Preparation of atomic Fock states by trap reduction

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by  M. Pons

Preparation of atomic Fock states by trap reduction
M. Pons,1, A. del Campo,2, J. G. Muga,3, and M. G. Raizen4,
∗ † ‡ §
1Departamento de F´ısica Aplicada I, E.U.I.T. de Minas y Obras Pu´blicas,
Universidad del Pa´ıs Vasco, 48901 Barakaldo, Spain
2Institute for Mathematical Sciences, Imperial College London, SW7 2PE, UK;
QOLS, The Blackett Laboratory, Imperial College London, Prince Consort Rd., SW7 2BW,UK
3Departamento de Qu´ımica-F´ısica, Universidad del Pa´ıs Vasco, Apartado 644, 48080 Bilbao, Spain
4Center for Nonlinear Dynamics and Department of Physics, University of Texas, Austin, Texas 78712 USA
(Dated: January 28, 2009)
We describe the preparation of atom-number states with strongly interacting bosons in one di-
mension, or spin-polarized fermions. The procedure is based on a combination of weakening and
9 squeezing of the trapping potential. For the resulting state, the full atom number distribution is
0 obtained. Starting with an unknown number of particles Ni, we optimize the sudden change in
0 the trapping potential which leads to the Fock state of N particles in the final trap. Non-zero
f
2
temperatureeffectsaswellasdifferentsmooth trappingpotentialsareanalyzed. Asimple criterion

n is provided to ensuretherobust preparation of theFock state for physically realistic traps.
a
J PACSnumbers: 32.80.Pj,05.30.Jp,05.30.Fk,03.75.Kk

8
2
I. INTRODUCTION when the process is sudden. From the expression of the

] number variance, a simple criterion for optimal perfor-
h
mance was obtained, namely, that the subspace spanned
p Given the importance of photon statistics in quantum
by the occupied levels in the initial trap configuration
- optics, the field of atom statistics is expected to develop
t containsthesubspaceoftheboundlevelsinthefinaltrap.
n vigorously in atom optics, fueled by the current ability
Starting from an unknown number of particles trapped
a
to measure the number of trapped ultracold atoms with
u at zero temperature, the mixed trap reduction method
nearlysingle-atomresolutionandwithoutensembleaver-
q assures that the final state indeed corresponds to the
[ agingoffluctuations[1,2,3]. Amongthepossibleatomic desired Fock state by avoiding the momentum or posi-
distributions, pure atom number (Fock) states form a tionspacetruncationsinherentinpuresqueezingorpure
1
fundamentalbasisandholduniqueandsimpleproperties
v weakening (the latter being called “culling” in [13]).
that make them ideal for studies of quantum dynamics
9 In [13, 14] the potential traps considered for simplic-
of few-body interacting systems [4], precision measure-
2 itywerefinitesquarewells,sodoubtcouldbecastonthe
5 ments [5], or quantum information processing [6, 7, 8]. validityoftheresultsinactualsmoothtraps. Otherlimi-
4 Efficient and robust creation, detection and manipula-
tationsweretheconsiderationofzerotemperatureinitial
. tion of atom number states are thus important goals in
1 states, and a statistical analysis limited to the first and
0 atomic physics. Several approaches have been proposed secondmomentofthenumberdistribution. Inthispaper
9 and explored recently with theoretical and experimental
we overcome these shortcomings by studying the mixed
0 workleadingtosub-Poissonianand,inthelimit,number
trap reduction process using smooth potentials, states
: states, such as atomic tweezers [9, 10], interferometric
v with finite temperature, and the full number distribu-
i methods [4, 11], Mott insulator states [12], or atomic tion.
X
culling [1, 13]. None of these methods is so far fully sat-
r isfying if the individual atoms have to be addressed (a
a
problem of the Mott insulator states in optical lattices),
II. THE TONKS REGIME AND POLARIZED
and if an arbitrary number of atoms is to be produced
FERMIONS
reliably and with small enough variance for the trapped
atom number, so further researchis still required.
