Cavity Optomechanics of Levitated Nano-Dumbbells: Non-Equilibrium Phases and Self-Assembly

2.2 MB·English

by  W. Lechner

Cavity Optomechanics of Levitated Nano-Dumbbells:
Non-Equilibrium Phases and Self-Assembly
W. Lechner1,2,∗ S. J. M. Habraken1,2, N. Kiesel3, M. Aspelmeyer3, and P. Zoller1,2
1Institute for Quantum Optics and Quantum Information,
Austrian Academy of Sciences, 6020 Innsbruck, Austria
2Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria and
3Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria
(Dated: January 18, 2013)
Levitated nanospheres in optical cavities open a novel route to study many-body systems out of
solutionandhighlyisolatedfromtheenvironment. Weshowthatproperlytunedopticalparameters
3
allowforthestudyofthenon-equilibriumdynamicsofcompositenano-particleswithnon-isotropic
1
optical friction. We find optically induced ordering and nematic transitions with non-equilibrium
0
analogs to liquid crystal phases for ensembles of dimers.
2

n
a Introduction - The interaction between light and mat- damped by the cavity-decay. The optomechanical inter-
J ter has been one of the central driving forces behind actiongivesrisetoanopticalpotentialandacoolingforce

7 recent developments in condensed matter physics with along the cavity axis. By compensating the potential
1 nanoparticles[1]. Opticaltweezers[2,3]andconfocalmi- withasecondopticalmode, theremainingoptomechani-

croscopy [4] have made it possible to study many-body caleffectisfrictionalongthecavityaxis. Fig.1illustrates
]
l systems of nanoparticles in solutions in real-time and the system we have in mind. The nanospheres are har-
l
a withsingle-particleresolution[5–8]. Recently,ithasbeen monicallytrappedandconfinedtothexyplane, wherex
h proposedtoopticallylevitateandcoolsinglenanospheres isthecavityaxis. Weshowthatthesteady-stateorienta-
- insideanopticalcavity[9–12]. Whileintherealmofsoft- tionofasingledimerisnon-uniforminpresenceofoptical
s
e matter physics, this may provide an alternative to con- friction. Remarkably, the full range of preferred orienta-
m finement in a solution [13], from a quantum-optics point tions from 0 to π/2 is accessible by appropriately tuning
. of view such a set-up provides a versatile alternative to the experimental parameters. In a many-body system,
t
a conventionaloptomechanicalsystems[14–16]. Combined thepresenceofadditionaldirectinteractionsbetweenthe
m with optomechanical cooling and trapping techniques of individualnanospheres,leadstocompetitionbetweenthe
- single particles, this may even open the possibility to natural triangular ordering of a two-dimensional crystal
d study fundamental aspects of quantum mechanics with anddissipation-inducedordering. Comparedtootherap-
n
mesoscopic objects [11, 17, 18]. Here, we focus on the
o
dynamicsofmany,interactingparticlesinthepresenceof
c
[ optomechanical cooling. While many-body systems with
2 nioonns-u[1n9i]f,otrhmepcooossliinbgilithyavtoecbreeeantesctoumdipeldexwsittrhucattuormesswaintdh (a) Ω z Vtrap κ (b) Dissipation
v γ γ
1 nanospheresofferscompletelynewopportunitiestostudy êz opt
9 pattern formation and self assembly. With novel synthe-
6 sismethodsitisnowpossibletodesigncompoundstruc- êy x
4 tures ranging from dimers to networks of nanospheres (c) Timescales
. connectedbyspring-likebiomolecules[20–26]. Adistinc- ν
2 y φ slow fast
1 tive feature of the self-assembly of composite particles x ν
2 is that the emerging patterns are characterised not only 0γopt γ-1 γ-1
1 by their positions, but also by their individual orienta- opt
: tions [27, 28]. The non-equilibrium self-assembly of such
v
i nano-structures in the presence of non-isotropic optical FIG. 1: (a) Levitated dimers composed of dielectric nano-
X coolingisanopenquestion,andholdsthepromiseofnew spheres in a laser driven optical two-mode cavity with laser
r means to optical control of pattern formation and novel strengthsΩ1,2,adecayrateκandanadditionalharmonictrap
a V . The orientation of the dimers is denoted φ, their equi-
non-equilibrium liquid crystal phases. trap
librium separation x and their frequency ν. The direct pair
0
In this Letter, we study the dynamics and self-
interaction between the nanospheres leads to liquid crystal
assembly of levitated nanosphere-dimers in presence of phaseswithrichdimerpatterns. (b)Theparticlesareincon-
optomechanical friction inside a two-mirror cavity. The tactwithathermalbathwithcouplingγ. Thecavity-particle
particles are subject to thermal forces and coupled to interactionresultsinstrongadditionaldissipationinonespa-
a cavity mode which is driven by an external laser and tial direction only, which acts as an effective zero tempera-
ture bath γopt. (c) Comparison of the relevant time scales:
We assume that γ is faster than γ and find 3 qualitatively
opt
differentregimes,whichgiverisetodifferentorientations,de-
pending on the relative timescales.
