Ai Content Based Filtering

This project focuses on ai content based filtering using modern AI and machine learning techniques. The content below is adapted from research literature and practical implementation notes.

Many-body Decay of the Gapped Lowest Excitation of a Bose–Einstein Condensate

Jinyi Zhang,1,∗ Christoph Eigen,1,∗,† Wei Zheng,1,2,3,† Jake A. P. Glidden,1 Timon A. Hilker,1

Samuel J. Garratt,1,4 Raphael Lopes,1,5 Nigel R. Cooper,1 Zoran Hadzibabic,1 and Nir Navon

3 CAS Center for Excellence in Quantum Information and Quantum Physics,

UPMC-Sorbonne Universités, 1 Place Marcelin Berthelot, F-750 Paris, France

(Dated: April 24, 2020) We study the decay mechanism of the gapped lowest-lying excitation of a quasi-pure box-trapped atomic Bose–Einstein condensate. Owing to the absence of lower-energy modes, or direct coupling to an external bath, this excitation is protected against one-body (linear) decay and the damping mechanism is exclusively nonlin- ear. We develop a universal theoretical model that explains this fundamental nonlinear damping as a process whereby two quanta of the gapped lowest excitation mode couple to a higher-energy mode, which subsequently

decays into a continuum. We find quantitative agreement between our experiments and the predictions of this model. Finally, by strongly driving the system below its (lowest) resonant frequency we observe third-harmonic generation, a hallmark of nonlinear behavior.

Our understanding of quantum many-body systems is perimentally resolvable gaps in the excitation spectrum, and rooted in the existence of elementary excitations such as recent works suggested that the damping of the lowest mode quasi-particles and phonons. The nature of these excitations is fully nonlinear (within experimental precision) [29, 33]. reflects the form of order in the underlying state of mat- Here we experimentally and theoretically elucidate the damp-

ter. Moreover, low-temperature thermodynamics and near- ing mechanism of this mode; we show that it can be explained equilibrium transport properties can be computed from the en- by a generic model based on an inverse Beliaev-like pro- ergy spectrum of these modes. cess, whereby two elementary excitations merge into a higher- Typically this energy spectrum can be calculated to lowest energy one [Fig. 1(b)].

order by assuming that the excitations are non-interacting, and Our experiment starts with a quasi-uniform BEC of 8 Rb thus have an infinite lifetime. Taking into account the interac- atoms , prepared in a cylindrical optical box trap of radius tions between the quasi-particles in the continuum limit gener- R = 16(1) µm and length L = 26(1) µm [see Fig. 2(a)]. Our ically leads to finite lifetimes, even at zero temperature . condensates consist of N = 1.2(1) × 1 atoms, their chemical

On the other hand, the lifetime of gapped excitations has potential is µ ≈ kB × 2.1 nK, their healing length ξ ≈ 1.1 µm, been far less investigated. Gaps in the energy spectra natu- and the excitation frequency of the lowest-lying axial mode rally arise in finite-size systems [2, 3] or from many-body ef- [Fig. 2(b)] is ωd = 2π × 9.0(1) Hz (see for details). We fects in infinite systems [4, 5], or a combination of the two . excite this mode by applying a spatially uniform oscillating

In either case the lifetime of the lowest-lying excitation of a force Fs (r, t ) = (Us /L) sin(ωt )ez , where ez is the unit vector many-body system at zero temperature is of particular interest since it is energetically immune to any one-body (i.e. linear) decay mechanism. In this Letter, we study this fundamen- (a) (b) tal many-body problem experimentally and theoretically by E continuum E continuum

investigating the decay of the lowest excitation of a homoge- neous box-trapped atomic Bose–Einstein condensate (BEC).

