Observation of quasiparticle pair-production and quantum entanglement in atomic quantum gases quenched to an attractive interaction
OObservation of quasiparticle pair-production and quantum entanglement in atomicquantum gases quenched to an attractive interaction
Cheng-An Chen, Sergei Khlebnikov,
1, 2 and Chen-Lung Hung
1, 2, ∗ Department of Physics and Astronomy, Purdue University, West Lafayette, IN 47907, USA Purdue Quantum Science and Engineering Institute,Purdue University, West Lafayette, IN 47907, USA (Dated: February 23, 2021)We report observation of quasiparticle pair-production and characterize quantum entanglementcreated by a modulational instability in an atomic superfluid. By quenching the atomic interactionto attractive and then back to weakly repulsive, we produce correlated quasiparticles and monitortheir evolution in a superfluid through evaluating the in situ density noise power spectrum, whichessentially measures a ‘homodyne’ interference between ground state atoms and quasiparticles ofopposite momenta. We observe large amplitude growth in the power spectrum and subsequentcoherent oscillations in a wide spatial frequency band within our resolution limit, demonstratingcoherent quasiparticle generation and evolution. The spectrum is observed to oscillate below aquantum limit set by the Peres-Horodecki separability criterion of continuous-variable states, therebyconfirming quantum entanglement between interaction quench-induced quasiparticles.
Coherent pair-production processes are enabling mech-anisms for entanglement generation in continuous vari-able states [1, 2]. In many-body systems, quasiparti-cle pair-production presents an interesting case, as in-teraction creates entanglement shared among collectivelyexcited interacting particles. Entanglement distributionthrough quasiparticle propagation is a direct manifesta-tion of transport property in a quantum many-body sys-tem [3, 4]. Controlling quasiparticle pair-production anddetecting entanglement evolution thus opens a door toprobing quantum many-body dynamics, enabling funda-mental studies such as information propagation [5, 6],entanglement entropy evolution [7], many-body thermal-ization [8], as well as Hawking radiation of quasiparticlesand thermodynamics of an analogue black hole [9–11].In atomic quantum gases, coherent quasiparticle pair-production can be stimulated through an interactionquench, which results in a rapid change of quasiparti-cle dispersion relation that can project collective exci-tations, from either existing thermal or quantum pop-ulations, into a superposition of correlated quasiparti-cle pairs [12–14]. This has led to prior observation ofSakharov oscillations in a quenched atomic superfluid[13, 15]. An intriguing case occurs when the atomic inter-action is quenched to an attractive value, where quasipar-ticles become unstable. In this case, the early time dy-namics is governed by a modulational instability (MI),which stimulates exponential growth of density waves.This growth leads eventually to wave fragmentation andsoliton formation [16]. Although consequences of MI havebeen observed in a number of recent quantum gas exper-iments [17–21], the early-time evolution itself has onlybeen recently studied [21], and the direct verification ofcoherent quasiparticle pair-production has remained anopen question. ∗ [email protected] In this letter, we demonstrate coherent quasiparti-cle pair-production by inducing MI in a homogeneous2D quantum gas quenched to an attractive interac-tion. Subsequent quasiparticle evolution is monitoredthrough quenching the interaction back to a positivevalue (Fig. 1). Through in situ imaging, we analyze thedynamics of density observables by a method analogous (a) (i) (iii) 𝒌−𝒌 (ii) + k -spaceImaging 𝛿𝑛% 𝒌 I n t e r ac ti on ∆𝜏 𝜏𝑔 )* 𝑔 + (i) (ii) (iii) 𝜏 = ms ms 𝑛 ( 𝜇 m - ) 𝜏 𝑔 - 𝑎% /𝒌 𝑎% 𝒌0 (b) Time(c) (d) (e) FIG. 1. Experiment scheme for quasiparticle pair-productionand detection. (a) A homogeneous 2D superfluid (red square)undergoes an interaction quench protocol from (i) g = g i > g MI < τ ; (ii) A second interaction quench to g = g f > τ ; (iii) In situ density noise in spatialfrequency domain, δn k , is essentially a ‘homodyne’ measure-ment of excitations in opposite momentum states interferingwith ground state atoms. (b-e) Single-shot density