∗Electronicaddress: [email protected]
2
proaches to dynamical ordering of dimers, such as shear the detuning from the bare cavity resonance. The equa-
[29] or static electric fields [30], this method offers ad- tion of motion for the transformed optical amplitude is
ditional advantages: (a) the orientation depends on the a˙(cid:48) = (i∆¯ − κ)a(cid:48) + i(∆(x ... x ) − ∆¯)(α + a(cid:48)). Sub-
1 j
frequencyofthevibrationalmodeofthedimerswhichal- stitution of a → α + a(cid:48) in the first term in Eq. (2)
lows for individual ordering in multi-species systems and gives rise to four terms. One of them corresponds
(b) in addition to ordering at the level of single parti- to an effective optical potential along the cavity axis
clesnon-uniformfrictionalsoleadstonovelliquidcrystal V(x ... x ) = −|α|2∆(x ... x ), which can be compen-
1 j 1 j
phases at the many-body level (see Fig. 3). We identify sated by V of the second optical mode. For the choice
c
three relevant time scales in this system: the frequency of a mode separated by one free spectral range from ω,
ofthevibrationalmodeofthedimersν,theoptomechan- the mode function in the focal range of the cavity is ap-
(cid:112)
ical damping rate γ and the rate of thermalization γ. proximatelygivenbyG(x)(cid:39)η 2/Ccos(kx),whereηis
opt
For limiting cases we present analytic results on the or- the polarization. When (cid:15) and η are orthogonal, the two
dering of individual dimers and numerical results on the modes do not interfere and the effective potentials add
non-equilibrium many-body dynamics. up, i.e. V(x ) = −g(|α |2sin2(kx ) + |α |2cos2(kx )).
j s j c j
Model-WeconsiderasystemofN/2dimersconsisting For |α | = |α |, the potential is independent of x and
s c j
ofN nanospheres,trappedinsideandopticalcavity. The the forces cancel. The second mode is driven on reso-
Hamiltonian is decomposed as H =H +H with nance and therefore does not lead to additional damping
sys om
(or amplification).
