Orthogonality Catastrophe in Quantum Sticking
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec Orthogonality Catastrophe in Quantum Sticking
Dennis P. Clougherty ∗ and Yanting Zhang Department of PhysicsUniversity of VermontBurlington, VT 05405-0125 (Dated: November 15, 2018)
Abstract
The probability that a particle will stick to a surface is fundamental to a variety of processesin surface science, including catalysis, epitaxial growth, and corrosion. At ultralow energies, howparticles scatter or stick to a surface affects the performance of atomic clocks, matter-wave interfer-ometers, atom chips and other quantum information processing devices. In this energy regime, thesticking probability is influenced by a distinctly quantum mechanical effect: quantum reflection, aresult of matter wave coherence, suppresses the probability of finding the particle near the surfaceand reduces the sticking probability. We find that another quantum effect can occur, further shap-ing the sticking probability: the orthogonality catastrophe, a result of the change in the quantumground state of the surface in the presence of a particle, can dramatically alter the probability forquantum sticking and create a superreflective surface at low energies.
PACS numbers: 68.43.Mn, 03.65.Nk, 68.49.Bc, 34.50.Cx H = H p + H b + H c (1)where H p = p m + V ( z ) , (2) H b = X q ω q a † q a q , (3) H c = − V ( z ) X q σ ( ω q ) ( a q + a † q ) (4)We might estimate the rate of sticking by one-phonon emission using Fermi’s goldenrule. We first consider the transition matrix element h b, q | H c | k, i of the dynamical particle-surface interaction, where | k, i denotes a state with the particle in the continuum state2ith incident energy E = ~ k m and the surface in its ground state with no excitations; | b, q i denotes the particle in a bound state of V and the surface has one excitation with wavenumber q .If we choose the normalization of the particle wave function such that it has unit ampli-tude far from the surface, for a potential V ( z ) that decays faster than z − as z → ∞ , it iswell-known [6–11] that the amplitude of the wave function near the surface scales as k . Thisis a result of quantum reflection, a wave phenomenon where the incident particle is reflectedfrom a surface without ever having reached a classical turning point. Thus the amplitude ofthe wave function tends to vanish near the surface as the particle’s incident energy tends tozero. We conclude that the transition matrix element for sticking vanishes in proportion to k at low energies. This result follows for all potentials that fall off faster than z − for large z and is universal in this sense.Within lowest order perturbation theory, the sticking probability s ( E ) of a particle withincident energy E varies as the square of the transition matrix element and inversely withthe incident particle flux. Hence, s ∝ k /k ∼ √ E . This low-energy threshold law forquantum sticking was implicit in work by Lennard-Jones [12] in the pioneering years ofquantum theory. With improvements in the cooling and trapping techniques of ultracoldatoms, this √ E threshold law was found to be consistent with experiment in the case ofhydrogen sticking to the surface of superfluid helium [13].It has been asserted [6, 10, 14] that the √ E law holds regardless of the form of H c . Wewill show that the √ E law only holds for a class of dynamical couplings H c , determinedby the low frequency behavior, in the same fashion as models of quantum dissipation areclassified by their spectral functions. For superohmic H c , the √ E threshold law holds forneutral particles impinging on zero temperature surfaces; however, for ohmic couplings, wewill show a different threshold law results. In essence, H c contains a final-state interactionthat can alter the ground state of the surface. We have recently found that this final-state interaction is responsible for an orthogonality catastrophe for ohmic coupling [15] thatsubsequently alters the threshold law for quantum sticking.At sufficiently low energies, we can ignore inelastic scattering and approximate the particlestate space by the initial state | k i and the final state | b i . In this truncated state space, the3amiltonian becomes H = Ec † k c k − E b b † b + X q ω q a † q a q − ( c † k b + b † c k ) V kb X q σ ( ω q ) ( a q + a † q ) − c † k c k V kk X q σ ( ω q ) ( a q + a † q ) − b † bV bb X q σ ( ω q ) ( a q + a † q ) (5)where V kb = h k | V | b i etc. The effects of the V kk term in Eq. 5 are of higher order in k thanthe V bb term. We consequently neglect the V kk term in what follows.The V bb term is the final-state interaction responsible for the orthogonality catastrophe fora certain class of frequency-dependent couplings. The Hamiltonian for the surface excitationshas a different form in the initial particle state compared to the final particle state. Hencewe need to include in the transition matrix element that the surface excitation in the finalstate is created from a different ground state from the initial ground state of the surface.The final state Hamiltonian of the surface, H s,f = P q ω q a † q a q − V bb P q σ ( ω q ) ( a q + a † q ),can be put in the form of that of the initial state by a displaced oscillator transformation.Such a transformation reflects that the ground state of the surface in the presence of thebound particle is polarized relative to the surface in isolation. To within an arbitrary phasefactor, the overlap of the ground state of H s,f with that of the isolated surface is S ≡ h f | i = e − F = exp − V bb X q σ ( ω q ) ω q ! (6)In the continuum limit, this overlap vanishes when D ( ω ) σ ( ω ) ∼ ω, ω → D ( ω ) is the density of surface excitations. In the language of models of quantumdissipation, this condition describes ohmic coupling [5]. We have previously shown [15] thisform of coupling applies to the dynamical interaction of the particle with phonons in anelastically isotropic surface; for example, in the case of Rayleigh