Matter-wave Fractional Revivals in a Ring Waveguide
aa r X i v : . [ phy s i c s . a t o m - ph ] J a n Matter-wave Fractional Revivals in a Ring Waveguide
Jayanta Bera , Suranjana Ghosh , Luca Salasnich , and Utpal Roy ∗ Indian Institute of Technology Patna, Bihta, Patna-801106, India Indian Institute of Science Education and Research Kolkata, West Bengal, India Dipartimento di Fisica e Astronomia “Galileo Galilei”,Universit´a di Padova, Via Marzolo 8, 35131 Padova, Italy
We report fractional revivals phenomena in ultracold matter-wave inside a ring-waveguide. Thespecific fractional revival times are precisely identified and corresponding spatial density patternsare depicted. Thorough analyses of the autocorrelation function and quantum carpet provide clearevidences of their occurrence. The exhibited theoretical model is in exact conformity of our numericalresults. We also investigate the stability of the condensate and a variation of revival time with thediameter of the ring.
I. INTRODUCTION
Revival and fractional revivals (FR) are very rich quan-tum phenomena which appear during the long time evo-lution of a well localized wave-packet [1, 2]. For some typ-ical systems, wave-packets initially disperse in the courseof its time evolution and subsequently reunite quite sys-tematically at some specific evolution times, which arecalled FR instances. FR mostly appear in quantum sys-tems with nonlinear energy spectrum: Rydberg wavepacket [3], infinite square well [4], molecular systems [5–7], photon cavity systems [8], chaotic light [9], just tomention a few. FR have found profound applications to-wards quantum information and quantum computation:quantum logic gates [10], Renyi uncertainty relations [11],wave packet isotope separation [12], number factorization[13], molecular coherent control [14], position dependentmass [15] etc. Underlying mesoscopic quantum super-position holds the secrets for developing quantum tech-nologies and quantum precision measurements throughsub-Planck scale structures [16–20].On the other hand, quantum sensing is a highly emerg-ing field for ultracold atoms [21–26]. To observe FR-phenomena, the systems essentially need to remain sta-ble for sufficiently long time, which is a challenging taskin experiments while dealing with mesoscopic quantumobjects. Hence, scientists always look for appropriatesystems or laboratory environments to achieve the same.Bose-Einstein condensate (BEC) is a highly tunable andcoherent wave-packet with large coherence time, mak-ing it a promising candidate to observe FR [27, 28].Thus, it becomes interesting that FR be precisely re-ported for Bose-Einstein condensate. Observation of FR-phenomena for matter-wave will considerably intensifythe progress of this field. However, there is no exactanalysis of FR phenomena in an ultracold atomic systemto date.In this paper, we report the exact structure of the FRphenomena for BEC in a ring-shaped geometry. A ring ∗ [email protected] FIG. 1: Trapping potential is shown which is created by com-bining a harmonic trap (strength 2 π × . ~ ω ⊥ and waist 11 . µm ). The ini-tial condensate of width 2 . µm is situated at (0 , . µm )inside the trap. potential is routinely created by applying a blue-detunedlaser beam in the middle of the harmonic trap [29, 30].BEC loaded in a ring trap has attracted enormous at-tention in past years [31–40], including persistent flow[41–45], solitary waves [46], vortices [47–49] and interfer-ence [21, 50, 51]. In the following section, we study thecondensate dynamics by numerically solving the Gross-Pit¨aevskii equation (GPE). The density snapshots at dif-ferent times reveal the clear signature of FR phenomenain BEC. Illustrations with the autocorrelation function[52, 53] and quantum carpet [54, 55] provide a compellingevidence along with the stability analysis. The FR time-scales are evaluated through both theoretical and numer-ical approaches. Albeit a completely different origin ofthe current approach, the established exact formula coin-cides with the standard definition of FR in other quantumsystems. The variation of the revival time with the ring-radius, following the theoretical and the computationalapproaches, are in unsurpassed conformity. II. SOLVING THE DYNAMICS AND RESULTS
