Dynamical excitation of maxon and roton modes in a Rydberg-Dressed Bose-Einstein Condensate
DDynamical excitation of maxon and roton modes in a Rydberg-DressedBose-Einstein Condensate
Gary McCormack , Rejish Nath and Weibin Li School of Physics and Astronomy, and Centre for the Mathematics and TheoreticalPhysics of Quantum Non-Equilibrium Systems, University of Nottingham, NG7 2RD, UK Indian Institute of Science Education and Research, Pune, 411008, India
We investigate the dynamics of a three-dimensional Bose-Einstein condensate of ultracold atomicgases with a soft-core shape long-range interaction, which is induced by laser dressing the atoms to ahighly excited Rydberg state. For a homogeneous condensate, the long-range interaction drasticallyalters the dispersion relation of the excitation, supporting both roton and maxon modes. Rotons aretypically responsible for the creation of supersolids, while maxons are normally dynamically unstablein BECs with dipolar interactions. We show that maxon modes in the Rydberg-dressed condensate,on the contrary, are dynamically stable. We find that the maxon modes can be excited throughan interaction quench, i.e. turning on the soft-core interaction instantaneously. The emergenceof the maxon modes is accompanied by oscillations at high frequencies in the quantum depletion,while rotons lead to much slower oscillations. The dynamically stable excitation of the roton andmaxon modes leads to persistent oscillations in the quantum depletion. Through a self-consistentBogoliubov approach, we identify the dependence of the maxon mode on the soft-core interaction.Our study shows that maxon and roton modes can be excited dynamically and simultaneously byquenching Rydberg-dressed long-range interactions. This is relevant to current studies in creatingand probing exotic states of matter with ultracold atomic gases.
I. INTRODUCTION
Collective excitations induced by particle-particle in-teractions play an important role in the understandingof static and dynamical properties of many-body sys-tems. The ability to routinely create and precisely con-trol properties of ultracold atomic gases opens excitingprospects to manipulate and probe collective excitations.In weakly interacting Bose-Einstein condensates (BECs)with s-wave interactions [1–4], phonon excitations reducethe condensate density, giving rise to quantum deple-tion [5]. It has been shown [6] that quantum deple-tion can be enhanced by increasing the s-wave scatteringlength through Feshbach resonances [7, 8]. By dynami-cally changing the s-wave scattering length [9], phononexcitations can alter the quantum depletion, the momen-tum distribution [10], correlations [11], contact [12, 13],and statistics [14] of the condensate. Moreover thephonon induced quantum depletion plays a vital role inthe formation of droplets in BECs [15].When long-range interactions are introduced, the dis-persion relation corresponding to the quasiparticle spec-trum of a BEC is qualitatively different, where the ex-citation energies of the collective modes depend non-monotonically on the momentum. Previously BECs withdipole-dipole interactions have been extensively exam-ined [16–22]. In two-dimensional (2D) dipolar BECs [23], roton and maxon modes emerge, where roton (maxon)modes correspond to local minima (maxima) in the dis-persion relation. The strength of dipolar interactionscan be tuned by either external electric or magneticfields [19]. When instabilities of roton modes are trig-gered, a homogeneous BEC undergoes density modula-tions such that a supersolid phase could form. The ex-istence of roton modes has been supported by a recent
FIG. 1. (color online)
Soft-core interaction and quenchscheme . (a) The soft-core interaction as a function of theinteratomic distance r . Energy is scaled by R /C with R and C to be the soft-core radius and dispersion coefficient.The interaction is constant when r (cid:28) R , and becomes a vdWtype when r (cid:29) R . (b) Fourier transformation of the soft-coreinteraction. The minimum of the interaction locates at k r ≈ π/ R , where the interaction is attractive. (c) The quenchscheme. A weakly interacting BEC with s-wave interactionsis first prepared. The laser dressing is applied at g ≤
