Restricted Thermalization for Two Interacting Atoms in a Multimode Harmonic Waveguide

0.22 MB·English

by  V. A. Yurovsky

Restricted Thermalization for Two Interacting Atoms in a Multimode Harmonic
Waveguide
V. A. Yurovsky
School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel
M. Olshanii
Department of Physics, University of Massachusetts Boston, Boston MA 02125, USA
(Dated: January 1, 2010)
0 In this article, we study the thermalizability of a system consisting of two atoms in a circular,
1 transversely harmonic waveguide in the multimode regime. While showing some signatures of the
0 quantum-chaotic behavior, the system fails to reach a thermal equilibrium in a relaxation from
2 an initial state, even when the interaction between the atoms is infinitely strong. We relate this
phenomenon to the previously addressed unattainability of a complete quantum chaos in the Sˇeba
n
billiard [P. Sˇeba, Phys. Rev. Lett., 64, 1855 (1990)], and we conjecture the absence of a complete
a
J thermalizationtobeagenericpropertyofintegrablequantumsystemsperturbedbyanon-integrable
but well localized perturbation.
1

PACSnumbers: 67.85.-d,37.10.Gh,05.45.Mt,05.70.Ln
]
h
p
- Introduction.–Theultracoldquantumgaseshavebeen upperboundforapossibledeviationoftherelaxedvalue
m
long proven to be an ideal testbench for studying the of an observable from its thermodynamical expectation,
o fundamental properties of quantum systems. One of the foranyinitialstateinprinciple. However,recentlyanew
at most interesting and frequently addressed topics is the direction of research has emerged: quantum quench in
. behavior of quantum systems in the vicinity of an inte- many-body interacting systems [1, 2, 9, 10]. In this class
s
c grable point. Experimental results already include the of problems, the initial state is inevitably decomposed
si suppression of relaxation of the momentum distribution intoalargesuperpositionoftheeigenstatesoftheHamil-
y [1] and the modified decay of coherence [2]. Note that tonian governing the dynamics, and the discrepancy be-
h
the fully developed quantum chaos with ultracold atoms tween the result of the relaxation and thermal values is
p
has been investigated in experiments [3]. expected to be diminished. In our article we show that
[
Traditionally, the underlying integrable system is rep- inthecaseoftwoshort-range-interactingatomsinahar-

1 resented by the Lieb-Liniger gas [4] that is well suited monic waveguide, a complete thermalization can never
v
to describe the dynamics of Van-der-Waals interacting be reached under either scenario.
5
bosons in a monomode atomic guide (see [5] for a re- We relate the absence of thermalization in our system
2
2 view). The nontrivial integrals of motion there are inti- tothepreviouslyaddressedunattainabilityofacomplete
0 mately related to both the one-dimensional character of quantumchaosintheSˇebabilliard–flattwo-dimensional
1. theatomicmotionandtotheeffectivelyzerovalueofthe rectangular billiard with a zero-range scatterer in the
0 two-body interaction range. The causes for lifting inte- middle [11]. Similarly to our system, the Sˇeba billiard
0 grability include the virtual excitation of the transverse does show some signatures of the quantum-chaotic be-
1 modes during the collision [6] and the coupling between havior, although both the level statistics [12–14] and the
:
v two parallel weakly interacting Lieb-Liniger gases [2]. momentum distributions in individual eigenstates [15]
i One would expect that in the multimode regime, when showsubstantialdeviationsfromthequantumchaospre-
X
the motion becomes substantially multidimensional, no dictions. Both the Sˇeba billiard (first suggested and
r
a remnants of the former integrals of motion will survive. solved in Ref. [11]) and our system allow for an exact
However, inthisarticle weshowthat forthecaseofonly analytic solution, in spite of the absence of a complete
two interacting atoms a strong separation between the set of integrals of motion.
transverse and longitudinal degrees of freedom remains, The system.– Consider two short-range-interacting
even in the case of the infinitely strong interaction be- atoms in a circular, transversally harmonic waveguide.