H =(cid:88)N (cid:32)p2j +V (x )(cid:33)+ (1) Wefindtheequationsofmotionofthenanospherepo-
sys 2m trap j sitions and momenta
j=1
N(cid:88)/2mν2 (cid:88) x˙j =pj/m
2 (|x2j−1−x2j|−x0)2+Γ0 Vint(|xi−xj|), p˙ =−∂Hsys +(cid:0)αa∗+α∗a+|a|2(cid:1)∂∆(x1 ... xj) . (3)
j=1 i(cid:54)=j j ∂x ∂x
j i
the system Hamiltonian. Here, m is the mass of the
The cavity decay rate κ sets a finite time scale for the
nanospheres, x and p are the position and momentum
j j cavity to respond to changes in the particle positions so
ofparticlej, respectively, V isthetrappingpotential,
trap that the optomechanical feedback does not only depend
ν thefrequencyofthevibrationalmodeofthedimers,x
0 on the particle positions, but also on the particle mo-
istheequilibriumseparationandV isthedirectdipolar
int menta along the cavity axis [14]. This gives rise to am-
pair interaction, which can be tuned by the parameter plification (for ∆¯ > 0) or damping (for ∆¯ < 0), respec-
Γ [6]. In a frame, rotating with the laser drive, the
0 tively. Optomechanical cooling of nanospheres has been
optomechanical Hamiltonian is given by
studied previously in the Lamb-Dicke regime [9], while
H =−∆(x ... x )|a|2+Ω(a+a∗)+V , (2) here, we focus on a different regime, similar to Ref. [33],
om 1 j c
in which the particles move almost freely and the rele-
√
where a(t) [ Js] is a normal variable, which describes vant frequency stems from the modulation of the optical
(cid:112) feedbackforceatω =2kp/m. Thedampingforcehas
the dynamics of the optical mode, Ω [ J/s] character- mod
resonances for ∆¯ = ±|ω |, while in the regime we are
izes the drive strength, V is a second compensating po- mod
c interestedin, |p|(cid:28)m∆¯/(2k), thecoolingrateisapprox-
tential, and ∆ = ω − ω(x ... x ) is the detuning of
d 1 j
imately constant
the laser drive from the optical resonance. The inter-
action derives from the electric polarizability αp of the 2g2|α|2k2∆¯κ
nanospheres [9, 10], which, for sub-wavelength particles, γ (cid:39) . (4)
opt m(∆¯2+κ2)2
gives rise to a position-depended cavity resonance fre-
quencyω(x ...x )=ω −(gC/2)(cid:80) |F(x )|2,whereω
1 j 0 j j 0 Finally, the equations of motion take the form of a mod-
isthebarecavityfrequency. F(x)isthenormalizedmode
ified Langevin equations with
function of the cavity mode, C is the mode volume and
g = α ω /C is the optomechanical coupling strength, ∂H
p 0 mx¨ =− sys −m(γ+γ )x˙ +ξ
which is proportional to the volume of the particle. We ∂x opt x
assume that the cavity mode can be approximated by ∂H
(cid:112) my¨ =− sys −mγy˙ +ξ . (5)
a standing wave F(x) (cid:39) (cid:15) 2/Csin(kx), where k is the ∂y y
wave number and (cid:15) is the polarization in the yz plane,
so that ∆(x ... x )=∆(x ... x ). Here, γ is the rate of thermalization and ξ and ξ are
1 j 1 j x y
The full dynamics of the optical part, including decay mutually uncorrelated Langevin forces, as characterized
from the cavity, is described by the equations of motion by (cid:104)ξ (t)ξ (t(cid:48))(cid:105)=k Tmγδ(t−t(cid:48)).
x,y x,y B
for the optical amplitude a˙ = (i∆(x ... x )−κ)a+iΩ, Results - We first focus on the dynamics of a single
1 j
where κ is the field decay rate. The drive can be elim- dimer described by Eq. (5). Separating the dynam-
inated by the transformation a → α + a(cid:48) with α = ics into the trivial center-of-mass motion and the rel-
√
Ω/(∆¯ + iκ) the average amplitude and ∆¯ = (ω − ω) ative coordinates (x − x )/ 2 ≡ (x,y), and to the
d 1 2
3
π(a) γopt/γ = 10 π(b) γopt/γ = 100 π(c) γopt/γ = 1000 ntioornmgailvizeendastfiexaeddy-xst.atCeodnissetqriubeunttiloyn, tohfethaevefraasgteyednierregcy-
2 2 2 (cid:82)∞
(cid:104)V(cid:105) = V(x,y)P(y|x) determines the distribution
x −∞
φπ φπ φπ P(x) for the x direction via a Markov process with the
4 4 4 scaled temperature T =Tγ/(γ+γ ), thus
x opt
0 0 1 2 3 0 0 1 2 3 0 0 1 2 3 P(x,y)= e−V(x,y)/(kBT) e−(cid:104)V(cid:105)x/(kBTx) .
