base was just a smudge onthe untouched hundred - mile coast of Manila Bay. 那基地只是马尼拉湾一百英里长安然无恙的海岸线上一个硝烟滚滚的污点. 6 来自辞典例句 198. You can't base an operation on the presumption that miracles are going to happen. 你不能把行动计划建筑在可能出现奇迹的假想基础上. 英语造句大全English sentence 在句子中，更好的记忆单词！ 1、(1)、able adj. 能 句子：We are able to live under the sea in the future. (2)、ability n. 能力 句子：Most school care for children of different abilities. (3)、enable v. 使。。。能句子：This pass enables me to travel half-price on trains. 2、(1)、accurate adj. 精确的句子：We must have the accurate calculation. (2)、accurately adv. 精确地 句子：His calculation is accurately. 3、(1)、act v. 扮演 句子：He act the interesting character. （2）、actor n. 演员 句子：He was a famous actor. (3)、actress n. 女演员 句子：She was a famous actress. (4)、active adj. 积极的 句子：He is an active boy. 4、add v. 加 句子：He adds a little sugar in the milk. 5、advantage n. 优势 句子：His advantage is fight. 6、age 年龄n. 句子：His age is 15. 7、amusing 娱人的adj. 句子：This story is amusing. 8、angry 生气的adj. 句子：He is angry. 9、America 美国n. 主语+谓语 1. 理解主谓结构 1) The students arrived. The students arrived at the park. 2) They are listening. They are listening to the music. 3) The disaster happened. 2．体会状语的位置 1) Tom always works hard. 2) Sometimes I go to the park at weekends.. 3) The girl cries very often. 4) We seldom come here. The disaster happened to the poor family. 3. 多个状语的排列次序 1) He works. 2) He works hard. 3) He always works hard. 4) He always works hard in the company. 5) He always works hard in the company recently. 6) He always works hard in the company recently because he wants to get promoted. 4. 写作常用不及物动词 1. ache My head aches. I’m aching all over. 2. agree agree with sb. about sth. agree to do sth. 3. apologize to sb. for sth. 4. appear (at the meeting, on the screen) 5. arrive at / in 6. belong to 7. chat with sb. about sth. 8. come (to …) 9. cry 10. dance 11. depend on /upon 12. die 13. fall 14. go to … 15. graduate from 16. … happen 17. laugh 18. listen to... 19. live 20. rise 21. sit 22. smile 23. swim 24. stay (at home / in a hotel) 25. work 26. wait for 汉译英： 1.昨天我去了电影院。 2.我能用英语跟外国人自由交谈。 3.晚上7点我们到达了机场。 4.暑假就要到了。 5.现在很多老人独自居住。 6.老师同意了。 7.刚才发生了一场车祸。 8.课上我们应该认真听讲。9. 我们的态度很重要。 10. 能否成功取决于你的态度。 11. 能取得多大进步取决于你付出多少努力。 12. 这个木桶能盛多少水取决于最短的一块板子的长度。 【it's time to和it's time for】 ——————这其实是一个句型,只不过后面要跟不同的东西. ——————It's time to跟的是不定式（to do）.也就是说,要跟一个动词,意思是“到做某事的时候了”.如： It's time to go home. It's time to tell him the truth. ——————It's time for 跟的是名词.也就是说,不能跟动词.如： It's time for lunch.(没必要说It's time to have lunch) It's time for class.(没必要说It's time to begin the class.) They can't wait to see you Please ask liming to study tonight. Please ask liming not to play computer games tonight. Don’t make/let me to smoke I can hear/see you dance at the stage You had better go to bed early. You had better not watch tv It’s better to go to bed early It’s best to run in the morning I am enjoy running with music. With 表伴随听音乐 I already finish studying You should keep working. You should keep on studying English Keep calm and carry on 保持冷静继续前行二战开始前英国皇家政府制造的海报名字 I have to go on studying I feel like I am flying I have to stop playing computer games and stop to go home now I forget/remember to finish my homework. I forget/remember cleaning the classroom We keep/percent/stop him from eating more chips I prefer orange to apple I prefer to walk rather than run I used to sing when I was young What’s wrong with you There have nothing to do with you I am so busy studying You are too young to na?ve I am so tired that I have to go to bed early 《The Kite Runner》追风筝的人--------------------------------美句摘抄 1.I can still see Hassan up on that tree, sunlight flickering through the leaves on his almost perfectly round face, a face like a Chinese doll chiseled from hardwood: his flat, broad nose and slanting, narrow eyes like bamboo leaves, eyes that looked, depending on the light, gold, green even sapphire 翻译：我依然能记得哈桑坐在树上的样子，阳光穿过叶子，照着他那浑圆的脸庞。