PRL 112, 233201 (2014)
week ending 13 JUNE 2014
PHYSICAL REVIEW LETTERS
Photon-Assisted Confinement-Induced Resonances for Ultracold Atoms 1
Vicente Leyton,1 Maryam Roghani,1 Vittorio Peano,2 and Michael Thorwart1
I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany Institute for Theoretical Physics II, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany (Received 3 March 2014; published 10 June 2014)
2
We solve the two-particle s-wave scattering for an ultracold-atom gas confined in a quasi-onedimensional trapping potential which is periodically modulated. The interaction between the atoms is included via Fermi’s pseudopotential. For a modulated isotropic transverse harmonic confinement, the atomic center of mass and relative degrees of freedom decouple and an exact solution is possible. The modulation opens additional photon-assisted resonant scattering channels. Applying the Bethe-Peierls boundary condition, we obtain the general scattering solution of the time-dependent Floquet-Schrödinger equation which is universal at low energies. The effective one-dimensional scattering length can be controlled by the external driving. DOI: 10.1103/PhysRevLett.112.233201
PACS numbers: 34.50.-s, 03.65.Nk, 05.30.Jp, 37.10.Jk
The ability to accurately control the effective atomic interactions in ultracold-atom gases has opened the doorway to novel exciting physics. The available experimental tools permit an implementation of analog quantum simulators realized by cold-atom assemblies [1,2]. For strong atomic interactions, the quantum gas becomes scale invariant and shows universal behavior quantified in terms of a few dimensionless coefficients. For instance, the threedimensional (3D) s-wave scattering length a characterizes the atomic interactions and has to be compared to the mean interatomic distance, which is of the order of the particles’ inverse momenta k−1 . It can be controlled over several orders of magnitude by a magnetic field tuned across a Feshbach resonance [3]. In 3D, the tunability of the scattering length reveals the crossover from the weakly interacting superfluid state, where 1=ðkaÞ → −∞, to the strongly interacting Bose-Einstein condensate of dimer molecules, where 1=ðkaÞ → þ∞ [1]. In quasi-one-dimensional (1D) gases, another relevant length scale appears in the form of the transverse confinement length a⊥ . The scattering of two tightly confined quantum particles induces universal low-energy features in the form of confinement-induced resonances (CIRs) [4–6]. Then, only the transverse ground state of the confining potential is significantly populated, whereas the higher transverse states can only be virtually populated during the elastic collisions. The remaining scattering processes in the longitudinal direction can be characterized by the effective 1D interaction strength g1D . It is governed by a single parameter, being the ratio of a and a⊥ . By tuning the confinement strength (or the 3D scattering length via a Feshbach resonance [3]) across a CIR, it is possible to cross over from repulsive to attractive interactions. CIRs have been observed in a confined 1D gas of bosonic Cs atoms [7], and the confinement-induced formation of a two-atom molecular state of fermionic 0031-9007=14=112(23)=233201(5)
K atoms has been reported [8]. By this, it became possible to investigate the crossover from a repulsive TonksGirardeau gas to an attractive super-Tonks-Girardeau gas [7]. CIRs have also been observed in a 2D Fermi gas [9] and in mixed dimensions [10]. In analogy to a Feshbach resonance, the CIR occurs when the continuum threshold for the lowest transverse state (the open channel) has the same energy as a bound state formed by two particles being in some transverse excited states (the closed channels) [5]. Put differently, the transverse orbital degrees of freedom of the confined atoms play the same role as the internal atomic spin degrees of freedom for a Feshbach resonance [3]. When the transverse confinement of two equal atom species is harmonic, only one such bound state exists, leading to a single universal CIR [4–6]. This feature can be traced back to the separability of the center of mass and relative coordinates [11]. In turn, a multitude of CIRs appears when these degrees of freedom are no longer separable, i.e., for a mixture of different species [11], anisotropic [12–14] and anharmonic confinement [11–16], and in mixed dimensions [10,17]. Dipolar CIRs have also been predicted in the presence of long-range anisotropic interactions between different atomic angular momentum states [18]. Coupled CIRs have also been predicted at higher energies [19]. As an alternative to the “orbital” Feshbach resonance to control the atomic scattering, a time-dependent modulation of internal atomic states generates an optical Feshbach resonance [20–22]. It occurs when the optical radiation resonantly couples two atoms in their electronic ground state to a molecular state formed by electronically excited states, i.e., the relevant closed channels. In this Letter, we propose an alternative concept to control the interactions of atoms in a quasi-1D trap. It is based on the time-dependent rf modulation of the trapping
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PRL 112, 233201 (2014)