The strongly interacting regime of ultracold bosonic
In a previous paper [14] a method was proposed in
atomscanbedescribedbytheso-calledTonks-Girardeau
whichthe trappingpotentialis simultaneouslyweakened
(TG) gas [15], which is achieved at low densities and/or
and squeezed so that the final trap holds a desired num-
large one-dimensional scattering length [16, 17]. It has
ber state. For a Tonks-Girardeaugas, it was shown that
been argued [14] that this regime is optimal for the cre-
this mixed trap reduction yields optimal results, even
ation of atomic Fock states by mixed trap reduction.
The TG gas and its “dual” system of spin-polarized
ideal fermions behave similarly, and share the same
one-particle spatial density as well as any other local-
∗Electronicaddress: [email protected]
correlation function, while differ on the non-local corre-
†Electronicaddress: [email protected]
‡Electronicaddress: [email protected] lations.
§Electronicaddress: [email protected] The fermionic many-body ground state wavefunc-
2
tion of the dual system is built at time t = 0 as a where
Slater determinant for N particles, ψ (x ,...,x ) =
√N1i!detNn,ik=1ϕin(xk), wheire ϕin(x) is Fthe1n−th Neiigen- Λi = Ni |ϕinihϕin| (4)
state of the initial trap, whose time evolution will nX=1
be denoted by ϕ (x,t) when the external trap b
n
is the projector onto the bound subspace occupied by
is modified. The bosonic wave function, sym-
the initial state. We may thus conclude that trap reduc-
metric under permutation of particles, is obtained
tioncanactuallyleadto the creationofFock states with
from ψ by the Fermi-Bose (FB) mapping [15, 18]
F hN i = C and σ2 = 0 quite simply when the initial
Aψ(x1=,...,xNi) = As(gxn1(,x...−,xNxi))ψFis(xt1h,e...“,axnNtiis)y,mwmheerte- staftes spanf the finNalfones,
ric unitQf1u≤njc<tkio≤nN”i. Nokting tjhat |ψ|2 = |ψ |2 it is
F Λ ⊂Λ . (5)
f i
clear that both systems obey the same counting statis-
tics. Moreover, since A does not include time explic- Infactthefullatomnubmberdbistribution[20]isaccessible
itly, the mapping is also valid when the trap Hamilto- intheatomcullingexperiments[1]andwenextfocusour
nian is modified, and the time-dependent density pro- attention on it. Consider the characteristic function of
file resulting from this change can be calculated as thenumberofparticlesintheboundsubspaceofthefinal
[19] ρ(x,t) = Ni |ψ(x,x2,...,xNi;t)|2dx2···dxNi = trap,
Ni |ϕ (x,t)|2. BRy reducing the trap capacity (maxi-
Pmunm=1numnber of bound states and thus particles that it F(θ)=Tr[ρˆeiθΛbfnˆΛbf]. (6)
canholdintheTGregime)someoftheN atomsinitially
i
Following [21], the atom number distribution can be ob-
confined may escape and only N will remain trapped.
tained as its Fourier transform,
To determine whether or not sub-Poissonianstatistics
or a Fock state are achievedin the reduced trapwe need 1 π
to calculate the atom-number fluctuations. p(n)= e−inθF(θ)dθ, (7)
2π Z
π
−
with n=1,...,C .
f
III. THE SUDDEN APPROXIMATION: FULL The characteristic function of spin-polarized fermions
COUNTING STATISTICS
or a Tonks-Girardeau gas restricted to a given subspace
was studied in [22, 23, 24]. Using the projector for the
We shall now describe the preparation of Fock states bound subspace in the final configuration,
by an abrupt change of the trap potential to reduce its
capacity. Consider a trap with an unknown number of F(θ)=det[1+(eiθ−1)ΛfΛi]. (8)
particles N , which supports a maximum of C bound
i i For computational purposes it is cobnvbenient to use the
states. Generally N is smaller than the capacity of the
i basis of single-particle eigenstates |ϕf i which spans the
trapC . Thetrappingpotentialisabruptlymodifiedtoa m
i final bound subspace so that F(θ)=detA, where A is a
final configuration of smaller capacity C . Similarly the
f
C ×C matrix with elements
final number of trapped particles will be N ≤ C . We f f
f f
are interested in the optimal potential change such that
A =δ +[exp(iθ)−1]hϕf|Λ |ϕf i. (9)
N =C to prepare the atomic Fock state |N =C i. nm nm n i m
f f f f
Let α = i,f stand for initial and final configuration. Clearly, if Λ ⊂ Λ , A = exp(biθ)δ , F(θ) =
f i nm nm
The Hilbert space associated with the Hamiltonian of a
exp(iθC ) and the Kronecker-delta atom-number distri-
f
particlemovinginanyrealistictrapV ,isthedirectsum b b
α bution associated with the Fock state |N = C i is ob-
f
H = B ⊕R of the subspace spanned by the bound
α α α tained,
states B ={|ϕαi|j =1,...,C }, and that of scattering
α j α
states Rα = {|χαki|k ∈ R}. Consider the projector onto p(n)=δn,Cf. (10)
the final bound states, B defined as
f
For completeness we note that the cumulant-generating
Cf function logF(θ) admits the expansion
Λ = |ϕfihϕf|. (1)
f j j
Xj=1 ∞ (iθ)n
b logF(θ)= κ , (11)
n
WithintheTGregimeandforspin-polarizedfermions, nX=1 n!