tween the atoms. We attribute this effect to the short- In this case the center-of-mass and the relative motion
range nature of the interaction. canbeseparated. Notethattheharmonicapproximation
For our system we study both the degree of the eigen- for the transverse confinement has been proven to model
state thermalization and the actual thermalization in an reallifeatomwaveguideswithhighaccuracy. Predictions
expansion from a class of realistic initial states. The of the analogous model involving an infinite waveguide
eigenstate thermalization [7–9] – the suppression of the [16,17]hasbeenconfirmedbyadirectexperiment-theory
eigenstate-by-eigenstate variance of quantum expecta- comparison of the energies of the quasi-one-dimensional
tion values of simple observables – provides an ultimate dimers [18] and numerical calculations involving anhar-
2
monic potentials (see review [5] for references). [24]). Higher partial wave scatterers were analyzed in
The unperturbed Hamiltonian of the relative motion the Ref. [25].
is given by the sum of the longitudinal and transverse At rational values of λ/π2, the unperturbed energy
kinetic energies and the transverse trapping energy Uˆ = spectrum shows degeneracies that are not fully lifted in
µω2ρ2/2, the full [deduced from (4)] spectrum. To minimize the
⊥
effect of the degeneracies, we, following Ref. [11], fix the
Hˆ0 =−2¯hµ2 ∂∂z22 − 2¯hµ2∆ρ+Uˆ −¯hω⊥ , (1) lwenhgertehφof t=he(1gu+id√e5t)o/2(Lis/ath⊥e)2go≡ldeλn=rat2iφog.rπ7 ≈ 9774,
gr
Standard quantum-chaotic tests.– The results of the
whereω⊥isthetransversefrequency,∆ρisthetransverse
study of the level spacing distribution in our system are
two-dimensional Laplacian, and µ is the reduced mass.
fully consistent with the analogous results for the Sˇeba
In our model, the transverse and longitudinal degrees of
freedomarecoupledbyapotentialofaFermi-Huangtype billiard [12, 13]. For the energy range E > 100h¯ω⊥
[5]: and the aspect ratio (L/a⊥)2 = 2φgrπ7, th∼e distribu-
tion quickly converges to the Sˇeba distribution [12] at
Vˆ = 2π¯hµ2asδ3(r)(∂/∂r)(r·), (2) asms a>∼ll l1ev0e−l1sap⊥a.cinTghsibsudtisftarilisbuttoiornepdroodesucsehtohwe aGaguaspsiaant
tailpredictedbytheGaussianOrthogonalEnsemble. At
were, as is the three-dimensional s-wave scattering small as the level spacing distribution tends to the Pois-
length. This approximation is valid at low collision en- sonone. Wehavealsoverifiedthatintheunitaryregime,
ergies (e.g., below 300 mK for Li, see a recent review as a⊥, the statistics of the point-by-point variations
≫
[19], while the ultracold atom energies lie well below of the eigenstate wavefunction in the spatial representa-
1 µK). The model (2) stands in a excellent agreement tionisclosetotheGaussianone,accordingtothegeneral
with the experimentally measured equation of state of prediction [26] and a particular observation in the case
the waveguide-trapped Bose gas [20] and with numerical of the Sˇeba billiard [11].
calculations with finite-range potentials [17, 21]. Eigenstate thermalization.– According to the eigen-
The ring geometry of the waveguide imposes period- state thermalization hypothesis [7–9], the ability of an
L boundary conditions along z-direction. In what fol- isolated quantum system (with few or many degrees of
lows, we will restrict the Hilbert space to the states that freedom regardless) to thermalize follows from the sup-
have zero z-component of the angular momentum and pression of the eigenstate-by-eigenstate fluctuations of
that are even under the z z reflection; the inter- the quantum expectation values of relevant observables.