potential by parametrically modulating the laser intensity. We consider a tight trap where bosonic atoms have been initially cooled to their transverse ground state. The trap eigenfrequency ω0 is modulated periodically such that ω2 ðtÞ ¼ ω20 − F cos ωex t. This can be easily realized experimentally by an acousto-optic modulator (Bragg cell) [23]. The modulation is switched on adiabatically to keep all atoms in the same transverse Floquet state. We show that a new type of CIR is induced by the virtual molecular recombination of the atoms (in their electronic ground state) during the collision. This process is mediated by the emission of m virtual photons by the scattering atoms at the continuum threshold ℏω0 ; see Fig. 1. The virtual transition to the bound state with energy EB becomes resonant when ℏðω0 − mωex Þ ¼ EB , leading to a series of photon-assisted CIRs. This process is fundamentally different from a Feshbach resonance, as it does not involve a bound state formed by the closed channels. By tuning the modulation parameters, the photon-assisted CIRs can easily be tuned. In a frame of reference where the center of mass of the two atoms is at rest, the two-body problem can be mapped to the scattering of a trapped single particle, with the reduced mass μ, by a central potential UðrÞ. The center of mass and relative degrees of freedom also decouple for a harmonic trap in the presence of a parametric timedependent modulation. The Hamiltonian is Hðr; tÞ ¼ H0 ðtÞ þ UðrÞ;
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PHYSICAL REVIEW LETTERS
H0 ðr; tÞ ¼
p2 1 2 þ μω ðtÞr2⊥ : 2μ 2 ð1Þ
We set the z direction as the direction of free evolution. The confinement is defined over the x-y plane with r⊥ ¼ ðx; yÞ. Furthermore, we choose the frequency ωðtÞ such that the
solutions of the classical equation of motion ẍ þ ω2 ðtÞx ¼ 0 are stable [24]. In a tight trap and at low temperature, the atoms are initially in the transverse ground state [25] and their de Broglie wavelength is determined by the oscillator length a⊥ ¼ ðμω0 Þ−1=2 in the static trap (ℏ ¼ 1). When it is much larger than the microscopic interaction range UðrÞ, the latter can be described by the zero-range Fermi pseudopotential [26] 2πa ∂ δðrÞ r: μ ∂r
UðrÞ ¼
ð2Þ
The interaction is characterized by a single parameter, i.e., the 3D scattering length a. The validity of this approach in this context has been verified numerically based on a microscopic finite-range interaction potential [6,27]. The wave function of the incoming atoms in the adiabatic transverse ground state is ψ in ðtÞ ¼ exp½−iεt exp½ikzu0 ðx; tÞu0 ðy; tÞ;
ð3Þ
where k is the longitudinal momentum and ε is the quasienergy of the atoms ε ¼ k2 =2μ þ ν. Here, the timeperiodic functions un ðx; tÞ are the Floquet eigenstates of the parametrically driven harmonic oscillator with quasienergy ðn þ 1=2Þν [28–30]. When the driving is switched off adiabatically, ν → ω0 and the un ðxÞ become the eigenstates of the harmonic oscillator. Our goal is to compute the solution ψðtÞ ¼ ψ in ðtÞ þ ψ out ðtÞ which includes the scattered wave ψ out ðtÞ. It is convenient to introduce the time-periodic Floquet state ϕðtÞ ¼ exp½iεtψðtÞ as the solution of the eigenvalue problem Hðr; tÞϕðr; tÞ ¼ εϕðr; tÞ, where H ≡ H − i∂ t is the Floquet Hamiltonian [29]. With T ¼ 2π=ωex , we obtain Z ϕðr; tÞ ¼ ϕin ðr; tÞ þ
T
0
dt0 fðt0 Þ ; Gε ðr; 0; t; t0 Þ 2μ T
ð4Þ
where ϕin ðtÞ ¼ exp½iεtψ in ðtÞ and the second term on the rhs represents the outgoing scattered wave function ϕout ðtÞ. The integral kernel Gε ¼ ðH0 − ε − i0Þ−1 is the retarded Floquet-Green’s function with H0 ¼ H0 − i∂ t . This wave function has to fulfill the Bethe-Peierls boundary condition FIG. 1 (color online). Sketch of the energy levels. A standard CIR occurs when the energy of the scattering atoms at the continuum threshold (horizontal blue line) matches the energy EB of a bound state formed by the closed channels (dashed lightgreen line). The energy EB of the real bound state is shown as a continuum line (solid dark-green curve). A photon-assisted CIR occurs when the binding energy ℏω0 − EB matches the energy of m photons. The photon energy ℏωex is indicated by wavy lines at the photon-assisted CIRs for the single-photon and two-photon resonances. The quasienergy is the energy folded on an energy interval ℏω ex .
r fðtÞ ϕðr → 0; tÞ ≃ 1 − ; a 4πr
ð5Þ
with the Bethe-Peierls amplitude fðtÞ yet to be determined. For large z, the asymptotic scattered wave can be decomposed into partial waves as ϕout ðrÞ ≈
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X n
Sm n
m¼open
k knm
1=2
eiknm jzj unx ðx; tÞuny ðy; tÞ: ð6Þ
PHYSICAL REVIEW LETTERS
PRL 112, 233201 (2014)
2 Here, jSm n j is the probability of the atoms to be excited into the transverse state with quantum number n ¼ ðnx ; ny Þ after absorbing m > 0 photons and thereby acquiring the momentum knm determined by
k2nm þ ðn þ 1Þν ¼ ε þ mωex ; 2μ
n ¼ nx þ ny :
ð7Þ
The number of available open channels depends on m. The S-matrix elements are determined by Eq. (5) (see Ref. [30] for details) as Z T i ffiffiffiffiffiffiffiffiffi ffi p Sm ¼ dt0 eimωex t unx ð0; t0 Þuny ð0; t0 Þfðt0 Þ: ð8Þ n 2 kknm T 0 Next, we insert Eq. R(4) into Eq. (5) and define a scalar product hfjgi ¼ T −1 0T dtf ðtÞgðtÞ on the Hilbert space of time-periodic functions. Then, an inhomogeneous linear equation for the Bethe-Peierls amplitude fðtÞ follows: a⊥ ζε þ jfi ¼ −4πN −1=2 jini: a
ð9Þ
g1D ¼ −1=ðμa1D Þ. Its imaginary part refers to the loss of atoms in the excited transverse states due to inelastic scattering processes into other channels provided by the modulation. From this, we obtain the elastic cross section, which is the probability of an elastic scattering event as σ l ¼ jS000 j2 and its inelastic counterpart as σ r ¼ 1 − σ l − j1 þ S000 j2 . From Eq. (13), we find σ l ¼ ð1 þ k2 ja1D j2 − 2kIma1D Þ−1 ; σ r ¼ −2σ l kIma1D
