Cross-Kerr nonlinearity in optomechanical systems

0.55 MB·English

by  R. Khan

Cross-Kerr nonlinearity in optomechanical systems
Rapha¨el Khan,1,∗ F. Massel,2,† and T. T. Heikkil¨a2,‡
1Low Temperature Laboratory, Aalto University, P.O. Box 15100, FI-00076 AALTO, Finland
2Department of Physics and Nanoscience Center, University of Jyva¨akyla¨,
P.O. Box 35 (YFL), FI-40014 University of Jyva¨skyl¨a, Finland
Weconsidertheresponseofananomechanicalresonatorinteractingwithanelectromagneticcavity
via a radiation pressure coupling and a cross-Kerr coupling. Using a mean field approach we solve
the dynamics of the system, and show the different corrections coming from the radiation pressure
and the cross-Kerr effect to the usually considered linearized dynamics.
5
1 I. INTRODUCTION duces a change to the refractive index of the cavity de-
0
pending on the number of phonons in the resonator,
2
Cavity optomechanics offers a framework to study the whereas the radiation pressure coupling gives rise to an
n couplingbetweenanelectromagneticfieldandthevibra- analogous effect, but depending on the displacement of
a
J tionsofamechanicalresonator. Theinteractionbetween the mechanical resonator.
these two systems is usually mediated by a radiation- In this paper we solve the dynamics of the cavity and
9
pressuretypecouplingproportional–throughacoupling the mechanical resonator in the presence of the cross-
] constantg–tothenumberofphotonsncinthecavityand Kerr and the radiation pressure coupling. We determine
ll the displacement of the mechanical resonator. The radi- theeffectsofthecross-Kerrcouplingontheredandblue
a ation pressure coupling offers the possibility of altering sidebands within a mean field approach. In particular,
h
the resonant frequency of the mechanical resonator and we demonstrate that the sideband peak is shifted due
-
s its damping. The latter can be used for cooling [4–6] or to the cross-Kerr coupling. In addition, the cross-Kerr
e amplification [7]. Moreover, the nonlinearity of the in- coupling induces a nonmonotonuous response of the ef-
m
teraction may allow for the observation of macroscopic fective mechanical damping as a function of the number
t. quantum phenomena such as quantum superpositon of of photons pumped into the cavity.
a
states [1, 2] or quantum squeezed states [3]. The re-
m
quirements for observing these quantum phenomena are
- the necessity of being close to the ground state and be- Mechanical
d Cavity
n ing in the strong coupling regime [8, 9] where g is larger resonator
o thanthecavityandthemechanicalresonatordecayrate.
c However, g is usually weak and to bypass this constraint g
[ astrongdrivetothecavityisappliedatthecostoflosing x^

1 the nonlinear property of the interaction. n
v Our recent proposal [10], in which the cavity and the c
n
2 resonatorarecoupledtoaJosephsonjunction,showsthat g m
9 the interaction between the cavity and the resonator can ck
0
2 bQeueandhraatnicceadnvdiahtihgheenro-onr-dlienreainritteyraocfttiohnesJoinsetphhesodnisepfflaeccet-. ωc κ ωm γ
0
ment have been investigated also in different setups such
.
1
as membrane in the middle geometries [11–13].
0
Analogously to the setup mentioned above, the non-
5
linearity of the Josephson effect leads to an additional
1 FIG. 1. Schematic picture of the system. A cavity and a
: nonlinear interaction, namely a cross-Kerr coupling gck mechanical resonator coupled via a radiation type coupling
v
between the cavity and the resonator. The difference g and a cross-Kerr coupling g . The number of photons in
i ck
X resides in the fact that in the Josephson junction setup the cavity n is coupled to the oscillations of the mechanical
c
the relative value of g and g depends on the value of resonator xˆ and the number of phonons in the mechanical
r ck
a the gate charge to a superconducting island, whereas in resonator nm.
[12,13], itgenerallyreflectsthepositionoftheresonator
within the cavity.