imagestaken prior to (b) or after the interaction quench (c-e) andheld for the indicated time τ . Image size: 77 × µ m . a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b k ( µ m − )0246 D e n s it y no i s e po w e r s p ec t r u m FIG. 2. Growth of density noise during the MI period. Den-sity noise power spectra measured before, S ( k ) (open circles),and right after the MI period, S ( k, τ ≈ T = 8 ± C k at g = g i ≈ . k c , below (above) which quasiparticles areexpected to be unstable (stable) at g = g MI ≈ − . to the well-established homodyne detection technique inquantum optics [22–24] and confirm non-classical cor-relations, that is, quantum entanglement in interactionquench-produced quasiparticle pairs.Our analyses are based on the time evolution of in situdensity noise, which is a manifestation of interferencebetween quasiparticle excitations and the ground stateatoms that serve as a coherent local oscillator [25]. InFourier space, the density noise operator can be writtenas δ ˆ n k ≈ √ N (ˆ a † k + ˆ a − k ), where N (cid:29) a ( † ) ± k are the annihilation (creation) operatorsfor ± k single-particle momentum eigenstates. They arerelated to quasiparticle operators ˆ α †± k by the Bogoliubovtransformation.To characterize non-classical correlation betweenquasiparticles, we study the density noise power spec-trum S ( k ) = (cid:104)| δn k | (cid:105) /N , where (cid:104)· · ·(cid:105) denotes ensembleaveraging. Within our resolution limit ( | k | (cid:46) . /µ m),the power spectrum conveniently measures the variancesof two-mode quadrature of quasiparticles of all ± k mo-mentum pairs, ˆ x k + ˆ x − k and ˆ p k − ˆ p − k , where ˆ x k =(ˆ α † k + ˆ α k ) / √ p k = i (ˆ α † k − ˆ α k ) / √ S ( k ), and use ± k to denote opposite momenta. It suffices to prove inseparability or quantum entan-glement in quasiparticles of opposite momenta by violat-ing the Peres-Horodecki separability criterion, which – inthe continuous-variable version [27, 28] and adapted toour case – reads (cid:2) (cid:104) (ˆ x k + ˆ x − k ) (cid:105) + (cid:104) (ˆ p k − ˆ p − k ) (cid:105) (cid:3) ≥
2. Interms of the density power spectrum, this becomes [26] S ( k ) = C k (cid:2) (cid:104) (ˆ x k + ˆ x − k ) (cid:105) + (cid:104) (ˆ p k − ˆ p − k ) (cid:105) (cid:3) ≥ C k , (1)where C k = (cid:15) k /(cid:15) ( k, g ) is a squeezing parameter de-termined by the ratio between the single-particle en-ergy (cid:15) k and the phonon dispersion relation (cid:15) ( k, g ) = (cid:113) (cid:15) k + 2 (cid:126) m ¯ ng(cid:15) k , ¯ n is the mean density, g is the inter-action at the time of the measurement, m is the atomicmass, and (cid:126) is the reduced Planck constant. For non-interacting particles, we would have C k = 1, represent-ing the limit of atomic shot-noise. Proving quasiparticle(phonon) entanglement in a superfluid ( g > C k < g >
0) [26, 29, 30]: S ( k, τ ) = C k (cid:2) N k + ∆ N k cos φ k ( τ ) (cid:3) , (2)where ¯ N k = (cid:104) ˆ α † k ˆ α k (cid:105) + (cid:104) ˆ α †− k ˆ α − k (cid:105) is the mean phononnumber in ± k modes, while ∆ N k and φ k ( τ ) are relatedto the amplitude and argument of the pair-correlationobservable (cid:104) ˆ α † k ˆ α †− k (cid:105) (and (cid:104) ˆ α k ˆ α − k (cid:105) ), respectively. Thepair-correlation observable oscillates as time evolves dueto a dynamical phase factor φ k ( τ ) accumulating betweenphonons of opposite momenta. Violating Eq. (1) is equiv-alent to having ∆ N k > ¯ N k , resulting in maximal two-mode squeezing S ( k ) /C k < φ k ≈ (2 l + 1) π andanti-squeezing S ( k ) /C k > φ k ≈ lπ ( l is an integer)– a key entanglement signature that we demonstrate inthis letter.To carry out the experiment, we prepare uniform su-perfluid samples formed by N ≈ . × nearly pureBose-condensed cesium atoms loaded inside a quasi-2Dbox potential, which compresses all atoms in the har-monic ground state along the imaging ( z -) direction[21] with l z = 184 nm being the harmonic oscillatorlength. A time-of-flight measurement estimates the sam-ple temperature T (cid:46) n ≈ /µ m is approximately uniform within a horizon-tal box size of ≈ × µ m . The interaction strength ofthe quasi-2D gas g = √ πa/l z is controlled by the s-wavescattering length a via a magnetic Feshbach resonance[31], giving an initial interaction strength g = g i ≈ . g ( δg ≈ ± . . g MI ≈ − .