the asymptotic mean number and variance of trapped
atoms are [14] from which the mean κ1 =hNfi in Eq. (2) and variance
κ =σ2 in Eq. (3) are just the first two orders.
2 Nf
hNfi = Tr(ΛiΛf) (2) Different regimes of interactions for ultracold Bose
gases in tight waveguides can be characterized by a sin-
and b b
gle parameter γ = mg L/~2N, where g is the one-
1D 1D
σN2f = Tr ΛiΛf −(ΛiΛf)2 , (3) dimensional coupling strength, L the size of the system,
(cid:2) (cid:3)
b b b b
3
1
0
(a) (b) (c)
0,8
-0,2
n)0,6
Vi -0,4 p(0,4
V/ -0,6
0,2
-0,8 (a) (b) 0
1 2 3 4 5 6 7 8 910 1 2 3 4 5 6 7 8 910 1 2 3 4 5 6 7 8 910
-1 n n n
-1 -0,5 0 0,5 1 -1 -0,5 0 0,5 1
x/L x/L FIG. 2: Atom number distribution p(n) in a trap reduction
i i setup combining both squeezing and weakening techniques,
for a bathtub potential given by Eq.(13) with σα = 0.03Lα
FIG. 1: Schematic potential change for trap weakening (a) and the parameters of the trap being Ni = Ci = 100 and
and squeezing (b), relative to the initial width Li and depth Cf = 10 (Ui = 104π2, Uf = 102π2). Plot (a) corresponds
Vi, for σ=0.05L. to almost pure squeezing, with Lf/Li = 0.04, while plot (c)
corresponds to pure weakening, Lf/Li = 1, showing that it
isthecombination of bothtechniques,Lf/Li =0.5, plot(b),
m and N the mass and number of atoms respectively. γ themost efficient way for our purposes.
can be varied [17] allowing to explore the physics from
themean-fieldregime(γ ≪1)tothe TGregime(γ ≫1)
[16]. We note that for the system to remain in the TG square potential U ≈ C 2π2. For the Gaussian poten-
α α
regime, it suffices to keep or decrease the density, since tial a single parameter U =V δ2 defines an isospectral
α α α
family.
n
γ =γ i. (12) Pure trap weakening corresponds to V < V , while
f i f i
n
f L = L , and pure trap squeezing to L < L , keeping
f i f i
V =V (Fig. 1). We shallnextdescribethe efficiencyof
Inparticular,trapweakeningclearlyleadstoareduction f i
atomic Fock state preparation by mixed trap reduction
of the density so that the system goes deeper into the
(V < V and L < L ) keeping constant the relative
TG regime. f i f i
smoothness parameter σ˜ = σ˜ , and going from the U
i f i
to the U families of traps. This procedure allows us to
f
apply squeezing of the potential up to any desired value
IV. DEPENDENCE ON THE TRAPPING
POTENTIAL of Lf keeping its bathtub shape.
Figure2illustratesthefullcountingstatisticsofthere-
sultingstateindifferentlimitsofatrapreductionscheme.
