Inelastic chaotic scattering on a Bose-Einstein condensate
Stefan Hunn, Moritz Hiller, Andreas Buchleitner, Doron Cohen, Tsampikos Kottos
IInelastic chaotic scattering on a Bose-Einsteincondensate
Stefan Hunn , Moritz Hiller , , Doron Cohen , TsampikosKottos , and Andreas Buchleitner Physikalisches Institut, Albert-Ludwigs-Universit¨at Freiburg, Hermann-Herder-Str.3, 79104 Freiburg, Germany Institute for Theoretical Physics, Vienna University of Technology, WiednerHauptstraße 8-10/136, A-1040 Vienna, Austria Department of Physics, Ben-Gurion University of the Negev, Beer-Sheva 84105,Israel Department of Physics, Wesleyan University, Middletown, Connecticut 06459, USAand MPI for Dynamics and Self-Organization, Am Faßberg 17, D-37077 G¨ottingen,Germany
Abstract.
We devise a microscopic scattering approach to probe the excitationspectrum of a Bose-Einstein condensate. We show that the experimentally accessiblescattering cross section exhibits universal Ericson fluctuations, with characteristicproperties intimately related to the underlying classical field equations. a r X i v : . [ qu a n t - ph ] M a r nelastic chaotic scattering on a Bose-Einstein condensate
1. Introduction
Bose-Einstein condensates (BEC) in optical lattices provide a versatile tool to addressexperimentally a variety of questions that emerge in diverse fields ranging fromquantum information and many-body quantum phase transitions to solid-state transportand atomtronics. An important element of these studies is the development andimplementation of methods which allow for an accurate measurement of the properties ofthe condensate. Among the most popular ones are time-of-flight and Bragg-spectroscopy[1, 2, 3] techniques which result in the destruction of the BEC, whereas only few worksconsider a scattering setup that leaves the condensate intact [4]. Specifically, the mainfocus of the present literature is on photon-atom scattering, while only very recentlymatter-wave scattering was experimentally implemented to infer the spatial ordering ofultracold atomic crystals [5], and theoretically proposed to probe the Mott insulatorto superfluid transition of the condensate ground state [6]. However, excited states arenaturally populated in experiments which probe non-trivial BEC dynamics [7]. Therapidly emerging complexity of the many-body dynamics – manifest, e.g., in dynamicalinstabilities [3] – is a direct fingerprint of the complex underlying spectral structure,which is itself reflected in the – in general chaotic – classical limit of the Bose-Hubbardmodel (achieved in the limit of large particle numbers). It is therefore timely to explorepossible experimental strategies to probe these spectral features, in a non-destructivemanner. In our present contribution, we show how an inelastically scattered probeparticle can reveal the state of a BEC target in the parameter regime of spectral chaos.Due to the inherent sensitivity of spectral cross sections under such conditions, a robustcharacterization requires a statistical approach, which can be further sharpened bysemiclassical considerations.