log(ν/γ) log(ν/γ) log(ν/γ) (cid:82)∞ dy e−V(x,y)/(kBT)(cid:82)∞ dxe−(cid:104)V(cid:105)x/(kBTx)
(d) (e) (f) −∞ −∞
π (6)
0.2
5 ν = 0.2 2 rc = 1 In regime (i), the spatial fluctuations are much larger
ρ()φ243 νν == 32.52.6 φπ4 00..115 ρ()φ1 rrcc == 11000 tthegarnalxs0c,a(cid:82)sno∞btheaetvVal(uxa,tyed) (cid:39)anmalyνt2i(cxa2lly+. yT2)h/e2dainsdtritbhuetiionn-
0.05 P(φ) = rdrP(rcosφ,rsinφ) for the orientation of
1 0
0 0 0 0 the dimer φ is found as
0 π/4 π/2 0 1 2 0 π/4 π/2
φ log(ν/γ) φ (cid:112)
γ(γ+γ )
opt
P(φ)= (7)
FIG. 2: Steady-state probabilities of the orientation φ as a 2π((γ+γopt)cos2φ+γsin2φ)
functionofthedimerfrequencyfromnumericalintegrationof
the second order Langevin equation Eqs. (5) for cooling rate with the maximum for φ=0. In case (ii), the harmonic
ratios γ /γ = 10 (a), γ /γ = 100 (b), and γ /γ = 1000 approximation of V(x,y) breaks down, and Eq. (6) is
opt opt opt
(c). The comparison of the analytic expressions Eq. (6) for evaluated numerically, with the results shown in Fig.2(d
smallfrequencies(ddashedlines)andnumericalresults(dfull ande)whichareinagreementwithnumericalintegration
lines) show good agreement over a wide range of frequencies. of Eq. (5) shown in Fig.2(b).
(e) Analytic results from Eq. (6) for a range of frequencies
In the rigid-rotor regime (iii), motion in the radial di-
with γopt/γ = 100 predicts the transition from φ = 0 orien- rection is suppressed, so that (cid:112)x2(t)+y2(t) = x and
tation to φ = π/2. For larger frequencies the model breaks 0
downasitpredictsφ=π/2forlargeν. (e). Intherigid-rotor we can derive a Langevin equation for φ alone: mx0φ¨=
regime, the analytic approximation Eq. (8) is in agreement −m(γ+γ cos2φ)φ˙ +ξ where ξ =ξ sinφ+ξ cosφ
opt φ φ x y
with the numerical simulation as shown for various cooling so that (cid:104)ξ (t)ξ (t(cid:48))(cid:105) = k Tmγδ(t − t(cid:48)) is independent
φ φ B
rates rc =γopt/γ (f). of φ. This equation describes thermal motion of the
orientation of the dimer with angle-dependent damping.
Since there is no conservative force, the motion is over-
damped,sothatφ¨(cid:39)γ(φ)φ˙,whereγ(φ)=γ+γ cos2φ,
extent that optomechanical coupling between the par- opt
ticles can be neglected, the non-linear force is ∂Hsys = and the Fokker-Planck equation reduces to ∂P/∂t =
−rmν2(cid:0)1−(2x /(x2+y2))1/2(cid:1), where r = (x,y∂)r and (γ+γoptcos2φ)−1∂P/∂φ. The stationary solution is
0
x0 is the dimer separation. Fig. 2 (a-c) show the steady γ+γ cos2φ
state solutions of the orientation of single dimers as a P(φ)= opt (8)
2πγ+πγ
function of the dimer frequency for various cooling rates opt
fromnumericalintegrationofEq. (5). Remarkably,when
and has a maximum at φ=0.