In this section we shall discuss the efficiency of the
Both pure weakening and squeezing fail to produce an
trap-reduction procedure at zero-temperature, focusing
atom-number state since the condition in Eq. (5) is not
on the relevance of the shape of the confining potential.
fulfilled. It was shown in [14] that this limitation arises
In particular, instead of the idealized square potentials
astheresultoftruncationofthe finalstatebothincoor-
used in [13, 14] we shall study here the family of “bath-
dinate (pure weakening) or momentum (pure squeezing)
tub” potentials
space,withrespecttothedesirableFockstate|N =C i.
f
1 |x|−L However, this state, whose full-distribution reduces to a
α
Vα(x;Vα,Lα,σα)=−2Vα(cid:20)1−tanh(cid:18) σα (cid:19)(cid:21)(13) Kronecker delta δn,Cf (see Eq. (10)), can be obtained
by combining both strategies, as shown in the middle
as well as the inverted Gaussian potential panel. The final trap is then perfectly filled by the pure
atom-numberstate. Inwhatfollowsweshallcharacterize
x2 the efficiency of the method just by the mean and atom
V(x)=−Vαe−2δ2. (14)
number variance of the prepared state, see Eqs. (2) and
For the bathtub Lα and Vα play respectively the role of (3). The normalized variance, σN2f/hNfi allows us to
the width and depth of the trap, while σ is an addi- distinguish between the sub-Poissonian, Poissonian, and
α
tional parameter describing the smoothness of the po- super-Poissonianstatistics whenever it is lower, equal or
tential trap. greater than 1, respectively.
The spectrum and eigenfunctions can be found nu- Let us now consider different trap geometries. Gen-
merically by a standard technique, first differencing the erally, the effect of the smoothness of the potential is
Hamiltonian and then diagonalizing the tridiagonal ma- to increase the density of states near the brims, where
trix obtained by such difference scheme [25]. the spacing between adjacent energy states is reduced,
For the bathtub potential, a given U = 2mV L /~2 see Fig. 3. As a consequence, a higher control of the
α α α
and σ˜ = σ /L defines a family of isospectral poten- depth of the potential would be required. Nonetheless
α α α
tials. In dimensionless units, their eigenvalues are the Fig. 4 shows that by increasing the smoothness, the
same and so are their eigenfunctions. In the limit of a Fock state creation condition (5) is actually satisfied for
4
0
0 10
-0,2 9,6
-30
-0,4 2
V/Vi-0,6 !E/n > 9,2 T=0K
-60 N 8,8 µ/k T=14, k T/E=4.7 102
-0,8 < B B i
-1 (a) (b) 8,4 µ/kBT=5.7, kBT/Ei=1.1 103
-90
-1,2-0,8-0,4 0 0,4 0,8 1,2 0 0,1 0,2 0,3 0,4 8 (a) µ/k T=3, k T/E=2.2 103
x/L "/L B B i
i i
10
FIG. 3: Effect of the smoothness of the trapping potential
(a) on thespectrum (b). As σ increases the density of states 9,6
concentratesnearthebrimofthetrapandmoreboundstates
9,2
appear. The spectrum in the low panel is obtained for a
>
potential V =(10π)2 (in units of ~2/2mL2) . N (b)
8,8
<
8,4 Ci=100, µ/kBT=3, kBT/Ei=2.2 103
10
8 C=130, µ/k T=4.8, k T/E=2.2 103
i B B i
9,98
0 0,2 0,4 0,6 0,8 1
> 9,96 ! =0 L/L
N " f i
< ! =0.03 L
9,94 " "
! =0.05 L FIG. 5: Effect of temperature on the asymptotic mean value
" "
9,92 gaussian (a) oftheatomnumberdistributionofaTonks-Girardeaugasob-
tainedbysuddenweakening-squeezingforthecaseofasquare
0,025 potential as a function of the width ratio between the initial
2 2
!=0 (b) and final trap. In plot (a) Ui =(100π) and Uf =(10π) so
0,02 ! =0.03 L the capacities Cα are the same as in Fig. 4, but the initial
" " occupation is now chosen to be Ni = 0.8Ci. As we increase
N> 0,015 ! =0.05 L the temperature, µ/kBT ≤ 5 the method starts to fail, but
" "