and the normalized vector jini via htjini ¼ N 1=2 a⊥ ϕin ð0; tÞ. For small initial momenta k ≪ knm , the scattering is dominated by elastic collisions. In this regime, it is convenient to divide the kernel ζ ε into a smooth part ζ~ ε , that can be evaluated for k ¼ 0, and the contribution of the channel of the incoming atoms, where the k dependence is retained. This yields [30] jfi ¼ −4πN −1=2
−1 ka1D ~ζν þ a⊥ jini: ka1D − i a
ð11Þ
Here, it is convenient to introduce the effective 1D scattering length a1D for the longitudinal scattering as a⊥ a⊥ −1 −1 ~ ¼ −2πN hinj ζ ν þ jini: a1D a
ð14Þ
(Ima1D < 0). The effective quasi-1D scattering cross sections for jIma1D j=ja1D j ¼ 0.15 are shown in Fig. 2. The probability of an elastic scattering event tends to 1 for small relative momenta k ≪ 1=ja1D j. On the other hand, the scattering is dominated by inelastic scattering events for larger momenta k ≫ jIma1D j−1 . Hence, when ja1D j becomes smaller than the typical longitudinal de Broglie wavelength, a scattering resonance results. Formally, the 1D scattering length a1D can be expressed in terms of the spectrum fλm g and the right and left eigenvalues jvRm i and jvLm i of the kernel ζ~ ε : X hinjvRm ihvLm jini a⊥ ¼ −2πN −1 : λm þ a⊥ =a a1D m
Here, we have introduced the regularized integral kernel 2πa⊥ 0 0 0 Tμ htjζε jt i ¼ ð10Þ Gϵ ðr; 0; t; t Þ − δðt − t Þ 2πr r→0 μ
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ð15Þ
Hence, if several eigenvectors significantly overlap with the vector j ini of the incoming wave, more than one scattering resonance may occur. The resonances in ja⊥ =a1D j are well resolved Lorentzian peaks with their center determined by the resonance condition a⊥ =a ¼ Reλm and with their linewidth given by Imλm, if their mutual distance exceeds the corresponding widths. The kernel ζ~ ε can be computed analytically only for F ¼ 0; see Ref. [30]. Then, the left and right eigenvectors jvLm i and jvRm i are plane waves htjvRm i ¼ hvLR jti ¼ exp½imωex t, and the incoming wave is given by jini ¼ jvR0 i. Thus, the overlap is 0 for m ≠ 0 and we recover the standard result: there is only one CIR for a⊥ =a ¼ −λ0 ¼ −ζð1=2; 1Þ [ζðx; yÞ is the Hurwitz zeta function] when the energy of the virtual bound state formed
ð12Þ
Since the operator ζ~ ν is not Hermitian, it acquires an imaginary part whose meaning is discussed below. Using Eq. (11) in Eq. (8), we obtain the S-matrix element S000 ¼ −
i ka1D − i
:
ð13Þ
This is the probability amplitude for the reflection of a 1D particle due to the effective scattering potential U1D ¼ g1D δðzÞ with complex interaction strength
FIG. 2 (color online). Effective scattering cross sections in quasi-1D: total (σ) and elastic (σ l ) cross sections as a function of the relative longitudinal momentum k for jIma1D j=ja1D j ¼ 0.15.
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(a)
(a)
(b)
(b)
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(c)
FIG. 3 (color online). Photon-assisted broadening of the standard CIR. The imaginary and real parts of a⊥ =a1D are shown in (a) and (b), respectively, for ω ex ¼ 1.2ω0 and F ¼ 10−4 ω20 . The dashed vertical line indicates the resonance position in the absence of driving a⊥ =a ¼ −ζð1=2; 1Þ. The resonance is broadened by driving-induced inelastic transitions to excited transverse channels. (c) The width of the CIR has a quadratic dependence on the driving amplitude Imλ0 ∝ F2 .
by the closed channels coincides with the continuum threshold [4]. For off-resonant weak driving F ≪ ω2ex ;
j2ω0 − mωd j ≫ ωd ð4F=ω2d Þm ;
ð16Þ
we expect the eigenvalues fλm g of the kernel ζ ε to only smoothly deviate from their values for F ¼ 0. To obtain a specific result, we consider [14] Cs atoms transversely confined in an optical lattice formed by counterpropagating laser beams with a wavelength of 1064 nm and a well depth V 0 ¼ 30ER , where ER is the photon recoil energy. A trap frequency of ω0 ¼ 2π × 14.5 kHz arises. We assume a modulation frequency of ωex ¼ 1.2ω0 and vary the amplitude such that the potential depth varies only weakly (F ∼ 10−4 ω20 ≪ V 0 ). Then, we diagonalize the kernel ζ~ ε with a semianalytical procedure outlined in Ref. [30]. The standard CIR for a weak nonresonant driving is shown in Figs. 3(a) and 3(b). The zero-photon resonance is clearly broadened by the inelastic collisions. In Fig. 3(c), we show that the width Imλ0 ∝ F2 . For a finite driving, there is no selection rule preventing the remaining eigenvectors of ζ~ ε to yield a finite contribution in Eq. (15) to a1D . For weak nonresonant driving, we can label the eigenvalues λm with the number m of radio frequency photons which are virtually absorbed (emitted for m < 0) and later reemitted (reabsorbed) during an elastic collision. From Eq. (15), we see that the contribution to a1D from the processes where m photons are virtually absorbed is largest for a⊥ =a ¼ − Reλm. In Ref. [30], we show that, for m > 0, this occurs when the m-photon transition from the continuum threshold ℏω0 to the virtual bound state with energy EB ðmÞ formed by the transverse
FIG. 4 (color online). Photon-assisted CIRs [imaginary part of a⊥ =a1D in (a) and real part in (b)] due to a one-photon absorption from and subsequent emission into the driving field (m ¼ −1) for ωex ¼ 1.2ω0 and for varying modulation amplitudes F.