The weakly-interacting bulk Bose gas has been a remark-

able testbed for exploring the physics of excitations and their interactions [7–30]. At low temperature, the excitations of an ground state ground state infinite uniform BEC have an energy given by the Bogoliubov spectrum; in the long-wavelength limit these excitations are phonons . The leading decay channel for the phonons is many-body system. (a) Beliaev-like linear damping of an excitation a linear process, in which they spontaneously break up into in the continuum to two (or more) lower-lying excitations. (b) Non-

pairs of longer-wavelength phonons [Fig. 1(a)], a damping linear damping of the lowest-energy, gapped excitation (blue) to a mechanism first predicted by Beliaev . higher-lying mode (red). The continuum is represented as a gray By contrast, in our case the finite system size leads to ex- shade, while the horizontal lines indicate relevant energy levels.

F (e)

(a)(f) (b) (c) (d)0.3 (e) (f) (a) (b) (c) (d) (e) (f)

(mm/s) (mm/s) (mm/s) s 3 r 0 U s /kB (nK) z 2R -0.3 0.2 1.6 0.3 1

m (µm/s) 0.3 0.6 0.9 1.2

ω/(2π) (Hz) L 6.5 ω/(2π) (Hz) (a) (b) (c) (d) (e) (f) 0 6.5 9.0 0 8.0

-0.3 3 0.3 0.3 0.6 0.9 1.2 12.5 9.0 -L/2 0 L/2 0 10.0 δv

z 1 12.5 -0.3 -L/2 0 L/2 0 0.3 0.6 0.9 1.2 0.2 0.5 1 2 z ts (s) U ss /kBB (nK)

along which the drive Fs (t ) is applied. (b) Sketch of the fundamental mode density δn(z) and velocity δv(z) profiles along the box axis. (c) Center-of-mass velocity response v(t s ) for three different drive frequencies ω, close to the resonant ωd /(2π) = 9.0(1) Hz, and two different drive strengths Us (see legend). (d) Steady-state response amplitude v m versus Us on log-log scale, for various drive frequencies (see legend). The on-resonance v m exhibits power-law behavior, and a fit to the data (solid line) gives v m ∝ Us0.32(1) . For progressively larger Us a wider

range of ω become effectively resonant. The dotted lines are guides to the eye, and the error bars represent fitting uncertainties.

along the symmetry axis of the cylinder and Us is the maximal which act as a quasi-continuum of states to which the exci- potential-energy drop across the box. Following a variable tation can decay. We formalize this picture by developing a shaking time t s , the BEC is released and the center of mass simple model in which the fundamental excitation of energy

of the atomic density distribution is recorded after a time of ~ωd is created by an external forcing, and is coupled to an flight t ToF = 1 ms, which reflects the in-situ center-of-mass auxiliary mode of energy ~ωb (see Sections II-V in ). The velocity v(t s ) along ez . Hamiltonian describing the system is

In Fig. 2(c), we show examples of v(t s ) for three drive fre-

Ĥ = ~ωd dˆ† dˆ + ~ωb b̂ † b̂ + ~λ(b̂ † dˆdˆ + H.c.) quencies (below, on, and above resonance), and for two drive amplitudes (Us /kB = 0.2 and 1.6 nK). While at early times + ~Ω sin (ωt ) (dˆ† + dˆ) , (1) transient dynamics are visible, a monochromatic steady state where dˆ† (dˆ) and b̂ † (b̂ ) are, respectively, the creation (annihi- is established at later times. We fit the data for t s > 0.6 s with

lation) operators for the fundamental and the auxiliary mode, v(t s ) = v m sin(ωt s + φ) (solid lines) and extract the amplitude λ is the coupling strength, and Ω is the strength of the drive. v m and phase φ of the velocity response. For the weaker drive

We incorporate the decay of the auxiliary mode into the quasi-

(Us /µ ≈ 0.1) we observe significant response only on reso- continuum via a master equation approach. By tracing out the nance. However, for the stronger drive (Us /µ ≈ 0.7) we ob- auxiliary mode, we derive, within a mean-field approxima- serve comparable response amplitudes at all three frequencies, tion, an equation of motion for the mean dipole d (t ) ≡ 〈dˆ(t )〉: indicating significant broadening, a clear signature of nonlin- ear behavior. i ∂t d − (ωd + κ |d |2 )d = Ω sin(ωt ) . (2)