In the context of optomechanical systems, the cross-
Kerr coupling between the resonator and the cavity in- II. MEAN FIELD APPROACH
We consider an electromagnetic cavity with frequency
∗ raphael.khan@aalto.fi ωc and linewidth κ coupled to a mechanical resonator
† francesco.p.massel@jyu.fi with frequency ω and linewidth γ. The number of
m
‡ tero.t.heikkila@jyu.fi phonons in the cavity n is coupled to the vibration am-
c
2
plitude of the mechanical resonator xˆ via a radiation- Here we have written the cavity operator a in a frame
pressuretypecouplingg. Inadditionthenumberofpho- rotating with frequency ω neglecting the fast oscillati-
p
tons n is coupled to the number of phonons n in the ing terms. We define b to be the thermal input of
c m in
mechanical resonator through a cross-Kerr coupling gck theresonatorsatisfying(cid:104)bin(t)(cid:105)=0and(cid:104)b†in(t)bin(t(cid:48))(cid:105)=
(Fig. 1). The Hamiltonian of the system is ((cid:126)=1) nthδ(t−t(cid:48)), where nth is the number of phonons in the
thermal bath damping the resonator. We split the cav-
H =ω a†a+ω b†b−ga†a(b†+b)−g a†ab†b, (1)
c m ck ity and the mechanical operators into a sum of coherent
and fluctuation parts, i.e., a ≡ δa+α and b ≡ δb+β
whereaandbaretheannihilationoperatorsofthecavity
with α = (cid:104)a(cid:105), β = (cid:104)b(cid:105) and (cid:104)δa(cid:105) = (cid:104)δb(cid:105) = 0. As usual,
andthemechanicalresonator, respectively. Wetreatthe
we assume that α and β oscillate at the same frequency
interactions with a mean-field (MF) approach. Within
as the coherent drive so that α˙ = β˙ = 0. With these
this frame, the radiation-pressure interaction becomes
approximations, the solutions of Eqs. (5, (6)) are
(cid:20) √
ga†a(b†+b)=g ((cid:104)a†(cid:105)a+(cid:104)a(cid:105)a†−(cid:104)a†a(cid:105))(b†+b) α= κfp , (7)
κ −i[∆−g (cid:104)b†b(cid:105)−(G∗β+Gβ∗)]
(cid:21) 2 ck
+(a†a−(cid:104)a(cid:105)a†−(cid:104)a†(cid:105)a)(cid:104)b†+b(cid:105) , (2) i(2G−g)|α|2−ig(cid:104)δa†δa(cid:105)
β = . (8)
γ/2+i(ω −g (cid:104)a†a(cid:105)))
m ck
where (cid:104)A(cid:105) stands for the average of A over the static
In the derivation of Eqs. (7, 8), we have assumed, in
nonequilibrium state of the system (mean field). The
agreementwithwhatisusuallydoneintheoptomechan-
negative terms in Eq. (2) are included to suppress dou- ical literature (see e.g. [18]), ∆+g(cid:104)b† +b(cid:105) ≈ ∆. The
blecounting. ThefirstlineofEq.(2)describesexchange
equations of motion for the fluctuations in the Fourier
processesbetweentheresonatorandthecavitywhilethe
space are given by
second line gives a frequency shift of the cavity which
(cid:104)κ (cid:105)
is proportional to the average displacement of the res- −i(ω+∆˜) δa=iGαδb†+iG∗αδb (9)
onator. This decomposition allows us to find the usual 2
(cid:104)γ (cid:105) √
resultsoftheweakradiationpressurecoupling[7,16]. In −i(ω−ω˜ ) δb=iGα∗δa+iGαδa†+ γb (,10)
MF, the cross-Kerr coupling becomes 2 m in
where ∆˜ = ∆+g (cid:104)b†b(cid:105) and ω˜ = ω −g (cid:104)a†a(cid:105). The
(cid:20) ck m m ck
gcka†ab†b=gck (cid:104)a†a(cid:105)b†b+(cid:104)b†b(cid:105)a†a effect of the thermal drive bin on the response of the
cavity is mediated by the coupling G. Through this cou-
(cid:21) plingtheoscillationsofthemechanicalresonatorproduce