< this can be improved increasing the capacity of the initial
/ 0,01 gaussian
2 well, as shown in plot (b), where the weakening-squeezing
N
! process is applied at the same temperature but two differ-
0,005
ent initial traps: Ui =(100π)2 used in the previous plot and
0 Ui =(130π)2, both with Ni =0.8Ci. (Ei=~2π2/2mLi2.)
0 0,2 0,4 0,6 0,8 1
L/L Wemightconcludethataninvariablyefficientstrategy
f i
forthesuddentransitionbetweenaU andU trapfami-
i f
lies,isachievedbyreducingtohalfthewidthoftheinitial
FIG. 4: Asymptotic mean value (a) and variance (b) of the trap and reducing the depth to the trap accordingly, so
atomnumberdistributionofaTonks-Girardeaugasobtained as to achieve the desired U and capacity C ,
f f
by sudden weakening-squeezing as a function of the width
ratio between the final and initial trap Lf/Li. The process L ≈ Li, V ≈ Uf Vi, (15)
is robust for different values of the smoothing parameter σ. f 2 f U 4
i
Theinitialstateisassumedtobeinthegroundstate. Uα are
chosen such that Ci =100 and Cf =10, this means that for which warrants the preparation of the Fock state |Cfi
the bathtub potential Ui = (100π)2 and Uf = (10π)2, while corresponding to the Uf-family.
for the Gaussian, Ui = (28π)2 and Uf = (8π)2. In all cases
we assume Ni =Ci=100.
V. NON-ZERO TEMPERATURE
a broader range of parameters which includes conditions
The above formalism can be generalized in a straight-
nearer pure weakening and pure squeezing. This is be-
forwardwaytoaccountfortheatomnumberdistribution
cause the initial state is spread out along the same re-
resulting from an arbitraryinitial state at non-zerotem-
gioninconfigurationspaceasthefinalone;moreoverthe
perature. It suffices to redefine
looser confinement reduces the momentum components
of the final state, which can be resolved more easily by Λ = π |ϕiihϕi| (16)
i n n n
the initial state.
Xn
b
5
which,ingeneral,isnotaprojectornow,whereπ isthe turecanbecompensatedbyenlargingthecapacityofthe
n
occupation probability of the state |ϕii. In the ground initialtrap. However,itisstillanexperimentalchallenge
n
state of the TG gas π = 1 ∀n = 1,...,N and π = 0 to get to the strong TG limit which must translate into
n i n
otherwise. For a thermal state, the Fermi-Dirac weights a correction to the fidelity. By contrast, non-interacting
π ={exp[β(Ei −µ)]+1} 1 (withβ =1/k T wherek polarized Fermions would be an ideal system for Fock
n n − B B
is the Boltzmann constant and T the absolute tempera- state preparation. For ultracold fermions, due to the
ture) result due to the effective Pauli exclusion principle wavefunction antisymmetry, s-wave scattering is forbid-
mimicked by bosons in the TG regime [23]. Notice the denandgenerallyp-waveinteractionscanbeneglectedso
normalization π = N . The preparation of a Fock that the gas is non-interacting to a good approximation.
n n i
state by a suddPen change of the trap will still be feasi- Such type of gases can be prepared in the laboratory
ble as long as Λf ⊂ Λi. The numerical results in Figure with linear densities of the order 0.2−2 µm−1 for which
5 (upper panel) illustrate the degradation of the quality the polarization remains constant in a given experiment
b b
of final state with increasing temperature. However, the [26]. For such gases the trap reduction technique can be
lowerpanelshowsthatthisnegativeeffectoftemperature directly extended to two and three dimensions.
can be compensated by starting from a “bigger” initial
trap with larger capacity.
Acknowledgments
VI. DISCUSSION AND CONCLUSION
We acknowledge discussions with Martin B. Plenio
We conclude thatthe controlledpreparationofatomic and Ju¨rgen Eschner. This work has been supported by
Fockstatesinthestronglyinteracting(Tonks-Girardeau) Ministerio de Educaci´on y Ciencia (FIS2006-10268-C03-
regime can be achieved by combining weakening and 01, FIS2008-01236) and the Basque Country University
squeezingofthetrappingpotential. Theprocessisrobust (UPV-EHU, GIU07/40). The work of MGR was sup-
withrespecttothesmoothnessofthepotentialtrap,and ported by the R. A. Welch Foundation and the National
moreover the deteriorating effect of increasing tempera- Science Foundation.
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