channels which are still closed after the absorption of m photons (with transverse energy E>EM ¼2ω0 ðMþ1=2Þ, where M ¼ Int ½mωex =ð2ω0 Þ þ 1) is resonant: EB ðmÞ ¼ ω0 þ mωex . We do not expect that these processes lead to a scattering resonance because the corresponding bound states leak very quickly into the open channels. In fact, the imaginary part Imλm is finite even for F → 0, Imλm ≳ 1 [30]. On the other hand, the contribution to a1D from processes where the photons are first emitted (m < 0) is largest for [30] a⊥ ¼ −Reλm ≈ −ζð1=2; jmjωex =2ω0 Þ: a
ð17Þ
The energy EB of the molecular bound state (for F ¼ 0) is given by a⊥ =a ¼ −ζ½1=2; ðEB − 1Þ=2ω0 [5]. Hence, the processes where jmj photons are virtually emitted lead to the largest enhancement of scattering when the molecular recombination accompanied by the emission of m photons is resonant: ω0 − jmjωex ≈ EB . Since limF→0 Imλm ¼ 0 [30] (the molecular bound state can only dissociate because of the driving), these processes induce sharp CIRs. The case of resonant emission of a single photon (m ¼ −1) is shown in Fig. 4 for different values of F (see also the one-photon line in Fig. 1). Conclusions.—We have shown that the s-wave scattering of two atoms confined in a tight quasi-1D trap can be coherently controlled by a rf modulation of the transverse confinement. The scattering of the atoms in the adiabatic transverse ground state can be efficiently described via a short-range interaction. The coupling constant g1D acquires an imaginary part and incorporates inelastic scattering into the transverse excited states. The results are universally valid for low energies, as they depend only on the
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(time-dependent) curvature at the bottom of the confining potential and the 3D scattering length. All other details of the trapping and interaction potentials are irrelevant. The photon-assisted CIR reminds one of Kapitza’s pendulum, where the slow motion in the longitudinal direction can be controlled by a modulation of the fast transverse motion. A more obvious analogy can be drawn to Feshbach-like resonances. However, the scattering resonances investigated here involve the true molecular bound state and not a virtual bound state formed by the closed channels. Hence, they can be understood as the dynamical equivalent of shape resonances (the scattering resonances that occur when a generic potential has a bound state close to the continuum threshold) [31]. The dynamical CIRs are another example in which photon-assisted processes carry signatures of atomic interparticle interactions [32,33]. Future investigations could focus on binary interactions of mixtures of species, anisotropic or anharmonic potentials, and fermionic atoms. Also, advanced numerical approaches [6,27] valid beyond the s-wave limit are available. We acknowledge support from the DFG SFB 925 “Light induced dynamics and control of correlated quantum systems” (Project C8) and the ERC Starting Grant OPTOMECH (V. P.).
[1] I. Bloch, J. Dalibard, and S. Nascimbène, Nat. Phys. 8, 267 (2012). [2] J. Struck, C. Ölschläger, R. Le Targat, P. Soltan-Panahi, A. Eckardt, M. Lewenstein, P. Windpassinger, and K. Sengstock, Science 333, 996 (2011). [3] E. Timmermanns, P. Tommasini, M. Hussein, and A. Kerman, Phys. Rep. 315, 199 (1999). [4] M. Olshanii, Phys. Rev. Lett. 81, 938 (1998). [5] T. Bergeman, M. G. Moore, and M. Olshanii, Phys. Rev. Lett. 91, 163201 (2003). [6] M. G. Moore, T. Bergeman, and M. Olshanii, J. Phys. IV France 116, 69 (2004). [7] E. Haller, M. Gustavsson, M. J. Mark, J. G. Danzl, R. Hart, G. Pupillo, and H.-C. Nägerl, Science 325, 1224 (2009). [8] H. Moritz, T. Stöferle, K. Guenter, M. Köhl, and T. Esslinger, Phys. Rev. Lett. 94, 210401 (2005). [9] B. Fröhlich, M. Feld, E. Vogt, M. Koschorreck, W. Zwerger, and M. Köhl, Phys. Rev. Lett. 106, 105301 (2011). [10] G. Lamporesi, J. Catani, G. Barontini, Y. Nishida, M. Inguscio, and F. Minardi, Phys. Rev. Lett. 104, 153202 (2010).
