Sagnac Interferometry Using Bright Matter-Wave Solitons
We use an effective one-dimensional Gross–Pitaevskii equation to study bright matter-wave solitons held in a tightly confining toroidal trapping potential, in a rotating frame of reference, as they are split and recombined on narrow barrier potentials. In particular, we present an analytical and numerical analysis of the phase evolution of the solitons and delimit a velocity regime in which soliton Sagnac interferometry is possible, taking account of the effect of quantum uncertainty.
pacs:05.45 Yv, 03.75 Lm, 67.85 De
A Bose–Einstein condensate (BEC) with attractive inter-atomic interactions can support soliton-like structures referred to as bright solitary matter-waves Khaykovich et al. (2002); Strecker et al. (2002); Cornish et al. (2006); Marchant et al. (2013); Nguyen et al. (2014). These propagate without dispersion Morgan et al. (1997), are robust to collisions with other bright solitary matter-waves and with slowly varying external potentials Parker et al. (2009); Billam et al. (2011), and have center-of-mass trajectories well-described by effective particle models Martin et al. (2007, 2008); Poletti et al. (2008). Such soliton-like properties are due to the mean-field description of an atomic BEC reducing to the nonlinear Schrödinger equation in a homogeneous, quasi-one-dimensional (quasi-1D) limit, which for the case of attractive interactions supports the bright soliton solutions well-known in the context of nonlinear optics Zakharov and Shabat (1971); Satsuma and Yajima (1974); Gordon (1983); Haus and Wong (1996); Helczynski et al. (2000). The quasi-1D limit is experimentally challenging for attractive condensates Billam et al. (2012), but solitary wave dynamics remain highly soliton-like outside this limit Cornish et al. (2006); Billam et al. (2011).
A bright solitary wave colliding with a narrow potential barrier is a good candidate mechanism to create two mutually coherent localised condensates, much as a beam-splitter splits the light of an optical interferometer. This has been extensively investigated in the quasi-1D, mean-field description of an atomic BEC Helm et al. (2012); Kivshar and Malomed (1989); Ernst and Brand (2010); Lee and Brand (2006); Cao and Malomed (1995); Holmer et al. (2007a, b); Polo and Ahufinger (2013); Damgaard Hansen et al. (2012); Minmar (2012); Abdullaev and Brazhnyi (2012), and sufficiently fast collisions do lead to the desired beam-splitting effect Holmer et al. (2007a, b). Consequently, bright solitary matter-waves, with their dispersion-free propagation, present an intriguing candidate system for future interferometric devices Helm et al. (2012); Strecker et al. (2002); Cornish et al. (2009); Weiss and Castin (2009); Streltsov et al. (2009); Billam et al. (2011); Al Khawaja and Stoof (2011); Martin and Ruostekoski (2012); McDonald et al. (2014). Previous work Martin and Ruostekoski (2012); Polo and Ahufinger (2013); Helm et al. (2014) considered a Mach–Zehnder interferometer using a narrow potential barrier to split harmonically trapped solitary waves, based on the configuration of a recent experiment Nguyen et al. (2014). These demonstrated one can also recombine solitary waves if they collide at the barrier; the collision dynamics are explained more fully in Helm et al. (2012). In these collisions, the relative atomic populations within the two outgoing solitary waves are governed by the relative phase between the incoming ones. The mean-field nonlinearity can lead to the relative populations of the outgoing waves exhibiting greater sensitivity to small variations in the phase ; however, simulations including quantum noise in the initial condition Helm et al. (2014) or via the truncated Wigner method Blakie et al. (2008), demonstrated that enhanced number fluctuations counteract this improvement Martin and Ruostekoski (2012).
We extend the framework of soliton interferometry to measurement of the Sagnac effect, first observed in an atom interferometer by Riehle et al. Riehle et al. (1991). In this experiment the observation manifested as a shift in the Ramsey fringes produced by passing an atomic beam of Ca through four travelling waves in a Ramsey geometry, producing an atomic beam interferometer. What we present differs from the Riehle setup in two ways. Firstly, in Riehle et al. (1991) some phase information is transported optically. In our system atom-light interactions serve only to coherently split the condensate; any resulting phase dynamics are incidental. Secondly, our system results, not in an interference fringe shift, but a population shift between the positive and negative domains of the interferometer. The Sagnac effect is inferred from measurements of particle numbers Halkyard et al. (2010) in the spatially distinct condensates on either side of the barrier, and not the structure of those condensates. (which are expected to remain soliton-like). We consider an experimental configuration, contained entirely within a rotating frame, where there is a smooth ring-shaped trapping potential (implemented by, e.g., using a spatial light modulator Moulder et al. (2010), time-averaging with acousto-optic deflectors Henderson et al. (2009), or imaging an intensity mask Corman et al. (2014)) and narrow barriers realised with optical light sheets, focussed using high numerical aperture lenses. Solitons, initially produced in an optical trap, can be adiabatically transferred into the ring, with the initial velocity set by moving the optical potential during the transfer Rakonjac et al. (2012). Key sources of error include: uncertainty in the value of the soliton velocity relative to the barrier strength and, in turn, the barrier transmission level Helm et al. (2012); initial particle number, which determines the ground-state energy and so sets the low-energy splitting threshold, close to which the system becomes sensitive to otherwise small fluctuations in the velocity Martin and Ruostekoski (2012); and measurement of final particle number.
