7 0 Time modulation of atom sources 0 2 n A. del Campo1, J. G. Muga1 and M. Moshinsky2 a J E-mail: [email protected],[email protected],[email protected] 1 1 1 Departamento de Qu´ımica-F´ısica,Universidad del Pa´ıs Vasco, Apdo. 644, Bilbao, Spain ] 2 Instituto de F´ısica, Universidad Nacional Aut´onoma de M´exico, Apartado Postal r e 20-364,01000 M´exico D.F., M´exico h t o . Abstract. Suddenturn-onofamatter-wavesourceleadstocharacteristicoscillations t a of the density profile which are the hallmark feature of diffraction in time. The m apodization of matter waves relies on the use of smooth aperture functions which - suppress such oscillations. The analytical dynamics of non-interacting bosons d n arising with different aperture functions are discussed systematically for switching- o on processes, and for single and many-pulse formation procedures. The possibility c and time scale of a revival of the diffraction-in-time pattern is also analysed. Similar [ modulations in time of the pulsed output coupling in atom-lasers are responsible for 2 the dynamical evolution and characteristics of the beam profile. For multiple pulses, v differentphaseschemesandregimesaredescribedandcompared. Stronglyoverlapping 7 8 pulses lead to a saturated, constant beam profile in time and space, up to the revival 4 phenomenon. 1 1 6 0 PACS numbers: 03.75.Pp, 03.75.-b, 03.75.Be / t Coherent matter-wave pulse formation and dynamics have been studied traditionally a m in the context of scattering and interferometric experiments with mechanical shutters. - d Suddenly switching-on and off matter-wave sources leads to an oscillatory pattern in n the particle density which was discovered by one of the authors in 1952 and dubbed o c diffraction in time [1, 2] because of the analogy with the diffraction of a light beam from : v a semi-infinite plane. The (sudden) “Moshinsky shutter” describes the evolution of a i X truncatedplane wave suddenly released andadmitsananalyticalsolutionwhich remains r abasicreferenceforanalysingmorecomplexandrealisticcases[3,4,5,6,7]. Awideclass a of experiments have reported diffraction in time in neutron [8] and atom optics [9], and with electrons as well [10]. It was also observed in a Bose-Einstein condensate bouncing off from a vibrating mirror, proving that the effect survives even in the presence of a mean-field interaction [11]. A more recent motivation for studying matter-wave pulses is the development of atom lasers: intense, coherent and directed matter-wave beams extracted from a Bose-Einstein condensate. One of the prototypes uses two-photon Ramanexcitationtocreate“outputcoupled”atomicpulseswithwelldefinedmomentum which overlap and form a quasi-continuous beam [12, 13] (For other approaches see e.g. [14] or [15]). Clearly, the features of the beam profile are of outmost importance Time modulation of atom sources 2 for applications [16, 17, 18], and will depend on the pulse shape, duration, emission frequency, and initial phase. While many studies on matter-wave pulse formation and dynamics deal with single or double pulses, more research is needed to understand and controlmultipleoverlappingpulsesrelevantforcurrentatom-laserexperiments. Weshall undertake this objective within the analytical framework of the elementary Moshinsky shutter, here at the level of the Schr¨odinger equation for independent bosonic atoms. Non-lineareffectswillbenumericallyconsideredelsewhere. Thisformulationmayindeed be adapted also to other important features of pulse formation in atom lasers, which are different from the mechanical shutter scenario. Instead of sudden switching we shall consider smooth aperture functions, i.e., “apodization”, a