Table Of ContentQuantum time of flight distribution for cold trapped atoms
Md. Manirul Ali∗1, Dipankar Home†2, A. S. Majumdar‡1, and Alok K. Pan§2
1 S. N. Bose National Centre for Basic Sciences, Salt Lake, Calcutta 700098, India and
2Department of Physics, Bose Institute, Calcutta 700009, India
(Dated: February 1, 2008)
Thetimeofflightdistributionforacloud ofcoldatomsfalling freely undergravityisconsidered.
Wegeneralisetheprobabilitycurrentdensityapproachtocalculatethequantumarrivaltimedistri-
butionforthemixedstatedescribingtheMaxwell-Boltzmanndistributionofvelocitiesforthefalling
atoms. We findan empirically testable differencebetween the time of flight distribution calculated
usingthequantumprobabilitycurrentandthatobtainedfromapurelyclassicaltreatmentwhichis
usually employed in analysing timeof flight measurements. Theclassical time of flightdistribution
matches with thequantumdistribution in the large mass and high temperature limits.
7
0 PACSnumbers: 03.65.Ta,03.65.Xp,32.80.Pj
0
2
I. INTRODUCTION trajectory models[9]. It is thus desirable that some of
n
the conceptually sound theoretical formulations of the
a
quantummechanicalarrivaltimedistribution[10]becon-
J Inrecenttimeslasercoolingandtrappingofatomshas
fronted with accurate experimental data. If such quan-
1 become an area of active research[1]. The measurement
1 oftheinitialtemperature ofthe cloudofatomsiscrucial tum mechanical approaches are employed for analysing
experimentsusingTOFmeasurements,itshouldnotonly
forcharacterisingthepropertiesofatomtraps. Thetem-
1 enable to determine the empirical viability of various
perature of the cloud can be inferred from the velocity
v competingarrivaltimemodels[10],butalsopossiblyshed
0 distribution of atoms in the cloud. A well-known tech- newlightontheconventionalinterpretationoftheresults
6 nique of measuring this velocity distribution is the time
of these experiments.
0 of flight (TOF) method. Measurements of the TOF dis-
In this paper we employ the probability current
1 tribution have been employed to analyse various experi-
0 mental data suchas those involving ions andisotopes[2], approach[11]towardsobtainingthequantumarrivaltime
7 distribution of cold trapped atoms. The probability cur-
andalsoinperformingmassspectroscopyofbiomolecules
0 rent approach for computation of the mean arrival time
like DNA[3].
/ of a quantum ensemble not only provides an unambigu-
h
ThetheoreticaltreatmentoftheTOFdistributionthat
p ous definition of arrivaltime at the quantum mechanical
can be obtained using for instance, the Green’s func-
- level[11, 12, 13], but also adresses the issue of obtaining
t tion method[4], however, turns out to produce perfect
n the proper classical limit of the time of flight of massive
agreement with the TOF distribution obtained by us-
a quantum particles [14, 15]. Here we derive the quantum
u ing Newton’s equations for ballistic motion of particles arrival time distribution for the case of initially trapped
q accelerated by the earth’s gravitational field[5]. Thus,
atomiccloudsthataresubsequentlyallowedtofallfreely
: the interpretation of the results of the various TOF
v under gravity[4, 16]. We compute the mean time of ar-
i experiments[1, 2, 3] where classical trajectories are in- rival for these atoms and compare it with the mean ar-
X
ferred from Newtonian mechanics[6] remains debatable,
rival time obtained through the classical time of flight
r especially in the domain of small atomic masses and low
a analysis[5] that has frequently been employed for such
temperatures where quantum mechanical effects should
experiments[1, 2, 3]. Our analysis predicts the mass and
be significant.
temperature range of the atomic clouds where the quan-
Thoughthereexistsnouniqueprescriptionforthedef- tum mechanical treatment alters the arrival time distri-
inition of time of flight and arrival time in quantum me- bution and the mean arrival time from that obtained
chanics, experimentalists measure arrival times of ele- through the classical analysis.