phonons, σ is independentof frequency [16], while D ( ω ) ∝ ω for these two dimensional surface modes. Hence, forsticking via the emission of Rayleigh phonons, the interaction is ohmic. We also concludethat sticking via emission of bulk phonons (or “mixed mode” phonons) has an interactionthat is ohmic, since D ( ω ) ∝ ω and σ ∝ ω − .4n the golden rule estimate, the relevant transition matrix element should have an excita-tion created out of the final-state vacuum. This reduces the transition matrix element by theFranck-Condon factor of Eq. 6, which in the ohmic case vanishes, signaling the breakdown ofperturbation theory. (We note that the dynamical coupling in the case of ultracold atomichydrogen sticking to superfluid helium by ripplon emission is superohmic and consequentlygives a non-vanishing Franck-Condon factor. We expect the √ E law to hold in this case.)In previous work [15], we calculated the sticking probability in the ohmic case usinga non-perturbative variational scheme and found that the logarithmic divergence in theFranck-Condon factor is cut-off by a frequency scale ω that depends linearly on the incidentparticle energy E at sufficiently low energies. The low frequency cutoff ω might be thoughtof as coming from the finite time needed for the particle to make a transition to the boundstate. Excitations with frequencies below ω do not have adequate time to adjust to thepresence of the bound particle and do not contribute to the Franck-Condon factor. For weak V bb , F ≈ α Z ω c c E dωω (8)where α = lim ω → V bb D ( ω ) σ ( ω ) /ω , c is a dimensionless constant, independent of E and ω c is the high-frequency cutoff of the bath.The truncation of the logarithmic divergence gives rise to a new behavior for the stickingprobability at threshold for ohmic systems. s ( E ) ∝ E · E α (9)We have considered a variety of experimental conditions to assess the likelihood thatthis new threshold law might be observed. Unfortunately we have found that the shift inexponent α is typically much smaller than one. Thus, this new threshold law might provevery challenging to verify experimentally. However, in the case of charged particles stickingto surfaces, we are optimistic that the effects of the orthogonality catastrophe on the stickingprobability will be accessible to experiment.The threshold law for charged particles differs from that of neutral particles. Chargedparticles are influenced by a long-range attractive Coulomb interaction due to the particle’simage charge, in contrast to the van der Waals interaction exerted on neutral particles.The Coulomb potential decays sufficiently slowly that a low-energy charged particle doesnot experience quantum reflection [8]. The amplitude of the wave function of the incident5article near the surface scales as √ k as k →
0. Hence, a na¨ıve application of Fermi’sgolden rule would predict that the sticking probability s ( E ) of a charged particle withincident energy E behaves as s ∝ ( √ k ) /k ∼ E , a constant.For the case of an ohmic dynamical coupling, the orthogonality catastrophe modifies thisna¨ıve threshold law for charged particles. The absence of quantum reflection for chargedparticles affects the energy-dependence of the low-frequency cutoff for the Franck-Condonfactor, with ω scaling as √ E at low energies. For small α , F ≈ α Z ω c c √ E dωω (10)where c is a constant, independent of E . Thus, for charged particles, we obtain s ( E ) ∝ E α/ (11)In contrast to the na¨ıve threshold law where the sticking probability approaches a non-vanishing constant as E →
0, the orthogonality catastrophe drives the sticking probabilityto zero.It is a straightforward matter to extend this theory to surfaces at finite temperature, andthere are several new features in the sticking probability that result. The Franck-Condonfactor is altered by thermally excited excitations in the bath S = exp − V bb X q σ ( ω q ) ω q coth βω q ! (12)There is a critical incident energy E c , dependent on the temperature T , below whichthe low-frequency cutoff ω sharply drops to zero. As a result, the sticking probability is avictim of the orthogonality catastrophe and vanishes for E < E c . Consider the exponent F of the Franck-Condon factor at finite temperature for vanishing cutoff ω F ∼ αT Z dωω → ∞ (13)Thus the Franck-Condon factor sharply drops to zero and the sticking probability vanishesfor E < E c , creating what might be termed a “superreflective” surface with perfect reflec-tivity below the critical incident energy. This sharp change in the sticking probability hasanalogy with the behavior of the tunneling probability in the spin-boson model. There,the tunneling probability is renormalized to zero beyond a temperature-dependent criticalcoupling to the bath, defining the localization phase boundary.6or the case of low surface temperature T ≪ ω , we recover the zero-temperature resultsof Eqs. 9 and 11 to leading order in T ; for the case of intermediate surface temperaturewhere T c ( E ) > T ≫ ω , we find that F ∝ ω − / for sufficiently low α and energy. Howeverat finite temperature, our variational calculations show that ω ∝ √ E for neutral particlesand √ E for charged particles. Thus, we find for low-energy neutral particles with E > E c and intermediate surface temperatures s ( E ) ∝ √ E exp (cid:16) − p E /E (cid:17) (14)while for low-energy charged particles, s ( E ) ∝ exp (cid:16) − p E /E (cid:17) (15)where E is an energy-independent constant and T c ( E ) is the critical temperature abovewhich ω vanishes (see Fig. 2).The numerical results from our variational mean-field theory for sticking are presented inFig. 1 for the case of ultracold electrons sticking to porous silicon at finite temperature. Wechoose highly porous silicon for two reasons: to remain in the regime where sticking occurspredominantly through one-phonon processes, the binding energy must be small comparedto ω c , the high frequency cutoff of the excitations; thus, we require a low dielectric constant.Silicon with a porosity of 92 .