026 and holding for a short time ∆ τ ≈ ∼ g f ≈ . τ before we perform in situabsorption imaging. Figures 1(b-e) show sample imagesmeasured before and after we initiate the quench pro-tocol. We evaluate δn k for each sample through Fourieranalysis [32] and obtain their density noise power spectra.Typically around 50 experiment repetitions are analyzedfor each hold time τ . Each power spectrum has beencarefully calibrated with respect to the atomic shot-noisemeasured from high temperature normal gases [26, 32].To confirm quasipaticle generation during the MI pe-riod, in Fig. 2 we compare the density noise power spectrameasured before and immediately after the MI period,that is, for hold time τ = 0. Before MI, the initial spec-trum S ( k ) is mostly below the atomic shot-noise dueto low temperature T (cid:46) C k <
1. Excessive noise in k (cid:46) . /µ mmay be due to technical heating in the box potential.After the MI time period ∆ τ , we indeed find signifi-cant increase in the density noise S ( k, > k (cid:46) k c = 2 (cid:112) ¯ n | g MI | ≈ . /µ m in the instability bandshowing hyperbolic growth due to a purely imaginary dis-persion relation (cid:15) ( k, g MI ) [21], and k (cid:38) k c in the stableregime showing quasiparticle production due to variationof the interaction strength [13].Our measured spectra can be well-captured by amodel S ( k,
0) = e − Γ k ∆ τ S coh ( k ) + S inc ( k ), which incor-porates the Bogoliubov theory and quasiparticle dis-sipation through coupling to a single-particle bath[26]. The first term is a damped coherent sig-nal S coh ( k ) = S ( k )[1 + (cid:15) ( k,g i ) − (cid:15) ( k,g MI ) (cid:15) ( k,g MI ) sin (cid:15) ( k,g MI )∆ τ (cid:126) ][13, 21], while the second term is referred to as anincoherent signal S inc ( k ) = { η − Γ k Γ k +4 (cid:15) ( k,g MI ) / (cid:126) [1 − e − Γ k ∆ τ (cos (cid:15) ( k,g MI )∆ τ (cid:126) − (cid:15) ( k,g MI ) (cid:126) Γ k sin (cid:15) ( k,g MI )∆ τ (cid:126) )]+ η + (1 − e − Γ k ∆ τ ) } , where η ± = 1 ± (cid:15) k /(cid:15) ( k, g MI ) . Our theory fits(solid curves in Fig. 2) suggest a k -dependent dissipationrate Γ k ∼ . (cid:15) k / (cid:126) [26], of the same order of magnitudeas the decay rate extracted from the subsequent time-evolution measurements at g = g f (Fig. 3).To demonstrate phase coherence and pair-correlationin these interaction quench-induced quasiparticle pairs,we plot the complete time and momentum dependenceof the density noise power spectrum S ( k, τ ), as shown inFig. 3(a). Here, oscillatory behavior is clearly visible overthe entire spectrum. The oscillations are a manifestationof the interference between coherent quasiparticles of op-posite momenta ± k , as suggested by Eq. (2), with therelative phase winding up in time as φ k ( τ ) = 2 γ k,f τ + φ ,where γ k,f = (cid:15) ( k, g f ) / (cid:126) is the expected Bogoliubovphonon frequency and φ is an initial phase difference. InFig. 3(b-d), we plot S ( k, ˜ τ ) in the rescaled time ˜ τ = γ k,f τ and confirm that all spectra oscillate synchronously witha time period ≈ π , thus validating the phonon interfer- k ( µ m − )051015 τ ( m s ) S ( k, τ ) (a) 0135 (c) With MI ∆ τ ≈ ms ˜ τ = γ k,f τ D e n s it y no i s e po w e r s p ec t r u m S (d) With MI ∆ τ ≈ ms No MI ∆ τ = 0 A k (e)0.00.51.0 φ / π (f)0.5 1.0 1.5 2.0 2.5 k ( µ m − ) ˜ Γ k (g) FIG. 3. Coherent oscillations in the density noise powerspectrum. (a) Full evolution of the power spectrum S ( k, τ )with ∆ τ ≈ k -space. (b-d) Synchronized oscillations of S ( k, ˜ τ )plotted in the rescaled time unit ˜ τ = γ k,f τ for various k ≈ (1 , . , . , . , . , . /µ m (Gray circles from brightto dark). Horizontal dashed lines mark the atomic shot-noise limit. Solid lines are sinusoidal fits. Fitted amplitude A k , phase offset φ , and decay rate ˜Γ k from samples with∆ τ ≈ ence picture. For comparison, we also plot the evolutionof samples with a direct interaction quench from g i to g f without an MI period (∆ τ = 0). Oscillations in S ( k, ˜ τ )can also be observed, albeit with smaller amplitudes andphase offsets φ ≈
