Experimental measurement of efficiency and transport coherence of a cold atom Brownian motor in optical lattices

2.2 MB·English

by  M. Zelan

Experimental measurement of efficiency and transport coherence of a cold atom
Brownian motor in optical lattices
M. Zelan,1,∗ H. Hagman,1 G. Labaigt,1 S. Jonsell,2 and C. M. Dion1
1Department of Physics, Ume˚a University, SE-90187 Ume˚a, Sweden
2Department of Physics, Stockholm University, SE-10695 Stockholm, Sweden
(Dated: January 27, 2011)
The rectification of noise into directed movement or useful energy is utilized by many different
systems. ThepeculiarnatureoftheenergysourceandconceptualdifferencesbetweensuchBrownian
motor systems makes a characterization of the performance far from straightforward. In this work,
1 where the Brownian motor consists of atoms interacting with dissipative optical lattices, we adopt
1 existing theory and present experimental measurements for both the efficiency and the transport
0 coherence. We achieve up to 0.3% for the efficiency and 0.01 for the P´eclet number.
2
PACSnumbers: 05.40.-a,05.60.Cd,37.10.Jk
n
a
J
Brownian motors (BMs) are devices that can rectify eitherfromatime-asymmetricperiodicdrivingforcewith
5
noise into work or directed motion in the absence of ex- zeroaverage(rockedratchet),orbyflashinganasymmet-
2
ternal forces. They are of interest for the understanding ric potential (flashed ratchet). However, as shown in our

] of fundamental principles in statistical physics and ther- system [17], rectification can be achieved by switching
h
modynamics, and several studies have shown that they between two symmetric potentials.
c
play a crucial part in transport phenomena in nature; The model for the BM used in our experiment was
e
m see, for example, [1–3]. Since BM’s utilize noise, they introduced in [20]. Briefly, particles with mass m move
can work in regions where the inherent noise is large in two symmetric potentials, U = A cos(kx) and U =
- 1 1 2
t compared to other interactions. Applications of BM’s, A cos(kx + φ), phase shifted by φ, and are randomly
a 2
therefore, reach into the nano-scales, where they make transferred between the two with unequal transfer rates
t
s ideal tools for powering up nano-machines [4–6]. Recent Γ (cid:54)= Γ . In addition, the particles experience a
. 1→2 2→1
t reviews of the subject can be found in [7–10]. friction force −α x˙ and a diffusive force ξ (t) in either
a i i
m lattice i=1,2. This gives the equations of motion
Of particular interest for any motor is the quantifica-
- tion of its efficiency, usually defined as the ratio of pro-
d duced work to input energy. Due to the peculiar nature mx¨=−∇xUi(x)−αix˙ +ξi(t). (1)
n
of the energy source of BMs, determination of efficiency
o Here, ξ (t) satisfies the relations (cid:104)ξ (t)(cid:105) = 0 and
i i
c is not straightforward. There have been several theo- (cid:104)ξ (t)ξ (t(cid:48))(cid:105) = 2D δ(t−t(cid:48)). Thus the atom is both sub-
i i i
[ retical discussions on the efficiency of BM’s [11–15], and
ject to work from the potential Ui and to fluctuations
differentperformancecharacteristicshavebeendiscussed
2 and dissipation given by the diffusion coefficient Di and
in [16]. We present here experimental measurements of
v the friction coefficient α .
i
8 two performance characteristics of a BM realized with For an atom moving in a single periodic potential, the
0 ultracold atoms in double optical lattices [17]: the effi-
long-timeaverageoftheworkgoestozero,andtheatoms
8 ciency,thatis,thefractionofinputpowerdrivingthedi-
reach a steady state with kinetic temperature D /α .
0 rectedmotion,andthetransportcoherence,ortheP´eclet i i
This changes when it is transferred between the poten-
.