9% has a dielectric constant of only κ = 1 .
2. Secondly, tomaximize α in the case of coupling to Rayleigh phonons, we seek materials that have a lowshear modulus and mass density. We expect highly porous silicon to have a shear modulusof 230 MPa.Our numerical calculations reveal a sharp transition in the sticking probability at a criticalenergy of E c ≈ . E c are predicted to be perfectly reflectedby the surface. Electrons with energy above E c stick to the surface with a probability reducedby roughly a factor of five compared to the na¨ıve golden rule result.In summary, on the basis of a variational mean-field theory for the sticking of ultracoldparticles on a finite temperature surface, we predict new scaling laws of the sticking proba-bility with incident energy at intermediate surface temperatures. We also predict a dramaticdownturn of the sticking probability, with the probability vanishing below a critical energy E c . This new feature in the sticking probability is a consequence of a bosonic orthogonalitycatastrophe, where the Franck-Condon factor, resulting from the surface polarization in the7 −4 −3 −2 −1 s ( E ) FIG. 1. (color online). The sticking probability of an electron of energy E to the surface of poroussilicon by the emission of a Rayleigh phonon. The surface temperature is taken to be T = 2 K. Theperturbative result using Fermi’s golden rule with a Franck-Condon factor of 1 is given by (green)circles, while the variational mean-field result is given by (blue) stars. The variational mean-fieldmethod gives a sharp transition at an incident energy E ≈ . P = 92 . κ = 1 .
2. The shear modulus of G = 230 MPa and Poisson’s ratio σ = 0 .
03 are calculated using Ref. [17] . presence of the particle, vanishes for
E < E c . We predict that this effect is experimentallyaccessible in the case of low energy electrons impinging on porous silicon.We gratefully acknowledge support of this work by the National Science Foundation underDMR-0814377. ∗ [email protected][1] M. Greiner, O. Mandel, T. Esslinger, T. W. H¨ansch, and I. Bloch, Nature , 39 (2002).[2] M. W. Zwierlein, A. Schirotzek, C. H. Schunck, and W. Ketterle, Science , 492 (2006).[3] W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. F¨olling, L. Pollet, and −6 −4 −2 s ( T ) FIG. 2. (color online). The sticking probability of an electron ( E = 1 mK) to the surface of poroussilicon by the emission of a Rayleigh phonon as function of the surface temperature T . The (green)circles result from using Fermi’s golden rule with a Franck-Condon factor S = 1. The variationalmean-field result is given by (blue) stars. There is a dramatic downturn in the probability at asurface temperature of T ≈ . , 547 (2010).[4] G. D. Mahan, Many-Particle Physics (Plenum Press, New York, 1981) p. 761.[5] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, and A. Garg, Rev. Mod. Phys. , 1 (1987).[6] W. Brenig, Z. Phys. B , 227 (1980).[7] D. S. Zimmerman and A. J. Berlinsky, Can. J. Phys. , 50 (1983).[8] D. P. Clougherty and W. Kohn, Phys. Rev. B , 4921 (1992).[9] T. W. Hijmans, J. T. M. Walraven, and G. V. Shlyapnikov, Phys. Rev. B , 2561 (1992).[10] C. Carraro and M. Cole, Prog. Surf. Sci. , 61 (1998).[11] D. P. Clougherty, Phys. Rev. Lett. , 226105 (2003).[12] J. E. Lennard-Jones and A. F. Devonshire, Proc. R. Soc. London, Ser. A , 29 (1936).[13] I. A. Yu, J. M. Doyle, J. C. Sandberg, C. L. Cesar, D. Kleppner, and T. J. Greytak, Phys. ev. Lett. , 1589 (1993).[14] S. G. Chung and T. F. George, Surf. Sci. , 347 (1988).[15] Y. Zhang and D. P. Clougherty, arXiv:1012.4405 (2010).[16] M. E. Flatt´e and W. Kohn, Phys. Rev. B , 7422 (1991).[17] A. Doghmane, Z. Hadjoub, M. Doghmane, and F. Hadjoub, Semiconductor Phys., QuantumElectronics and Optoelectronics , 4 (2006)., 4 (2006).