0, as these oscillations result solelyfrom the interference of in-phase quasiparticle projectionsfrom suddenly decreasing the Bogoliubov energy [13]. Ineither case, with or without MI, we observe that phasecoherence is lost in a few cycles and the density noisespectra reach new steady-state values.To quantify phase coherence and dissipation at final g = g f , we perform simple sinusoidal fits S ( k, ˜ τ ) = S f − S o e − ˜Γ k ˜ τ − A k e − ˜Γ k ˜ τ cos(2˜ τ + φ ) to the data to ex-tract ( A k , φ , ˜Γ k ), as shown in Fig. 3(e-g) (the steady-state values S f and S o are not shown). The larger oscil-lation amplitudes A k found in samples with ∆ τ ≈ φ (cid:38) π/ k (cid:38) . /µ m in Fig. 3(f),which is coherently accumulated during the MI period.Furthermore, in Fig. 3(g), we observe a nearly con- P honon s p ec t r u m ˜ S = S / C k (b)0.52.04.0 (a)0 5 ˜ τ = γ k,f τ ˜ S m i n (d)1.0 1.5 2.0 2.5 k ( µ m − )012 ¯ N k , ∆ N k (e) FIG. 4. Testing two-mode squeezing and quantum entan-glement in the phonon basis. (a-c) Rescaled phonon spec-trum ˜ S ( k, ˜ τ ) for k ≈ (1 . , . , . , . , . , . /µ m (filled cir-cles from bright to dark), evaluated using data as shown inFigs. 3(b-d). Solid curves are guides to the eye. (d) Firstminima ˜ S min in the phonon spectra of various wavenumber k ,at ∆ τ ≈ N k (filled symbols) and pair-correlation ampli-tude ∆ N k (open symbols) extracted using the first minimaand maxima identified in (a, circles), (b, squares), and (c,triangles), respectively. Blue (red) shaded areas mark the re-gion where ∆ N k > ¯ N k (∆ N k < ¯ N k ). Error bars representstatistical errors. stant decay rate ˜Γ k ≈ . ± .
08 at k (cid:38) . /µ m forthese MI-induced oscillations. This is close to the de-cay rate ˜Γ k ≈ . ± .
04 in samples without an MIperiod (∆ τ = 0), suggesting that the short MI dynamicsdoes not heat up the sample significantly to increase thephonon dissipation rate.We now focus on identifying a key signature of non-classical correlations. To search for entanglement in thefinal phonon basis, we evaluate the squeezing parameter C k = (cid:15) k /(cid:15) ( k, g f ) at g = g f and plot the rescaled phononspectra ˜ S ( k, ˜ τ ) = S ( k, ˜ τ ) /C k , as shown in Figs. 4(a-c). Inthis basis, the phonon spectra at momenta k (cid:38) . /µ mcan be observed to oscillate above and below the rescaledquantum limit ˜ S = 1, showing signatures of two-mode squeezing and anti-squeezing as time evolves. The firstminimum ˜ S min identified at various momenta k is plot-ted in Fig. 4(d), in which we find that ˜ S min violates theinequality Eq. (1) in a wider range for the MI samplewith ∆ τ ≈ τ . The strongest violation isin the range of 2 . /µ m (cid:46) k (cid:46) . /µ m and has average˜ S min ≈ . <
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We thank M. Kruczenski and Q. Zhou for discussions.This work is supported in part by DOE QuantISED pro-gram (Grant
SUPPLEMENTARY INFORMATIONCalibration procedure
We calibrate our imaging system (effective numerical aperature N.A. ≈ .