+(cid:104)b†a(cid:105)ba†+(cid:104)ba†(cid:105)b†a+(cid:104)ba(cid:105)b†a†+(cid:104)b†a†(cid:105)ba . (3)
sideband peaks at ω ±ω˜ in the cavity response. They
d m
allow for the exchange of energy between the cavity and
The term (cid:104)a†a(cid:105)b†b ((cid:104)b†b(cid:105)a†a) describes a Hartree-like in- the resonator when ∆ ≈ ±ω [5, 17]. These processes
m
teraction between the resonator and the cavity while the aredepictedinFig.2. For∆≈−ω thesystemisinthe
m
othertermsdescribeexchangeprocessesbetweentheres- redsidebandregimeandonecantransferenergyfromthe
onatorandthecavity. ThuswecanrewritetheHamilto- resonator to the cavity, thus the mechanical resonator is
nian as damped and cooled. For ∆ ≈ ω , the system is in the
m
blue sideband regime and one can transfer energy from
H =[ω −g (cid:104)b†b(cid:105)]a†a+[ω −g (cid:104)a†a(cid:105)]b†b thecavitytotheresonator,thusthemechanicalresonator
c ck m ck
is excited and heated. In order to find the correction to
−G[(cid:104)a†(cid:105)ab†+(cid:104)a(cid:105)a†b†]−G∗[(cid:104)a†(cid:105)ab+(cid:104)a(cid:105)a†b]
the damping, we solve the response function of δa for
+g[(a†a−(cid:104)a(cid:105)a†−(cid:104)a†(cid:105)a)(cid:104)b†+b(cid:105)−(cid:104)a†a(cid:105)(b†+b)], (4) the thermal input δb . We find that it is a Lorentzian
in
function peaked at ω˜ +ω with
m shift
where the expectation values of the different operators
have to be determined self-consistently within the MF
|G|2|α|2(∆˜2−ω˜2 + κ2)
pmiecntutarel saintudaGtio=ngw+hegrcekω(cid:104)bc(cid:105).(cid:29)Wωemasasnudmwehthereeutshueaclaevxiptyeriis- ωshift =− ω˜m m 4
(cid:32) (cid:33)
driven with a coherent field of strength fp oscillating at 1 1
frequency ω = ω +∆. Using the input-output formal- − , (11)
p c κ2 +(ω˜ +∆˜)2 κ2 +(ω˜ −∆˜)2
ism [16] the equations of motion are 4 m 4 m
κ √ and whose linewidth is γ+Γ with
a˙ =−i[−∆−g (cid:104)b†b(cid:105)]a− a+ κf opt
ck 2 p
+iG∗(cid:104)a(cid:105)b+iG(cid:104)a(cid:105)b†−ig(cid:104)b†+b(cid:105)[a−(cid:104)a(cid:105)] (5) Γ =|G|2|α|2κ
opt
γ √ (cid:32) (cid:33)
b˙ =−i[ω −g (cid:104)a†a(cid:105)]b− b+ γb 1 1
m ck 2 in − . (12)
+iG(cid:104)a†(cid:105)a+iG(cid:104)a(cid:105)a†−ig(cid:104)a†a(cid:105), (6) κ42 +(ω˜m+∆˜)2 κ42 +(ω˜m−∆˜)2
3
n +1n+1
m c
n n+1 γ
m c
nm-1nc+1 ωm
γ+Γopt
ω
ωd c nm+1 nc ωp Δ=-ωm ωshift|ωgck=c0ωshift|gck<0
n n ωshift|gck>0
m c
n -1 n FIG.3. Schematicpictureoftheredsidebandwithandwith-
m c
out cross-Kerr coupling for ∆<0. For g >0 the sideband
ck
FIG. 2. Cooling (heating) process. The cavity is driven with peak is shifted to lower values while forgck <0 the sideband
a frequency ω = ω −ω (ω = ω +ω ). The drive does peak is shifted to higher values.
d c m d c m
notallowatransitionfrom|n ,n (cid:105)→|n ,n +1(cid:105)butallow
m c m c
the transition from |n ,n (cid:105) → |n −1,n +1(cid:105) (|n ,n (cid:105) →
m c m c m c
|n +1,n +1(cid:105)). The cavity relaxes then to the state |n −
m c m
1,n (cid:105) (|n +1,n (cid:105)) resulting into cooling (heating) of the up only in the frequency shift which now depends on the
c m c
mechanical resonator. number of coherent and thermal photons in the cavity
Eq. (16). In Fig. 3 we show a schematic picture of what
happenstothesidebandinthepresenceofthecross-Kerr
Integrating the Lorentzian function obtained above, we
coupling.