0, whichinduces the soft-core interaction. experiment [24]. Maxon modes, on the other hand, nor-mally appear at lower momentum states [23]. It wasshown however that the maxon modes in dipolar BECsare typically unstable and decay rapidly through the Be-liaev damping [20, 21].Strong and long-range interactions are also found ingases of ultracold Rydberg atoms [25–29]. Rydbergatoms are in highly excited electronic states and interactvia long-range van der Waals (vdW) interactions. Thestrength of the vdW interaction is proportional to N with N to be the principal quantum number in the Ryd-berg state. For large N (current experiments exploit N typically between 30 and 100), the interaction between a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b two Rydberg atoms can be as large as several MHz at aseparation of several micrometers [30]. However lifetimesin Rydberg states are typically 10 ∼ µ s, which is notlong enough to explore spatial coherence. As a result, Rydberg-dressing , in which a far detuned laser coupleselectronic ground states to Rydberg states, is proposed.The laser coupling generates a long-range, soft-core typeinteraction between Rydberg-dressed atoms [31–40]. Thecoherence time and interaction strength can be controlledby the dressing laser [35]. With this dressed interaction,interesting physics, such as magnets [41], transport [42],supersolids [31, 34, 43, 44], etc, have been studied. Signa-tures of the dressed interaction have been experimentallydemonstrated with atoms trapped in optical lattices andoptical tweezers [41, 45].In this paper, we study excitations of roton and maxonmodes in three dimensional (3D) Rydberg-dressed BECsin free space at zero temperature. Three dimensionaluniform trapping potential of ultracold atoms have beenrealized experimentally [46]. When the soft-core inter-action is strong, both the roton and maxon modes arefound in the dispersion relation of the collective excita-tions. Starting from a weakly interacting BEC, roton andmaxon modes are dynamically excited by instantaneouslyswitching on the Rydberg-dressed interaction. Through aself-consistent Bogoliubov calculation, we show that theroton and maxon modes leads to non-equilibrium dynam-ics, where the quantum depletion exhibits slow and fastoscillations. Through analyzing the Bogoliubov spectra,we identify that the slow oscillation corresponds to theexcitation of the roton modes, while the fast oscillationcomes from the excitation of the maxon modes. The de-pendence these modes have on the quantum depletion inthe long time limit is determined analytically and numer-ically.The paper is organized as follows. In Sec. II, the Hamil-tonian of the system and properties of the soft-core in-teraction are introduced. Bogoliubov methods, that arecapable to study static as well as dynamics of the ex-citation, are presented. In Sec. III, dispersion relationsare found using the static Bogoliubov calculation, whereroton and maxon modes are identified. We then ex-amine the dynamics of the quantum depletion due tothe interaction quench. Excitations of the roton andmaxon modes are studied using a self-consistent Bogoli-ubov method. The asymptotic behavior of the BEC atlong times is also explored. Finally, with Sec. IV we con-clude our work.
II. HAMILTONIAN AND METHODA. Hamiltonian of the Rydberg-dressed BEC
We consider a uniform 3D Bose gas of N atoms thatinteract through both s-wave and soft-core interactions. The Hamiltonian of the system is given by ( (cid:126) ≡ H = (cid:90) ψ † ( r ) (cid:18) − ∇ m − µ (cid:19) ψ ( r ) d r + 12 (cid:90) ψ † ( r ) ψ † ( r (cid:48) )˜ g ( r − r (cid:48) ) ψ ( r ) ψ ( r (cid:48) ) d r d r (cid:48) , (1)where ψ ( r ) is the annihilation operator of the bosonicfield, µ is the chemical potential, m is the mass of aboson, and ∇ is the 3D nabla operator on coordinate r = { x, y, z } . The interaction potential is described by˜ g ( r − r (cid:48) ) = g δ ( r ) + ˜ V ( r − r (cid:48) ), where g = 4 πa s /m isthe short-range contact interaction controlled by the s-wave scattering length a s [3]. ˜ V ( r − r (cid:48) ) is the long-rangesoft-core interaction,˜ V ( r − r (cid:48) ) = C R + | r − r (cid:48) | , (2)where C is the strength of the dressed interaction po-tential and R is the soft-core radius [35]. Both theseparameters can be tuned independently by varying thedressing laser [35]. The interaction potential saturatesto a constant, i.e. ˜ V ( r ) → C /R when | r | (cid:28) R , andapproaches to a vdW type at distances of | r | (cid:29) R , i.e.˜ V ( r ) → C / | r | . An example of the soft-core potentialis shown in Fig. 1(a). The Fourier transformation of thesoft-core potential is V ( k ) = U f ( k ), where U = C /R determines the strength and f ( k ) has an analytical form f ( k ) = 2 π e − kR kR (cid:34) e − kR − (cid:32) π − √ kR (cid:33)(cid:35) , which characterizes the momentum dependence of theinteraction. Though the interaction is repulsive in realspace, i.e. ˜ V ( r ) >
0, it contains negative regions in mo-mentum space, as shown in Fig. 1(b). The negative partof V ( k ) appears at momentum around kR ∼ π/
3. Pre-viously, it was shown that the attractive interaction inmomentum space is crucially important to the formationof roton instabilities [23].