↔ −
action has no effect on the rest of the Hilbert space. Scattered dots at the Fig. 1 show the quantum expec-
The unperturbed spectrum is given by (see [22]) Enl = tation values of the transverse trapping energy, αUˆ α
2h¯ω⊥n+¯h2(2πl/L)2/(2µ), where n 0 and l 0 are as a function of the eigenstate energy. The vhari|ati|oni
≥ ≥
thetransverseandlongitudialquantumnumbers,respec- of αUˆ α remains (in contrast to the quantum-chaotic
h | | i
tively. billiards [27]) of the order of the mean, even for the en-
The interacting eigenfunctions with the eigenenergy ergies much larger than any conceivable energy scale of
Eα can be expressed as (see [22]) the system. Here and below, we work in the regime of
∞ cos 2√ǫ λnζ the infinitely strong interactions, as =106a⊥.
hρ,z|αi=Cαn=0√ǫα−(cid:0)λnsiαn−√ǫα−(cid:1)λne−ξ2L(n0)(ξ)(,3) ofFtohre aocgciuvpeantioeingepnrsotbataeb,iliqtuieasntfourmtheexptercatnastvieornseva(Fluiegs.
X
2a) and longitudinal (Fig. 2b) modes show no conver-
where the rescaled energy ǫ is given by ǫ
zλ/ELα/(12/h¯2ω,⊥ξ), λ(ρ≡/a⊥()L2/,a⊥L)(n20)(iξsα)tahree tahsepeLcetgernatdiroe,αpζoly≡≡- gBedonltchδe(dEtoinstltrh−iebiEurtm)ioaincnsrdoacPraen(loh)ne∝iacvaillydenxdpδoe(mcEtinanltai−toendEv)baylrueeossnp,leyPct(ainvf)eel∝yw.
nom−ials, a⊥ =≡(h¯/µω⊥)1/2 is the size of the transverse Rpeaks(cf. scarsinthemomeRntumspaceforSˇebabilliards
groundstate, andCα isthenormalizationconstant. The [15]).
eigenenergies are solutions of the following transcenden- Relaxation from an initial state.– Now we are going
tal equation (see [22]): to address directly the ability of our system to ther-
malize from an initial state ψ(t = 0) . In this case,
√λ ∞ cot√ǫα−λn+i ζ 1, ǫα = a⊥ , (4) the infinite time average of |the quantium expectation
√ǫ λn − 2 − λ a value of an observable Aˆ will be given by A
nX=0 α− (cid:18) (cid:19) s limtmax→∞(1/tmax) 0tmaxhψ(t)|Aˆ|ψ(t)i = α|hαre|lψax(.t =≡
where ζ(ν,x) is the Hurwitz ζ-function (see [5]). Similar 0)2 αAˆα , where α are the eigenstates of the system
| h | | i |Ri P
solutions were obtained for two atoms with a zero-range (see [8]). For a quantum-chaotic system, it is expected
interaction inacylindrically-symmetric harmonic poten- that the above infinite time average coincides with the
tial [23] (that system was analysed numerically in Ref.
3
m.