technique well-known in Fourier optics to avoid the diffraction effect at the expense of broadening the energy distribution [19]. Manifold applications of this technique have also been found in time filters for signal analysis [20]. In more detail, motivated by the recent realisation of an atom laser in a waveguide [21], we shall focus on the longitudinal beam features and explore the space-time evolution of effective one dimensional sources represented by “source” boundary conditions of the form ψ(x = 0,t) = χ(t)e−iω0t, t > 0. (1) ∀ The aperture function χ(t) modulates the wave amplitude at the source and is therefore responsible for the apodizationin time, that is, the suppresion of the fringes in the beam profile. However we shall show that, for long enough pulses or for multiple overlapping pulses, such effect holds only in a given time domain because of the revival of the diffraction in time. Localised sources [22, 23] represented by an amplitude ψ(x = 0,t) e−iω0t have ∼ been used for understanding tunnelling dynamics or front propagation and related time quantities such as the tunnelling times [24]. Moreover, this approach has shed new light on transient effects in neutron optics [25, 26, 8] and diffraction of atoms both in time andspace domains[27,28]. The connection andequivalence between “source” boundary conditions, in which the state is specified at a single point at all times and the usual initial value problems, in which the state is specified for all points at a single time, was studied in [29]. Notice that we shall use the source conditions (1) for simplicity, but the pulses could also be examined with the initial value approach. In section 1 we shall review the sudden switching on of the source since any other case is acombination of thiselementary dynamics. Single pulse formationwith standard apodizing functions is examined in sec. 2, and we describe the dynamics for a smoothly switched-on source in sec. 3; Section 4 is devoted to the multiple pulse case treated with different aperture functions, and the paper ends with a discussion. 1. Turning-on the source Weshallconsider thedynamicsofasourceoffrequencyω = ~k2/2m,whichisturnedon 0 0 in a horizontal waveguide. This has the advantage of cancelling the effect of gravity off, Time modulation of atom sources 3 avoiding the decrement of the de Broglie wavelength λ = 2π/k due to the downward dB 0 acceleration [21]. Note that the effects of an external linear potential on the diffraction in time phenomenon can also be taken into account following [30]. At the same time the dynamics is effectively one-dimensional provided that ω0 ω⊥ where ω⊥ is the ≪ transverse frequency of the waveguide. The sudden approximation for turning on a source is well-known [1, 23, 29, 24] but briefly reviewed here for completeness. The aperture function is then taken to be the Heaviside step function, χ (t) = Θ(t). The inverse Fourier transform of the source 0 ψ (x = 0,k ,t) = e−iω0tΘ(t) is given by 0 0 ∞ 1 ψ(x = 0,k ,ω) = F−1[ψ ](x = 0,k ,ω) ψ (x = 0,k ,t)eiωtdt 0 0 0 0 0 ≡ √2π −∞ Z i 1 b = . (2) √2π ω ω +i0 0 − For x,t > 0, the wave function evolves according to i ∞ eikx−iωt ψ (x,k ,t) = dω , 0 0 2π ω ω +i0 −∞ 0 Z − where Imk 0. This can be written in the complex k-plane by deforming the contour ≥ of integration to Γ which goes from to passing above the poles, + −∞ ∞ i eikx−i~2km2t ψ (x,k ,t) = dk2k 0 0 2π k2 k2 ZΓ+ − 0 = i dk 1 + 1 eikx−i~2km2t. 2π k +k k k ZΓ+ (cid:18) 0 − 0(cid:19) Since one of the definitions of the Moshinsky function M(x,k,t) is precisely i ′eik′x−ik′22t M(x,k,t) = dk , (3) 2π k′ k ZΓ+ − (The Moshinsky function can be related to the complementary error function, see Appendix A.), the resulting state can then be simply written as ~t ~t ψ (x,k ,t) = M x,k , +M x, k , . (4) 0 0 0 0 m − m (cid:18) (cid:19) (cid:18) (cid:19) Such combination of Moshinsky functions is ubiquitous when working with quantum sources and will appear throughout the paper. The source density profile corresponding to Eq.