B. Time-independent Bogoliubov approach
In momentum space, we expand the field operators us-ing a plane wave basis, ψ ( r ) = 1 / √ Ω (cid:80) k e i k · r ˆ a k . Themany-body Hamiltonian can be rewritten asˆ H = (cid:88) k ( (cid:15) k − µ )ˆ a † k ˆ a k + (cid:88) q , k , k (cid:48) g k
2Ω ˆ a † k + q ˆ a † k (cid:48) − q ˆ a k ˆ a k (cid:48) , (3)where ˆ a † k (ˆ a k ) is the creation (annihilation) operator ofthe momentum state k , and Ω volume of the BEC. Thekinetic energy is (cid:15) k = k / m with k = | k | , while theFourier transformation of the atomic interaction ˜ g ( r − r (cid:48) )is given by g k = g + V ( k ).For a homogeneous condensate and in the stationaryregime, we apply a conventional Bogoliubov approach[47, 48] to study the excitation spectra. At zero tempera-ture we assume a macroscopic occupation in the conden-sate, which allows us to replace ˆ a ≈ √ N with N beingthe number of condensed atoms. We then apply a canon-ical transformation on the bosonic operators of the non-zero momentum states [3], ˆ a k (cid:54) =0 = ¯ u k ˆ b k − ¯ v ∗ k ˆ b †− k where b k (ˆ b †− k ) is the annihilation (creation) operator for bosonicquasiparticles and ¯ u k and ¯ v k are complex numbers suchthat | ¯ u k | −| ¯ v k | = 1, which satisfies the bosonic commu-tation relation [3]. The excitation spectra of the Bogoli-ubov modes for different momentum components givesthe dispersion relation,¯ E k = (cid:112) (cid:15) k [ (cid:15) k + 2 g k n ] , (4)with n = N / Ω being the density of the condensedatoms. The coefficients in the Bogoliubov transforma-tion are [3] ¯ u k = (cid:115) (cid:20) (cid:15) k + g k n ¯ E k + 1 (cid:21) ¯ v k = − (cid:115) (cid:20) (cid:15) k + g k n ¯ E k − (cid:21) . (5)The distribution of the non-condensed atoms is givenby n k = (cid:104) a † k a k (cid:105) = | ¯ v k | . Taking into account contri-butions from all non-condensed components, the quan-tum depletion in the stationary state is evaluated as¯ n d = 1 / Ω (cid:80) k (cid:54) =0 | ¯ v k | . C. Self-consistent Bogoliubov approach for thequench dynamics
The quench of the soft-core interaction consists of twosteps. The system is initially in the ground state of aweakly interacting BEC, i.e. U = 0 when t <
0. Attime t ≥ g k = (cid:26) g when t ≤ g + U f ( k ) when t ≥ . (6)We assume that the s-wave interaction is not affectedduring the quench. Hence we use parameter α = U /g to characterize the strength of the soft-core interactionwith respect to the s-wave interaction.A time-dependent Bogoliubov approach is applied tostudy the dynamics induced by the interaction quench.It is an extension of the conventional Bogoliubov approx-imation, where the canonical transformation becomestime-dependent, ˆ a k (cid:54) =0 ( t ) = u k ( t )ˆ b k − v k ( t ) ∗ ˆ b †− k where u k ( t ) and v k ( t ) are time-dependent amplitudes with therelation | u k ( t ) | − | v k ( t ) | = 1, which preserves thebosonic commutation relation. This approach has beenwidely used to study excitation dynamics in BECs with or without long-range interactions [10–12, 14, 20]. It pro-vides a good approximation when the condensate has notundergone significant depletion.Using the Heisenberg equation of the bosonic opera-tors, we obtain equations of motion of u k ( t ) and v k ( t ), i (cid:20) ˙ u k ( t )˙ v k ( t ) (cid:21) = (cid:20) (cid:15) k + g k n c ( t ) g k n c ( t ) − g k n c ( t ) − (cid:15) k − g k n c ( t ) (cid:21) (cid:20) u k ( t ) v k ( t ) (cid:21) , (7)where n c ( t ) is the time-dependent condensate density.The total density consists of the condensate and deple-tion densities as n = n c ( t ) + n d ( t ) with the total densityof the excitation, i.e. quantum depletion given as n d ( t ) = 1Ω (cid:88) k n k ( t ) , (8)where n k ( t ) ≡ (cid:104) ˆ a † k ˆ a k (cid:105) = | v k ( t ) | is the distribution of allpossible momentum states.For a particle conserving system, both the depleteddensity and the condensate density are time dependent.In practice, the quantum depletion as a function of timeis difficult to calculate, as the differential