1 number, that is, the comparison between the drift and tials, changing instantaneously its potential energy. The
0 the diffusion. Usually, the efficiency is defined in terms
total work on an atom is therefore equal to the changes
1 oftheamountofworkobtainedfromthemotoragainsta
in potential energy summed over all jumps between the
1 load. As no load is present in our case, we instead follow
: potentials. For identical potentials (A1 =A2, φ=0), no
v the convention [11, 12] of defining “useful energy” as the
energy is gained by an atom transferred between the po-
i energy needed to drive the directed motion of the atoms
X tentials; see Fig. 1(a). In this situation, both potentials
against friction. It has also been argued that including
satisfy the symmetry condition U (−x) = U (x), which
r i i
the dissipation due to friction against the directed mo-
a entailsthat(cid:104)x˙(cid:105)=−(cid:104)x˙(cid:105), andhencenoBMeffectispossi-
tion provides a better definition of efficiency even when
ble[21]. Introducinganonzeroφbetweenthelattices,the
a load is present [11].
system still possesses glide reflection [22], but because of
For a BM to be able to function, it has to (i) present theunequaltransferratesbetweenthepotentials,thereis
an asymmetry [18] and (ii) be out of thermal equilib- no symmetry condition requiring (cid:104)x˙(cid:105) = 0, and therefore
rium [19]. In most cases, the symmetry breaking arises there will in general be a rectification [18]. An exception
is the point φ=π, where again U (−x)=U (x), leading
i i
to zero current, with the input energy gained from the
transfer between potentials only appearing as a heating
∗[email protected] of the atoms.
2
(a) (b) interaction between the induced atomic dipole moment
andtheperiodiclightfields[26]. Thetwopotentialscor-
respond to two different hyperfine levels, F = 3 and
F = 4, of the electronic ground state of cesium. Each
atom will be transferred between the two potentials at
random times through optical pumping, with rates for
transfer, scattering, andcoolingsetbytheparametersof
FIG.1: (Coloronline)Schematicdrawingoftheprocessdriv- the laser fields (intensity and detuning). To collect data,
ingourBM.Ineachpotential,aninherentfrictionanddiffu- weuseabsorptionimagingtomeasurethemeanmomen-
sion is present. Vertical arrows indicate the transfer between tumpaswellasthesizeoftheatomiccloud. Theimaging
the potentials, horizontal ones the total diffusion. (a) φ=0: isdoneinthehorizontalplanetoavoidanyeffectsofgrav-
atoms are transferred between the lattices without any gain ity [27]. To access the mean momentum spread (the ki-
in potential energy. No symmetry is broken and no drift is
netic temperature) δp2 in Eq. (4), we use a time-of-flight
induced. (b) φ = 3/2π: the symmetry is broken and the
technique that enables fast and accurate measurements
transfer adds energy to the system, as a leftward drift.
of the distribution of the momentum, δp2 [28].
z
In the experiment we, adjust the potential depths by
Following the discussion in [12], we can derive a de- controlling the intensities in the lattice beams such that
tailed balance relation for the input power Pin acting on A1 =A2. To assess the quantity D/α, we study the sys-
a particle governed by the equation of motion (1), tematφ=0,wherethepotentialsareidenticalandthere
is no BM effect since the transfer between the potentials
αp2 αδp2 ND doesnotchangetheenergyofthesystem(Pin =0). From
P = + − . (2)
in m2 m2 m theenergybalance(2),wethenobtain,inagreementwith
the equipartition theorem,
Here α, etc., signifies the time average (over the two lat-
(cid:12)
tpi(cte)satraotuens)d, aitnsdloδnpg(t-)timisethaevevraargiaetipo.nTofhtehfiermstoamnedntsuemc- Ekin(cid:12)(cid:12)φ=0 = δ2pm2(cid:12)(cid:12)(cid:12) = N2 Dα. (5)
(cid:12)
ond terms on the right-hand side of Eq. (2) are the en- φ=0
ergy loss due to dissipation associated with the directed
The association of D/α with the kinetic temperature at
and the random motion, respectively. The third term
φ = 0 assumes that the diffusion and friction constants
represents the energy gained through diffusive processes
areindependentoftherelativephaseofthelattices. Fol-
internal to either of the two lattices, as represented by
lowing the standard model of Sisyphus cooling [29], our
the constants of momentum diffusion D . The factor N
i model (1) assumes that diffusion and friction are spa-
is the dimensionality of the system, in our case N = 3,
tially homogeneous. It should be noted, though, that in
and δp2 = δp2 +δp2 +δp2 (cid:39) 3δp2. In our system, the
x y z z a more accurate model these coefficients are dependent
temperature might be slightly different in the different
on the position x of the atom in the lattice. The spatial
directions [23]. However, this approximation will only
distribution of atoms in either lattice will have some de-
have a minor effect on the results.
