Bright soliton quantum superpositions: signatures of high- and low-fidelity states
Scattering quantum bright solitons off barriers has been predicted to lead to nonlocal quantum superpositions, in particular the NOON-state. The focus of this paper lies on signatures of both high- and low-fidelity quantum superposition states. We numerically demonstrate that a one-dimensional geometry with the barrier potential situated in the middle of an additional – experimentally typical – harmonic confinement gives rise to particularly well-observable signatures. In the elastic scattering regime we investigate signatures of NOON-states on the -particle level within an effective potential approach. We show that removing the barrier potential and subsequently recombining both parts of the quantum superposition leads to a high-contrast interference pattern in the center-of-mass coordinate for narrow and broad potential barriers. We demonstrate that the presented signatures can be used to clearly distinguish quantum superpositions states from statistical mixtures and are sufficiently robust against experimentally relevant excitations of the center-of-mass wave function to higher lying oscillator states. For two-particle solitons we extend these considerations to low-fidelity superposition states: even for strong deviations from the two-particle NOON-state we find interference patterns with high contrast.
pacs:03.75.Gg, 03.75.Lm, 34.50.Cx, 67.85.-d
The experimental realization of mesoscopic entanglement is in the focus of current research: interest stems both from fundamental aspects as tests of decoherence mechanisms Zurek (2003) and the relevance for quantum-enhanced interferometry Giovanetti et al. (2004). Recent suggestions for realizations of nonlocal mesoscopic superpositions include Bose-Einstein condensates (BECs) Carr et al. (2010), cavity quantum optomechanical systems Romero-Isart et al. (2011) and topological defects Dziarmaga et al. (2012). Bright solitons111In the following the term “soliton” is used synonymously with “solitary wave”., self-bound matter-waves generated from Bose-Einstein condensates Pethick and Smith (2002); Pitaevskii and Stringari (2003) are, in their quantum version Lai and Haus (1989); Castin and Herzog (2001); Sykes et al. (2007), a particularly promising system to generate quantum superpositions Weiss and Castin (2009); Streltsov et al. (2009); Gertjerenken et al. (2013).
Landmark experiments have already realized bright solitons which behave close to Gross-Pitaevskii solitons Pethick and Smith (2002); Helm et al. (2012); Cuevas et al. (2013): both single bright matter-wave solitons Khaykovich et al. (2002) and soliton trains Strecker et al. (2002); Cornish et al. (2006) have been created in (quasi-)one-dimensional attractive Bose gases. Scattering bright solitons off barrier potentials is currently investigated in a new generation of experiments Pollack et al. (); Marchant et al. (2013).
The stability of solitons Khaykovich et al. (2002); Billam et al. (2013) makes them a candidate system for matter-wave interferometry Martin and Ruostekoski (2012); Helm et al. (2012); Cuevas et al. (2013). Theoretical investigations of one-dimensional attractive Bose gases also comprise soliton localization via disorder Müller (2011), polaritonic solitons in an optical lattice Dong et al. (2013), effects of higher dimension Billam et al. (2013); Cuevas et al. (2013), macroscopic quantum tunneling Glick and Carr (2011), resonant trapping in a quantum well Ernst and Brand (2010), collision-induced entanglement of indistinguishable solitons Lewenstein and Malomed (2009), the creation of Bell states via collisions of distinguishable solitons Gertjerenken et al. (2013), and collision dynamics and entanglement generation of two initially independent and indistinguishable boson pairs Holdaway et al. (2013).
While a mean-field description has been found to successfully describe many aspects in the scattering dynamics of bright solitons for high kinetic energies Akkermans et al. (2008); Cuevas et al. (2013); Marchant et al. (2013) considerably deviating behavior has been predicted for lower kinetic energies. In this regime scattering bright solitons off a barrier potential gives rise to a continuously varying reflection/transmission coefficient on the -particle quantum level while a discontinuous behavior has been observed on the Gross-Pitaevskii level Wang et al. (2012); Gertjerenken et al. (2012). This indicates the formation of quantum superposition states at the -particle level which are not allowed by the nonlinear Gross-Pitaevskii equation Gertjerenken et al. (2012).
where all the particles are placed in a coherent superposition with a distance between the two parts of the wave function that is much larger than the soliton size. Here, and in the following, the notation signifies that () particles are situated to the left (right) of the barrier potential. The NOON-state is very sensitive to decoherence: a single atom loss suffices to destroy the quantum superposition. In this view particle numbers on the order of have been suggested Weiss and Castin (2009); Streltsov et al. (2009). Experimental requirements for the realization of the suggested protocols are low temperatures Leanhardt et al. (2003), a good vacuum Anderson et al. (1995) and the particle-number control available in experiments like Gross et al. (2010), cf. Sec. III.