equations (7)become non-autonomous. Here we will apply a self-consistent procedure as used in Ref. [12]. First, we forcethe pre-quench density to be the total density, meaning n c (0) = n , i.e. assuming that the non-condensed occupa-tion is negligible. This is a valid assumption so long asthe s-wave interaction is weak. Eq. (7) is solved exactly,yielding solutions (cid:20) u k ( t ) v k ( t ) (cid:21) = (cid:20) cos( E k ( t ) t ) I − i sin( E k ( t ) t ) E k ( t ) × (cid:18) (cid:15) k + g k n c ( t ) g k n c ( t ) − g k n c ( t ) − (cid:15) k − g k n c ( t ) (cid:19) (cid:21) (cid:20) u k (0) v k (0) (cid:21) , (9)where I is the identity matrix, and the initial values of u k ( t ) and v k ( t ) are [3], u k (0) = (cid:115) (cid:20) (cid:15) k + g nE k (0) + 1 (cid:21) v k (0) = − (cid:115) (cid:20) (cid:15) k + g nE k (0) − (cid:21) . (10)The evolution of coefficients u k ( t ) and v k ( t ) dependson the dispersion relation E k ( t ) = (cid:112) (cid:15) k [ (cid:15) k + 2 g k n c ( t )],which is assumed to change adiabatically with timethrough the condensate density n c ( t ). We can then cal-culate the momentum distribution as n k ( t ) = | v k (0) | + g k n c ( t ) (cid:20) g k n c ( t ) − g n (cid:21) × (cid:15) k [1 − cos(2 E k ( t ) t )] E k ( t ) E k (0) . (11)Taking into account all of the momentum components,the quantum depletion is evaluated through, n d ( t ) = 12 π (cid:90) ∞ n k ( t ) k d k, (12)where we have replaced the summation by the integrationover momentum space. The angular part in the integra-tion has been integrated out in the above equation. Withthe quantum depletion at hand, the condensate fractionis found to be n c ( t ) = n − n d ( t ). We then reinsert theresult back into Eq. (11) and iterate the procedure untilthe calculation converges self-consistently.In the following calculations, we will scale the ener-gies, lengths, and times with respect to the interactionenergy g n , coherence length ζ = ( mg n ) − / , and coher-ence time τ = tg n of the initial condensate. The zerorange interaction strength is fixed by the s-wave scatter-ing length, which is set to a s = 0 . n − / throughout thiswork. III. RESULTS AND DISCUSSIONSA. Stationary dispersion relation
The soft-core interaction drastically alters the disper-sion relation of the Bogoliubov excitations. To illustratethis, we first examine dispersion relations of a static BECby assuming that the soft-core interaction is present.When the soft-core interaction is weak, i.e. α is small,the dispersion relation resembles that of a weakly inter-acting BEC. The excitation energies increase monotoni-cally with momentum k [3] [see Fig. 2(a)]. By increasing α , the shape of the Bogoliubov spectra changes signifi-cantly. A local maximum and minimum can be seen inthe dispersion relation [Fig. 2(a)]. At the maximum, spe-cial modes called maxon modes form, while roton modesemerge around the minima [23]. In the following, we willdenote the energies of the maxons and rotons with γ m and γ r , as the local maximal and minimal values of thedispersion relation.The roton and maxon modes depend on the soft-coreinteraction non-trivially. When increasing α , γ r decreaseswhile γ m increases, as given by the examples shown inFig. 2(a). For sufficiently large α , the roton gap vanishesas the energies become complex, i.e. the roton is unsta-ble. The roton instability can drive the system out of auniform condensate, leading to the formation of super-solids [35, 49, 50]. It should be noted that the instabilityhere is induced by stronger, isotropic interactions. Indipolar BECs, instabilities are caused by angular depen-dent interactions [16].It is important to obtain the critical value at whichthe roton mode becomes unstable. From Fig. 1(b), theFourier transform of the soft-core potential has the mostnegative value around k r ≈ π/ R . The roton minimumtakes place around this momentum. By substituting k r into the dispersion relation, we can identify the critical α at which the roton energy becomes complex, α r = 5e π/ (cid:0) R + 25 π (cid:1) πR (cid:104) π/ sin (cid:16) π − π √ (cid:17) − (cid:105) . (13) FIG. 2. (color online)