(4) exhibits the characteristic oscillations of diffraction in time [1, 2]. We are interested in describing more general switching procedures. However, given their dependence on pulse formation results, its discussion will be postponed to section 3. 2. Single pulse In this section we study the formation of a single-pulse of duration τ from a quantum sourcemodulatedaccordingtoagivenaperturefunctionχ(1)(t),thesuperscriptdenoting Time modulation of atom sources 4 the creation of a single pulse. Notice that some apodizing functions were already explored in the quantum domain [28], but we shall present a more general description. (1) In particular, let us consider the family of aperture functions χ (t) = n sinn(Ωt)Θ(t)Θ(τ t) with Ω = π/τ, see Fig. 1(a). The inverse Fourier transform − of the source amplitude at the origin is given by 1 2−nei(ω−ω0)τ/2τΓ(1+n) ψ(1)(x = 0,k ,ω) = , (5) n 0 √2πΓ (2+n)π+(ω−ω0)τ Γ (2+n)π−(ω−ω0)τ 2π 2π b (cid:20) (cid:21) (cid:20) (cid:21) where Γ(z) is the Gamma function [35]. The energy distribution of the associated wavefunction [29] is proportional to ω1/2 ψ(1)(x = 0,ω) 2. As shown in Fig.1(b), the n | | smoother the aperture function the wider is the energy distribution, which will affect the spacetime profile of the resulting pulseb. We shall next illustrate the dynamics for the n = 0,1,2 cases. The solution for an arbitrary aperture function is presented in Appendix B. 2.1. Rectangular aperture function (1) For the case χ (t) = Θ(t)Θ(τ t) corresponding to a single-slit in time the dynamics 0 − of the wave function is given by ψ(1)(x,k ,t;τ) = ψ (x,k ,t) Θ(t τ)e−iω0τψ (x,k ,t τ). (6) 0 0 0 0 − − 0 0 − Note that if t < τ, before the pulse has been fully formed, the problem reduces to that discussed in the previous section, being χ(t) = Θ(t). Equation (B.4) is also valid for these times, in contrast to previous works restricted to t > τ [25, 26, 27, 28]. The same will be true for the following pulses. 2.2. Sine aperture function (1) Let us consider now χ1 (t) = sin(Ωt)Θ(t)Θ(τ −t). Then, defining ω± = ω0 ± Ω and the corresponding wavenumbers k± = 2mω±/~, i (1) p(1) ψ (x,k ,t;τ) = αψ (x,k ,t;τ). (7) 1 0 2 0 α α=± X 2.3. Sine-square aperture function The sine-square (Hanning) aperture function is given by 1 1 χ(1)(t) = sin2(Ωt)Θ(t)Θ(τ t) = 1 cos(2Ωt) Θ(t)Θ(τ t). (8) 2 − 2 − 2 − (cid:20) (cid:21) Introducing k = 2m(ω 2Ω)/~ with β = , leads to β 0 ± ± p 1 1 (1) (1) (1) ψ (x,k ,t;τ) = ψ (x,k ,t;τ) ψ (x,k ,t;τ) , (9) 2 0 2 0 0 − 2 0 β " # β=± X Time modulation of atom sources 5 1.5 2.0 a) b) n=0 n=1 1.0 1.5 n=2 ω)0 (1)χ(t) 0.5 (1)χω−](n1.0 0.0 −1F[ 0.5 −0.5 −0.5 0.0 0.5 1.0 1.5 0.0 t/τ −1 0 1 2 3 ω/ω 0 Figure 1. a) Family χ (t) of single-pulse aperture functions. The smoothness at n the edges increases with in this order: rectangular (n = 0, continuous line), sine (n = 1, dashed line), sine-square (n = 2, dot-dashed line). b) The aperture function in the frequency domain, F−1[χ(1)](ω ω ), broadens as a result of apodization, with n 0 − increasing n. wheretheeffectoftheapodizationistosubstracttothepulsewiththesourcemomentum two other matter-wave trains associated with k , all of them formed with rectangular β aperture functions. The effect of a short τ on the energy distribution has already been studied in [2, 28] from which a time-energy uncertainty relation was inferred. Indeed, the uncertainty product reaches an approximate minimum of 2~ if the pulse duration is shorter than = π/ω . The space-time dynamics for a source apodized with a rectangular aperture 0 0 ⊤ function is illustrated in Fig. 2, where it is shown that for τ > the quantum average 0 ⊤ of the position operator follows the classical free-particle trajectory whereas if τ . 