Another interesting class of states, also relevant for quantum-enhanced interferometry, are general superposition states involving contributions like
Such a quantum superposition is less sensitive to decoherence via atom losses: it would be turned into a statistical mixture of still entangled states. Additionally, in the presented protocols low-fidelity quantum superpositions, enabling higher initial CoM kinetic energies, could be realized on shorter time-scales than the NOON-state.
An unambiguous experimental demonstration requires clear, experimentally measurable signatures that distinguish quantum superpositions from statistical mixtures, which is the focus of this work. In Weiss and Castin (2009) it has been suggested to switch off the scattering potential and let both parts of the quantum superposition recombine and interfere. These interference patterns are particularly well-observable in the center-of-mass (CoM) density Gertjerenken and Weiss (2012)
while in general222For particle numbers as low as interference patterns in the single-particle density can be observed but with reduced contrast in comparison to the CoM density. For larger particle numbers only the CoM density gives rise to high-contrast interference patterns, cf. Gertjerenken and Weiss (2012). they vanish in the single-particle density Gertjerenken and Weiss (2012)
Here, denotes the expectation value, the spatial coordinate, the positions of the particles and the CoM of the system. The CoM has to be treated quantum mechanically Weiss and Castin (2009); Müller (2011) and is measurable with larger precision than the width of the soliton. In contrast to the interference of two Bose-condensates where a single experiment yields an interference pattern Andrews et al. (1997), the CoM density (3) is determined in a series of measurements: each run gives a single point and an interference patterns builds up as for single-photon interference Grangier et al. (1986). In the following we assume sufficient experimental stability to guarantee shifts of the interference pattern from run to run to be smaller than the distance of neighboring interference maxima to avoid washing out of the pattern.
Here, we investigate the scattering of a bright soliton on a barrier potential that is located in the middle of an experimentally typical Strecker et al. (2002); Cornish et al. (2006); Pollack et al. (); Marchant et al. (2013) slight harmonic confinement and show that this set-up leads to advantageous signatures of quantum superposition states. First, we investigate the elastic scattering regime within the effective potential approach Weiss and Castin (2009); Sacha et al. (2009); Weiss and Castin (2012). In contrast to previous work Gertjerenken et al. (2012), where scattering twice off the barrier potential has been investigated, we focus on interference patterns in the CoM density after scattering once off the barrier potential. While the first has been identified as a clear signature of quantum superpositions particularly suited for not-too-broad effective potentials, we demonstrate high-contrast interference patterns in the CoM density both for narrow and broad effective potentials. We also show that both signatures are sufficiently robust to experimentally relevant excitations in the CoM coordinate for realistic parameters.
An interesting question is if interference patterns with suitable contrast are still observable for general quantum superposition states (2), requiring -particle methods beyond the effective potential approach. At the example of a two-particle soliton we demonstrate sufficiently high values of the contrast via discretization of the two-particle Schrödinger equation.
The paper is structured as follows: Section II introduces the underlying -particle methods and the numerical implementation via discretization of the Schrödinger equation. In Sec. III experimental requirements for the creation of quantum superposition states in this set-up are described. In Sec. IV the resulting interference patterns in the CoM density are investigated for high-fidelity NOON-states (1) within the effective potential approach. A model for the effect of excitations in the CoM coordinate is outlined in Sec. V. In Sec. VI the discussion is extended to general, low-fidelity quantum superposition states for two-particle solitons. Section VII concludes the paper.