2. Model
The scattering setup that we have in mind is shown in Fig. 1: A probe particle withmomentum k moves in a waveguide which is placed in the proximity of a BEC confinedby an optical lattice. When the particle approaches the condensate, it interacts with thelatter – much as the condensate particles between themselves – leading to an exchangeof energy. The particle energy on exit from the waveguide defines the scattering crosssection. The dynamics of the process is generated by H tot = H TB ⊗ ˆ1 + ˆ1 ⊗ H BH + H int , (1)where ˆ1 denotes the identity operator. In (1), H BH represents the BEC target’s Bose-Hubbard-Hamiltonian [8] H BH = U L (cid:88) i =1 ˆ n i (ˆ n i − − K (cid:88) i (cid:104) ˆ b † i ˆ b i +1 + h . c . (cid:105) (2)of N interacting bosons on an L -site optical lattice, with ˆ b ( † ) i the bosonic annihilation(creation) operators, and ˆ n i = ˆ b † i ˆ b i the particle number at site i . U and K parameterize nelastic chaotic scattering on a Bose-Einstein condensate N → ∞ ( U N fixed), the dynamics of the condensate is well describedby mean-field theory, i.e. the discrete Gross-Pitaevskii equation. In this limit, thequantum operators b ( † ) i are replaced by L complex amplitudes A ( ∗ ) i of a single-particlefield. The Hamiltonian (2) then reads H GP N = U N L (cid:88) i =1 | A i | − K (cid:88) i [ A ∗ i A i +1 + c . c . ] , (3)where the A i are time-dependent, and obey the canonical equations i∂A i /∂t = ∂ H GP /∂A ∗ i .The waveguide in our scattering scheme is (for mathematical convenience) modeledby two semi-infinite tight-binding (TB) leads with hopping term J and lattice spacing a = 1. The particle’s energy in the momentum eigenstate | k m (cid:105) of each lead is (cid:15) m = − J cos( k m ), with corresponding velocity v m = 2 J sin( k m ) [9]. These two leadsare coupled with strength J to the central site j = 0, which is closest to the BEC. J thus controls the effective coupling of the projectile-target interaction region to theasymptotically free states of the lead. The projectile Hamiltonian is: H TB = (cid:104) − J (cid:88) j (cid:54) = − , ˆ c j ˆ c † j +1 − J (cid:88) j = − , ˆ c j ˆ c † j +1 (cid:105) + h . c . , (4)with ˆ c ( † ) j the annihilation (creation) operators of the probe particle at site j of the TBlead.Finally, the probe-target interaction H int is assumed to be of similar type (i.e. shortrange) as the bosonic inter-particle interaction in the condensate: H int = α ˆ c † ˆ c ⊗ ˆ n . (5)For non-vanishing tunneling coupling K , H int induces transitions between differenteigenmodes of the condensate, what renders the scattering process inelastic. In thissense, α > H int /N = α | A | c † c .
3. Scattering matrix
Given the total Hamiltonian (1) and the asymptotic freedom of the probe particle,we can now define the scattering matrix of our problem, as the fundamental buildingblock for our subsequent observations: For the BEC initially prepared in an energyeigenstate | E m (cid:105) , and the probe particle injected with an energy (cid:15) m , the total systemenergy is E = E m + (cid:15) m . The open channels (modes) of the scattering process are thendetermined by energy conservation and characterized by the kinetic energy (cid:15) n = E − E n of the outgoing probe particle. The transmission block of the scattering matrix can be nelastic chaotic scattering on a Bose-Einstein condensate Figure 1. (color online). Scattering setup: The probe particle is injected intoa wave guide, and locally exchanges energy with a BEC confined by a three-siteoptical potential, in the contact region between wave guide and site one of the lattice.The inelastic scattering cross section measured on the wave guide’s exit bears directinformation on the state of the BEC. derived from the Green’s function of a particle at site j = 0, with two semi-infinite leadsattached, and reads ‡ :[ ˆ S T ]( E ) = √ ˆ v i γ (1 − γ )[ E − ˆ H BH ] − α ˆ n + iγ ˆ v √ ˆ v , (6)where γ ≡ ( J /J ) , and ˆ v is the velocity operator. In the eigenbasis of the BHHamiltonian, [ H BH ] nm = E n δ nm and v nm = v n δ nm are diagonal matrices, while Q nm ≡ (cid:104) E n | ˆ n | E m (cid:105) is not. For γ = 1, Eq. (6) coincides with the S -matrix for inelasticelectronic scattering derived in [10]. In our setup, γ < γ = 0 the latter is isolated and the probe particle is perfectly reflected). As γ is increasedfrom zero, one observes a crossover from a regime of well-resolved, narrow resonances toa regime of overlapping resonances. Whereas the former regime would in principle allowus to directly infer the BH spectrum, it puts very hard demands on the experimentalprecision as far as preparation and measurement are concerned. Instead, we find thatin the latter regime one obtains an experimentally robust, fluctuating scattering signalthat bears information on the target’s spectrum (around γ = 0 . γ becomes very large compared to the mean resonance spacing such that ‡ The derivation of this formula in the present work is a tight-binding generalization of the continuumlimit that has been analyzed in [10]. Details on the derivation can be found in [11]. nelastic chaotic scattering on a Bose-Einstein condensate T E Figure 2. (color online). Transmission T m , averaged over the channels m = 400 − E in units of K , for γ = 0 . N = 38bosons, and α = 5 . K . these characteristic fluctuations are washed out.