her 1.4 l (n) 1.2 (a) initial
Ut P
/x. 1.2 l ty, 1 eigerenlastxaetde
ela 1 n bili 0.8 microcanon. x 100
Ur n ba
y, 0.8 l ro 0.6 l
g p
ener 00..46 * l n ation 0.4 n
g p 0.2
pin 0.2 n cu
ap 0 Oc 0
r 0 20 40 60 80 100
T 100 200 300 400 500 600
Energy, E//hω Transverse mode, n
perp
FIG. 1. The time average of the transverse trapping energy
Urelax. ≡limtmax→∞(1/tmax) 0tmaxhψ(t)|Uˆ|ψ(t)i after relax- ) 0.4
ation from an initial state as a function of the energy of the (l (b) initial
P
initialstateE [seeHamiltoniaRn(1)]. Fourfamiliesoftheini- 0.35 relaxed
,
wtiaitlhstna0te=sa0re, cδo=nsi0d.e9r9e,da.nLdontgh-edaensheergdy(gbreeienng)clionnet:rostllaetdev(5ia) bility 0.3 microcaenigoenn.s xta 1te0
scanning l0; short-dashed (blue) line: the same as the previ- a 0.25
b
ous one, except for δ=.1; dotted (purple) line: the state (5) o
with δ = 0.1 and l0 = 0 and the energy control via n0; solid pr 0.2 l
(red) line: state (6) with κ2 =2κ1, δ =0.99, l0 =0 and the n 0.15 beamsplitter
o
energycontrolviaκ1. Legendsschematicallyshowthedistri- ati 0.1 n
butionoftheinitialstatesoverthequantumnumbers;arrows p
indicatethedirectioninwhichtheparameterscontrollingthe u 0.05
c
energy are scanned. The asterisk (*) corresponds to the set c
O 0
of parameters used to produce the Fig. 2. Grey dots show
0 50 100 150 200 250 300 350
the quantum expectation value of the trapping energy U for
every hundreds eigenstate of the full Hamiltonian (1,2). All
Longitudinal mode, l
the values of the trapping energy are shown with respect to
its microcanonical expectation value U ≈ E/3. For all
therm.
the points, (L/a⊥)2 =2φgrπ7 ≈9774 and as =106a⊥. FIG. 2. Distribution of the mean occupation numbers for
the transverse (a) and longitudinal (b) modes of the unper-
turbedHamiltonian. Thelong-dashed(green)andsolid(red)
lines show the initial and time-averaged distributions for an
thermal prediction.
initial state (5) with n0 = 0, δ = 0.99, and l0 = 292.8. En-
Consider the following initial state:
ergyoftheinitialstateisE ≡(2/λ)¯hω⊥ǫα =173.2h¯ω⊥. The
short-dashed(blue)linecorrespondstothe17096’seigenstate
πζ δ
(closest in energy to E from the above, with a relative dis-
z ψ(t=0) cos θ ζ cos(2πl ζ) n . (5)
h | i∝ δ 2 −| | 0 | 0i crepancyof10−5)oftheinteractingHamiltonian. Thedotted
(cid:18) (cid:19)
(purple) line shows the microcanonical prediction. The rest
Longitudinally, the state is represented by the ground of the parameters is the same as at the Fig. 1.
state of a length Lδ hard-wall box split initially by an
ideal beamsplitter with momenta 2πl /L; the box is
0
centered at the maximal interatomi±c distance. The state the thermal prediction, thus being seemingly correlated
is distributed among approximately π/δ axial modes lo- with its initial, negative value. Note that the correla-
calized about l = l . The initial transverse state is tion between the final and initial values of observables
0
limited to a single m±ode n . Note that for n = 0 and in quantum systems is addressed in Ref. [28]. Also, note
0 0
δ 1 the state is conceptually similar to the initial state thattheinitialstatesofthesecondsequencehaveamuch
us≈ed in the equilibration experiments [1]. greater spreadover the longitudinal modes than the first
After relaxation, both transverse (Fig. 2a) and longi- one; consistently, the energy-to-energy variation of the
tudinal (Fig. 2b) distributions remain very far from the relaxedvaluesofU islessthanforthefirstsequence. The
equilibrium predictions. Fig. 1 shows the equilibrium third sequence, with low longitudial energy, shows devi-
value of the transverse trapping energy U (see (1)) for ations from the thermal prediction that range between
three sequences of the initial states of the type (5). The +50% and +5% but never reach zero.
first two, with low transverse energy, show an approx-
imately -40% deviation of the relaxed value of U from
4
Another type of the initial state We are grateful to Felix Werner for the enlightening
discussions on the subject. This work was supported by
πζ δ
ρ, z ψ(t=0) cos θ ζ cos(2πl ζ) a grant from the Office of Naval Research (N00014-09-1-
0
h | i∝ δ (cid:18)2 −| |(cid:19) 0502).
e−κ1ξ e−κ2ξ (6)
× −
allowsfor(cid:0)anadditionals(cid:1)preadoverthetransversemodes.