Within the Gross–Pitaevskii equation (GPE) framework, we consider bosonic atoms of mass and scattering length , in an effective 1D configuration due to a tightly confining (frequency ) harmonic trapping potential in the degrees of freedom perpendicular to the direction of free motion, implying an interaction strength of per particle. We use “soliton units” Martin et al. (2008) (equivalent to ), where position, time and energy are in units of , , and Note1 (). To describe a tightly confining toroidal trap geometry (or ring trap), we introduce periodic boundary conditions over the domain , where is the dimensionless form of the circumference Helm et al. (2014). It is common to discuss Sagnac interferometry and ring systems in terms of an angle coordinate Halkyard et al. (2010); Kanamoto et al. (2010); we choose not to, making it easier to draw on earlier work on splitting solitons at narrow barriers Helm et al. (2012); Holmer et al. (2007a, b); Helm et al. (2014). Considering the dynamics within a frame rotating with dimensionless angular frequency results in the following GPE:
where [which we can also write in terms of the dimensional circumference and angular frequency as ], and is the (unit norm) condensate wave function. Note the two barrier terms; presence or absence of the second barrier implies two different forms of Sagnac interferometry: one where both solitons perform full circumnavigations of the ring, enclosing the area within the ring twice; and one where each soliton circumnavigates a different half of the ring, enclosing the area once. These cases are distinguished by for the first (single barrier) case and for the second (two antipodal barriers) case; the second barrier term is zero for [see Fig. 1(a)] and identical to the other barrier term, up to a spatial offset, for [see Fig. 1(b)]. All simulations were carried out with ; this width is suitably narrow to approximate a delta function for collisional velocities up to Note2 (), corresponding to a (variable, depending on the ring circumference) dimensionless angular velocity of .
We obtain soliton solutions to Eq. (1) (in the absence of splitting potentials and periodic boundary conditions), i.e., the usual nonlinear Schrödinger equation in a frame moving with velocity , by the Galilean invariance of the standard soliton profile Gordon (1983). The (amplitude ) invariant soliton solution is ; the tilde notation denotes the stationary frame of reference. A soliton moving with velocity in a frame moving with velocity is moving at velocity in the stationary frame. In the moving frame, where , we obtain
We now outline the three-step process of soliton Sagnac-interferometry, common to both () configurations; later we will analyse the system phase evolution fully. First, a ground state soliton is split into two secondary solitons, of equal size and a specific relative phase, at a narrow potential barrier [time in Fig. 1(a)(ii) and (b)(ii)]. We obtain an equal split by selecting the barrier’s strength Note3 () for a given incident velocity Loiko et al. (2010) and barrier width . It is the velocity in the frame comoving with the barrier that must be known; the value of the frame velocity (itself due to the angular frequency ) does not affect the outcome. In the second step the secondary solitons accumulate a further relative phase difference . This is the -dependent quantity we wish to measure, gained as a result of the differing path lengths travelled by counter propagating waves in a moving frame [time in Fig. 1(a)(ii) and (b)(ii)]. Finally, the two solitons collide at a narrow barrier [time in Fig. 1(a)(ii) and (b)(ii)]. After this collision the wave-function integrals on either side of the barrier, , allow us to determine the value of Helm et al. (2012, 2014), where and are the positive and negative domain populations. These are ideally determined with an atom number variance below one particle, i.e., exact particle counting at output. This is a challenge facing the whole field of atom interferometry, particularly for experiments pursuing Heisenberg-limited measurements. Single atom resolution has been achieved using a variety of techniques Ockeloen et al. (2010); Muessel et al. (2010); Buëcker et al. (2010); Hu and Kimble (1994); Alt et al. (2010); Serwane et al. (2011); Grünzweig et al. (2010); Puppe et al. (2007); Gehr et al. (2010); Goldwin et al. (2011) for small numbers () and has recently Hume et al. (2013) been extended to mesoscopic ensembles () typical of the output states of the soliton interferometer.