4. Chaotic scattering
In our present contribution, we will investigate the properties of a probe particlescattering on a BEC that is described by the Hamiltonian (2). In contrast toRef. [6], we are not concerned with the well-understood ground-state properties andthe associated superfluid to Mott-insulator transition, which takes place at rather largeinteraction strengths U . Instead, we focus on the complex properties of excited statesat intermediate interaction strengths.Namely, for L > (cid:46) u (cid:46)
12) of the control parameter u = U N/ K ,the classical Hamiltonian H GP (3) generates chaotic dynamics [12, 13, 14, 15, 16].Quantum manifestations thereof were investigated in a series of publications withemphasis on the statistical properties of the energy spectra [17, 18, 19]. For a two-site lattice ( L = 2) [20], inelastic scattering revealed immediate fingerprints of the fullyintegrable mean-field dynamics. Here, we consider a three-site BH Hamiltonian ( L = 3),intermediate values of the control parameter around u = 5, what corresponds to themaximally chaotic regime [12], large filling factors of the lattice (i.e. N = 38 bosons),and a condensate that is initially prepared in an energy eigenstate | E m (cid:105) in the bulk ofthe BH spectrum, where the target dynamics is predominantly chaotic. The presence ofchaos qualitatively alters the resulting physics: In stark contrast to the integrable case nelastic chaotic scattering on a Bose-Einstein condensate Figure 3. (color online).
Left:
Logarithmically color-coded snapshot of the underlyinginteraction matrix Q nm , for u = 5. Right:
Distribution P ( ˜ Q nm ) of the unfolded off-diagonal interaction matrix elements ˜ Q nm taken from a sub-matrix in the bulk of Q nm (indicated by the white box in the left panel; the corresponding indices run from n = m = 360 to n = m = 500). The distribution P ( ˜ Q nm ) nicely resembles the dashedred line which represents the Gaussian distribution with unit variance N (0 , L = 2, the scattering quantities will strongly fluctuate (depending, e.g., on the probeparticle’s energy), what requires a statistical analysis of the scattering signal. On theother hand, chaos is expected to yield universal behavior, i.e., the obtained results shouldnot depend on the details of the target system. As for the other scattering parameters,we set J = E max − E min = 211 K , equal to the (numerically inferred) spectral width ofthe BH Hamiltonian, such that all scattering channels are open, and fix γ = 0 . T m ( E ) = (cid:80) n | [ S T ] nm | . It denotes the probability that a probe particlewith incoming energy (cid:15) m exits the scattering area in any of the outgoing channels (cid:15) n .In Fig. 2, we show T m versus the total energy E , averaged over 30 incoming channels (cid:15) m in the middle of the spectrum, i.e. in the chaotic regime, where approximately 180outgoing channels contribute to T m :Strong fluctuations dominate the transmission signal – a first indicator for thecomplexity of the target. In order to understand how this complexity enters in thescattering process, we turn to the scattering matrix (6). In the {| E n (cid:105)} basis, in which thelatter is evaluated, the interaction operator ˆ n becomes the only non-diagonal quantityon the r.h.s. of (6), and thus gives rise to inelastic scattering processes. A closerinspection of the corresponding matrix Q nm ≡ (cid:104) E n | ˆ n | E m (cid:105) (see Fig. 3a)) shows thatfor intermediate energies corresponding to the center of the matrix (i.e., in the chaoticenergy regime) the matrix is banded and its off-diagonal elements look rather erraticallydistributed. We analyze these elements, taken from the box indicated in Fig. 3a),and consider their distribution P ( ˜ Q nm ). Here, ˜ Q nm = Q nm / (cid:112) (cid:104)| Q nm | (cid:105) represent the nelastic chaotic scattering on a Bose-Einstein condensate -2 -1 α -6 -5 -4 -3 σ n m σ∼ nm -3 -2 -1 P ( σ ∼ n m ) a) b) c) Figure 4. (color online).