Roton and maxon mode . (a) Bo-goliubov spectra in the stationary state for α = 0 (dashed),1 (blue), 6 (green), and 7 . R = 15. Theenergy gaps γ r and γ m indicating respectively the roton andmaxon energies are marked for the green curve. For α > . α r vs R . Analytical calculates (black) agree with the numericaldata (red dots). (c) Roton energy γ r . Increasing α , the ro-ton energy decreases. For large α , the analytical (black solid)and numerical (dot) results agree. At small α , roton minimabecome weak and eventually disappear, which leads to thedeviation. The data points in red are the energies taken nu-merically from the dispersion. (d) Maxon energy γ m increaseswith α . The analytical (black solid) and numerical data agreenicely. In (c) and (d) R = 15. To check the accuracy of this critical value, we numeri-cally find the instability point from the dispersion rela-tion. Both numerical and analytical values are shown inFig. 2(b). The analytical result agrees with the numeri-cal values very well. This supports the assumption thatthe roton minimum happens around momentum k r .Knowing the momentum k r , we can obtain the rotonenergies by inserting it into Eq. (4). It is found thatthe roton energy γ r decreases with increasing α [see Fig.2(c)]. The roton energy from the numerical calculationsagrees with the analytical data, especially when the soft-core interaction is strong. Decreasing the soft-core inter-action, the roton modes disappear for sufficiently small α , as our numerical calculations indicate. We notice largedeviations between the two methods in this regime.On the other hand, the location of the maxon modesin momentum space is difficult to find. By analyzing thedispersion relation, the momentum corresponding to themaxon mode is approximately given by k m ≈ k r /
2. Usingthis approximation, we substitute this momentum value
FIG. 3. (color online)
Excitation of the roton and maxon mode . In the upper panels, (a) gives the dispersion for astatic BEC. The momentum of the roton and maxon modes decreases with increasing soft-core radius R . Without soft-coreinteractions, the excitation energy monotonically increases with momentum (dashed). In (b)-(c), the interaction quench isapplied. Momentum densities at time τ = 30 are shown in (b). The black dashed curve shows the momentum distributionof the initial state. Fast oscillations are found in the quantum depletion (c), which leads to sharp peaks in the respectiveFourier transformation (d). The frequency µ m at the major peaks is determined by the maxon frequency. Minor peakscorresponding to other frequencies are almost invisible. In (a)-(d), three different soft-core radius R = 8 (red), 10 (green),and 12 (blue) are considered. In the lower panels, the dispersion (e), momentum distribution (f), quantum depletion (g) andFourier transformation of the quantum depletion (h) for R = 10 and α = 5 (blue), 6 . .
99 (red) are shown.Approaching to the roton instability (e), the momentum distribution (f) develops a large occupation around modes at k r at τ = 30. The depletion dynamics maintains a slower oscillation (g) as the interaction strength is increased, which can be seenfrom the Fourier transformation of the quantum depltion (h). The lower peak frequency ν r is determined by the roton mode.The major peaks at higher frequencies are due to the excitation of maxons. When α = 7 .
99, both the roton and maxon modeare dynamically stable, giving narrow Fourier spectra. into Eq. (4) and calculate the maxon energy. The re-sult is shown in Fig. 2(d), where the approximate valuematches the numerical values with a high degree of ac-curacy.Recently, the stationary state of 2D and 3D Rydberg-dressed BECs have been examined [51]. It was shownthat the increased occupation around the roton modesleads to instabilities in the ground state in the form ofdensity waves. It was also seen that the strong interpar-ticle interactions lead to a large depletion of the conden-sate.