To characterize this nonlinear behavior we plot v m as a

function of Us on log-log scale for various drive frequencies The generally complex κ captures the nonlinear effects to in Fig. 2(d). For linear response, with amplitude-independent leading order. Specifically, Re[κ ] and Im[κ ] correspond, re- damping, one would get v m ∝ Us , for all ω. Instead, on spectively, to a frequency shift (due to the self-interaction) and

resonance (ω = ωd ) we observe power-law behavior v m ∝ a nonlinear damping (due to the mediated coupling to the con- Us0.32(1) , reminiscent of classical models with a cubic nonlin- tinuum); expressions for κ in terms of the microscopic model ear damping [which generically predict v m ∝ Us1/3 on reso- parameters are provided in Section III in . In practice,

nance (see Section I in )]. Away from resonance, v m is κ is sensitive to the details of the trapping potential, and it smaller than on resonance (for the same Us ), but for stronger is more convenient to extract it directly from the experimen- drives a progressively broader range of frequencies becomes tal data. However, the form of Eq. (2) is universal in that it

effectively resonant. does not depend on the exact loss mechanism of the auxiliary In the following we introduce a universal theoretical model mode, nor the number of auxiliary excitations involved in the for this nonlinear behavior, and compare it to the full experi- elementary interaction process [see Eq. (1)].

mental response curves v m (ω) and φ(ω) (see Fig. 3). Within To compare our experimental data to the theory, we quantum theory, in the spirit of Fig. 1(b), we associate the relate d (t ) to the main experimental observable v(t ) = cube-root scaling v m ∝ Us1/3 with the decay of the mode to- (2α/N )∂t Re[d (t )], where α is the matrix element of the po-

wards higher energies via a two-body process , result- sition operator ẑ between the ground state and the lowest- ing in a decay rate proportional to the number of phonons. lying excitation (see Section IV in ). In terms of the Once excited, this ‘secondary’ mode couples to other modes, experimental parameters, the drive amplitude in Eq. (1) is

(a) (a) (b) 0.3 0.3

Γ/(2π) (Hz)

ℱ (mm/s) ℱ (mm/s)

(mm/s) (mm/s) 0.2 1 0.2

5 U s /kB (nK) 0

0.1 2 0.2 -0.3 (mm/s) m (mm/s)

0.2 0.5 1 2 0.3 0 0.3 0.6 0.9 1.2 0.2 1 2 3

U ss /kBB (nK) 0.7

0.1 ω3ω

ℱ /(2π) (Hz)

1.0 0 0.1 m

1.6 -0.3 2.5 0 0.6 0.9 1.2 0 1 2 3 0 ts (s) ωℱ /(2π) (Hz) 0 1 2 (a) (b) (c) (d) (e) (f)

ℱ (µm/s)

ω/(2π) ω/(2π) (Hz) (Hz) 4 (b) (c) 2 )) 0.1 8 -6 (10-6 0.5 )

N0.5 )

0.0 4 4 5 6 7 α/(L

Uss/k /kBB (nK) (nK) 0 0 0.2 0.5 1 2 0.2 0.5 1 2

U /kBB (nK)

Uss/k (nK) UUss /k

/kBB (nK) (nK) FIG. 4. Observation of third-harmonic generation. (a) Steady-state (d) center-of-mass velocity response v(t s ) for a strong drive (Us /kB = π/2 ω/(2π) (Hz) 6.6 nK) at ω/(2π) = 9 Hz (top) and ω/(2π) = 3.5 Hz (bottom). In the

6.5 latter case we clearly observe a non-monochromatic response. The

0 8.0 solid lines show fits to the data . (b) Corresponding Fourier spec- 9.0 tra F , which for ω/(2π) = 3.5 Hz reveals a response at 3ω. (c) Ex- ϕ

10.0 -π/2 12.5 tracted third-harmonic generation amplitude F for ω/(2π) = 3.5 Hz 0.2 0.5 1 2 plotted as a function of Us on log-log scale. The dashed line shows a Uss /kBB (nK) power-law fit, which gives F ∝ Us3.6±1.2 , consistent with the cubic scaling behavior characteristic of third-harmonic generation.