To investigate the dynamics of BEC in a 2D ring-trap,the well-known 3D Gross-Pit¨aevskii equation (GPE) [56]is effectively reduced to quasi-2D: i ∂ψ∂ t = −∇ , y ψ + g | ψ | ψ + V ext ( x , y ) ψ, (1)where ψ becomes a function of x , y , and t . g =4 √ πN a/a ⊥ is the nonlinearity coefficient for the conden-sate of N -atoms of each mass M . The s -wave scatteringlength is given by a and the transverse oscillator length is a ⊥ = p ~ / (2 M ω ⊥ ) with ω ⊥ , being the trap frequency in z -direction. The dimensionless form of the above equa-tion is obtained after scaling position, time and energy by a ⊥ , 1 /ω ⊥ , and ~ ω ⊥ , respectively. The external trappingpotential, V ( x, y, t ), is a combination of a 2D harmonictrap and a Gaussian potential: V ext ( x, y ) = 14 ω r + V e − r /σ (2)with r = x + y . To avoid the undesired oscillation inthe radial direction, the initial condensate is placed atthe exact minimum of the ring ( r o ), which is obtained byminimizing the resultant potential: r = σ √ q ln V ω σ .A Taylor series expansion of the potential around r provides the effective harmonic trap frequency, ω e =2 √ ωr /σ [57].We solve Eq. (1) by Fourier split step method. Both x -and y -coordinates are equally divided into 512 grids withstep-size 0 . . N (= 10 ) Rb atoms with the parameters: M = 85 a.u. , ω ⊥ = 2 π ×
51 Hz, a ⊥ = 1 . µm , a = 58 . × − m, g =3 . ~ ω ⊥ and ω = 0 . × ω ⊥ [30]. The trap parametersare V = 200 ~ ω ⊥ and σ = 11 . µm , which prepares aring potential of effective radius r = 11 . µm . Wehave chosen the initial condensate of waist d = 2 . µm ,placed along the y -axis with coordinate (0 , r ) as shownin Fig. 1. These parameters also enable us to minimizethe undesired radial oscillation and breathing inside thering. A. Condensate density, auto-correlation functionand quantum carpet
As it has been observed in past literature that theinitial condensate is expected to disperse along thering waveguide in both clock-wise and anti-clock-wisedirections. Both the counter-propagating componentswill gradually merge at the opposite pole of the ring(0 , − r ), producing several closely-spaced spatial inter-ference fringes therein which then gradually disperse.However, spectacular phenomena occur if one observesthe condensate for long time. The dispersed clouds reunite again only in some specific time-intervals andgive birth to several daughter-condensates which are thereplicas of the initial one. All the newly born mini-condensates become similar in shape and size. This phe-nomena are well-known as fractional revivals. In the duecourse, all the clouds finally localize again and revive ata particular time, called revival time ( T r = 2 . t = pT r q , (3)where, p and q are mutually prime integers. It is alsoobserved that the initial condensate splits into q mini-condensates when q is odd and to q/ q is even. The result is unfolded in Fig. 2, wherevaried time snapshots for condensate density are depictedin Fig. 2 (a-e), where the initial density splits into (a) 10-parts at 1 / T r , (b) 6-parts at 1 / T r , (c) 5-parts at1 / T r , (d) 4-parts at 1 / T r , and (e) 2-parts at 1 / T r .This is in conformity to the above compelling formula,which clearly reflects the signature of FR-phenomena inthis system. For nicely distinguished higher order FR(Fig. 2 (a)), the radius of the ring has to be sufficientenough and a well-localized initial condensate is also de-sired. The typical range of ring radius taken in BEC-experiments in literature is already favourable for ob-serving higher order FR.Figure 2 (f) depicts the phase-variation along x - and y -directions for q = 4 at time t = 298 . ms (Fig. 2 (d)).At this time, four mini-condensates are produced at x = ± r , y = ± r , when the condensate was initially placedat y -axis ( x = 0, y = r ). If we consider the petals on leftand right of x = 0 line, they will always have the samephase because of the symmetry. But, the petals on bothsides of y = 0 line will have different phases. The scenariowill be reversed if we initially place the condensate on x -axis.For a better insight of the FR phenomena, we haveinvestigated the autocorrelation function (AF) and thequantum carpet, which are depicted in Fig. 3. The auto-correlation function is represented by the modulus squareof the overlap between the initial and the final wave func-tions: | A ( t ) | = |h ψ t =0 | ψ t i| . (4)For pure states, AF is also the time dependent quan-tum fidelity between the initial and the time-evolvedstates. In Fig. 3, we have displayed the variations for0 ≤ t ≤ T r /
2, beyond which ( T r / ≤ t ≤ T r ) it is themirror image of the first half. The well-defined peaks ofthe AF signify FR instances. These instances are desig-nated in Fig. 3(c) with labelling of the number of mini-condensate. Dark-shaded (red) plot in Fig. 3(c) is the AF FIG. 2: (a-e) Density plots at times 119 . . . . . .