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[11] V. Peano, M. Thorwart, C. Mora, and R. Egger, New J. Phys. 7, 192 (2005). [12] S.-G. Peng, H. Hu, X.-J. Liu, and P. D. Drummond, Phys. Rev. A 84, 043619 (2011). [13] S. Sala, P.-I. Schneider, and A. Saenz, Phys. Rev. Lett. 109, 073201 (2012). [14] E. Haller, M. J. Mark, R. Hart, J. G. Danzl, L. Reichsöllner, V. Melezhik, P. Schmelcher, and H.-Ch. Nägerl, Phys. Rev. Lett. 104, 153203 (2010). [15] J. P. Kestner and L. M. Duan, New J. Phys. 12, 053016 (2010). [16] S. Sala, G. Zürn, T. Lompe, A. N. Wenz, S. Murmann, F. Serwane, S. Jochim, and A. Saenz, Phys. Rev. Lett. 110, 203202 (2013). [17] P. Massignan and Y. Castin, Phys. Rev. A 74, 013616 (2006). [18] P. Giannakeas, V. S. Melezhik, and P. Schmelcher, Phys. Rev. Lett. 111, 183201 (2013). [19] P. Giannakeas, F. K. Diakonos, and P. Schmelcher, Phys. Rev. A 86, 042703 (2012). [20] P. O. Fedichev, Y. Kagan, G. V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 77, 2913 (1996). [21] M. Theis, G. Thalhammer, K. Winkler, M. Hellwig, G. Ruff, R. Grimm, and J. Hecker Denschlag, Phys. Rev. Lett. 93, 123001 (2004). [22] T. Zelevinsky, M. M. Boyd, A. D. Ludlow, T. Ido, J. Ye, R. Ciurylo, P. Naidon, and P. S. Julienne, Phys. Rev. Lett. 96, 203201 (2006). [23] O. Morsch and M. Oberthaler, Rev. Mod. Phys. 78, 179 (2006). [24] We do not consider combinations of F and ωext for which the Mathieu equation, which underlies the classical parametric oscillator, has a diverging solution; see M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965). [25] The modulation is weak enough such that the low-energy limit is still realized. [26] K. Huang and C. N. Yang, Phys. Rev. 105, 767 (1957). [27] V. S. Melezhik, J. I. Kim, and P. Schmelcher, Phys. Rev. A 76, 053611 (2007). [28] L. S. Brown, Phys. Rev. Lett. 66, 527 (1991). [29] M. Grifoni and P. Hänggi, Phys. Rep. 304, 229 (1998). [30] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.233201 for details of the calculation. [31] L. D. Landau and E. M. Lifshitz, Quantum Mechanics: NonRelativistic Theory, Course of Theoretical Physics Vol. 3 (Pergamon, Oxford, 1958). [32] A. Eckardt, T. Jinasundera, C. Weiss, and M. Holthaus, Phys. Rev. Lett. 95, 200401 (2005). [33] A. Eckardt and M. Holthaus, Phys. Rev. Lett. 101, 245302 (2008).
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PHYSICAL REVIEW LETTERS
Photon-Assisted Confinement-Induced Resonances for Ultracold Atoms 1
Vicente Leyton,1 Maryam Roghani,1 Vittorio Peano,2 and Michael Thorwart1
I. Institut für Theoretische Physik, Universität Hamburg, Jungiusstraße 9, 20355 Hamburg, Germany Institute for Theoretical Physics II, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstraße 7, 91058 Erlangen, Germany (Received 3 March 2014; published 10 June 2014)
2
We solve the two-particle s-wave scattering for an ultracold-atom gas confined in a quasi-onedimensional trapping potential which is periodically modulated. The interaction between the atoms is included via Fermi’s pseudopotential. For a modulated isotropic transverse harmonic confinement, the atomic center of mass and relative degrees of freedom decouple and an exact solution is possible. The modulation opens additional photon-assisted resonant scattering channels. Applying the Bethe-Peierls boundary condition, we obtain the general scattering solution of the time-dependent Floquet-Schrödinger equation which is universal at low energies. The effective one-dimensional scattering length can be controlled by the external driving. DOI: 10.1103/PhysRevLett.112.233201
PACS numbers: 34.50.-s, 03.65.Nk, 05.30.Jp, 37.10.Jk
The ability to accurately control the effective atomic interactions in ultracold-atom gases has opened the doorway to novel exciting physics. The available experimental tools permit an implementation of analog quantum simulators realized by cold-atom assemblies [1,2]. For strong atomic interactions, the quantum gas becomes scale invariant and shows universal behavior quantified in terms of a few dimensionless coefficients. For instance, the threedimensional (3D) s-wave scattering length a characterizes the atomic interactions and has to be compared to the mean interatomic distance, which is of the order of the particles’ inverse momenta k−1 . It can be controlled over several orders of magnitude by a magnetic field tuned across a Feshbach resonance [3]. In 3D, the tunability of the scattering length reveals the crossover from the weakly interacting superfluid state, where 1=ðkaÞ → −∞, to the strongly interacting Bose-Einstein condensate of dimer molecules, where 1=ðkaÞ → þ∞ [1]. In quasi-one-dimensional (1D) gases, another relevant length scale appears in the form of the transverse confinement length a⊥ . The scattering of two tightly confined quantum particles induces universal low-energy features in the form of confinement-induced resonances (CIRs) [4–6]. Then, only the transverse ground state of the confining potential is significantly populated, whereas the higher transverse states can only be virtually populated during the elastic collisions. The remaining scattering processes in the longitudinal direction can be characterized by the effective 1D interaction strength g1D . It is governed by a single parameter, being the ratio of a and a⊥ . By tuning the confinement strength (or the 3D scattering length via a Feshbach resonance [3]) across a CIR, it is possible to cross over from repulsive to attractive interactions. CIRs have been observed in a confined 1D gas of bosonic Cs atoms [7], and the confinement-induced formation of a two-atom molecular state of fermionic 0031-9007=14=112(23)=233201(5)