The system can be modeled on the -particle level with the exactly solvable Lai and Haus (1989); Castin and Herzog (2001); Sykes et al. (2007) Lieb-Liniger(-McGuire) Hamiltonian Lieb and Liniger (1963); McGuire (1964) with additional external potential :
Here, denotes the particle number, the particle mass and attractive contact interaction with coupling constant is assumed.
For zero external potential eigensolutions of the resulting Schrödinger equation are (up to a phase factor) translationally invariant:
with . Eigenenergies are given by the CoM kinetic energy
of the quantum soliton
Taking a delta-function for the CoM wave function leads in the limit to a single-particle density (4) of the quantum soliton (9) identical to the mean-field density Calogero and Degasperis (1975); Castin and Herzog (2001)
While in this case mean-field and -particle solutions agree, mesoscopic quantum superpositions which are in the focus of this work cannot be described by the nonlinear Gross-Pitaevskii equation.
ii.1 Effective potential approach
In the low-energy regime scattering bright solitons off barrier potentials can be described on the -particle level within the mathematically rigorous Weiss and Castin (2012) effective potential approach Weiss and Castin (2009); Sacha et al. (2009). This approach is particularly suited to obtain physical insight into the scattering dynamics and gives an effective Schrödinger equation for the CoM motion (as in Gertjerenken et al. (2012) with additional harmonic confinement):
Here denotes the axial trapping frequency. The effective potential is the convolution of the internal density profile of the soliton with the barrier potential :
with introduced in equation (12) and
The effective potential thus has the form of the soliton. Narrow effective potentials imply small soliton and barrier widths. If the effective potential is narrow enough, it can be approximated with a delta function
where we also allow for shifts of the barrier potential out of the middle of the harmonic confinement. Broad potentials of the form (15) would also be obtained by broader scattering potentials as in current set-ups Marchant et al. (2013). For the effective potential approach to be valid these have to be sufficiently smooth which for using a laser focus as a scattering potential will always be the case.
ii.2 Numerical implementation
The situation can be modeled with a Bose-Hubbard Hamiltonian with additional harmonic confinement,
both on the -particle level within the effective potential approach from Sec. II.1 (in this case without interaction term) and for a two-particle soliton. Here denotes the tunneling strength, the on-site interaction strength, the strength of the harmonic confinement and the strength of the delta-like barrier potential. The operators annihilate (create) a particle at lattice site , and is the particle number operator for lattice site . The time-evolution corresponding to the Hamiltonian (18) is computed via the Shampine-Gordon routine Shampine and Gordon (1975) for sufficiently small lattice spacing : in the limit the Lieb-Liniger model with additional harmonic confinement is recovered.
Iii Scattering bright solitons off barrier potentials in additional harmonic confinement
In current experiments scattering bright solitons off barrier potentials is investigated in additional harmonic confinement Pollack et al. (); Marchant et al. (2013). Here, we model such a protocol for the low-energy regime necessary for the production of quantum superposition states Weiss and Castin (2009); Streltsov et al. (2009).
Initially the many-particle ground state is prepared in the harmonic trap Holdaway et al. (2012): throughout this work, we assume that the internal degrees of freedom are described by the Lieb-Liniger soliton (9) while the CoM motion is determined by the harmonic confinement. The center of the trap is then (quasi-)instantaneously shifted and the scattering potential in the middle of the trap is switched on.
In the following we specify the experimental requirements necessary for the creation of mesoscopic quantum superpositions. The initial state has to be prepared carefully: With probability
the initial CoM wave function is given by the ground state of the harmonic oscillator. To avoid excitations to higher lying oscillator states temperatures in the range of 450 pK, as in Leanhardt et al. (2003), are required. The temperature should also be small in comparison with , as defined in Eq. (II), to avoid excitations of single particles out of the soliton. Additionally, to ensure elastic scattering we assume low CoM kinetic energies Weiss and Castin (2009); Gertjerenken et al. (2012), corresponding to weak harmonic confinement. To reduce decoherence by single-particle losses a very good vacuum Anderson et al. (1995) is required. No thermal rest gas should be present in the harmonic trap. To ensure clear signatures a particle number post selection as in Gross et al. (2010) is assumed.