pendenceonφ,whichwilltranslateintoadependenceon
In [11, 12], the efficiency of a BM without a load is
the spatially averaged friction and diffusion coefficients.
definedastheenergydissipatedbyfrictionactingagainst
This is ignored in our model, introducing a degree of ap-
the directed motion over the total energy input, that is,
proximation in Eq. (4) for efficiency.
αp2/m αp2/m Absorption images were taken for five different poten-
η = = . (3) tialdepths. InFig.2,weshowtypicalrawdataforatoms
Pin αp2/m+αδp2/m−ND
kept 150 ms in the lattices. A clear drift is seen in im-
Assuming that α is uncorrelated to the lattice state, so ages 2 and 4, while images 1, 3, and 5 show no drift,
that αδp2 =αδp2, this expression is simplified to as expected. Images such as Fig. 2 have been taken for
0≤φ≤2π for potential depths between 40 and 200 µK,
p2/m and the analyzed data can be seen in Fig. 3 in terms of
η = . (4)
p2/m+δp2/m−ND/α the drift velocity. The induced drifts are expected to be
symmetric around φ=π. However, slightly larger drifts
Simulations indicate that, even for a relatively large dif- areobservedforφ=2π/3thanforφ=4π/3,mostlikely
ferenceinthefrictioncoefficientsbetweenthetwolattices duetoexperimentallimitationsinthealignmentandthe
(α = α /2), this assumption introduces an error in P intensity balance of the lattice beams. Also shown in
2 1 in
of only about 2%. Fig. 3 is the kinetic temperature for the same parame-
The experiment has been described in detail in [17, ters. Wefindthatthebaselineofthekinetictemperature
24, 25]. In short, we use laser cooling to trap and cool increases with the potential depth, while the amplitude
cesium atoms and transfer them into a double optical of its variation with φ is roughly unchanged. As dis-
lattice [24, 25]. These are potentials realized from the cussed earlier, the φ-dependent kinetic temperature is
interferencepatternoflaserbeamsduetoasecond-order represented by the second term on the right-hand side
3
mm 0.25 40 µK
1 80 µK
120 µK
0.20
160 µK
200 µK
%)0.15
φ=0 φ=2π/3 φ=π φ=4π/3 φ=2π η(
0.10
FIG. 2: Absorption images of atoms in the double optical
lattice for five different relative spatial phases where the po- 0.05
tentialdepthis200µK. Theatomsarekeptinthelatticesfor 0.00
150 ms. For φ=0 and 2π, the drift is zero and the diffusion
0.0 0.5 1.0 1.5 2.0
is small. For φ = π, the drift is also zero, but the diffusion
φ/π
is much larger. Images 2 and 4 show maximum drifts in the
two different directions.
FIG.4: (Coloronline)Theefficiency,accordingtoEq.(4),of
theBrownianmotorasafunctionoftherelativespatialphase
10 for five different potential depths. The greatest efficiency is
1.0 K)8 achieved for φ equal to 2π/3 and 4π/3 and drops to zero for
T (µ6 φ=π.
4
0.5
2
s) -1 0 1 2
v (mm/0.0 40 µK φ/π 1 mm
80 µK
-0.5 120 µK
160 µK
200 µK 0 ms 75 ms 150 ms 225 ms 300 ms
-1.0
0.0 0.5 1.0 1.5 2.0
FIG. 5: Time evolution of the atomic sample for a potential
φ/π
depth of 200 µK. Both the drift and the diffusion are seen.