35) by using in situ density noise powerspectrum measured from thermal gases confined in the box potential. We first load a 2D gas using the standard loadingprocedure described in [21]. We then quench the scattering length to a large and negative value at a ≈ − a , where a is the Bohr radius, and hold for >
100 ms. During this time, the 2D sample suffers three-body recombination lossand heating, with N ≈ ,
000 atoms remaining in the box and at a temperature up to T = 170 ±
15 nK independentlymeasured in time-of-flight measurements. Following this heating procedure, thermal de Broglie wavelength λ dB < . µ m is much smaller than the image resolution. We record the density fluctuations by quenching the scatteringlength back to a nearly non-interacting value and perform in situ absorption imaging. We evaluate the density noisepower spectrum of the high temperature normal gases, which can be used to accurately determine the modulationtransfer function of our imaging system [32] that terminates at k (cid:38) . /µ m. All the density noise power spectrapresented in this letter are normalized by the atomic shot-noise calibrated modulation transfer function to removesystematic image aberrations. Separability criterion
In this section, we derive the separability criterion for quasiparticles of opposite momenta ± k using the densityobservables discussed in the main text. We define ˆ α † k , ˆ α k as the quasiparticle creation and annihilation operators.The associated coordinate and momentum operators can then be written asˆ x = 1 √ α k + ˆ α † k )ˆ x = 1 √ α − k + ˆ α †− k )ˆ p = i √ α † k − ˆ α k )ˆ p = i √ α †− k − ˆ α − k ) . These have canonical commutation relations (with the reduced Planck constant (cid:126) = 1). Next, we consider variancesof ˆ u = ˆ x + ˆ x = ˆ y + ˆ y ˆ v = ˆ p − ˆ p = i (ˆ y − ˆ y ) , where ˆ y and ˆ y are non-Hermitian but momentum-conserving operatorsˆ y = 1 √ α k + ˆ α †− k )ˆ y = 1 √ α − k + ˆ α † k ) , which commute with each other. By momentum conservation (in the states we consider), we have (cid:104) ˆ y (cid:105) = (cid:104) ˆ y (cid:105) = 0,and (cid:104) ˆ u (cid:105) = 2 (cid:104) ˆ y ˆ y (cid:105) = (cid:104) ˆ α k ˆ α − k + ˆ α k ˆ α † k + ˆ α †− k ˆ α − k + ˆ α †− k ˆ α † k (cid:105)(cid:104) ˆ v (cid:105) = 2 (cid:104) ˆ y ˆ y (cid:105) = (cid:104) ˆ u (cid:105) The total variance is (cid:104) ˆ u (cid:105) + (cid:104) ˆ v (cid:105) = 2 (cid:104) ˆ α k ˆ α − k + ˆ α k ˆ α † k + ˆ α †− k ˆ α − k + ˆ α †− k ˆ α † k (cid:105) . For a separable state, by the theorem proven in [27, 28], the total variance must satisfy the following inequality[ (cid:104) ˆ u (cid:105) + (cid:104) ˆ v (cid:105) ] sep ≥ . (S4)If we take a thermal state of ˆ α ± k , ˆ α †± k for an example, such as when a quenched superfluid has fully equilibrated, wehave [ (cid:104) ˆ u (cid:105) + (cid:104) ˆ v (cid:105) ] therm = 2( (cid:104) ˆ α k ˆ α † k (cid:105) + (cid:104) ˆ α †− k ˆ α − k (cid:105) ) = 2(2 n B + 1) > , satisfying the separability criterion. Here n B > S ( k ). We begin by considering the density noise and phase of the order parameter in the momentumspace. The density noise is calculated as δn ( x , t ) = n ( x , t ) − ¯ n at an arbitrary time t , where ¯ n is the average density ofthe superfluid, assumed uniform and time-independent. We define the Fourier components ˆ n k and ˆ θ k of the densitynoise and phase operators as follows: δ ˆ n ( x , t ) = 1 V (cid:88) k (cid:54) =0 ˆ n k ( t ) e i k · x (S5)ˆ θ ( x , t ) = 1 V (cid:88) k (cid:54) =0 ˆ θ k ( t ) e