K atoms has been reported [8]. By this, it became possible to investigate the crossover from a repulsive TonksGirardeau gas to an attractive super-Tonks-Girardeau gas [7]. CIRs have also been observed in a 2D Fermi gas [9] and in mixed dimensions [10]. In analogy to a Feshbach resonance, the CIR occurs when the continuum threshold for the lowest transverse state (the open channel) has the same energy as a bound state formed by two particles being in some transverse excited states (the closed channels) [5]. Put differently, the transverse orbital degrees of freedom of the confined atoms play the same role as the internal atomic spin degrees of freedom for a Feshbach resonance [3]. When the transverse confinement of two equal atom species is harmonic, only one such bound state exists, leading to a single universal CIR [4–6]. This feature can be traced back to the separability of the center of mass and relative coordinates [11]. In turn, a multitude of CIRs appears when these degrees of freedom are no longer separable, i.e., for a mixture of different species [11], anisotropic [12–14] and anharmonic confinement [11–16], and in mixed dimensions [10,17]. Dipolar CIRs have also been predicted in the presence of long-range anisotropic interactions between different atomic angular momentum states [18]. Coupled CIRs have also been predicted at higher energies [19]. As an alternative to the “orbital” Feshbach resonance to control the atomic scattering, a time-dependent modulation of internal atomic states generates an optical Feshbach resonance [20–22]. It occurs when the optical radiation resonantly couples two atoms in their electronic ground state to a molecular state formed by electronically excited states, i.e., the relevant closed channels. In this Letter, we propose an alternative concept to control the interactions of atoms in a quasi-1D trap. It is based on the time-dependent rf modulation of the trapping
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© 2014 American Physical Society
PRL 112, 233201 (2014)
potential by parametrically modulating the laser intensity. We consider a tight trap where bosonic atoms have been initially cooled to their transverse ground state. The trap eigenfrequency ω0 is modulated periodically such that ω2 ðtÞ ¼ ω20 − F cos ωex t. This can be easily realized experimentally by an acousto-optic modulator (Bragg cell) [23]. The modulation is switched on adiabatically to keep all atoms in the same transverse Floquet state. We show that a new type of CIR is induced by the virtual molecular recombination of the atoms (in their electronic ground state) during the collision. This process is mediated by the emission of m virtual photons by the scattering atoms at the continuum threshold ℏω0 ; see Fig. 1. The virtual transition to the bound state with energy EB becomes resonant when ℏðω0 − mωex Þ ¼ EB , leading to a series of photon-assisted CIRs. This process is fundamentally different from a Feshbach resonance, as it does not involve a bound state formed by the closed channels. By tuning the modulation parameters, the photon-assisted CIRs can easily be tuned. In a frame of reference where the center of mass of the two atoms is at rest, the two-body problem can be mapped to the scattering of a trapped single particle, with the reduced mass μ, by a central potential UðrÞ. The center of mass and relative degrees of freedom also decouple for a harmonic trap in the presence of a parametric timedependent modulation. The Hamiltonian is Hðr; tÞ ¼ H0 ðtÞ þ UðrÞ;
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H0 ðr; tÞ ¼
p2 1 2 þ μω ðtÞr2⊥ : 2μ 2 ð1Þ
We set the z direction as the direction of free evolution. The confinement is defined over the x-y plane with r⊥ ¼ ðx; yÞ. Furthermore, we choose the frequency ωðtÞ such that the
solutions of the classical equation of motion ẍ þ ω2 ðtÞx ¼ 0 are stable [24]. In a tight trap and at low temperature, the atoms are initially in the transverse ground state [25] and their de Broglie wavelength is determined by the oscillator length a⊥ ¼ ðμω0 Þ−1=2 in the static trap (ℏ ¼ 1). When it is much larger than the microscopic interaction range UðrÞ, the latter can be described by the zero-range Fermi pseudopotential [26] 2πa ∂ δðrÞ r: μ ∂r
UðrÞ ¼
ð2Þ
The interaction is characterized by a single parameter, i.e., the 3D scattering length a. The validity of this approach in this context has been verified numerically based on a microscopic finite-range interaction potential [6,27]. The wave function of the incoming atoms in the adiabatic transverse ground state is ψ in ðtÞ ¼ exp½−iεt exp½ikzu0 ðx; tÞu0 ðy; tÞ;
ð3Þ
where k is the longitudinal momentum and ε is the quasienergy of the atoms ε ¼ k2 =2μ þ ν. Here, the timeperiodic functions un ðx; tÞ are the Floquet eigenstates of the parametrically driven harmonic oscillator with quasienergy ðn þ 1=2Þν [28–30]. When the driving is switched off adiabatically, ν → ω0 and the un ðxÞ become the eigenstates of the harmonic oscillator. Our goal is to compute the solution ψðtÞ ¼ ψ in ðtÞ þ ψ out ðtÞ which includes the scattered wave ψ out ðtÞ. It is convenient to introduce the time-periodic Floquet state ϕðtÞ ¼ exp½iεtψðtÞ as the solution of the eigenvalue problem Hðr; tÞϕðr; tÞ ¼ εϕðr; tÞ, where H ≡ H − i∂ t is the Floquet Hamiltonian [29]. With T ¼ 2π=ωex , we obtain Z ϕðr; tÞ ¼ ϕin ðr; tÞ þ
T
0
dt0 fðt0 Þ ; Gε ðr; 0; t; t0 Þ 2μ T
ð4Þ
where ϕin ðtÞ ¼ exp½iεtψ in ðtÞ and the second term on the rhs represents the outgoing scattered wave function ϕout ðtÞ. The integral kernel Gε ¼ ðH0 − ε − i0Þ−1 is the retarded Floquet-Green’s function with H0 ¼ H0 − i∂ t . This wave function has to fulfill the Bethe-Peierls boundary condition FIG. 1 (color online). Sketch of the energy levels. A standard CIR occurs when the energy of the scattering atoms at the continuum threshold (horizontal blue line) matches the energy EB of a bound state formed by the closed channels (dashed lightgreen line). The energy EB of the real bound state is shown as a continuum line (solid dark-green curve). A photon-assisted CIR occurs when the binding energy ℏω0 − EB matches the energy of m photons. The photon energy ℏωex is indicated by wavy lines at the photon-assisted CIRs for the single-photon and two-photon resonances. The quasienergy is the energy folded on an energy interval ℏω ex .