mentary particles, atoms and molecules using the TOF
methods. In spite of the difficulties to give time an ob-
servable status in quantum mechanics, several logically II. CLASSICAL ANALYSIS OF TIME OF
consistent schemes for the treatment of the arrival time FLIGHT MEASUREMENTS
distributionhavebeenformulated,suchasthosebasedon
axiomatic appraches[7], opearator constructions[8], and
We beginwith a brief descriptionofthe classicalanal-
ysis of TOF measurements of trapped atoms. A probe
laser, focussed in the form of a sheet, is placed under-
neath the atomic cloud. When the trapping forces are
∗[email protected]
turned off, the cold atom cloud falls through the laser
†[email protected]
‡[email protected] probe under the influence of gravity. It is then possi-
§[email protected] ble to detect the fluorescence from the atoms as they
2
reach the sheet. The fluorescence is measured as a func- quantum probability current without using any classical
tion of time and the initial temperature of the cloud is ingredients. To that end, let the initial state of each of
determined by fitting the experimental resultto the the- the atomsbe representedbya onedimensionalGaussian
oretically predicted TOF signal of the cloud[4, 16]. A wave function of the form
detailed derivation of the TOF signal recorded by the
dperotebcetolars(etrhaastaisf,utnhcetionnumofbetrimoef)awtoams dsearrivrievdinbgyaYtatvhine ψ(z,0)=(2πσ02)−1/4exp(im~vz)exp −4σz202 (5)
(cid:18) (cid:19)
et al[5].
The cloud of atoms consisting of noninteracting parti- centered at z = 0 and moving with a group velocity v.
cleshasaMaxwell-Boltzmannvelocitydistributiongiven The Schro¨dinger time evolved wave function under the
(in one dimension) by Hamiltonian H =p2/2m+mgz is given by
Π(v)dv = m 1/2exp −mv2 dv (1) 2 −1/4 z−vt+ 21gt2 2
ψ(z,t)= 2πs exp
(cid:16)2πkT(cid:17) (cid:18) 2kT(cid:19) t "(cid:0) 4stσ0 (cid:1) #
(cid:0) (cid:1)
where T is the initial temperature of the cloud, and m m 1
×exp i( ) (v−gt)(z−vt/2)− g2t3 (6)
is the atomic mass. Using the Newton’s equations for ~ 6
ballistic motion of a particle accelerated by the earth’s (cid:20) (cid:26) (cid:27)(cid:21)
gravitationalfield (inthe verticalz-direction),the veloc- where st =σ0 1+i~t/2mσ02 .
ity is obtained in terms of the time of flight as
Consideringthefreefalloftheatomsundergravity,the
(cid:0) (cid:1)
1 expressionfortheSchro¨dingerprobabilitycurrentdensity
2
v =(z+ gt )/t (2)
2 i~ ∂ψ∗ ∂ψ
J(z,t)≡ (ψ −ψ∗ ) (7)
Substituting the above expression for v from Eq.(2) in 2m ∂z ∂z
Eq.(1), one can obtain the time of flight distribution at
an arbitrary distance z, given by for the time evolved state is calculated using the initial
state given by Eq.(5) to be
m 1/2 m(z+ 1gt2)2
ΠC(t)dt = (cid:16)×2(π−kzT+(cid:17)21gte2x)pdt(cid:18)− 2kT2t2 (cid:19) (3) J(z,t)=P(z,t)(cid:20)(v−gt)+ 4m2~σ20t2σ2(z−vt+ 12gt2)(cid:21)(8)
t2
where the expression for the position probability distri-
The corresponding classical mean time of flight or mean bution is given by
arrival time τ for the atomic cloud calculated using
Π (t) as the tiCme of flight distribution is given by 1 z−vt+ 1gt2 2
C P(z,t)= exp − 2 (9)
∞Π (t)tdt (2πσ2)1/2 " (cid:0) 2σ2 (cid:1) #
0 C
τ = (4)
C ∞
R0 ΠC(t)dt with σ = σ0 1+~2t2/4m2σ04 1/2. The modulus of the
For simplicity, here we rRestrict ourselves to the case of a probability current density J(z,t) given by Eq.(8) pro-
(cid:0) (cid:1)
point-sizedcloud. Thethree-dimensionalcalculationwas videsthearrivaltime distributionforapurewavepacket
done by Yavin et al[5] using a simple coordinate trans- falling under gravity. Note that the quantum probabil-
formation and the same expression was obtained for the ity current as defined by Eq.(7) is formally ambiguous
TOFdistribution. Theseauthors[5]haveclaimedperfect up to a total divergence term[17]. However, J(z,t) can
agreement of their results with a previous calculation[4] be uniquely defined through relativistic wave equations
wheretheTOFdistributionwasderivedusingasophisti- which impart appropriate spin-dependent corrections to
cated Green’s function technique. We call this TOF dis- it that persist even in the non-relativistic limit[12, 18].
tributiongivenbyEq.(3)astheclassicaltimeofflightdis- Theensuingarrivaltimedistributiondefinedthroughthe
tribution,andEq.(4)denotes thecorrespondingclassical probability current thus contains a spin-dependent cor-
meanarrivaltimesince theclassicalNewtonianequation rection for particles with spin[13].