FIG. 3: (Color online) The drift velocity versus the phase
shift for a lattice holding time of 150 ms. Inset: the kinetic
temperature for the same data. calculated from the expansion of the atomic cloud in the
optical lattices, where the size of the cloud is given by
(cid:113)
of Eq. (2), while the baseline (or kinetic temperature at σ = σ 2+2D˜t, (8)
t 0
φ=0) is represented by the third term. Hence their dif-
ferenceisthevariationofthekinetictemperaturewithφ. with σ the root-mean-square radius at time t.
t
Our data show that this variation is approximately the In order to quantify the performance in terms of the
same for different potential depths. Hence, the greater P´eclet number, series of absorption images of the time
efficiencyforlargerpotentialdepthsismainlyduetothe evolution of the atomic cloud, such as shown in Fig. 5,
increase in the drift momentum. Using the data from have been taken. The phase is set to achieve maximum
Fig. 3 in Eq. (4), we obtain the efficiency as a function drift (φ=2π/3).
of φ, where the maximum efficiency is close to 0.3%; see In Fig. 6(a), a series of such images have been ana-
Fig. 4. lyzed and the drift is plotted against the holding time
An alternative way to characterize the rectified mo- in the lattice. In Fig. 6(b), the width of the sample is
tion is by the coherence of the transport, where the lin- shown against the holding time, from which the diffu-
ear transport is compared to the diffusion. This can be sion constant D˜ can be extracted by fitting to Eq. (8).
eff
quantified using the P´eclet number [16], Combining the result with the measured average veloc-
ity of the sample, according to Eq. (6), gives the P´eclet
|(cid:104)v(cid:105)|l
Pe≡ , (6) number for different potential depths; see Fig. 7.
D˜eff In conclusion, we have adopted existing theory and
presented experimental results for two measures of the
where l is a characteristic length of the system, in our
case the lattice constant, and D˜ is the effective spatial performance of a Brownian motor, namely the efficiency
eff and the P´eclet number, in a system of ultracold atoms
diffusion given by
in a double optical lattice. The results indicate trends
(cid:104)x2(t)(cid:105)−(cid:104)x(t)(cid:105)2 that give higher efficiency and transport coherence for
D˜eff ≡t→li+m∞ 2t . (7) deeper potentials, and are in agreement with the val-
ues of the P´eclet number that were predicted for similar
For atoms in dissipative optical lattices, where thermal systems [31]. Although our BM prototype differs from
fluctuations play an important role, D˜ becomes the otherBM’s,thefundamentalprinciplesarethesame,and
eff
spatial diffusion constant D˜ = (cid:104)[δx(t) − δx(0)]2(cid:105)/(2t), hence these kinds of characteristic measurements allow
where δx(t) = x(t)−(cid:104)x(t)(cid:105) [30]. This quantity can be for interesting comparisons between BM systems from
4
(a) (b) different fields.
0.0 40 µK 0.5
80 µK
m) 160 µK
x(t)> (m-0.2 200 µK σ (mm)t0.4
<
-0.4
0.3
-0.6
0.0 0.1 0.2 0.3 0.0 0.1 0.2 0.3
t (s) t (s)
FIG. 6: (Color online) (a) Position of the center of mass
of the atomic sample. (b) Root-mean-square radius of the
atomic sample as a function of holding time for four differ-
ent potential depths. The center of mass moves linearly as
expected [17], and indicates faster drifts for higher potential
depths. Thesizeofthecloudgrowswithtimeduetodiffusion
according to Eq. (8).
0.0095
0.0090
Pe0.0085
Acknowledgments
0.0080
0.0075
200 400 600 800 1000
A/kB (µK)
FIG.7: (Coloronline)ThemeasuredP´ecletnumber,Eq.(6), This project was supported by the Swedish Research
asafunctionofpotentialdepth. Thedataindicatesagreater Council, Knut & Alice Wallenbergs Stiftelse, Carl Tryg-
coherence in the transport for higher potential depths. ger Stiftelse, and Ume˚a University.
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