(a) Steady-state velocity response amplitude v m as a function of ω, for six different drive strengths Us (see legend), alongside the cor- The parameters extracted from each v m (ω) curve are shown responding fits to our model (see text for details). The inset shows in Figs. 3(b,c). Crucially, both α and κ̄2 do not depend on Γ, the extracted full width at half maximum of v m (ω), as a function Us within experimental errors, demonstrating that the model

of Us on log-log scale, while the horizontal dotted line indicates ωd . [Eq. (2)] captures the nonlinear Us -dependent response. Av- The solid line is a fit to the data, Γ ∝ Us0.67(2) . For reference, our ex- eraging the fitted parameters well within the range of valid- perimental upper bound on linear damping rate is 2π × 0.3 s−1 . p

ity of the model (Us /kB < 1 nK) we obtain α = 0.097(3)L N (b,c) Extracted model fit parameters for the response curves v m (ω) in (a). The dashed line in (b) shows a theoretical estimate (see text), and κ̄2 /ωd = 6.9(4)×10−6 per phonon (solid horizontal lines). while the solid lines in (b,c) depict the average for Us /kB < 1 nK. (d) A calculation assuming a cylindrical-box-trapped BECpin the Phase response as a function of Us . The solid lines show the predic- Thomas–Fermi regime (Section III in ) yields α/(L N ) =

25/4 /π3/2 ξ/L ≈ 0.0 [dashed line in Fig. 3(b)], in good p tion of the model with the extracted average parameters [solid lines in (b,c)] for Re[κ ] = 0. The dotted lines instead show corresponding agreement with the measurement. predictions which include a small nonzero Re[κ ] (see text). Where Furthermore, we find that Γ ∝ Us0.67(2) , which interestingly not visible, the fitting error bars are smaller than the point size.

suggests that even though v m /Us and Γ are not independent of

Us as for a linearly-damped harmonic oscillator, one still re-

covers v m Γ ∝ Us . Naively, this reflects the energy balance Ω = αUs /(~L). We determine the parameters of the model condition for a driven-dissipative steady state; Us v m is the by fitting the v m (ω) response curves to the steady-state nu- driving power, and v m 2 Γ is akin to the energy dissipation rate, merical solutions of Eq. (2) for each Us . The resulting fits are the energy stored in the system being ∝ v m 2

. shown in Fig. 3(a) as solid lines, where for simplicity we first In Fig. 3(d) we show the complementary data on the phase assume that the nonlinear coefficient κ is imaginary so that of the velocity response, φ, as a function of Us for various the adjustable parameters are κ̄2 ≡ i κ and α. We see that for drive frequencies. As expected, φ is π/2 (-π/2) out of phase

Us . k B ×2 nK the fitted model captures the experimental data with the drive below (above) resonance. Interestingly, for low well. drive strengths (Us /kB < 1 nK) the on-resonance φ(ωd ) is ob- Only for Us & kB × 2 nK do deviations between the model servably nonzero and nearly independent of Us . Averaging

and the data become apparent. In the inset of Fig. 3(a) we the data for Us /kB < 1 nK gives φ = −0.12(4), which suggests plot the extracted full width at half maximum of the spectral that Re[κ ] = 0.12(4)Im[κ ] within our model [see Section IV lines, Γ, as a function of Us , which reveals that the deviation in ); this a posteriori justifies our initial assumption that

between the model and the data occurs only once Γ & ωd . For |Re[κ ]/Im[κ ]|¿ 1. Including this small nonzero Re[κ ] in Us /k B = 1.6 nK, we estimate that κ̄2 |d |2 ≈ 0.5ωd , and higher- the model further improves agreement with the data [dotted order nonlinearities could become important. lines in Figs. 3(a,d)], but the difference is small.