Left:
Integrated total inelastic cross section (cid:82) d E(cid:104) ρ m in ( E ) (cid:105) ,for the same parameters as in Fig. 2, averaged over the channels m = 400 −
430 inthe chaotic regime, versus the inelasticity parameter α in units of K . The integrationruns over the entire energy axis. Middle:
A representative inelastic cross section σ nm versus the total energy E in units of J , for the same parameters and α = 5 . K . Right:
Histogram of the normalized inelastic cross section P (˜ σ nm ), for fifteen differentchannels σ m ( m = 401 − off-diagonal matrix elements which are rescaled by their local standard deviation, thelatter being calculated in a small, moving energy window. This “unfolding” is necessaryto remove system-specific features and reveal universal properties of the interaction-operator ˆ n (see, e.g. [12]). The resulting distribution is shown in Fig.3b) togetherwith a Gaussian of unit variance. The good agreement with the latter indicates that(neglecting higher-order correlations) the matrix elements are essentially independentrandom numbers, in perfect agreement with the predictions for quantum systems thatpossess a chaotic classical counterpart [21, 22, 23]. We conclude that S T and thus allscattering quantities inherit their complexity from Q , since the latter represents the keyingredient in the scattering matrix.To gain insight in the role of the parameter α that controls the inelasticity inducedby Q , we next consider the total inelastic scattering cross section ρ m in ( E ) = 2 (cid:88) n (cid:54) = m | [ S T ( E )] nm | , (7)which essentially resembles T m , except for the direct processes. For a given value of α ,we integrate over the energy axis, to obtain robust results, unaffected by the sensitiveenergy dependence of ρ m in ( E ). Fig. 4a) shows that (cid:82) d E (cid:104) ρ m in ( E ) (cid:105) m takes its maximal valuefor intermediate values of the inelasticity parameter α , while it vanishes in the limit ofsmall and large α . In the former case, the probe particle is directly transmitted, since(6) with α ≈ α (cid:29) K in the denominator. Consequently, only for intermediate α -values around α ∼ K can we expect to infer information on the condensate from the nelastic chaotic scattering on a Bose-Einstein condensate
5. Ericson fluctuations
Beyond total cross sections there is nontrivial dynamical information encoded in the partial inelastic cross sections σ nm ( E ) ≡ | [ S T ( E )] nm | , which quantify the probabilityfor a transition from a state E m to a state E n of the target (or, equivalently, from anenergy (cid:15) m to an energy (cid:15) n of the projectile). In Fig. 4b) we show σ nm ( E ), for the sameparameter values as the transmission in Fig. 2. We observe much stronger fluctuationsthan for the total transmission, what is simply due to the fact that the latter impliesan additional effective averaging over many scattering channels. As we will show now,this sensitive dependence on the energy is an unambiguous trait of (universal) Ericsonfluctuations, that were hitherto only reported in the context of nuclear [24] and atomicphysics [25], as well as in microwave experiments [26], and are also connected to universalconductance fluctuations in mesoscopic physics [27, 28].The rapid fluctuations of the cross section are due to interference effects betweenoverlapping resonances: The scattering amplitudes [ S T ] nm can be represented by amany-resonance Breit-Wigner formula, where each individual term in the sum is arandom variable whose stochastic properties originate from the statistically independentGaussian distributed elements of the interaction matrix. Then, due to the centrallimit theorem, one expects that real and imaginary part of the scattering