0 ⊤ the quantum pulse is sped up. Fig. 3 shows for different χ (t) (n = 0,1,2) the effect n on the velocity distribution which is only centered at ~k /m for τ > . Concerning 0 0 ⊤ the apodization, the higher the value of n the narrower is the mean width of χ (t) and n the larger the shift on the mean velocity. The smoothing of the χ (t) is reflected on the n suppression of the sidelobes in the velocity distribution. 3. Smooth switching: apodization and diffraction in time Instead of forming a finite pulse as discussed in the previous section, some sources are prepared to reach a stationary regime with constant flux after the initial transient. The extreme case is the sudden, zero-time switching of Sec. 1. In this section we shall study how the matter-wave dynamics is modified for slow aperture functions with a given switching time τ. In order to do so, we introduce the family of switching functions 0, t < 0 χs(t) = χ(t), 0 t < τ  ≤  1, t τ ≥ where it is assumed that χ(τ) = 1, and which is represented in Fig. (4)a for  χ(t) = sinnΩ t with n = 0,1,2 and Ω = π/2τ. Note that for n = 0, one recovers s s Time modulation of atom sources 6 Figure 2. (color online) Spacetime effects of the time-energy uncertainty relation. A pulse of duration τ < is sped up with respect to the source frequency. The 0 ⊤ spacetimedynamicsisillustratedfora87Rbsourcewithτ =1mswith~k /m=1cm/s 0 (left column, = 0.0467ms) and ~k /m = 0.1cm/s (right column, = 4.67ms), 0 0 0 ⊤ ⊤ modulated with different apodizing functions to form a single pulse. The dashed line reproduces the classical trajectory. The grey scale changes from dark to light as the function values increase. Figures in the same column show the suppression of the sidelobes in the probability density with increasing n. the sudden aperture χ (t) = Θ(t) which maximises the diffraction in time fringes. For 0 n = 1 (n=2) the source amplitude follows half a sine (square)-lobe. The general result for the time-dependent wavefunction simply reads, ψs(x,k ,t;τ) = e−iω0τψ (x,k ,t τ)Θ(t τ)+ψ(1)(x,k ,t;τ), (10) χ 0 0 0 − − χ 0 where the superscript s stands for switching and ψ(1) refers to a pulse of length τ and χ apodizing function χ. Figure 4(b) shows the effect of a finite switching time on the oscillatory pattern at a given time. As a result ofthe smoothaperture functions new frequencies aresingled out, leading to a progressive suppression of diffraction in time. Moreover, for a given χ (t) n function, a larger switching time τ increases the apodization of the source, washing out Time modulation of atom sources 7 30 25 a) 2 20 )| n=0 v 15 ( n=1 (1)ψ 10 n=2 | 5 0 0.9 0.95 1 1.05 1.1 4 b) 3 2 )| v2 ( 1) (ψ 1 | 0 0 0.2 0.4 0.6 0.8 v(cm/s) Figure 3. (Color on-line) Distortion of the velocity distribution for pulse duration smaller than . The corresponding pulses to Fig. 2 are plotted in velocity space 0 ⊤ for a 87Rb source with τ = 1ms apodized with different aperture functions and a) ~k /m=1cm/s ( =0.0467ms)and b) 0.1cm/s ( =4.67ms). The triangle marks 0 0 0 ⊤ ⊤ the classical velocity. 