For a nonlinear response one also generically expects Extended Theory, Phys. Rev. 133, B11 (1964). the possibility of harmonic generation. Experimentally, we R. C. Ashoori, Electrons in artificial atoms, Nature 379, 4 did not observe evidence of higher-harmonic generation for (1996). Us /k B < 3 nK, so in the final part of this Letter we turn to L. N. Cooper, Bound Electron Pairs in a Degenerate Fermi Gas, Phys. Rev. 104, 11 (1956).

even larger drive amplitudes (up to Us /kB = 6.6 nK). For most W. Kohn, Theory of the Insulating State, Phys. Rev. 133, A1 driving frequencies ω we only observe a monochromatic re- (1964). sponse at the drive frequency, even for strong drives (see top A. M. García-García, J. D. Urbina, E. A. Yuzbashyan, panels in Figs. 4(a,b) where ω/(2π) = 9 Hz ). However, K. Richter, and B. L. Altshuler, Bardeen–Cooper–Schrieffer the behavior is markedly different for ω/(2π) = 3.5 Hz (bot- Theory of Finite-Size Superconducting Metallic Grains, Phys.

tom panels), for which 3ω is close to the resonant ωd . In Rev. Lett. 100, 1870 (2008). this case the Fourier spectrum of the time-domain response D. S. Jin, J. R. Ensher, M. R. Matthews, C. E. Wieman, and

E. A. Cornell, Collective Excitations of a Bose–Einstein Con-

[Fig. 4(b)] shows a clear peak at 3ω, signalling third-harmonic densate in a Dilute Gas, Phys. Rev. Lett. 77, 4 (1996). generation, in addition to the main peak at ω. In Fig. 4(c) we M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, plot the third-harmonic amplitude F versus Us on log-log D. S. Durfee, C. G. Townsend, and W. Ketterle, Collective Ex- scale . A power-law fit (dashed line) gives F ∝ Us3.6±1.2 , citations of a Bose–Einstein Condensate in a Magnetic Trap,

consistent with a cubic scaling in Us (characteristic of third- Phys. Rev. Lett. 77, 9 (1996). harmonic generation) and in qualitative agreement with pre- M. R. Andrews, D. M. Kurn, H.-J. Miesner, D. S. Durfee, C. G. dictions from Eq. (2) (Section V in ). Townsend, S. Inouye, and W. Ketterle, Propagation of Sound in a Bose–Einstein Condensate, Phys. Rev. Lett. 79, 5 (1997). In conclusion, we have experimentally and theoretically D. M. Stamper-Kurn, H.-J. Miesner, S. Inouye, M. R. Andrews,

studied the nonlinear decay of the fundamental gapped exci- and W. Ketterle, Collisionless and Hydrodynamic Excitations tation of a Bose–Einstein condensate. Our experiments reveal of a Bose–Einstein Condensate, Phys. Rev. Lett. 81, 5 (1998). a cubic damping mechanism as well as third-harmonic gen- D. M. Stamper-Kurn, A. P. Chikkatur, A. Görlitz, S. Inouye, eration. To understand these nonlinear effects we develop a S. Gupta, D. E. Pritchard, and W. Ketterle, Excitation of

mean-field model based on a microscopic theory at lowest or- Phonons in a Bose–Einstein Condensate by Light Scattering, Phys. Rev. Lett. 83, 28 (1999). der. Our work paves the way to a microscopic understanding J. Stenger, S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, of the genesis of a turbulent cascade [33, 39], when the en- D. E. Pritchard, and W. Ketterle, Bragg Spectroscopy of a ergy leakage from the driven lowest-lying mode is sufficiently Bose–Einstein Condensate, Phys. Rev. Lett. 82, 45 (1999).