他的脸很像木头刻成的中国娃娃，鼻子大而扁平，双眼眯斜如同竹叶，在不同光线下会显现出金色、绿色，甚至是宝石蓝。 E.g.: A shadow of disquiet flickering over his face. 2.Never told that the mirror, like shooting walnuts at the neighbor's dog, was always my idea. 翻译：从来不提镜子、用胡桃射狗其实都是我的鬼主意。E.g.:His secret died with him, for he never told anyone. 3.We would sit across from each other on a pair of high 一、翻译 1. The idea of consciously seeking out a special title was new to me., but not without appeal. 让我自己挑选自己最喜欢的书籍这个有意思的想法真的对我具有吸引力。 2.I was plunged into the aching tragedy of the Holocaust, the extraordinary clash of good, represented by the one decent man, and evil. 我陷入到大屠杀悲剧的痛苦之中，一个体面的人所代表的善与恶的猛烈冲击之中。 3.I was astonished by the the great power a novel could contain. I lacked the vocabulary to translate my feelings into words. 我被这部小说所包含的巨大能量感到震惊。我无法用语言来表达我的感情（心情）。 4，make sth. long to short长话短说 5.I learned that summer that reading was not the innocent(简单的) pastime(消遣) I have assumed it to be., not a breezy, instantly forgettable escape in the hammock(吊床)，( though I’ ve enjoyed many of those too ). I discovered that a book, if it arrives at the right moment, in the proper season, will change the course of all that follows. 那年夏天，我懂得了读书不是我认为的简单的娱乐消遣，也不只是躺在吊床上，一阵风吹过就忘记的消遣。我发现如果在适宜的时间、合适的季节读一本书的话，他将能改变一个人以后的人生道路。 二、词组造句 1. on purpose 特意，故意 This is especially true here, and it was ~. (这一点在这里尤其准确，并且他是故意的) 2.think up 虚构，编造，想出 She has thought up a good idea. 她想出了一个好的主意。 His story was thought up. 他的故事是编出来的。 3. in the meantime 与此同时 助记：in advance 事前in the meantime 与此同时in place 适当地... In the meantime, what can you do? 在这期间您能做什么呢？ In the meantime, we may not know how it works, but we know that it works. 在此期间，我们不知道它是如何工作的，但我们知道，它的确在发挥作用。 4.as though 好像，仿佛 It sounds as though you enjoyed Great wall. 这听起来好像你喜欢长城。 5. plunge into 使陷入 He plunged the room into darkness by switching off the light. 他把灯一关，房 The effective sentences:(improve the sentences!) 1.She hopes to spend this holiday either in Shanghai or in Suzhou. 2.Showing/to show sincerity and to keep/keeping promises are the basic requirements of a real friend. 3.I want to know the space of this house and when it was built. I want to know how big this house is and when it was built. I want to know the space of this house and the building time of the house. 4.In the past ten years,Mr.Smith has been a waiter,a tour guide,and taught English. In the past ten years,Mr.Smith has been a waiter,a tour guide,and an English teacher. 5.They are sweeping the floor wearing masks. They are sweeping the floor by wearing masks. wearing masks,They are sweeping the floor. 6.the drivers are told to drive carefully on the radio. the drivers are told on the radio to drive carefully 7.I almost spent two hours on this exercises. I spent almost two hours on this exercises. 8.Checking carefully,a serious mistake was found in the design. Checking carefully,I found a serious mistake in the design. M1 U1 一. 把下列短语填入每个句子的空白处（注意所填短语的形式变化）： add up (to) be concerned about go through set down a series of on purpose in order to according to get along with fall in love (with) join in have got to hide away face to face 1 We’ve chatted online for some time but we have never met ___________. 