obtain the number of phonons and photons coming from
the thermal vibrations of the resonator. We get [6] In the Doppler limit (ω ≤κ) the frequency shift and
m
optical damping are given by
γnth+Γ n
(cid:104)δb†δb(cid:105)= opt m0, (13)
γ+Γ
opt
(cid:104)δa†δa(cid:105)=G2|α|2(cid:104)δb†δb(cid:105) (14) ω −g (cid:104)a†a(cid:105)
ω =∓4|G|2|α|2 m ck (18)
(cid:32) 1 1 (cid:33) shift κ42 +4(ωm−gck(cid:104)a†a(cid:105))2
+
κ42 +(ω˜m+∆˜)2 κ42 +(ω˜m−∆˜)2 Γ =±4|G|2|α|2 4(ωm−gck(cid:104)a†a(cid:105))2 . (19)
opt κ κ2 +4(ω −g (cid:104)a†a(cid:105))2
with 4 m ck
(ω˜ +∆˜)2+ κ2
n =− m 4 . (15)
m0 4∆˜ω˜ Now both the frequency shift and the optical damping
m
depend on the cross-Kerr coupling. In Figs. 4-5 we plot
Eqs. (7), (8), (13) and (14) form a set of self-consistency
the number of phonons and the optical damping as a
equations and are solved in the next sections in order to
function of ω /κ for the red sideband in the Doppler
findthenumberofphononsintheresonatorandphotons m
limit. Sincethecross-Kerrcouplingshiftsthemechanical
in the cavity. We now focus on the sidebands.
frequency,thevalueofΓ isshiftedaswell. Thesignof
opt
theshiftisgivenbythesignofg . Otherwisewerecover
ck
the cooling of the resonator for the red sideband (Fig. 4)
III. OPTIMAL COOLING/HEATING
and the parametric instability when Γ = −γ for the
opt
blue sideband (Fig. 5).
In order to minimize/maximize the optical damping
Γ , we set ∆˜ = ∓ω˜ . The upper sign refers to the
opt m
red sideband (Γ > 0) and the lower sign to the blue
opt
sideband (Γ < 0). In the resolved sideband limit,
opt
ω (cid:29)κ(cid:29)γ, the frequency shift and the optical damp-
m
ing become IV. CASE WITH ∆=ωm.
|G|2|α|2 |G|2|α|2
ω =∓ =∓ , (16)
shift ω˜ ω −g (cid:104)a†a(cid:105) In experiments the parameter one can tune directly is
m m ck the detuning ∆ and not ∆˜ as it can be difficult to set
Γopt =±4|G|κ2|α|2. (17) ∆v˜ar=ied∓.ω˜Tmhefroerfoeraec,hanvaoltuheerorfe|gαim| aeswtehecopnusmidpersitsrethnegtchasies
The result for the optical damping Eq. (17), is identical where ∆ = ∓ω , i.e, setting ∆˜ = ∓ω +g (cid:104)b†b(cid:105). In
m m ck
to the one usually obtained in optomechanics in the ab- this case the frequency shift and optical damping in the
sence of the cross-Kerr coupling. The effect of g shows red (upper sign) and in the blue (lower sign) sideband
ck
4
2 2
10 10
0.4
umber ggcckk==00.1g 000.0..003345 umber ggcckk==00.1g ωm000..23.355
on n gck=−0.1g Γ/κopt 00.0.0225 on n101 /Γopt00.1.25
n 0.015 n 0.1
or pho101 00.0.00015.2 0.4 ω0m.6/κ 0.8 1 tor pho100 0.0050 20 40|α|260 80 100
t a
a n
n o
o s
s e
e R
R −1
10
0 0 20 40 60 80 100
10
0.2 0.4 0.6 0.8 1 |α|2
ω /κ
m FIG.6. Steady-statephononnumberintheresonatorandthe
optical damping Γ (inset) as a function of the number of
opt
FIG. 4. Steady-state phonon number in the resonator and photonspumpedintothecavityinthecasewhere∆=−ω .
m
Γfoorptth(iensoeptt)imasaalfcuansect∆i˜on=of−thω˜emrawtiiothωmγ/=κ1a0t−th3κe,regd=sid1e0b−a2nκd. Tgh=e1v0a−lu2eωsmfoarntdhethpearbaamthetteermspareeraγtu=re1c0o−r3rωesmp,oκnd=s1to0−n1tωhm=,
The number of photons pumped into the cavity is fixed to 100.
|α|2 = 100 and the bath temperature corresponds to nth =
100.
become
|G|2|α|2
ω =∓
shift ω˜
m
[κ2/4−2g ω ((cid:104)b†b(cid:105)−(cid:104)a†a(cid:105))]
3 ck m , (20)
r 10 [g2 ((cid:104)b†b(cid:105)−(cid:104)a†a(cid:105))2+κ2/4]
be 5x10−3 ck
|G|2|α|2κ
m
nu /Γκopt−50 Γopt =±gc2k((cid:104)b†b(cid:105)−(cid:104)a†a(cid:105))2+κ2/4. (21)
n In Figs. 6 and 7 the steady-state phonon number and
−10
o
the optical damping are plotted as a function of the
n
o 102 −105.05 0.1 0.15 number of photons pumped into the cavity, for the red
h ω /κ
p m and blue sidebands respectively. For the red sideband
r (Fig. 6) the optical damping increases with increasing α.
o
at gck=0 When gc2k|α|2 (cid:29) κ2/4 the optical damping becomes in-
n g =0.02g versely proportional to the number of photons pumped
ck
o into the cavity, consequently, the cooling deteriorates
es gck=−0.02g when pumping more phonons in the cavity.