exposed to uni-directional friction, a loosely connected
This motivates the following intuitive picture: The
dimer as well as a rigid rotor tends to align orthogonal
dominant mechanism in (i) is purely geometrical. Here,
to the direction of friction, whereas a dimer of moder-
the distribution P(x,y) is a Gaussian that is squeezed
ate stiffness aligns parallel to it. This can be understood
in the direction of cooling x and, therefore, the most
fromthreecompetingeffectswhichderivefromtheorder
likely orientation is φ = 0. In the rigid-rotor case (iii),
of the relevant time scales in the system (see Fig. 1c):
in which no orientation-dependent energies are involved,
γ−1,γ−1 andν−1. Whileγandγ setthescalesofther-
opt opt the anisotropy of the steady-state orientation is due to a
malizationandnon-uniformfriction,ν setsthetimescale
purelydynamicaleffect. Inthiscase,thedimerisdynam-
atwhichthedegreesoffreedommixduetothenon-linear
icallyattractedtoorientationsforwhichthefluctuations
nature of the force term. Assuming that γ >γ, there
opt intheangulardirectionaresuppressed. Theoppositeori-
are three limiting parameter regimes: (i) γ (cid:29) γ (cid:29) ν,
opt entation is reached in the intermediate regime (ii), when
(ii) γ (cid:29)ν (cid:29)γ and (iii) ν (cid:29)γ (cid:29)γ . In the follow-
opt opt a third, purely energetical effect, is dominant. The mo-
ingweanalyticallystudytheselimitingcasesandgivean
tion is mostly confined to configurations of constant x,
intuitive explanation of this remarkable non-equilibrium
forwhichthepotentialenergychangesfromadoublehar-
ordering phenomenon. monicwellatx=0toasingleanharmonicwell(V ∝x4)
In (i) and (ii), γ is the largest scale. Due to the at x = ±x . With the average kinetic energy (cid:104)T(cid:105) con-
opt 0
resulting separation of time scales of the motion in the stant and using the virial theorem (cid:104)V(cid:105) = 2/n(cid:104)T(cid:105), with
x and y directions, the steady-state distribution is of n the power of the external potential, we find that the
the form P(x,y) = P(y|x)P(x). Here, P(y|x) is the probability exp[−β (cid:104)V(cid:105)] is largest for φ=π/2.
x
4
Let us consider the experimental feasibility in a con- have been previously studied in equilibrium and non-
figuration as shown in Fig. 1. The confinement of the equilibrium [28]. Non-isotropic friction and the resulting
dumbbells to the xy plane can be provided by an ex- ordering may offer novel tools to guide the self-assembly
ternal standing wave optical trap crossing the Fabry- towards preferred structures and to study novel nematic
Perot cavity in z-direction. The three relevant time phases. We numerically study the system described by
scales can be controlled over a large range of parame- the Eq. (1) with the experimental parameters as given
ters: Dimer frequencies up to ν (cid:39) 2π ×1 kHz can be above. Fig. 3(a) depicts an ensemble of dimers without
reached e.g. with spring constant k (cid:39) 0.2 pN/µm of optical friction. For this choice of parameters, the sys-
DNA and silica nanospheres with a radius of r =50 nm tem is in the liquid phase and the dimer orientations are
(mass m = 1.2×10−18 kg). We note that spring con- distributed uniformly. Additional non-isotropic cooling
stants smaller by orders of magnitude are possible above with γ /γ = 100 induces a non-equilibrium transition
opt
the persistence length of 50nm [32]. The optical damp- to a phase characterized by the single- and many-dimer
ing is provided by a cavity with length L (cid:39) 10−2 m order parameters respectively shown in Fig. 3 (b) and
and mode waist of w (cid:39) 10−4 m. We find the optome- (c). The single dimers are still aligned. In addition, the
chanical coupling g (cid:39) 2π × 104 Hz via the mode vol- interplay between orientation and many-body dynamics
ume C = (π/4)Lw2. We further assume a cavity finesse leads to a remarkable phase with liquid order in the y-
of F (cid:39) 105, so that κ (cid:39) 2 × 105 Hz at a wavelength direction and solid order in the x-direction. With the
of λ = 1064 nm. When we further choose ∆¯ >∼ 5κ, frequency ν(cid:48) = 500ν and keeping all other parameters
a power for the cooling laser of P = 3 × 10−5 W fixed,adifferentpatternwithallindividualdimersreori-
drive
results in a cooling rate of γopt <∼ 2π × 4 Hz. For entatedorthogonaltothedirectionofcooling(Fig. 3(c))
the thermal environment we assume room temperature isfound. Againthemany-bodydynamicsleadstoorder-
T = 293 K and γ = 0.05 Hz, which corresponds for the ing along the x-direction, as measured by the directional
chosen nanospheres to an air pressure of approximately pair-correlation function g (x)=(cid:104)δ((x −x )−x)(cid:105) de-
x,y i j
10−5 mbar. Therefore, rates of up to γ /γ ≈ 100 are picted in Fig. 3(c). The patterns in Fig. 3 (b) and (c)
opt
possible. Note that even higher rates have recently been are non-equilibrium analogs of liquid crystal phases in
achievedexperimentallybyopticalfeedbackcooling[31]. equilibrium. ForlargeΓ abovethemeltingtemperature
0
Many-body phases - Liquid crystal phases of dimers [6], two-dimensional dipolar particles self-assemble into
a triangular lattice. The guided orientation of individ-
ualdimersinducedbynon-isotropicfrictioniscompeting
(a) (b) withtheorientationinthetriangularlatticeinwhichthe
crystalline
liq dimer orientations are random integer multiples of π/3.