i k · x = 1 V (cid:88) k (cid:54) =0 ˆ θ † k ( t ) e − i k · x , (S6)where V is the volume of the gas. The commutation relation is [ˆ n ( x ) , − ˆ θ ( x (cid:48) )] = iδ ( x − x (cid:48) ), where δ ( x ) is the Diracdelta-function. In Fourier components, it corresponds to [ˆ n k , − ˆ θ † k (cid:48) ] = iV δ k , k (cid:48) , where δ k , k (cid:48) is the Kronecker delta.In Fourier space, the density and phase operators are related to the single-particle operators ˆ a ( † ) ± k byˆ n k = √ N (ˆ a k + ˆ a †− k ) , (S7)ˆ θ k = iV √ N (ˆ a †− k − ˆ a k ) , (S8)where N = ¯ nV is the total particle number, assumed to be largely accounted for by ground state atoms. Through theBogoliubov transformation, we can further express the same quantities in terms of the quasiparticle operators ˆ α ( † ) ± k .We have ˆ n k = (cid:112) N C k (ˆ α k + ˆ α †− k ) (S9) − ˆ θ † k = iV √ N C k ( − ˆ α − k + ˆ α † k ) (S10)where C k = (cid:15) k /(cid:15) ( k ), (cid:15) ( k ) = (cid:113) (cid:15) k + 2 (cid:126) m ¯ ng(cid:15) k is the Bogoliubov energy (at the interaction g when the measurementtakes place), (cid:15) k = (cid:126) | k | m is the single particle energy, and m is the atomic mass.Using Eq. (S9), we can relate the density noise power spectrum to the variances of two-mode quadrature S ( k ) = (cid:104) ˆ n † k ˆ n k (cid:105) N = C k (cid:2) (cid:104) u (cid:105) + (cid:104) v (cid:105) (cid:3) . (S11)Thus, in a separable state we would have S ( k ) = (cid:104) ˆ n † k ˆ n k (cid:105) sep N ≥ C k , (S12)where we have applied the inequality (S4). Evolution of quasiparticles coupled to a bath
Both in the present study and former experiments [13, 15], quasiparticles are observed to dissipate. In this section, wecalculate the coherent evolution of density fluctuations following an interaction quench in the presence of dissipation.We consider an instantaneous quench after which the value of the interaction parameter g can be either positive ornegative. At the level of the quadratic Hamiltonian, we can consider each ( k , − k ) pair of modes separately. Thecorresponding system Hamiltonian isˆ H ( k , − k ) = 1 V (cid:20) n(cid:15) k ˆ θ − k ˆ θ k + 1¯ n (cid:16) (cid:15) k n ˜ g (cid:17) ˆ n − k ˆ n k (cid:21) , (S13)where ¯ n is the average dentity, which is assumed uniform and time-independent, and ˜ g = (cid:126) m g . Note that (cid:126) ˆ θ k is thecanonical momentum conjugate to ˆ n − k . In what follows, we set (cid:126) = 1.The quadratic approximation neglects nonlinear effects, which can change population of a given pair of modes.For near-equilibrium states, one may consider taking nonlinear terms into account via a version of the Boltzmannequation, written in terms of Bogoliubov quasiparticles. For negative g , however, the uniform equilibrium is unstable,and the usual definition of a quasiparticle does not apply. To qualitatively describe the effect of nonlinear terms inthis case, we consider a model in which the loss (gain) of particles in a given momentum mode is due to Markovianquantum noise. The model is specified by stating which of the system operators the noise couples to. Here, we assumethose to be the single particle operators ˆ a k , ˆ a † k .To compute the effect of the noise on the evolution of the system, we use the quantum Langevin equation (QLE) inthe form presented in Ref. [40]. In this formalism, each noise channel is described by time-dependent “in” operators,subject to commutation relations [ˆ b in ( t ) , ˆ b † in ( t (cid:48) )] = δ ( t − t (cid:48) ) (S14)(with other pairwise commutators equal to zero). If there is only one such channel, the evolution of any operator ˆ A characterizing the system is given by the QLE as follows [40]:˙ˆ A = − i [ ˆ A, ˆ H ] − [ ˆ A, ˆ O † ] (cid:26)