B. Roton and maxon excitation
Depending on parameters of the soft-core interaction,the stationary dispersion relation could support rotonand maxon modes. One example is displayed in Fig.3(a). Now if we quench the interaction, the dispersionrelation of the initial and final state is different. The sys-tem is driven out of equilibrium, such that momentumdistributions evolve with time. In Fig 3(b), snapshotsof the momentum distribution are shown. At τ = 0, theBEC is in a stationary state, which depends on the initialcondition, ¯ v k . The respective momentum distribution isa smooth function of k . At later times, different mo-mentum components are excited by the presence of thesoft-core interaction, causing dynamical evolution of the quantum depletion.The dynamics of the quantum depletion depends vi-tally on the parameter R and α in the soft-core interac-tion. After switching on the interaction, the excitationof the Bogoliubov modes significantly affects the momen-tum distribution. We will first investigate the oscillatorybehavior of the quantum depletion. For moderate soft-core interactions, many momentum modes are excited bythe soft-core interaction, as shown in Fig. 3(c). As a re-sult, the quantum depletion increases rapidly with time,and then oscillates around a constant value [Fig. 3(c)].The Fourier transformation ˜ n c ( ν ) of the quantum deple-tion, characterizing the spectra of the dynamics, shows asharp peak [Fig. 3(d)]. The peak positions, i.e. frequencyof the oscillations, decrease gradually when increasing thesoft-core radius.For stronger soft-core interactions, the roton modemoves towards the instability point [see Fig. 3(e)]. In thiscase, higher momentum components can be excited dur-ing the interaction quench [Fig. 3(f)]. Here a new, lowerfrequency pattern develops on top of the fast oscillationin the quantum depletion [Fig. 3(g)]. This changes theFourier spectra of the quantum depletion, where a newpeak is found at a lower frequency [Fig. 3(h)].It is important that the peak positions in ˜ n c ( ν ) aredetermined by the roton and maxon energies. In thequantum depletion, the fast oscillations are due to theexcitations of the maxon modes, while slow oscillations ν m α ν r
12 13 14 15 R FIG. 4. (color online)
Maxon frequency (a-b) and ro-ton frequency (c-d) . The dots are numerical data from theFourier spectra. The solid curves are analytical results 2 γ m in(a)-(b) and 2 γ r in (c)-(d) obtained from the Bogoliubov dis-persion. The maxon (roton) frequency increases (decreases)with increasing interaction strength. At the critical point α r ,the roton mode loses stability. Frequencies of both modestends towards 0 for larger R values as the soft-core interac-tion becomes weaker. In (a) and (c) R = 15. In (b) and (d) α = 4. are due to the roton modes. To verify this, we first ob-tain the maxon and roton frequencies by substituting thecorresponding momentum k m and k r in Eq. (4). We thencompare them with the frequency at the peak positions inthe Fourier spectra. Note that the oscillation frequency(i.e. peak frequency of the Fourier spectra) in the quan-tum depletion is twice the Bogoliubov energy, as can beseen in Eq. (11). As shown in Fig. 4, the numerical datafor both the maxon mode (a-b) and roton mode (c-d)agree with the analytical calculations. When varyingthe interaction strength, the maxon (roton) frequencyincreases (decreases) with increasing α . If we increasethe soft-core radius, frequencies of both modes decrease.The agreement between numerical and analytical cal-culation confirm that both roton and maxon modes areexcited via quenching the soft-core interaction. The dy-namically excited modes are stable, as both the fast andslow oscillations are persistent for a long time. In ournumerical simulations, the oscillations will not dampeneven when the simulation time τ > ∝ ζ − , while oscillatory patterns are notpresent in the depletion [11], due to the fact that lowenergy phonon modes dominate the quench dynamics.In dipolar BECs [19, 20, 24, 52, 53], on the other hand, FIG. 5. (color online)
Asymptotic quantum depletion .The asymptotic quantum depletion increases with increasing α (a), which is seen from both the analytical and numeri-cal calculations. The quantum depletion n ∞ d decreases withincreasing soft-core radius (b). The solid line is found ana-lytically using Eq. (15), while the data points are found bynumerically solving Eq. (12) and taking the mean value atlater times between τ ≈ → R = 3 (black) and 4 (red). Parameters in (b) are α = 1(black) and 3 . roton modes are formed due to the interplay betweenlong-range dipolar and s-wave interactions [19, 20, 24,52, 53]. These roton modes can be excited by quenchingthe dipolar interaction, while maxon modes are typicallyunstable in the dynamics [see Appendix A for examples]. C. Quantum depletion in the long time limit
In the long time limit τ (cid:29)