The transverse wavefunction vanishes at the waveguide
axis. Themodeoccupationhasaminimumatn=0,and [1] T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440,
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both transverse and longitudinal modes, the deviations ture Physics 4, 489 (2008).
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Supplementary material to: Restricted Thermalization for Two Interacting Atoms in
a Multimode Harmonic Waveguide
V. A. Yurovsky
School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel
M. Olshanii
Department of Physics, University of Massachusetts Boston, Boston MA 02125, USA
(Dated: January 1, 2010)
0 This supplementary material contains a detailed derivation of the eigenenergies and eigenstates
1 forasystemconsistingoftwoatomsinacircular,transverselyharmonicwaveguideinthemultimode
0 regime.
2
PACSnumbers: 67.85.-d,37.10.Gh,05.45.Mt,05.70.Ln
n
a
J
The waveguide system [1], as well as the Sˇeba billiard Fermi-Huang interaction (see Eq. (2) in [1]) can be writ-
1
[2], belongs to the class of problems where an integrable tenintheseparableform[3]withtheinteractionstrength

] system of Hamiltonian Hˆ0 is perturbed by a separable and formfactors given by
h rank I interaction V (see e.g. [3]). Eigenstates α
-p oftheinteractingsys|LteimhRa|resolutionsoftheSchr¨oding|eri V = 2π¯h2as, =δ (r), =δ (r) ∂ (r ). (5)
3 3
m equation µ |Li hR| ∂r ·
o
t [Hˆ0+V ]α =Eα α , (1) In what follows, we will restrict the Hilbert space to
a |LihR|| i | i the states of zero z-component of the angular mo-
.
s where E are the corresponding eigenenergies. Let us mentum and even under the z z reflection; the
c α ↔ −
expandthe eigenstate α overthe eigenstates ~n (of en- interaction has no effect on the rest of the Hilbert
i
s ergy E ) of the non-i|ntieracting hamiltonian|Hˆi. The space. The non-interacting eigenstates are products
y ~n 0
Schr¨odingerequationleadstothefollowingequationsfor of the transverse and longitudial wavefunctions. The
h
p the expansion coefficients transversetwo-dimensionalzero-angular-momentumhar-
[ monic wavefunctions,

(E E ) ~nα = ~n α ~n , (2)
1 α− ~n h | i h |LihR| i∝h |Li 1
v ρn = L(0)(ξ)exp( ξ/2), (6)
5 where the omitted factor in the second equality is inde- h | i √πa⊥ n −
2 pendent of ~n.
labeled by the quantum number n 0, are expressed
2 As a result, any eigenfunction α functionally co- ≥
0 | i in terms of the Legendre polynomials, L(0)(ξ), where
incides with the Green function of the non-interacting n
1. Hamiltonian taken at the energy of the eigenstate E , ξ = (ρ/a⊥)2 and a⊥ = (h¯/µω⊥)1/2 is the transverse os-
α
0 cillator range. The longitudial wavefunctions, labeled by
0 ~n ~n l 0, are the symmetric plane waves satisfying periodic
1 α | ih |Li. (3) ≥
| i∝ E E conditions with the period L:
: α ~n
v X~n −
i z k =(2/L)1/2cos2πkζ, z 0 =L−1/2, (7)
X The omitted factor is determined by the normalization
h | i h | i
r conditions.