12 Γ ˆ O + √ Γˆ b in ( t ) (cid:27) + (cid:26)
12 Γ ˆ O † + √ Γˆ b † in ( t ) (cid:27) [ ˆ A, ˆ O ] , (S15)where ˆ O is the system operator that couples to the noise, and Γ is the strength (width) of that coupling.When there are two or more independent noise channels, one can generalize Eq. (S15) by defining separate ˆ b in , ˆ O ,and Γ for each channel and summing up the corresponding noise terms in the equation. In the present case, we havetwo noise channels, which couple to the system operatorsˆ O = ˆ a k , ˆ O = ˆ a − k . We call the corresponding noise operators ˆ b in and (cid:101) b in , respectively. In what follows, we assume that the two widthsare equal: Γ = Γ = Γ. We allow the width to depend on k = | k | , even though for brevity we do not attach asubscript k to Γ.We now consider evolution of the system with the Hamiltonian (S13) after an instantaneous quench of the coupling˜ g from a positive initial value ˜ g i to a final value ˜ g f ; the latter can be positive or negative. It is convenient to define,instead of the phase component ˆ θ k , a related variable ˆΘ k ≡ − n ˆ θ k . Then, the QLEs for ˆΘ k and ˆ n k can be writtenin the matrix form as (cid:32) ˙ˆΘ k ˙ˆ n k (cid:33) = (cid:18) − Γ (cid:15) k + 2¯ n ˜ g − (cid:15) k − Γ (cid:19) (cid:18) ˆΘ k ˆ n k (cid:19) − √ N Γ (cid:18) i [ˆ b in ( t ) − (cid:101) b † in ( t )]ˆ b in ( t ) + (cid:101) b † in ( t ) (cid:19) . (S16)We search for solution as an expansion in the eigenvectors of the matrix in (S15), that is, (cid:18) ˆΘ k ˆ n k (cid:19) = ˆ c ( t ) (cid:18) − i ω k (cid:15) k (cid:19) + ˆ c ( t ) (cid:18) i ω k (cid:15) k (cid:19) , (S17)where ω k = (cid:113) (cid:15) k + 2¯ n ˜ g f (cid:15) k is Bogoliubov’s quasiparticle frequency, and ˜ g f is the coupling constant after the quench. For negative ˜ g f , there is aninstability band, corresponding to those k for which the argument of the square root is negative. For these k , ω k isimaginary; by convention we choose it to have a negative imaginary part.Equations satisfied by ˆ c , are ˙ˆ c = λ + ˆ c − √ N Γ ζ + ( t ) , ˙ˆ c = λ − ˆ c − √ N Γ ζ − ( t ) , where λ ± = −
12 Γ ± iω k are the eigenvalues of the evolution matrix, andˆ ζ ± ( t ) = 12 (cid:18) ∓ (cid:15) k ω k (cid:19) ˆ b in ( t ) + 12 (cid:18) ± (cid:15) k ω k (cid:19) (cid:101) b † in ( t ) . (S18)According to Eq. (S17), ˆ n k ( t ) = ˆ c ( t ) + ˆ c ( t ). The general solution for it isˆ n k ( t ) = ˆ c (0) e λ + t + ˆ c (0) e λ − t − √ N Γ (cid:90) t (cid:104) ˆ ζ + ( t (cid:48) ) e λ + ( t − t (cid:48) ) + ˆ ζ − ( t (cid:48) ) e λ − ( t − t (cid:48) ) (cid:105) dt (cid:48) . (S19)The operator constants ˆ c (0) and ˆ c (0) are determined from the initial conditions. Let ˆ β † k , ˆ β k be the creation andannihilation operators of Bogoliubov’s quasiparticles before the quench. Then, at t → − ,ˆΘ k (0 − ) = iV (cid:115) NC (cid:48) k ( ˆ β k − ˆ β † k ) , ˆ n k (0 − ) = (cid:113) N C (cid:48) k ( ˆ β k + ˆ β †− k ) , where C (cid:48) k ≡ (cid:15) k / Ω k , and Ω k = (cid:113) (cid:15) k + 2¯ n ˜ g i (cid:15) k is the initial-state quasiparticle frequency. Equating these ˆΘ k , ˆ n k to their t → + limits, obtained from Eq. (S17), wefind ˆ c , (0) = (cid:112) N C (cid:48) k (cid:20) ˆ β k (cid:18) ∓ Ω k ω k (cid:19) + ˆ β †− k (cid:18) ± Ω k ω k (cid:19)(cid:21) , where