1, the quantum depletionoscillates rapidly around a mean value [Fig. 3(c) and (g)].In the following, we will evaluate the asymptotic value ofthe quantum depletion. First we will derive an analyticexpression using the following approximations. In thelong time limit, the time averaged quantum depletion islargely determined by the low momentum modes. More-over, we will neglect the oscillation term in Eq. (11), asthey are related to the roton and maxons. Using theseapproximations, the asymptotic form of the momentumdistribution n ∞ k is obtained, n ∞ k ≈ (cid:32) k + 1 (cid:112) k ( k + 4) − (cid:33) + αf ( k )4 k n ∞ c n , (14)where n ∞ c is the asymptotic condensate density. Aftercarrying out the integral over momentum space, the ap-proximate quantum depletion when τ → ∞ is obtained, n ∞ d n ≈ (cid:18) R + απ R + 2 πα Γ (cid:19) , (15)where Γ = (2 π ζ n ) − . This result predicts thatthe quantum depletion approaches to a constant value n ∞ d /n → / R → ∞ . This resembles theresult of the weakly interacting BEC, i.e. the soft-coreinteractions plays no role in the quench.To verify the analytical calculation, we numericallyfind the mean value of the quantum depletion when timeis large. Both the numerical and analytical results areshown in Fig. 5. For small α , low momentum statesare populated by switching on the soft-core interaction.This is the regime where the approximation works. Wefind a good agreement between the numerical and ana-lytical calculations. Increasing the interaction strength,more and more higher momentum components are pop-ulated [see Fig. 3(b) and (f)], causing larger depletion.A clear deviation between the numerical and analyticaldata is found, as the approximation we made in evaluat-ing Eq. (15) becomes less accurate. On the other hand,the quantum depletion becomes smaller by increasing thesoft-core radius, as the strength of the soft-core interac-tion reduces. In this case the numerical and analyticalresults agree well [see Fig. 5(b)]. IV. CONCLUSION
We have studied dynamics of 3D BECs in free space,with Rydberg-dressed soft-core interactions. An interac-tion quench is implemented through turning on the soft-core interaction instantaneously, starting from a weaklyinteracting BEC. The Bogoliubov spectra of the BEC dis-plays local maxima and minima, which are identified asmaxon and roton modes. Through a time-dependent Bo-goliubov approach, we have calculated dynamics of thequantum depletion self-consistently. Our results showthat both roton and maxon modes are excited by switch-ing on the soft-core interaction. The excitation of rotonand maxon modes generate slow and fast oscillatory dy-namics in the quantum depletion. Our simulations showthat the excited roton and maxon mode are stable in thepresence of the soft-core interaction, which are observedfrom the persistent oscillations of the quantum depletion.We have found the frequencies of the roton and maxonmodes approximately, which are confirmed by the numer-ical simulations.Our study shows that exotic roton and maxon excita-tions can be created in Rydberg-dressed BECs throughthe interaction quench. Properties of the maxons androtons can also been seen from condensate fluctuations[see Appendix B for details)] and density-density cor-relations [see Appendix C]. The result studied in this work might be useful to identify the soft-core interac-tion through measuring frequencies and strength of thequantum depletion. In the future, it is worth studyingformation of droplets and spatial patterns in Rydberg-dressed BECs, which could be affected by the presenceof roton or maxon modes.
ACKNOWLEDGEMENTS
We thank Yijia Zhou and S Kumar Mallavarapu forfruitful discussions. The research leading to these re-sults has received funding from the EPSRC Grant No.EP/M014266/1, the EPSRC Grant No. EP/R04340X/1via the QuantERA project ERyQSenS, the UKIERI-UGC Thematic Partnership No. IND/CONT/G/16-17/73, and the Royal Society through the InternationalExchanges Cost Share award No. IEC \ NSFC \ Appendix A: Dynamics of 2D Dipolar Systems
Quench dynamics in BECs with dipolar interactionsare drastically different. The dipolar interaction is givenby ˜ V dd ( r − r (cid:48) ) = g δ ( r ) + d | r − r (cid:48) | [1 − ( θ )] , (A1)where d is the dipole moment, θ is the angle between thedipoles and molecular axis, and g is the short-range con-tact interaction as before. In 3D, the Fourier transformof the dipolar interaction has no momentum dependence[54]. In a 2D trapped dipolar Bose gas [17, 18], the in-teraction potential displays a strong momentum depen-dence [20].We consider a quasi-2D setup [20], where a strong con-finement is applied in the perpendicular z -direction whileleaving atoms free to move in the x − y plane. Thedipoles are polarized along this z -axis. This leads theaxial confinement as l z , which provides a natural rescal-ing of r (cid:55)→ r /l z [17, 18, 20–22]. After integrating Eq.(A1) in the z -axis, we obtain the Fourier transformationof the quasi-2D dipolar interaction [20] V dd ( k ) = 2 − k √ π Erfc( k )e k , (A2)where Erfc( k ) is the complimentary error function. Herewe define the dimensionless parameter α d = d /g tocharacterizing the strength of the dipolar interaction,such that the interaction after the quench is given as g d /g = 1 + α d V dd ( k ). The quench scheme for the dipo-lar case is similar to the procedure outlined in the maintext. We switch on the dipolar interaction instanta-neously, while keeping the s-wave interaction unchanged. k E k / g n k n k τ n d ( τ ) / n ν ˜ n d ( ν ) / n FIG. A1. (color online)
Quantum depletion in a dipolarBEC . Red curves are for α d = 2 . α d = 2 .
7. The axial confinement is set to l z = 0 . n − / .We show the dispersion relation in (a) while the momentumdistribution at time τ = 30 is shown for (b). The quantumdepletion and corresponding Fourier spectra are shown in (c)and (d) respectively. The inset shows a maxon mode is excitedfor α d = 2 .