is used from the outset to derive this TOF distribution. Theatomiccloudisrepresentedbyanensembleofpar-
ticles in thermalequilibrium with a thermal distribution
of initial velocities. Each particle has a wave function of
III. THE QUANTUM ARRIVAL TIME the form (5), with a Maxwell-Boltzmann distribution of
DISTRIBUTION THROUGH THE initialvelocitiesgivenbyEq.(1). Thustheinitialthermal
PROBABILITY CURRENT state of the atomic cloud we have described is a mixed
state. We obtain the corresponding position probabil-
Our aim here is to derive an expression for the time ity distribution by averaging the pure state distribution
of flight distribution for the atomic cloud through the (9) over a thermal distribution of initial velocities. The
3
80 150 6000
(a) (b) (a) (b)
1800
Class. and Quant. time distributions -1s 12345670000000 MTa=s 2s .=5 6x.94110- a6.m.Ku. (Li) -------->---Π----C->ΠQ Class. and Quant. time distributions -1s15000 MTa=s 2s .=5 2x2.9180-96 a.mK.u. (Na) --------->->ΠΠCQ Class. and Quant. time distributions -1s12345000000000000000 ---->--Π----TC->= 4Π.5Q0x10-9 K Class. and Quant. time distributions -1s11253690000000000 ----->---Π--->CΠTQ= 5.00x10-8 K
0 0
0 0
0.244 0.245 0.246 0.247 0.248 0.249 0.25 0.24 0.242 0.244 0.246 0.248 0.25 0.252 0.254
0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.22 0.23 0.24 0.25 0.26 0.27 0.28
Time in sec. Time in sec.
Time in sec. Time in sec.
400 125
300 (c) 400 (d) (c) ---->ΠC (d)
Class. and Quant. time distributions -1s11225050500000 MTa=s 2s .=5 8x5.4160-76 a.mK.u. (Rb) -------->---Π---C-->ΠQ Class. and Quant. time distributions -1s123000000 MTa=s 2s .=5 1x32.190-0654 aK.m.u. (Cs) ΠQ------>ΠC Class. and Quant. time distributions -1s123000000 ------->ΠQT= 1.10x10-6 K Class. and Quant. time distributions -1s150270055 ΠQ---->T=Π 1C.41 x10-5 K
0 0 0 0
0.232 0.236 0.24 0.244 0.248 0.252 0.256 0.26 0.232 0.236 0.24 0.244 0.248 0.252 0.256 0.26 0.24 0.242 0.244 0.246 0.248 0.25 0.252 0.254 0.256 0.23 0.235 0.24 0.245 0.25 0.255 0.26 0.265 0.27
Time in sec. Time in sec. Time in sec. Time in sec.
FIG.1: Theclassical(ΠC(t))andquantum(ΠQ(t))TOFdis- FIG. 2: The classical (ΠC(t)) and quantum (ΠQ(t)) TOF
tributionsof theatomiccloud falling freely undergravityare distributions of the atomic cloud falling freely under gravity
plotted for varyingmass of theatoms at a fixedtemperature are plotted for varying temperatures at a fixed mass of Rb
T =2.5×10−6K with σ0 =10−5cm, Z =−30cm. atom (m=85.4678a.m.u) with σ0 =10−5cm, Z =−30cm.
mal distribution of initial velocities. The result is
result is m 1/2 ∞ mv2
Π (t) = J(z,t)exp − dv
Q
2πkT 2kT
P (z,t) = m 1/2 ∞ P(z,t)exp −mv2 dv = (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) 1 (cid:17) eZx−p∞− z+ 21gt2(cid:18)2 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)
T (cid:16)2πkT(cid:17) Z−∞ (cid:18) 2kT(cid:19) (2πσT2)1/2 " (cid:0) 2σT2 (cid:1) #
= (2πσT12)1/2 exp"−(cid:0)z+2σ12Tg2t2(cid:1)2# (10) ×"(z+ 12gt2)(σkmTT2+ 4m~22σ02)t# (11)
In this way we generalise the probability currentdensity
where σ 2 = σ2 +(kT/m)t2. The peak of the position approachto calculate the quantumarrivaltime distribu-
T
tionfor the mixedstate atfinite temperature. Itmay be
probability distribution P (z,t) follows the classicaltra-
T
mentionedherethatinthepresentcalculationweneglect
jectory and the effect of the mass and temperature de-
the smallspin-dependentcorrection[13]thatmayappear
pendences of the position probability occurs essentially
in the probability current density, as mentioned above,
because the spreading of the wave packet is different for
andconsequentlythemeanarrivaltimethatwecompute
different atomic mass and temperature of the cloud.
for any fermionic atoms.