R 101
Inthebluesideband(Fig.7)themaineffectofasmall
0.05 0.1 0.15 0.2
cross-Kerr coupling is to limit the instability to a finite
ω /κ
(cid:112)
m number of phonons, (cid:104)b†b(cid:105) ≈ κ/γ|G||α|/|g | + |α|2.
ck
This thus competes with the usual limitation coming
FIG. 5. Steady-state phonon number in the resonator and
fromtheintrinsic(Duffing)nonlinearityoftheresonator.
Γ (inset) as a function of the ratio ω /κ at the blue
opt m InFigs.8-9weplotthefrequencyshiftasafunctionof
sideband for the optimal case ∆˜ = ω˜ with γ = 10−2κ,
m thenumberofphotonspumpedintothecavityforthered
g = 10−2κ. The number of photons pumped into the cavity
is fixed to |α|2 =100 and the bath temperature corresponds and blue sidebands. For the red sideband when gck > 0
to nth =10. The dashed lines indicate the onset of the para- (gck < 0) the frequency shift increases (decreases) as α
metric instability for Γ =−γ. increases until α ≈ (cid:104)b†b(cid:105) after which it increases (de-
opt
creases). For the blue sideband when g > 0 the fre-
ck
quency shift decreases while for for g < 0 it increases.
ck
The difference at small α between the red and blue side-
bandsarises fromthefact thatin the redsideband when
pumping more photons into the cavity the cooling im-
proves, thus the number of phonons in the mechanical
5
a) depending on the number of photons in the cavity and
phonons in the resonator. In addition, we have shown
er 104 that when the detuning of the pump is equal to the fre-
b g =0
m ck quency of the mechanical resonator the variation of the
u g =0.1g
n ck optical damping saturates instead of being linearly de-
g =0.01g
on 103 ck pendentonthenumberofphononspumpedintothecav-
on ity.
h
p This work was supported by the European Re-
or 102 2x 10−3
t
a
n 1
o
s
Re 101 0
0 10 20 30 40 50
2 −1
|α|
b) 0X10-3 ωshift ωm−−32 gck=0
g =0.01g
−4 ck
g =−0.01g
-0.5 −5 ck
m
ω
−6
/opt -1 ggck==00.1g 0 5 1|α0|2 15 20
Γ ck
g =0.01g
ck
-1.5 FIG. 8. Frequency shift as a function of the number of pho-
tons pumped into the cavity when ∆ = −ω with γ =
m
10−3ω κ = 10−1ω , g = 10−2ω and the bath tempera-
m m m
-2 ture corresponds to nth =100.
0 10 20 30 40 50
2
|α|
X10-3
1
FIG.7. a)Steady-statephononnumberintheresonatorand
b) the optical damping Γopt as a function of the number of g =0
ck
photons pumped into the cavity in the case where ∆ = ωm. 0.5 g =0.01g
Thevaluesfortheparametersareγ =10−3ω ,κ=10−1ω , ck
m m g =-0.01g
g=10−2ω and the bath temperature corresponds to nth = ck
10. m ωshift ωm 0
resonator decreases, making it possible to have a num- -0.5
ber of photons in the cavity of the same order and larger
than the number of phonons in the resonator.
-1
0 10 20 30 40 50
|α|2
V. CONCLUSION
FIG. 9. Frequency shift as a function of the number of pho-
tonspumpedintothecavitywhen∆=ω withγ =10−3ω
m m
In conclusion, we have solved the dynamics of a me- κ = 10−1ω , g = 10−2ω and the bath temperature corre-
m m
chanical resonator coupled to an electromagnetic cavity sponds to nth =100.
via a radiation pressure coupling and a cross-Kerr cou-
pling using a mean field approach. We have shown that
the cross-Kerr coupling shifts the frequency of the me- search Council (Grant No. 240362-Heattronics) and the
chanical resonator and of the optical cavity, the shift Academy of Finland.
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