1.5 1.5 a) b) n=0 n=1 1.0 n=2 1.0 sχ(t)n 0.5 2ψ(x,t)| n=0 | 0.5 0.0 n=1 n=2 −0.5 0.0 −1.0 0.0 1.0 2.0 140 160 180 200 220 t/τ x(µm) Figure 4. a) Family χ (t) of switching-on aperture functions. The smoothness at n the edges increases with n. b) Apodization of the 87Rb beam profiles with increasing smoothness of the aperture function (higher n), and fixed switching time τ = 0.5ms s at t=20ms, and ~k /m=1 cm/s. 0 the fringes in the probability density, see Fig. 5(a). (In this sense, the effect of a finite band-widthsourceistantamounttoatimedependent modulation,aswasshownin[23].) However, it is remarkable that the effect of the apodization is limited in time because of a revival of the diffraction in time. The intuitive explanation is that the intensity of the signal from the apodization “cap”, see Fig. 6, decays with time whereas the intensity of the main signal (coming from the step excitation) remains constant. For sufficiently large times, the main signal, carrying its diffraction-in-time phenomenon, overwhelms the effect of the small cap. One can estimate the revival time by considering that the Time modulation of atom sources 8 1.5 a) 2x,t)| 1.0 001...150mmmsss ψ( 0.5 2.0ms | 0.0 160 180 200 220 1.5 b) 0.1 t 2x,t)| 1.0 012...500 tttrrr ψ( 0.5 r | 0.0 0 100 200 300 x(µm) Figure 5. Revival of the diffraction in time. a) The oscillations in the beam profilecharacteristicofthediffractionintimearegraduallysuppressedwithincreasing switching time τ. The beam profile is shown at t = 20ms, and ~k /m = 1cm/s. For 0 increasing τ the amplitude of the oscillations diminishes and the signal is delayed. b) During the time evolution there is a revival of the diffraction in time which is again clearly visible for times greater than t = ω τ2 (τ = 1ms, ~k /m = 0.5cm/s, and r 0 0 t =16.8ms). Both a) and b) refer to a 87Rb source which is switched on following a r sine-square function. 1.5 1.0 sχ(t)n 0.5 0.0 −0.5 −1 0 1 2 t/τ Figure 6. Aperture function decomposition. In a switching proccess the aperture functionhasafinitetransientcomponentresponsiblefortheapodization(shadedarea). Such apodization cap is followed by a step excitation for t τ which leads eventually ≥ to the revival of diffraction in time. momentum width of the apodization cap will be of the order π ∆p 2m~(ω +Ω ) 2m~(ω Ω ) 2m~ω , (11) 0 s 0 s 0 ≈ − − ≈ 2ω τ 0 p p p and taking into account the, essentially linear with time , spread of the cap half-pulse (1) described by ψ in Eq. (10) (we have checked numerically that the classical dispersion χ relation ∆x(t) ∆x(0)+∆pt /m holds). r ≃ (1) The ratio of the spatial widths corresponding to the initial and evolved cap ψ is χ given by ∆x(0) ∆x(0) R = . (12) ∆x(t ) ≈ ∆x(0)+∆pt /m r r Time modulation of atom sources 9 Taking ∆x(0) = 2~ω /mτ and imposing R = 1/2 leads to the expression for a 0 revival time scale p t ω τ2, (13) r 0 ≈ which is analogous to the Rayleigh distance of classical diffraction theory but in the time domain [31]. The smoothing effect of the apodizing cap cannot hold for times much longer than t . This is shown in Fig. 5(b), where the initially apodized beam r profile develops eventually spatial fringes, which reach the maximum value associated with the sudden switching at t 2t . r ≈ 4. Multiple pulses We show next that the single pulse aperture functions χ(1)(t), described in section 2 are useful to consider matter wave sources periodically modulated in time. Indeed, the N-pulse aperture function can be written as a convolution of χ(1)(t) with a grating function. In particular, if the grating function g(t) is chosen to be a finite Dirac comb g (t) = N−1δ(t jT), it is readily found that N j=0 − N−1 P χ(N)(t) = χ(1)(t) g (t) = dt′χ(1)(t t′)g (t′) = χ(1)(t jT), (14) N N ∗ − − Z j=0 X which describes the formation of N consecutive χ(1)-pulses, each of them of duration τ and with an “emission rate” (number of pulses per unit time) 1/T. If we are to describe an atom beam