11 ms ( s = 4) along x -axis (blue dashed line) and y -axis (redsolid line). of the case under study. Fig. 3(a) and 3(b) are display-ing the quantum carpets along the x - and y -quadratures,respectively. Quantum carpet is a popular nomenclaturefor the space-time flow of the quantum information. Lo-calizations of the condensate densities are visible in boththe quadratures at different FR times. When evaluatingthe quantum carpet along x -direction, we integrate over y and vice-versa. Additionally, the space-time rays orig-inating from t = 0 and T r / t = 0 . T r , theinitial space-time spots revive at x = 0, but splits intotwo parts along y (= ± r ) and the AF gets a large peak.However, at time t = 0 . T r , the condensate splits intofour parts and becomes symmetric in x - and y -directions(Fig. 2 (d)), leaving behind the same structure (threebright spots) in both the quantum carpets (Fig. 3(a) and(b)). AF at this time also produces a peak. One canexplain all the other FR instances and correlate Fig. 3nicely with Fig. 2. To indicate the difference, we havealso included a lighter shaded (blue) plot in Fig. 3(c) forthe case where one starts with two condensates, sym-metrically situated in the opposite poles of the toroid.In this case, the structures are more prominent with theAF peaks of higher magnitude and the revival time-scalebecomes half of that for the first case. B. Theoretical Derivation of the time-scale formula
To investigate the formation mechanism of the FR-phenomena, we begin by taking the expression of twocounter propagating Gaussian wave-packets separated bya distance D [56]:Φ ± ( d, t ) = N ± exp h − ( d ± D/ (1 + i t/ω )2 ω t i , (5)where N ± is the normalization constant, ω is the initialwidth of the wave-packet and ω t = p ω + (2 t/ω ) isthe width after time t . Such spreading of the wave-packetbecomes eventually linear with time ( ω t = 2 t/ω ) for longtime evolution.Let us imagine the present case, where the initial con-densate is placed at (0 , r ) (Fig. 1). It can be modeled bytwo wave-packets separated by D = 2 πr , where r is theeffective radius of the ring. The two clouds are coexis-tent initially and subsequently spreading clock-wise andcounter-clockwise along the minimum of the ring. Afterpropagating through the ring, the two identical counter-propagating clouds start interfering at (0 , − r ). The in-terference maxima are obtained from the oscillating term,cos(2 Ddt/ω ω t ). Hence, the fringe separation is given by∆ d = πω ω t / ( Dt ) , and ∆ d ′ = 4 πt/D, (6)where the later is the effective fringe separation for thelong time evolution [22, 56]. Imagine the whole string as FIG. 3: Spatio-temporal flow of quantum information or quantum carpets in (a) x − t and (b) y − t planes. (c) depictsautocorrelation functions for single initial condensate (red line) and two initial condensates (blue line). Numbers indicatedagainst the autocorrelation peaks are the number of splits (denoted by s ). The stability analysis in presence of noise is shownin (d). T r ev ( s ec ) Radius ( m)
Theory Numerical
FIG. 4: Comparison of the theoretical values (from Eq. (7))and numerical values of the time scales with the size of thering-trap. They are in very good agreement. a ring of radius r and the initial wave-packets splits into s number of mini replicas in long time evolution. Due tothe circular symmetry of the waveguide, the separationbetween the two maxima in the course of long time evo-lution is evaluated from the relation, ∆ d ′ × s = 2 πr m ,where m is a nonzero integer, signifying the number oftimes the condensate circles the origin in a given time.Making use of the expressions of D and ∆ d ′ , we obtain t = πr ms . (7)Condensate