Taking the modulus of the quantum probability cur- The corresponding quantum mean arrival time τQ
rent density as determining the quantum arrival time which is an observable quantity, is given by
distribution[11], we obtain the arrival time probability
∞
distributionfortheatomiccloud(whichisamixedstate) 0 ΠQ(t)tdt
τ = (12)
Q ∞
by averaging the pure state distribution (8) over a ther- Π (t)dt
R0 Q
R
4
One may note that though the integral in the numera-
tor of Eq.(12) formally diverges, several techniques have
0.2485
been employed in the literature ensuring rapid fall off
for the probability distributions asymptotically[19], so
that convergent results are obtained for the integrated
arrival time. For our present purposes it is sufficient to (s)
employ the simple strategy of taking a cut-off (t = tc) me 0.248
in the upper limit of the time integral[14, 15] with al ti
toocff=tthhepe mw2(aeZvaen+pa6arrσciktvcea)tl/gatitwmtheimesreaerσtect.cva=TlidhσuTus(p,tot=uorttchc)oeims6tpσhuelteawvteiidoltnohsf Mean arriv (b)(a)
spread in the wave function. 0.2475
If we impose now the classical limit for the quan-
(c)
tum TOF distribution given by Eq.(11), then one can
check that under the large mass and high tempera-
ture limits, and when σ0 << (kT/m)t2, one can take
σ 2 ≈ (kT/m)t2. Thus, Π (t) = Π (t), i.e., the 0.247
T Q C 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
two distributions match in the limit of large mass and
Mass (m) in a.m.u.
high temperature. The probability current method of
computing the quantum arrival time distribution fur-
nishes an effective way of approaching the classical limit FIG.3: Themassvariation of themeanarrival times τQ and
of the distribution by smoothly varying the parame- τ(TC =are1.4sh1o×wn10i−n6Kth)ewfiigtuhrZe f=or−a30fixecdm.vaTluheeoqfuatenmtupmermateuarne
ters such as mass and temperature of the quantum arrival time τQ is plotted for two different values of σ0. 2(a)
distribution[14, 15]. Note also, that the mass depen- τQ for σ0 =10−5cm, 2(b) τQ for σ0 =2×10−5cm, 2(c) The
dence of arrival time distribution given by Eq.(11) and mean arrival time τC calculated through the classical TOF
consequently, the observable mean arrival time given by distributionΠC(t)withT =1.41×10−6K andZ =−30 cm.
Eq.(12), signifies the quantum mechanical violation of
the gravitational weak equivalence principle[15]. Thus
TOF measurements[1, 2, 3, 4, 16] offer a practical possi- rival times (τ and τ ) with varying temperature of the
Q C
bility for experimental demonstration of the equivalence cloud, and obtain similar results, as expected from the
principle violation at the quantum level[20]. temperature variation of the classical and quantum ar-
rival time distributions plotted in Fig.2. The mass and
temperature dependences of the quantum arrival time
IV. NUMERICAL RESULTS distribution andconsequently the quantummean arrival
timeariseessentiallyduetothespreadofthewavepacket
We perform a numerical study of the quantum arrival for the atoms. A smaller value ofthe initial width σ0 for
time distribution of the falling atomic cloud by varying the wave packetresults in its faster spread. The amount
itsmassandtemperatureseparately. WefirstplotΠ (t) of departure of the quantum distribution from its clas-
C
andΠ (t)forfixedtemperatureof2.5×10−6K inFig.1. sical counterpart is thus contingent on the magnitude
Q
Itis seenthe classicalandthe quantumdistributionsare of the ensemble spread (since ΠC(t) is independent of
clearly different for clouds of small atomic mass such as σ0). This is clearly depicted in Fig.4 where the classi-
Li and Na. However, as one increases the atomic mass, cal(ΠC(t)) and quantum (ΠQ(t)) TOFdistributions are
oneseesthatΠ (t)andΠ (t)begintooverlapforheavy plotted for three different values of σ0 at a fixed mass
C Q
atoms such as Rb for this value of temperature. The and temperature.