matrixelements are Gaussian distributed random numbers with zero mean. In other words,the Gaussian distribution of the interaction matrix elements gives rise to Gaussiandistributed real and imaginary part of [ S T ] nm . This results in an exponential distribution[24] P (˜ σ nm ) = exp[ − ˜ σ nm ] of the normalized inelastic cross section ˜ σ nm = σ nm / ¯ σ nm ,where ¯ σ nm denotes the average inelastic cross section in the energy interval ∆ E (that issmall compared to classical energy scales). This expectation is clearly confirmed by ournumerical data presented in Fig. 4c).The central figure of merit to identify Ericson fluctuations is the energyautocorrelation function C nm ( ε ) = (cid:90) ∆ E d E ( σ nm ( E + ε ) − ¯ σ nm )( σ nm ( E ) − ¯ σ nm ) . (8)A least-square fit of the numerically obtained autocorrelation as depicted in Fig. 5 showsthat it perfectly matches a Lorentzian C nm ( ε ) ∝ Γ ε + Γ , (9)with mean resonance width Γ = 3 . · − J § , which is one order of magnitude larger thanthe mean level spacing ∆ ≈ · − J , directly extracted from our numerical data. Thisis in perfect agreement with Ericson’s scenario of overlapping resonances, and can beunderpinned by a semiclassical picture [29]: The autocorrelation (9) can be interpreted § We verified that a preparation in the regular regime (i.e., u (cid:28) u (cid:29) C nm ( ε ) shows significantdeviations from a Lorentzian. nelastic chaotic scattering on a Bose-Einstein condensate ε C n m ( ε ) / C n m ( ) Γ β Figure 5. (color online). Autocorrelation function C nm ( (cid:15) ) (black ◦ ) calculated fromthe inelastic scattering signal shown in Fig. 4b). The curve nicely matches a Lorentzianfit (blue). Inset:
Semiclassical decay constant β versus the mean resonance width Γ(black (cid:3) ). The data points correspond to different values of the coupling constant γ . The semiclassical result is obtained after averaging over several initial conditionsin the chaotic regime. The predicted correspondence ( β = Γ) is confirmed by the fit β = 0 .
92 Γ (red). as the squared Fourier transform of the survival probability P ( t ) of the probe particleto stay a given time t in the scattering region, i.e. on the TB site j = 0, hence with P ( t ) = | c ( t ) | . That latter quantity is evaluated by direct solution of the classicalevolution equations derived from (3, 5) (with initial conditions P (0) = 1 and the GPsystem prepared at an energy corresponding to E m ), and exhibits an exponential decay P ( t ) ∝ e − βt . β thus determines the width of the (classical) autocorrelation function C clas ( ε ), by virtue of C clas ( ε ) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) dt P ( t ) exp( iεt ) (cid:12)(cid:12)(cid:12)(cid:12) , (10)which implies an average over all outgoing probe energies (cid:15) n . The inset of Fig. 5demonstrates a perfect match of this semiclassically extracted quantity β with the widthΓ of the autocorrelation extracted from the quantum mechanical cross section, and thusprovides an independent, semiclassical proof of the Ericson scenario in the present many-particle problem. nelastic chaotic scattering on a Bose-Einstein condensate
6. Conclusions
In the light of recent BEC experiments, the proposed scattering setup represents anexperimentally feasible and robust way to non-destructively probe the condensate. Forexample, a typical lattice depth of V = 10 E R recoil energies yields a tunneling strength K = 0 . E R (cf. Eq.(20) of [3]). The exemplary case studied above ( u = 5, N = 38)leads to a mean level spacing of ∆ = 2 . · − E R . This, in turn, corresponds to anenergy resolution of ∆ /E ≈ − needed to resolve single scattering channels, i.e. thefinal state of the system. As chaotic spectral statistics extend over a large parameterregime [19, 30] one could also choose larger lattices ( L >