2 It is nearly 11 o’clock yet he is not back. His mother ____________ him. 3 The Lius ___________ hard times before liberation. 4 ____________ get a good mark I worked very hard before the exam. 5 I think the window was broken ___________ by someone. 6 You should ___________ the language points on the blackboard. They are useful. 7 They met at Tom’s party and later on ____________ with each other. 8 You can find ____________ English reading materials in the school library. 9 I am easy to be with and _____________my classmates pretty well. 10 They __________ in a small village so that they might not be found. 11 Which of the following statements is not right ____________ the above passage? 12 It’s getting dark. I ___________ be off now. 13 More than 1,000 workers ___________ the general strike last week. 14 All her earnings _____________ about 3,000 yuan per month. 二.用以下短语造句： 1.go through 2. no longer/ not… any longer 3. on purpose 4. calm… down 5. happen to 6. set down 7. wonder if 三. 翻译： 1.曾经有段时间，我对学习丧失了兴趣。（there was a time when…） 2. 这是我第一次和她交流。（It is/was the first time that …注意时态） 3.他昨天公园里遇到的是他的一个老朋友。（强调句） 4. 他是在知道真相之后才意识到错怪女儿了。（强调句） M 1 U 2 一. 把下列短语填入每个句子的空白处（注意所填短语的形式变化）： play a …role (in) because of come up such as even if play a …part (in) 1 Dujiangyan(都江堰) is still ___________in irrigation(灌溉) today. 2 That question ___________ at yesterday’s meeting. 3 Karl Marx could speak a few foreign languages, _________Russian and English. 4 You must ask for leave first __________ you have something very important. 5 The media _________ major ________ in influencing people’s opinion s. 6 _________ years of hard work she looked like a woman in her fifties. 二.用以下短语造句： 1.make (good/full) use of 2. play a(n) important role in 3. even if 4. believe it or not 5. such as 6. because of English sentence 1、(1)、able adj. 能 句子：We are able to live under the sea in the future. (2)、ability n. 能力 句子：Most school care for children of different abilities. (3)、enable v. 使。。。能 句子：This pass enables me to travel half-price on trains. 2、(1)、accurate adj. 精确的 句子：We must have the accurate calculation. (2)、accurately adv. 精确地 句子：His calculation is accurately. 3、(1)、act v. 扮演 句子：He act the interesting character.（2）、actor n. 演员 句子：He was a famous actor. (3)、actress n. 女演员 句子：She was a famous actress. (4)、active adj. 积极的 句子：He is an active boy. 4、add v. 加 句子：He adds a little sugar in the milk. 5、advantage n. 优势 句子：His advantage is fight. 6、age 年龄n. 句子：His age is 15. 7、amusing 娱人的adj. 句子：This story is amusing. 8、angry 生气的adj. 句子：He is angry. 9、America 美国n. 句子：He is in America. 10、appear 出现v. He appears in this place. 11. artist 艺术家n. He is an artist. 12. attract 吸引 He attracts the dog. 13. Australia 澳大利亚 He is in Australia. 14.base 基地 She is in the base now. 15.basket 篮子 His basket is nice. 16.beautiful 美丽的 She is very beautiful. 17.begin 开始 He begins writing. 18.black 黑色的 He is black. 19.bright 明亮的 His eyes are bright. 20.good 好的 He is good at basketball. 21.British 英国人 He is British. 22.building 建造物 The building is highest in this city 23.busy 忙的 He is busy now. 24.calculate 计算 He calculates this test well. 25.Canada 加拿大 He borns in Canada. 26.care 照顾 He cared she yesterday. 27.certain 无疑的 They are certain to succeed. 28.change 改变 He changes the system. 29.chemical 化学药品on the contrary的解析
英语造句
学生造句--Unit 1
英语句子结构和造句
六级单词解析造句记忆MNO
base on的例句
英语造句大全
(完整版)主谓造句
初中英语造句
The Kite Runner-美句摘抄及造句
翻译加造句
改写句子练习2标准答案
用以下短语造句
英语造句