temperaturevariationofthearrivaltimedistributionsare
displayedinFig. 2whereoneseesthatevenforheavyRb
atoms,the twodistributions arequite distinct in the low V. SUMMARY AND CONCLUSIONS
(nanokelvin)temperaturerange. Itwouldbeinteresting
ifourpredictioninthelowtemperatureandloweratomic To summarize, in this work we have considered the
massregionwherethequantumtimeofflightdistribution analysisofthetimeofflightmeasurementsoffallingcold
sharply differs from the classicalTOF distribution could atomic clouds. The inference of the temperature of the
be verified in actual experiments. cloudinvariousexperiments[1,2,3]isusuallyperformed
The variation with mass of the quantum and classical throughaclassicalanalysisinwhichthe resultsobtained
mean arrival times at a particular detector location for are same as through the solution of Newton’s equations
the ensemble of falling atoms is depicted in Fig.3. One for ballistic motion of particles falling under gravity[5].
can see that in the limit of large mass the mean arrival Here we emphasize the relevance of employing a quan-
time τ asymptotically approaches the classical result. tum mechanicalarrivaltime distributionfor the analysis
Q
One can also investigate the variation of the mean ar- ofsuchexperiments. Weusetheprobabilitycurrentden-
5
80 80 in the literature[9, 11, 12, 13, 21]. Further, we gener-
(a) (b)
alise the probability current density approach to calcu-
-1s -1s late the quantum arrival time distribution for a mixed
butions 60 σ0=10-6cm butions 60 σ0=5x10-6cm ---->ΠC svtealotecitdieesscrfiobrintghethfealMlinagxwateollm-Bicoltczlomuadnsnindistthreiburetlieovnanotf
distri distri experiments[1, 4, 16]. We compute the TOF distribu-
me 40 me 40 tionandmeanarrivaltimethroughthisschemeandcom-
nt. ti nt. ti pare our results with those obtained through a classical
nd Qua 20 ---->ΠC nd Qua 20 analysis[6] for various atomic masses.
Class. a ------->ΠQ Class. a ------->ΠQ The obtained quantum arrival time distribution
matches with the classical TOF distribution in the high
0 0
temperature and the large mass limits, hence furnishing
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.1 0.15 0.2 0.25 0.3 0.35 0.4
anotherexampleofthesmoothemergenceoftheclassical
Time in sec. Time in sec.
80 80 limit[14] of the quantum arrival time in the framework
(c) (c)
of the probability current approach. However, a clear
ns -1s 60 ---------->->ΠΠCQ ns -1s 60 ΠQ----->ΠC dbiusttiionncstioisnebxehtiwbieteendtfhoerqeiutahnetrusmmaalnldattohmeicclamssaicssa,lodrisltorwi-
me distributio 40 σ0=10-5cm me distributio 40 σ0=5X10-5cm twaenamdvpewepiradacttkhuerotefsopthfreetahwdeainvcgleodpueadpc.keneTtdhifniosgrrtuehpseuolnattstohfmreosmm. aOsdusi,ffrvesercelhonectmiitayel
Quant. ti Quant. ti tohfutsheprporvoibdaesbialintyewcumrreetnhtoddefnosriteyxpaeprpimroeancthalfovrerciafilccautliaotn-
Class. and 20 Class. and 20 ipcnlrogocptkho[2see2ad].rrmFivieantlahtloliymd,ewudseiinwstgirsihtbhutetoisoepnmi,npinhraoastdiazdteiottrihoaansttmoaoaqrnueaeinantrvuleimesr-
tigationsofmodernexperimentsemployingtimeofflight
0 0
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.1 0.15 0.2 0.25 0.3 0.35 0.4 techniques should be performed using various quantum
Time in sec. Time in sec. mechanical schemes[10]. Such studies have the potential
toempiricallyresolveambiguitiesinherentinthetheoret-
FIG. 4: The classical (ΠC(t)) and quantum (ΠQ(t)) TOF icalformulationsofthequantumarrivaltimedistribution
distributions of the atomic cloud falling freely under gravity
using cold trapped atom experimental techniques, and
are plotted for four different values of σ0 at a fixed mass of
Be atom (m = 9.01a.m.u) and at a fixed temperature T = may also shed new light on the inference of the experi-
3.0×10−6K with Z =−30cm. mental data.
Acknowledgments
sity approach towards obtaining the arrival time or the This work was initiated while MMA was visiting
TOF distribution. Our definition of the quantum ar- theAtominstitut derO¨sterreichischenUniversita¨ten,Vi-
rival time distribution and the observable mean arrival enna. MMAwouldliketothankYujiHasegawaandHel-
time in terms of the modulus of the probability current mutRauchforthestimulatingdiscussionsandwarmhos-
density is particularly motivated from the equation of pitality at the Atomic Institute. DH acknowledges the
continuity, and other physical considerations discussed support of the JawaharlalNehru Fellowship.
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