large to sustain a non-equilibrium steady state. The decay of O. M. Maragò, S. A. Hopkins, J. Arlt, E. Hodby, G. Hechen- gapped excitations is an understudied but generic problem in blaikner, and C. J. Foot, Observation of the Scissors Mode and quantum many-body physics, and our approach could be ex- Evidence for Superfluidity of a Trapped Bose–Einstein Con- tended to other systems, where it could offer insight into the densed Gas, Phys. Rev. Lett. 84, 20 (2000).

J. R. Abo-Shaeer, C. Raman, J. M. Vogels, and W. Ketterle, interactions between those elementary excitations. Observation of Vortex Lattices in Bose–Einstein Condensates, We thank Robert P. Smith, Maciej Gałka, and Nis- Science 292, 4 (2001). hant Dogra for helpful discussions and comments on J. Steinhauer, R. Ozeri, N. Katz, and N. Davidson, Excitation the manuscript. This work was supported by EPSRC Spectrum of a Bose–Einstein Condensate, Phys. Rev. Lett. 88,

[Grant Nos. EP/N011759/1, EP/P009565/1, EP/N011759/1, 1204 (2002). EP/P009565/1 and EP/K030094/1], ERC (QBox), QuantERA J. M. Vogels, K. Xu, C. Raman, J. R. Abo-Shaeer, and W. Ket- terle, Experimental Observation of the Bogoliubov Transforma- (NAQUAS, EPSRC Grant No. EP/R043396/1), AFOSR, tion for a Bose–Einstein Condensed Gas, Phys. Rev. Lett. 88, and ARO. N. N. acknowledges support from Trinity Col- 0604 (2002). lege (Cambridge), the David and Lucile Packard Founda- N. Katz, J. Steinhauer, R. Ozeri, and N. Davidson, Beliaev

tion, the Alfred P. Sloan Foundation, and NSF CAREER Damping of Quasiparticles in a Bose–Einstein Condensate, (1945324). C. E. acknowledges support from Jesus Col- Phys. Rev. Lett. 89, 2204 (2002). lege (Cambridge). T. A. H. acknowledges support from the L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubi- EU Marie Skłodowska-Curie program [Grant No. MSCA-IF- zolles, L. D. Carr, Y. Castin, and C. Salomon, Formation of

a Matter-Wave Bright Soliton, Science 296, 12 (2002). 20 840081]. Z. H. acknowledges support from the Royal K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, Society Wolfson Fellowship. N. R. C. acknowledges the sup- Formation and propagation of matter-wave soliton trains, Na- port of the Simons Foundation. ture 417, 1 (2002).

I. Coddington, P. Engels, V. Schweikhard, and E. A. Cor-

nell, Observation of Tkachenko Oscillations in Rapidly Rotat- ing Bose–Einstein Condensates, Phys. Rev. Lett. 91, 1004 (2003). ∗ These two authors contributed equally. V. Bretin, P. Rosenbusch, F. Chevy, G. V. Shlyapnikov, and † To whom correspondence should be addressed: J. Dalibard, Quadrupole Oscillation of a Single-Vortex Bose– A. L. Fetter and J. D. Walecka, Quantum Theory of Many- Lett. 90, 1004 (2003). Particle Systems (Dover Publications, 2003). P. Engels, C. Atherton, and M. A. Hoefer, Observation of Fara

Frequently Asked Questions

What is this project about?

This project covers practical implementation and research aspects of the topic using AI/ML techniques.

This section provides additional detailed analysis and supporting information derived from the research paper content to ensure comprehensive coverage of the topic with expanded discussion on key concepts, methods, and findings presented in the source material.

Speaker Verification Xvector