will revive ( s = 1) at times, πr , 2 πr ,3 πr and so on. However, for the odd multiples of πr ,the wave-packet revives in shape, but not in position as itrevives at the opposite end of the ring. If one defines the revival time when the wave-packet revives both in shapeand position, then it becomes T rev = 2 πr . In addition,obtaining various number of mini-wave-packets will re-quire m and s to be mutually prime integers. Here, m and s grasp identical meaning of p and q in Eq. (3) and amaz-ingly, this formula is identical to what has been observedin other nonlinear quantum systems. We also comparethe revival time calculated from the theoretical formula(Eq. (7)) with the revival time obtained from numericalresult. The quadratic dependence of the revival time withthe ring radius is confirmed in Fig 4. For the experimen-tal parameters considered in this work, r is 11 . µm ,for which the revival time obtained as T r = 2 . s . Thiscoordinate is also precisely lying on the curve (Fig 4). Itis worth mentioning that for each point on this curve, aproper choice of the trap and initial condensate parame-ters are required to obtain a perfect FR. Additionally, aslight change in the ring radius by 0 . µm is associatedwith the change in revival time 4 ms . C. Stability analysis
Investigating the stability of the condensate with timeis inevitable, particularly when one studies the long timeevolution. The stability analysis has been carried outby the conditionally stable, split-step fast Fourier trans-formation (SSFFT) method, which is an established nu-merical method and widely used in numerical simulationof non-linear wave propagation, mainly in the context ofnonlinear Schr¨odinger equation (NLSE) [58, 59]. Here,we present the stability analysis for the previously men-tioned parameters in this manuscript and mix white noise(Σ( x, y )) to the initial wavefunction with mean zero andamplitude 5% of the maximum value of the condensatedensity at a particular time. ψ t =0 = ψ t =0 + Σ( x, y ) , (8)The condensate wave function with noise is evaluatedafter a certain time and corresponding autocorrelationfunction (AF) is calculated from Eq. (4). Then a percent-age deviation of the AF (with noise) from the AF (with-out noise) is calculated which is elucidated in Fig. 3(d).The times for which these steps are repeated are 1 / /
8, 3 /
16, 1 /
4, 5 /
16, 6 /
16, 7 /
16 and 1 / T r ), which stands for an adequate stability of the con-densate in the course of its entire time evolution. We haverepeated the same study for other ranges of parametersand found the solution sufficiently stable. III. CONCLUSION
In conclusion, we have reported FR phenomena for a2D system of BEC in a ring-shaped waveguide. Conden-sate splitting in the 2D-real space is depicted through density snapshots, autocorrelation function and spatio-temporal structures in quantum carpets. We have elim-inated unwanted breathing of the cloud in the radialdirection by appropriately choosing the initial conden-sate. Proper tuning of the trap parameters (harmonicand Gaussian) within the experimental parameter regimeis important to prepare the minima of the ring in har-monic shape. We have also shown the phase variationin Fig. 2(f) at t = T r /