r fðtÞ ϕðr → 0; tÞ ≃ 1 − ; a 4πr
ð5Þ
with the Bethe-Peierls amplitude fðtÞ yet to be determined. For large z, the asymptotic scattered wave can be decomposed into partial waves as ϕout ðrÞ ≈
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X n
Sm n
m¼open
k knm
1=2
eiknm jzj unx ðx; tÞuny ðy; tÞ: ð6Þ
PHYSICAL REVIEW LETTERS
PRL 112, 233201 (2014)
2 Here, jSm n j is the probability of the atoms to be excited into the transverse state with quantum number n ¼ ðnx ; ny Þ after absorbing m > 0 photons and thereby acquiring the momentum knm determined by
k2nm þ ðn þ 1Þν ¼ ε þ mωex ; 2μ
n ¼ nx þ ny :
ð7Þ
The number of available open channels depends on m. The S-matrix elements are determined by Eq. (5) (see Ref. [30] for details) as Z T i ffiffiffiffiffiffiffiffiffi ffi p Sm ¼ dt0 eimωex t unx ð0; t0 Þuny ð0; t0 Þfðt0 Þ: ð8Þ n 2 kknm T 0 Next, we insert Eq. R(4) into Eq. (5) and define a scalar product hfjgi ¼ T −1 0T dtf ðtÞgðtÞ on the Hilbert space of time-periodic functions. Then, an inhomogeneous linear equation for the Bethe-Peierls amplitude fðtÞ follows: a⊥ ζε þ jfi ¼ −4πN −1=2 jini: a
ð9Þ
g1D ¼ −1=ðμa1D Þ. Its imaginary part refers to the loss of atoms in the excited transverse states due to inelastic scattering processes into other channels provided by the modulation. From this, we obtain the elastic cross section, which is the probability of an elastic scattering event as σ l ¼ jS000 j2 and its inelastic counterpart as σ r ¼ 1 − σ l − j1 þ S000 j2 . From Eq. (13), we find σ l ¼ ð1 þ k2 ja1D j2 − 2kIma1D Þ−1 ; σ r ¼ −2σ l kIma1D
and the normalized vector jini via htjini ¼ N 1=2 a⊥ ϕin ð0; tÞ. For small initial momenta k ≪ knm , the scattering is dominated by elastic collisions. In this regime, it is convenient to divide the kernel ζ ε into a smooth part ζ~ ε , that can be evaluated for k ¼ 0, and the contribution of the channel of the incoming atoms, where the k dependence is retained. This yields [30] jfi ¼ −4πN −1=2
−1 ka1D ~ζν þ a⊥ jini: ka1D − i a
ð11Þ
Here, it is convenient to introduce the effective 1D scattering length a1D for the longitudinal scattering as a⊥ a⊥ −1 −1 ~ ¼ −2πN hinj ζ ν þ jini: a1D a
ð14Þ
(Ima1D < 0). The effective quasi-1D scattering cross sections for jIma1D j=ja1D j ¼ 0.15 are shown in Fig. 2. The probability of an elastic scattering event tends to 1 for small relative momenta k ≪ 1=ja1D j. On the other hand, the scattering is dominated by inelastic scattering events for larger momenta k ≫ jIma1D j−1 . Hence, when ja1D j becomes smaller than the typical longitudinal de Broglie wavelength, a scattering resonance results. Formally, the 1D scattering length a1D can be expressed in terms of the spectrum fλm g and the right and left eigenvalues jvRm i and jvLm i of the kernel ζ~ ε : X hinjvRm ihvLm jini a⊥ ¼ −2πN −1 : λm þ a⊥ =a a1D m
Here, we have introduced the regularized integral kernel 2πa⊥ 0 0 0 Tμ htjζε jt i ¼ ð10Þ Gϵ ðr; 0; t; t Þ − δðt − t Þ 2πr r→0 μ
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ð15Þ
Hence, if several eigenvectors significantly overlap with the vector j ini of the incoming wave, more than one scattering resonance may occur. The resonances in ja⊥ =a1D j are well resolved Lorentzian peaks with their center determined by the resonance condition a⊥ =a ¼ Reλm and with their linewidth given by Imλm, if their mutual distance exceeds the corresponding widths. The kernel ζ~ ε can be computed analytically only for F ¼ 0; see Ref. [30]. Then, the left and right eigenvectors jvLm i and jvRm i are plane waves htjvRm i ¼ hvLR jti ¼ exp½imωex t, and the incoming wave is given by jini ¼ jvR0 i. Thus, the overlap is 0 for m ≠ 0 and we recover the standard result: there is only one CIR for a⊥ =a ¼ −λ0 ¼ −ζð1=2; 1Þ [ζðx; yÞ is the Hurwitz zeta function] when the energy of the virtual bound state formed
ð12Þ
Since the operator ζ~ ν is not Hermitian, it acquires an imaginary part whose meaning is discussed below. Using Eq. (11) in Eq. (8), we obtain the S-matrix element S000 ¼ −
i ka1D − i
:
ð13Þ
This is the probability amplitude for the reflection of a 1D particle due to the effective scattering potential U1D ¼ g1D δðzÞ with complex interaction strength
FIG. 2 (color online). Effective scattering cross sections in quasi-1D: total (σ) and elastic (σ l ) cross sections as a function of the relative longitudinal momentum k for jIma1D j=ja1D j ¼ 0.15.