the upper signs are for ˆ c , and the lower for ˆ c .For computation of the power spectrum of density fluctuations, we will need both ˆ n k and ˆ n − k . The former isobtained by substituting the above expressions for ˆ c , (0) into Eq. (S19). For the latter, we need to exchange k with − k in the expressions for ˆ c , (0). In addition, recalling that untilded noise operators refer to mode number k , andtilded ones to − k , we need to exchange the tilded and untilded operators in Eq. (S18).The quasiparticle operators ˆ β , ˆ β † can in principle depend on noise, but only on that part of it that is representedby ˆ b in ( t ), (cid:101) b in ( t ) with t <
0. None of those appear in the solution (S19). Thus, we can consider ˆ c , (0) on the one hand,and the noise operators appearing in Eq. (S19) on the other to be distributed independently. Further, we assumethat all the one-point functions, i.e., (cid:104) ˆ β (cid:105) , (cid:104) ˆ b in (cid:105) , (cid:104) (cid:101) b in (cid:105) , of these operators are zero. As a result, the power spectrum willconsist of two separate parts, S ( k ) = (cid:104) ˆ n − k ˆ n k (cid:105) N = (cid:104) ˆ n − k ˆ n k (cid:105) coh N + (cid:104) ˆ n − k ˆ n k (cid:105) inc N , (S20)which we refer to as coherent and incoherent. The former corresponds to the first two terms in (S19) and the latterto the last term, due entirely to the noise. Outside the instability band, where ω k is real, the coherent part decaysat large times, while the incoherent part goes to a constant value. For k in the instability band, both parts can beamplified; that is the case when | Im[ ω k ] | > Γ /
2, where Im[ . ] denotes imaginary part.Finally, we assume that, among the two-point functions of the aforementioned operators, the only nonzero ones are (cid:104) ˆ β † k ˆ β k (cid:105) = N k , (cid:104) ˆ β k ˆ β † k (cid:105) = N k + 1 , (cid:104) ˆ b in ( t )ˆ b † in ( t (cid:48) ) (cid:105) = (cid:104) (cid:101) b in ( t ) (cid:101) b † in ( t (cid:48) ) (cid:105) = δ ( t − t (cid:48) ) . The latter is a simplifying assumption to the effect that the noise is in vacuum. One may reasonably hope it to workfor sufficiently short evolution times. With these assumptions, we obtain (cid:104) ˆ n − k ˆ n k (cid:105) coh N = e − Γ t S coh ( k ) = e − Γ t S i ( k ) (cid:20) (cid:18) Ω k ω k − (cid:19) sin ω k t (cid:21) , k ( µ m − )0246 D e n s it y no i s e po w e r s p ec t r u m Fig. SM1. Fitting the measured density noise power spectra (symbols) right after the MI period with ∆ τ ≈ S inc ( k ). Vertical dotted line marks the wavenumber k c , below (above) which quasiparticles are expectedto be unstable (stable). where S i ( k ) = C (cid:48) k (2 N k + 1) is the initial density noise power spectrum and (cid:104) ˆ n − k ˆ n k (cid:105) inc N = S inc ( k )= 12 (cid:26)(cid:18) − (cid:15) k ω k (cid:19) ΓΓ + 4 ω k (cid:2) Γ (cid:0) − e − Γ t cos 2 ω k t (cid:1) + 2 ω k e − Γ t sin 2 ω k t (cid:3) + (cid:18) (cid:15) k ω k (cid:19) (cid:0) − e − Γ t (cid:1)(cid:27) We compare the above theory calculation with the density noise power spectra measured after we quench theinteraction strength to g = g MI and hold for a time t = ∆ τ , as shown in Fig. 2. We assume a k -dependent coupling rateΓ k = η(cid:15) k and set η as a fit parameter. We then use the measured initial spectrum S i ( k ) and all other experimentallymeasured quantities to evaluate the power spectrum S ( k ) = e − Γ k ∆ τ S coh ( k ) + S inc ( k ). As shown in Fig. SM1, we finda reasonable fit with η ≈ . τ ≈≈