Experimentally realistic parameters Weiss and Castin (2009) are333In Weiss and Castin (2009) a slightly larger axial trapping frequency Hz was chosen. In contrast to this work, the axial trapping potential is used for the initial state preparation and subsequently opened.
a typical soliton size m and 100 Li atoms, implying an oscillator frequency m. For these parameters we find sufficiently short time scales of decoherence. While the other restrictions are fulfilled, the chosen parameters yield occupation probabilities , and for the ground, first and second excited oscillator state. Hence, excited states of the CoM are expected to be significantly occupied. In Sec. V we demonstrate that the signatures of quantum superposition states shown in Sec. IV are sufficiently robust against such excitations of the CoM wave function to higher lying oscillator states. An optimization of experimental parameters could be a topic for future research.
Iv Interference patterns for high-fidelity NOON-states
In the following we exemplary investigate scattering bright solitons off very narrow delta-like and broader barrier potentials. At this point we neglect excitations of the CoM wave function. Within the effective potential approach we numerically investigate the CoM motion as described in Secs. II.1 and II.2. The contrast of an interference pattern is defined as
where () denotes the maximum (minimum) value of the intensity on a suitably chosen interval.
Time scales are given in units of the dimensionless time , where
and lengths scales in units of the CoM oscillator length
First, we assume a delta-like effective potential (17). The time-evolution of the CoM density (3) is depicted in Fig. 1. Initially the CoM of the soliton is prepared in the shifted oscillator ground state. Scattering at the barrier potential in the middle of the harmonic confinement at approximately leads to the creation of a nonlocal mesoscopic quantum superposition (1). The barrier height throughout this work is chosen to ensure 50%-50%-splitting of the wave function. At the barrier potential is switched off, resulting in an interference of both parts of the wave function at . The interference pattern with fringe spacing and maximal contrast is stable under variations of the initial particle number, cf. Fig. 1 (a). Assuming the quantum superposition to be turned into a statistical mixture results in a clearly different time-evolution as shown in Fig. 1 (b).
The same is depicted in Fig. 2 for a broader effective potential (17) with and a larger shift of the initial wave function. Also in this case the interference pattern with high contrast (again under consideration of variations of the initial particle number) can be used to clearly distinguish the quantum superposition state from a statistical mixture. The larger CoM kinetic energy results in a smaller fringe spacing in the interference pattern. The fringe spacing can in general be enhanced by choosing a smaller trapping frequency or smaller particle numbers.
iv.1 Plane wave approximation of interference patterns
The numerically observed interference patterns in the CoM coordinate are well described in an approximation with plane waves, cf. Fig. 3.
with reflection and transmission coefficients
In order to realize 50%-50%-splitting the condition
must be fulfilled.
The leading order behavior is understood by considering only the mean momentum component. The envelope of the interference pattern is captured by considering a wave packet centered around the mean momentum component . To mimick the time-evolution induced by the harmonic confinement we assume for each component
where we include shifts of the barrier potential out of the middle of the harmonic confinement. For a wave packet with Gaussian envelope this results in
The absolute square then gives a good approximation of the interference pattern, cf. Fig. 3 (a).
V Excitations in center-of-mass coordinate
For the creation of quantum superposition states the restriction has to be fulfilled Weiss and Castin (2009). What happens if the much more stringent temperature restriction is relaxed, such that the CoM of the gas may be in an excited state?
Bose-Einstein condensates at finite temperature have been investigated in Sinatra et al. (2001) with a classical field method. Here, we simulate effects of finite temperature by assuming a statistical occupation of higher lying oscillator states for the CoM wave function while we assume that the internal degrees of freedom are still described by the quantum soliton (9).
In the following a renormalized occupation probability for a finite number of included oscillator states is assumed:
The statistically averaged CoM density distribution then is given by
Here is the occupation probability (19) of the -th occupied oscillator state and the CoM density distribution of the -th oscillator state.
How are the interference patterns from section IV affected by such excitations? For a very narrow, delta-like effective potential (17) this is illustrated in figure 4 (a), where the averaged density distribution (31) is displayed for and the parameters (20). The contrast (21) of the resulting interference pattern is only slightly reduced from one to . For the experimentally realistic parameters (20), corresponding to a width of the effective potential (15), the results are displayed in Fig. 4 (b). The contrast (21) is reduced from approximately one to , a value still allowing a clear distinction from a statistical mixture.