periodically modulated with such apodizing function, the space-time evolution simply reads, N−1 ψ(N)(x,k ,t) = e−iω0jTψ(1)(x,k ,t jT;τ)Θ(t jT) (15) χ 0 χ 0 − − j=0 X being a linear combination of the single-pulse wavefunction ψ(1)(x,t) studied in section 2. The Θ(t jT) corresponds to the condition that the j-th pulse starts to emerge only − after jT. A similar mathematical framework can be employed to describe an atom laser in theno-interactionlimit. Theexperimental prescription foranatomlaser depends onthe outcoupling mechanism. Here we shall focus on the scheme described in [12, 13]. The pulses areobtainedfroma condensate trappedina MOT, anda well-defined momentum is imparted on each of them at the instant of their creation through a stimulated Raman (1) process. The result is that each of the pulses ψ does not have a memory phase, the χ wavefunction describing the coherent atom laser being then N−1 φ(N)(x,k ,t) = ψ(1)(x,k ,t jT;τ)Θ(t jT), (16) χ 0 χ 0 − − j=0 X compare with (15). It is interesting to impose a relation between the outcoupling period T and the kinetic energy imparted to each pulse ~ω , such that ω T = lπ, where if l is chosen an 0 0 Time modulation of atom sources 10 0.03 0.03 a) 0.02 0.02 d) 0.01 0.01 0.00 0.00 0.004 8e−05 e) b) 4e−05 0.002 0e+00 0.003 0.000 0.002 f) 0.001 0.015 0.000 0.010 c) 0.06 0.04 g) 0.005 0.02 0.0000 50 100 150 0.00 x(µm) 0 200 400 600 x(µm) Figure 7. Beam density profile of a 23Na atom laser with τ = 0.833ms, T = 0.3ms, in the coherently constructive ω T = 10π (5.9 cm/s) (a), destructive ω T = 11π 0 0 (5.9 cm/s) (b), and incoherent case (c), with same conditions than a) with a sine- squareaperturefunction. Theconditionsequivalenttotheexperimentreportedin[12] (τ = 0.83ms, T = 0.05ms) are studied for the constructive (d), destructive (e) and incoherent case (f). If the pulse is not apodized the beam profile becomes noisy as shown in (g) for rectangular χ, in the same conditions than (d). In all cases the norm is relative to the incoherent case. (N) even (odd) integer the interference in the beam profile ψ is constructive (destructive), χ (N) see Fig. 7. However, in the case of a periodically chopped atom beam ψ , Eq. (15), χ the additional phase e−iω0jT singles out the constructive interference independently of the parity of l. So far we have assumed that the source is coherent. The interference pattern between adjacent pulses is affected by the presence of noise in the phases [32]. The average over many realisations in which the phase α of each pulse varies randomly in j [0,2π] leads to the density profile, N−1 2 ρ(x,t) = eiαjψ(1)(x,k ,t jT;τ)Θ(t jT) (17) χ 0 − − *(cid:12) (cid:12) + (cid:12)Xj=0 (cid:12) α (cid:12) (cid:12) (cid:12) (cid:12) (1) which reproduces the incoherent sum of the pulses, ρ(x,t) = ψ (x,k ,t (cid:12) (cid:12) j| n 0 − jT;τ) 2Θ(t jT). | − P One advantage of employing stimulated Raman pulses is that any desired fraction of atoms can be extracted from the condensate. Let us define s as the number of atoms (1) outcoupled in a single pulse, ψ . The total norm for the N-pulse incoherent atom laser χ is simply = sN. A remarkable fact for coherent sources is that the total number inc N of atoms outcoupled from the Bose-Einstein condensate reservoir depends on the c N nature of the interference [33]. Therefore, in Figs 7 and 8 we shall consider the signal relative to the incoherent case, namely, ψ(N) 2/ . χ inc | | N Note the difference in the time evolution between coherently constructive and incoherent beams in Fig. 8: At short times both present a desirable saturation in the probability density of the beam. However, in the former case a revival of the diffraction