u
id This frequency-dependent ordering can be used to cre-
ate patterns of dimers with different orientations. Fig.
3(d) depicts a mixture of dimers with ν = 500ν and
1
0.2 0.2 with ν = ν. In the absence of direct interactions these
ρ(φ)0.1 ρ(φ)0.1 species2order with almost orthogonal relative alignment.
00 πφ/4 π/2 00 πφ/4 π/2 Increasing the interactions Γ0 > Γmelt above the critical
(c) (d) meltingtemperature,thesystemformsacrystalwithtri-
102 xy 0.5 angular ordering and separate orientational order of the
0.4 two individual dimer species.
101
gx,y100 ρ(φ)00..23 icaIlnsesutumpmthaaryt,awlleowhasvfoerpnreosveenltdeidssaipraetailviestciconotprtoolmoveecrhtahne-
10-1 0.1 orientationofdimerscomposedofnanospheres. Wehave
shown that this approach can be used to prepare non-
10-20 1 2 3 00 π/4 π/2 equilibrium analogs of liquid crystals and to study tran-
r/x φ
0 sitionsinmixturesofmultiplespeciesofdimers. Theonly
relevant parameters that determine the non-equilibrium
FIG.3: (a)Ensembleofdimerswithfrequencyν =200γ (in-
ordering are the time scales of the vibrational mode of
termediate regime (ii)) in thermal equilibrium are uniformly
the dimers and the rates of thermalization and non-
orientated(inseta). (b)Thenon-isotropicopticalfrictionin-
isotropicfriction. ComplexstructuresofDNA-connected
ducesorientationalorderingofindividualdimers(insetb)and
nanospheresareofgrowinginterest,andwehopethatthe
in addition, many-body nematic ordering along the direction
approach discussed here will provide useful means to op-
of cooling. (c) The non-equilibrium phase analog to a liquid
crystal phase characterized by the pair-correlation function tical control of such systems, complementary to direct
g(x). The rigid rotors align orthogonal to cooling with addi- opticalmanipulationviaopticaltweezers. Thepresented
tionalnematicorderindirectionofcooling(black)andliquid mechanism does not rely on any specific properties of
order in y-direction (red). (d) Mixture of dimers self assem- nanospheres and applies, at least in principle, in general
bles into a crystal with dimer-orientations according to the tocomplexstructuresofdielectricobjects,suchasviruses
frequencies ν. and bacteria [34]. We speculate that, in the longer run,
5
it may also be applied to complex molecules, and may M. Gru¨nwald, and C. Dellago for fruitful discussions.
even prove fruitful as a microseeding technique for the WorkatInnsbruckissupportedbytheintegratedproject
nucleation of complex molecules. AQUTE, the Austrian Science Fund through SFB F40
Acknowledgments. We thank P. Rabl, C.W. Gardiner, FOQUS, and by the Institut fu¨r Quanteninformation.
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