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(a)
(a)
(b)
(b)
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(c)
FIG. 3 (color online). Photon-assisted broadening of the standard CIR. The imaginary and real parts of a⊥ =a1D are shown in (a) and (b), respectively, for ω ex ¼ 1.2ω0 and F ¼ 10−4 ω20 . The dashed vertical line indicates the resonance position in the absence of driving a⊥ =a ¼ −ζð1=2; 1Þ. The resonance is broadened by driving-induced inelastic transitions to excited transverse channels. (c) The width of the CIR has a quadratic dependence on the driving amplitude Imλ0 ∝ F2 .
by the closed channels coincides with the continuum threshold [4]. For off-resonant weak driving F ≪ ω2ex ;
j2ω0 − mωd j ≫ ωd ð4F=ω2d Þm ;
ð16Þ
we expect the eigenvalues fλm g of the kernel ζ ε to only smoothly deviate from their values for F ¼ 0. To obtain a specific result, we consider [14] Cs atoms transversely confined in an optical lattice formed by counterpropagating laser beams with a wavelength of 1064 nm and a well depth V 0 ¼ 30ER , where ER is the photon recoil energy. A trap frequency of ω0 ¼ 2π × 14.5 kHz arises. We assume a modulation frequency of ωex ¼ 1.2ω0 and vary the amplitude such that the potential depth varies only weakly (F ∼ 10−4 ω20 ≪ V 0 ). Then, we diagonalize the kernel ζ~ ε with a semianalytical procedure outlined in Ref. [30]. The standard CIR for a weak nonresonant driving is shown in Figs. 3(a) and 3(b). The zero-photon resonance is clearly broadened by the inelastic collisions. In Fig. 3(c), we show that the width Imλ0 ∝ F2 . For a finite driving, there is no selection rule preventing the remaining eigenvectors of ζ~ ε to yield a finite contribution in Eq. (15) to a1D . For weak nonresonant driving, we can label the eigenvalues λm with the number m of radio frequency photons which are virtually absorbed (emitted for m < 0) and later reemitted (reabsorbed) during an elastic collision. From Eq. (15), we see that the contribution to a1D from the processes where m photons are virtually absorbed is largest for a⊥ =a ¼ − Reλm. In Ref. [30], we show that, for m > 0, this occurs when the m-photon transition from the continuum threshold ℏω0 to the virtual bound state with energy EB ðmÞ formed by the transverse
FIG. 4 (color online). Photon-assisted CIRs [imaginary part of a⊥ =a1D in (a) and real part in (b)] due to a one-photon absorption from and subsequent emission into the driving field (m ¼ −1) for ωex ¼ 1.2ω0 and for varying modulation amplitudes F.
channels which are still closed after the absorption of m photons (with transverse energy E>EM ¼2ω0 ðMþ1=2Þ, where M ¼ Int ½mωex =ð2ω0 Þ þ 1) is resonant: EB ðmÞ ¼ ω0 þ mωex . We do not expect that these processes lead to a scattering resonance because the corresponding bound states leak very quickly into the open channels. In fact, the imaginary part Imλm is finite even for F → 0, Imλm ≳ 1 [30]. On the other hand, the contribution to a1D from processes where the photons are first emitted (m < 0) is largest for [30] a⊥ ¼ −Reλm ≈ −ζð1=2; jmjωex =2ω0 Þ: a
ð17Þ
The energy EB of the molecular bound state (for F ¼ 0) is given by a⊥ =a ¼ −ζ½1=2; ðEB − 1Þ=2ω0 [5]. Hence, the processes where jmj photons are virtually emitted lead to the largest enhancement of scattering when the molecular recombination accompanied by the emission of m photons is resonant: ω0 − jmjωex ≈ EB . Since limF→0 Imλm ¼ 0 [30] (the molecular bound state can only dissociate because of the driving), these processes induce sharp CIRs. The case of resonant emission of a single photon (m ¼ −1) is shown in Fig. 4 for different values of F (see also the one-photon line in Fig. 1). Conclusions.—We have shown that the s-wave scattering of two atoms confined in a tight quasi-1D trap can be coherently controlled by a rf modulation of the transverse confinement. The scattering of the atoms in the adiabatic transverse ground state can be efficiently described via a short-range interaction. The coupling constant g1D acquires an imaginary part and incorporates inelastic scattering into the transverse excited states. The results are universally valid for low energies, as they depend only on the
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(time-dependent) curvature at the bottom of the confining potential and the 3D scattering length. All other details of the trapping and interaction potentials are irrelevant. The photon-assisted CIR reminds one of Kapitza’s pendulum, where the slow motion in the longitudinal direction can be controlled by a modulation of the fast transverse motion. A more obvious analogy can be drawn to Feshbach-like resonances. However, the scattering resonances investigated here involve the true molecular bound state and not a virtual bound state formed by the closed channels. Hence, they can be understood as the dynamical equivalent of shape resonances (the scattering resonances that occur when a generic potential has a bound state close to the continuum threshold) [31]. The dynamical CIRs are another example in which photon-assisted processes carry signatures of atomic interparticle interactions [32,33]. Future investigations could focus on binary interactions of mixtures of species, anisotropic or anharmonic potentials, and fermionic atoms. Also, advanced numerical approaches [6,27] valid beyond the s-wave limit are available. We acknowledge support from the DFG SFB 925 “Light induced dynamics and control of correlated quantum systems” (Project C8) and the ERC Starting Grant OPTOMECH (V. P.).
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