a where ζ z/L 1/2. The unperturbed spectrum is
Substitution of this expression to the Schr¨odinger ≡ −
therefore given by
equation (1) leads to the following eigenenergy equation:
~n ~n 1 Enl =2h¯ω⊥n+h¯2(2πl/L)2/(2µ) (8)
hR| ih |Li = . (4)
E E V
α ~n Substitutingtheabovenon-interactingeigenstatesand
~n −
X
eigenenergies to Eq. (3) and using Eq. (1.445.8) in [8]
Similar expressions were obtained in Refs. [2, 4] for the for summation over l, one obtains Eq. (3) in [1] for the
case of the Sˇeba billiard and its generalizations. interacting eigenstates. The normalization factor C is
α
In the case of the relative motion of two short-range- determined by the condition
interacting atoms in a circular, transversely harmonic
waveguide, the derivation uses ideas of the analogous L ∞
model involving an infinite waveguide [5–7]. The unper- 2π dz ρ2dρ α′ ρ,z ρ,z α =δαα′. (9)
h | ih | i
turbedHamiltonianhereisgivenbyEq.(1)in[1]andthe Z Z
0 0
2
Using orthogonality of the Legendre polynomials it can 1 1 ǫ
= ζ , (13)
be expressed as −√λ 2 −λ
(cid:18) (cid:19)
∞
2 cot√ǫ λn
α
C = −
α a⊥√πL(cid:20)n=0(cid:18)(ǫα−λn)3/2
X (see [6]) where ζ(ν,α) is the Hurwitz zeta function (see,
1 −1/2
+ , (10) e. g.,[7,8]. Finallywearriveatthetranscendentalequa-
(ǫα−λn)sin2√ǫα−λn(cid:19)(cid:21) tion for the eigenenergies (4) in [1]. The summands in
the sums Eqs. (3) and (4) in [1] and in Eq. (10) decay
where the rescaled energy ǫ is given by ǫ
λEα/(2h¯ω⊥) and λ≡(L/a⊥)2 iαs the aspect ratio.α ≡ erixeps.oneNnottiaelltyhawtitthhen,imleaagdiinnagrytoptahretsfaosfttchoenvtewrogintgermses-
FortheFermi-Huanginteractiontheeigenenergyequa-
in the left hand side of Eq. (4) in [1] cancel each other
tion (4) attains the form
automatically.
∂ ρn z l n0 00 1
r h | ih | ih | ih | i = (11)
∂r  E E  V
nl
n,l −
X r=0
  [1] V. A. Yurovsky and M. Olshanii, Restricted Thermaliza-
In the limit r 0, the eigenstate is spherically symmet-
→ tionforTwoInteractingAtomsinaMultimodeHarmonic
ric. This allows us to deal with the z-axis only (see [6]). Waveguide.
Substitutionofthenoninteractingwavefunctions(6)and [2] P. Sˇeba, Phys. Rev. Lett. 64, 1855 (1990).
(7)andenergies(8)withthesubsequentsummationover [3] S. Albeverio and P. Kurasov., Singular perturbations of
l leads to differential operators : solvable Schr¨oinger type operators,
no.271inLondonMathematicalSocietylecturenoteseries
∂ ∞ cos 2√ǫ λn(z/L 1/2) a (University Press, Cambridge, 2000).
√λ z α− − = s. [4] S. Albeverio and P. Sˇeba, J. Stat. Phys. 64, 369 (1991).
∂z " nX=0 √(cid:0)ǫα−λnsin√ǫα−λn (cid:1)#z=0 a⊥ [5] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998).
(12) [6] M.G.Moore,T.Bergeman,andM.Olshanii,J.Phys.IV
Thissumcontainsboththeregularpartandtheirregular (France) 116, 69 (2004).
one, the latter being proportional to z−1. The regular [7] V. A. Yurovsky, M. Olshanii, and D. S. Weiss, in Adv.
part can be extracted using the identity At. Mol. Opt. Phys. (Elsvier Academic Press, New York,
2008), vol. 55, pp. 61–138.
∞ [8] I.S.GradshteynandI.M.Ryzhik,Table of Integrals, Se-
L
lim (λn ǫ)−1/2exp( 2√λn ǫz/L) ries, and Products (Academic Press, New York, 1994).
z→0" − − − − λz#
n=0
X