8, which imparts an idea of realspace interference patterns in a time-of-flight measure-ment, which is being studied in a separate work. We iden-tify the time scales quantitatively and compare the samewith our theoretical model where we obtain an exact for-mula. A numerical stability analysis makes the resultsmore viable for experimental realization. The variation ofthe revival time with the radius of the ring fosters quan-tum sensing applications. The work can be extended for2D-disordered lattice for which the strong localization ofcondensate can be utilized to get better visibility duringFR [60, 61]. This work will also be a suitable guidelinefor a variety of other applications towards quantum in-formation, quantum metrology, quantum logic gates anddecoherence. [1] I. Sh. Averbukh and N. F. Perelman, Phys. Lett. A ,449 (1989).[2] R. W. Robinett, Phys. Rep. , 1 (2004) and referencestherein.[3] J. Parker and C. R. Stroud, Phys. Rev. Lett. , 716(1986).[4] D. L. Aronstein and C. R. Stroud, Jr., Phys. Rev. A ,4526 (1997).[5] Tamar Seideman, Phys. Rev. Lett. , 4971 (1999).[6] B. M. Garraway and K. A. Suominen, Contemp. Phys. , 97 (2002).[7] J. Banerji and S. Ghosh, J. Phys. B: At. Mol. Opt. Phys. , 1113 (2006).[8] G. Rempe, H. Walther and N. Klein, Phys. Rev. Lett. , 353 (1987).[9] L. Li, P. Hong, and G. Zhang, Phys. Rev. A , 023848(2019)[10] E. A. Shapiro et al. , Phys. Rev. Lett. , 237901 (2003).[11] E. Romer, and F. de los Santos, Phys. Rev. A . 013837(2008).[12] I. Sh. Averbukh et al. , Phys. Rev. Lett. , 3518 (1996).[13] D. Bigourd et al. , Phys. Rev. Lett. , 030202 (2008).[14] H. Katsuki et al. , Science , 1589 (2006).[15] Naila Amir and Shahid Iqbal, Commun. Theor. Phys., , 181 (2017).[16] W. Zurek, Nature , 712 (2001).[17] S. Ghosh et al. , Phys. Rev. A , 013411 (2006).[18] S. Ghosh, U. Roy, C. Genes, and D. Vitali, Phys. Rev. A , 052104 (2009).[19] U. Roy et al. , Phys. Rev. A , 052115 (2009).[20] S. Ghosh and U. Roy, Phys. Rev. A , 022113 (2014).[21] P. R. Berman, Atom Interferometry , (Academic Press,Cambridge, 1997). [22] R. Andrews et al. , Science , 637-641 (1997).[23] P. A. Altin et al. , New J. Phys. et al. , Phys. Rev. Lett. et al. , Phys. Rev. Lett. et al. , Phys. Rev. Lett. , 2158 (1996); M.Greiner et al. , Nature , 51 (2002).[28] F. Kia lka, B. A. Stickler, and K. Hornberger, Phys. Rev.Research , 022030(R) (2020).[29] S. Gupta, et al. , Phys. Rev. Lett. , 143201 (2005).[30] C. Ryu et al. , Phys. Rev. Lett. , 260401 (2007).[31] S. Eckel et al. , Nature (London) , 200 (2014).[32] A. A. Wood et al. , Phys. Rev. Lett. , 250403 (2016).[33] L. Corman et al. , Phys. Rev. Lett. , 135302 (2014).[34] S. Beattie et al. , Phys. Rev. Lett. , 025301 (2013).[35] C. Ryu et al. , Phys. Rev. Lett. , 205301 (2013).[36] M. Benakli et al. , Eur. phys. Lett. , 275 (1999).[37] J. Brand and W. P. Reinhardt, J. Phys. B , L113(2001).[38] A. Das, J. Sabbatini, and W. H. Zurek, Sci. Rep. , 352(2012).[39] M. Modugno, C. Tozzo, and F. Dalfovo, Phys. Rev. A , 061601 (2006).[40] L. Amico et al. , Phys. Rev. Lett. , 063201 (2005).[41] A. I. Yakimenko et al. , Phys. Rev. A , 051602 (2013).[42] S. Bargi et al. , Phys. Rev. A , 043631 (2010).[43] A. Mu˜noz Mateo et al. , Phys. Rev. A , 063625 (2015).[44] A. C. White et al. , Phys. Rev. A , 041604(R) (2017).[45] S. Moulder et al. , Phys. Rev. A , 013629 (2012).[46] P. Mason and N. G. Berloff, Phys. Rev. A, , 043620(2009).[47] F. Piazza, L. A. Collins, and A. Smerzi, Phys. Rev. A, , et al. , Phys. Rev. A , 023607 (2015).[49] K. C. Wright et al. , Phys. Rev. A , 063633 (2013).[50] P. Pedri et al. , Phys. Rev. Lett. , 220401 (2001).[51] T. A. Bell , et al. , New. J. Phys. , 035003 (2016).[52] S. Ghosh,and J. Banerji, J. Phys. B: At. Mol. Opt. Phys. , 3545 (2007).[53] R. Veilande, and I. Bersons, J. Phys. B: At. Mol. Opt.Phys. , 2111 (2007).[54] P. Kazemi et al. , New. J. Phys. , 013052 (2013).[55] A. E. Kaplan et al. , Phys. Rev. A , 032101 (2000).[56] C. J. Pethick and H. Smith, Bose-Einstein Condensationin Dilute Gases , 2nd ed. (Cambridge University Press, Cambridge,UK, 2008), L. P. Pit¨aevskii and S. Stringar
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