1. However the signal is very weak and almostinvisible. The axes of the inset is same as panel (d).
The dispersion relation for the dipolar BEC is shownin Fig. A1(a), where both roton and maxon modes canbe seen.When the dipolar interaction is compared to theRydberg-dressed BEC [e.g Fig. 2(a)], the energies of thelow momentum modes remain small, as seen by directlycomparing the dispersion relations. The absence of theselarge maxon energies means that the mechanism behindthe dipolar interactions prevent the oscillations that wepreviously attributed to the maxon modes from reachinglarge amplitudes [Fig. A1(b)][20, 21, 55].We follow the same self-consistent process to obtainthe condensate fraction. We calculate the quantum de-pletion as before as n d /n = 1 / (2 πl z n ) (cid:82) ∞ n k k d k . When α d is small, the dynamics develops maxon oscillations,which dampens in short time scales, as shown in Fig.A1(c). When α d is large, the roton frequency completelyoverpowers the maxon frequency in the dynamics. Theabsence of a stable maxon mode is also seen in the Fourierspectra [Fig. A1(d)]. Appendix B: Condensate fluctuation
In this section, we evaluate the fluctuation of the con-densate for the Rydberg-dressed BEC. The condensate τ √ N ∆ n c ( τ ) / n α √ N ∆ n c ( ∞ ) / n R FIG. B1. (color online)
Condensate fluctuation . (a) Dy-namics of the condensate fluctuation. We fix R = 10, andevolve the system for α = 5 (blue), 6 . . α = 0. The inset shows fluctuations when α = 7 .
99 to highlight the low frequency oscillations due to ro-tons. Mean values of the fluctuations for different α (b) and R (c) when time τ → ∞ . We have considered R = 3 (black)and 4 (red) in (b) and α = 1 (black) and 3 . fluctuation is defined as∆ n c = (cid:113) (cid:104) n (cid:105) − (cid:104) n c (cid:105) = (cid:113) (cid:104) n (cid:105) − (cid:104) n d (cid:105) = 1Ω (cid:115) (cid:88) kk (cid:48) (cid:54) =0 (cid:104) (cid:104) ˆ a † k ˆ a k ˆ a † k (cid:48) ˆ a k (cid:48) (cid:105) − (cid:104) ˆ a † k ˆ a k (cid:105) (cid:104) ˆ a † k (cid:48) ˆ a k (cid:48) (cid:105) (cid:105) , where we have assumed the total density n is a constant.Using the Bogoliubov transformation, the fluctuation ofthe condensate is obtained,∆ n c = 1Ω (cid:115) (cid:88) k (cid:54) =0 n k (1 + n k ) (B1)One can numerically evaluate the fluctuation by insert-ing Eq. (11) into the above equation. For convenience,the relative fluctuation, √ N ∆ n c /n , will be calculated.Some examples are shown in Fig. B1(a). The fluctua-tion increases rapidly, and then saturates at an asymp-totic value when time is large. The fluctuation oscillatesaround the asymptotic value. The maxon modes lead tofast oscillations. When the roton mode is significantlypopulated, a slower oscillation is found.The asymptotic value of the fluctuation depends on thesoft-core interaction. Increasing α , the asymptotic value FIG. C1. (color online)
Density-density correlation . (a)the density-density correlations as a function of D and τ ,when R = 15 and α = 7 .
7. Correlations at D = 5 (blue), 15(orange), and 25 (green) are shown in (b). The correspondingFourier spectrum of the correlation function is shown in (c).In the Fourier spectra, the peaks at lower and higher frequen-cies are due to the excitation of roton and maxon modes. increases [see Fig. B1(a) and (b)]. We can estimate theasymptotic value of the density fluctuation by replacing n k with its asymptotic value Eq. (14), in Eq. (B1), whichyields √ N ∆ n ∞ c n = (cid:115) (cid:90) ∞ n ∞ k [1 + n ∞ k ] k d k. (B2)Further assuming the fluctuation depends solely on lowmomentum states, we obtain the approximate result ofthe fluctuation when τ → ∞ , √ N ∆ n ∞ c n ≈ (cid:115) π (cid:2) π α (cid:0) √ παC (cid:1)(cid:3) R , (B3)with the constant C = (cid:2) √ π − (cid:0) (cid:1)(cid:3) . The ap- proximation result shows that fluctuations of the con-densate decreases (increases) with increasing R ( α ). InFig. B1(b) and (c), numerical and approximate resultsare both shown. The two calculations agree when α issmall or R is large, where the depletion and fluctuationare both small. Though large discrepancy is found when α is large or R is small, the trend found from both nu-merical and analytical calculations are the same. Appendix C: Density-Density Correlation
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