3) and/or fewer bosons toincrease ∆, and still obtain comparable results. Let us also stress that, in the proposedmeasurement, it is not necessary to determine the boson number on the lattice: As longas the detector can resolve the energy of the probe particle, the target properties can beinferred from the scattering signal, without knowledge of the exact number of bosons inthe system.As expected, the observed chaotic scattering signals exhibit an inherent sensitivityon, e.g., the probe particle’s energy. To obtain a reliable characterization of the target,we employed a statistical analysis based mainly on the autocorrelation function C ( ε )and the probability distribution P ( σ ij ) of the inelastic cross section. On the otherhand, the robustness of the obtained results against unavoidable fluctuations in theexperimental control parameters is corroborated by averaging our results over ∆ m = 30channels (see, e.g., Fig. 4). This averaging, together with the statistical independenceof the S -matrix elements, implies that our observations are robust against uncertainties,e.g., in the initially prepared state | E m (cid:105) of the condensate or in the particle number N ,as well as in the resolution of the scattering channels. For a typical laser wavelengthof 1064 n m, N = 38 Cs atoms, and u = 5, this average corresponds to a residualtemperature of ≈ n K what is readily achievable with state-of-the-art experimentsthat reach temperatures as low as 0 . n K [31].Hence, measurements of the partial inelastic cross section can identify anunambiguous case of Ericson fluctuations, what yields robust information on the many-body spectrum. In contrast to compound nuclear reactions, here the latter is underperfect control, through the accurate experimental manipulation of the underlying Bose-Hubbard Hamiltonian, via the control parameter u . This would allow to experimentallyinvestigate the fate of the Ericson fluctuations at the transition from the chaotic to theregular regime. By virtue of this control, our proposed setup could also add to therecent debate on complex many-body scattering [32, 33, 34]. Finally, the possibility torecord single scattering channels could help the understanding of a related phenomenon:Summing up an increasing number of contributing channels would then allow to studyin a controlled way the crossover from the Ericson regime to the multi-channel regimeof universal conductance fluctuations. nelastic chaotic scattering on a Bose-Einstein condensate Acknowledgments
We acknowledge financial support by DFG
Research Unit 760 , the US-Israel BinationalScience Foundation (BSF), Jerusalem, Israel, a grant from AFOSR No. FA 9550-10-1-0433, and a grant from the Deutsch-Israelische Projektkooperation (DIP).
References [1] Stenger J et al
Phys. Rev. Lett. et al Phys. Rev. Lett. Rev. Mod. Phys. Nature Phys. arXiv:1104.2564 [6] Sanders S N, Mintert F and Heller E J 2010 Phys. Rev. Lett. et al
Phys. Rev. Lett. et al Phys. Rev. Lett. Electronic Transport in Mesoscopic Systems (Cambridge University Press)[10] Bandopadhyay S and Cohen D 2008
Phys. Rev. B Probing a Bose-Hubbard system with a scattering particle,
Phys. Rev. A Phys. Rev. A Phys. Rev. A Phys. Rev. B Phys. Rev. A Nonlinearity Phys. Rev. E Phys. Rev. Lett. EPJ D J. Phys. A Europhys. Lett. J. Phys. A L319[24] Ericson T and Mayer-Kuckuk T 1966
Annu. Rev. Nucl. Sci. Phys. Rev. Lett. et al J. Phys. A Nucl. Phys.
A512 et al
Europhys. Lett. Phys. Rev. Lett. Ultracold bosons in tilted optical lattices – impact of spectral statisticson simulability, stability, and dynamics
Phys. Rev. Lett. et al
Phys. Rev. Lett.
Phys. Rev. Lett. et al
Phys. Rev. Lett.106