To determine how the Sagnac effect manifests in GPE soliton interferometry, we must describe the phase dynamics more fully. After the initial split at time , the transmitted soliton (in the positive domain) has peak phase (value of the phase at the position of the soliton’s peak amplitude), while that reflected (in the negative domain) has peak phase . We wish to determine the phase difference between the solitons before they collide with one another at a barrier at time , i.e., [the prefactor changes the sign of the phase difference to account for the solitons approaching the collisional barrier from different directions depending on the value of ]. In both cases we choose (the initial soliton starts at ). If the solitons created by the splitting event must both fully circumnavigate the ring before colliding at the same barrier, while for the solitons only travel half the distance; hence . In the limiting case of a -function barrier, the first (splitting) step causes the transmitted soliton to be phase shifted by ahead of the (equal amplitude) reflected soliton, as shown analytically in Helm et al. (2014). We use this figure as an estimate of the phase difference accumulated by splitting on a Gaussian barrier Helm et al. (2012); see Polo and Ahufinger (2013) for a discussion of phase shifts accumulated with finite-width barriers. We select a Gaussian profile for the barrier, as is typical for experimental setups involving off-resonant sheets of light Marchant et al. (2013), and take . We obtain the phase evolution at the peak of an individual soliton by taking the imaginary part of the exponent of Eq. (2) and setting , giving (up to an initial offset) . Hence, , , and the final phase difference between the solitons is
Without a second barrier (), the solitons mutually collide at the point antipodal to the splitting barrier. As this occurs in the absence of any axial potentials or barriers, the solitons are unaffected beyond asymptotic shifts to position and phase Gordon (1983); Zakharov and Shabat (1972), given by
where and . The quantities and are asymptotic position and phase shifts associated with the th soliton, while and describe that soliton’s velocity and amplitude. Associating the soliton transmitted through the barrier at time with , we obtain the correct sign for our asymptotic shifts. In our case, noting that we determine the relative phase shift, and the relationship between the position shifts which arise as a result of this collision to be:
Both results use the standard complex logarithmic identity . Equation (5) shows us that can be omitted from the calculation of , that rapidly as , and also that whatever the size of the asymptotic position shift, the solitons are always shifted by equal amounts in opposite directions, and so will always meet at the collisional barrier situated at . Hence, the antipodal collision in the absence of a barrier does not affect the outcome of Sagnac interferometry if we can assume that the solitons’ accelerations during the collision do not affect the Sagnac phase accumulation. The analysis supporting this assumption is beyond the scope of the current work but can be verified numerically. A potential experimental advantage of the single-barrier configuration is that there is no need to locate a second barrier with great precision relative to the first; that both splitting products traverse exactly the same path before recombining is also likely to “smooth over” effects of small asymmetries in the trapping potential.
where as , and is the Sagnac phase we wish to determine. We show results of numerical GPE simulations in Fig. 2(a,b). For very high velocities, , the interference follows our prediction [Eq. (6)] closely, with very small skews arising from nonlinear effects during the final barrier collision, i.e., we can consider in this regime. The (c,d) and (a,b) cases have similar structures, however for the phase varies twice as quickly, as the interrogation time per shot is twice as long. Otherwise, the similarity of the structures supports the assumption that accelerations during barrier-free collisions do not affect the Sagnac phase accumulation. As we reduce the velocity, and the necessary (to avoid complicating nonlinear effects arising from a slow interaction with the barrier) assumption of high initial kinetic energy Holmer et al. (2007a, b) breaks down, our numerics show that the preceding analysis no longer holds, and so we conclude that Sagnac interferometry is not practicable in the regime. This is consistent with previous work delimiting as the high-to-low-energy transitional regime Helm et al. (2014), and the results shown here are comparable to those obtained for the Mach–Zehnder configuration Helm et al. (2014).
Figure 3 shows results of Monte Carlo simulations following the methodology described in Helm et al. (2014), which accounts for quantum uncertainty in the initial soliton’s center of mass (CoM) position and velocity by adding Gaussian random offsets to the classical soliton’s initial velocity and peak position. Here we consider a two-barrier system where the soliton is initially accelerated by a harmonic trap, with frequency and its minimum at . The soliton is prepared and released from a position (before quantum fluctuations in the CoM are considered). This harmonic trap is then switched off once the soliton reaches , and its velocity is . The CoM position and velocity uncertainties contribute to velocity uncertainty at the point of collision, giving collisional velocities that follow a Rician distribution Helm et al. (2014). Increasing reduces the widths of the outcome distributions by reducing the relative significance of quantum fluctuations, hence making the transmission curves [Fig. 3(a-d)(i)] steeper. As the gradients of these curves asymptote upward, the distributions of the simulation outcomes [Fig. 3(a-d)(ii)] become narrower. The distributions for the and sets of simulations should, ideally, be centered on ; these distributions do not have the same location, but approach the ideal () with increasing . This is due to the nonlinear skew interfering with the phase evolution during the final collision at time , as described in Helm et al. (2012), and predicted by the GPE.
In conclusion, we have employed a GPE treatment to show how, using a moving bright matter-wave soliton as the initial condition, a matter-wave Sagnac interferometer can be realized within a quasi-1D toroidal trapping configuration (ring trap), in combination with one or two narrow Gaussian barriers due to off-resonant sheets of light. Although both configurations are in principle equally effective, we note that the single-barrier case is likely less susceptible to systematics due to small asymmetries in an experimental configuration. We have also explored the effects of quantum fluctuations in the atomic matter-wave’s center-of-mass position and velocity; we find that, so long as the initial soliton velocity is sufficiently fast, particle numbers of suffice to give sharp transmission responses, which can then be interpreted to deduce a Sagnac phase.
Acknowledgements.We thank D. I. H. Holdaway, A. L. Marchant, and C. Weiss for useful discussions, and the UK EPSRC (grant no. EP/K030558/1) for support.
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