In Gertjerenken et al. (2012) another protocol, particularly suited for not-too-broad effective potentials, was investigated: Leaving the barrier potential switched on after the creation of the NOON-state (1) and scattering twice off the barrier potential in a Mach-Zehnder like set-up leads to a particularly well-observable signature of quantum superposition states. We again assume a delta-like effective potential (17). For the CoM wave function initially in the ground state of the harmonic confinement the time-evolution is displayed in Fig. 5 (a): After one oscillation period all the particles would be found with a probability to the right of the barrier potential, if initially situated to the left side of the barrier potential444This behavior can also be understood in an approximation with plane waves Gertjerenken et al. (2012). The probability is sensitive on the width of the barrier potential and shifts of the barrier potential out of the middle of the harmonic confinement. This could be used in an interferometric application to measure small potential gradients along the center of the harmonic trap, cf. Figs. 2 and 3 from Gertjerenken et al. (2012). (including variations of the initial particle number as in Fig. 1 the value is ). In the case of a statistical mixture the particles would be found with equal probability at either side of the barrier potential, allowing a clear distinction between quantum superposition and statistical mixture in a series of measurements. Figure 5 (b) and (c) show the time-evolution for the first and second excited eigenstate. The statistically averaged density distribution (31) is displayed in Fig. 5 (d) for included oscillator states and parameters (20). This yields a statistically averaged probability to find the particles to the right of the barrier potential. Figure 5 (e) shows the averaged CoM density distribution when the quantum superposition is turned into a statistical mixture at : As at zero temperature the particles would be found with equal probability to either side of the barrier potential.
For a broader effective potential with the probability to find the particles to the right of the barrier potential is for the quantum superposition and for the statistical mixture (cf. Fig. 3 from Gertjerenken et al. (2012).). For the parameters (20) we obtain and . To allow for a good experimental distinguishability a large number of runs is required both for zero and finite temperature.
Vi Two-particle solitons: interference patterns for high- and low-fidelity quantum superposition states
Research on two-particle bound states includes experimental realizations Winkler et al. (2006) and theoretical investigations Piil and Mølmer (2007); Petrosyan et al. (2007); Javanainen et al. (2010); Weiss (2010); Sørensen and Mølmer (2012); Gertjerenken and Weiss (2012); Fogarty et al. (2013). The creation of the two-particle NOON-state
via scattering at a barrier potential has been demonstrated numerically Weiss (2010); Gertjerenken and Weiss (2012); Fogarty et al. (2013). The subsequent recombination of both parts of the wave function again gives rise to an interference pattern Gertjerenken and Weiss (2012); Fogarty et al. (2013) that is particularly well-observable in the CoM density, cf. Gertjerenken and Weiss (2012) and footnote 2.
The former considerations for high-fidelity NOON-states are extended in the following: we show that lowering the interparticle interaction can lead to the generation of low-fidelity quantum superposition states
with complex coefficients and . The occupation probabilities of the states , are denoted in the following. We show that an interference pattern can still be visible in this generalized situation, even for strong deviation from the two-particle NOON-state (32). A high ratio of kinetic energy to interaction energy finally leads to the creation of product states, cf. the observed behavior for Gertjerenken et al. (2012): for non-interacting particles the set-up corresponds to the action of a single-particle beam-splitter, yielding occupation probabilities , and .
Effects of the interparticle interaction have also been investigated in Fogarty et al. (2013) with a focus on single-particle densities. Along the lines of Gertjerenken and Weiss (2012) we focus in the following in particular on interferences in the CoM density.
vi.1 Numerical results
We investigate scattering a two-particle soliton off a delta-like barrier potential, situated in the middle of an additional harmonic confinement. Lengths scales are given in units of
The presented results are obtained via discretization of the two-particle Schrödinger equation with sufficiently small lattice spacing, cf. Sec. II.2. With the parameter the ratio of interparticle interaction strength to kinetic energy can be adjusted. A lattice with 201 sites has proven suitable to allow for a satisfying spatial resolution in reasonable computing time. The initial state, the two-particle ground state, is determined by imaginary time evolution Glick and Carr (2011), on a lattice where the harmonic confinement is shifted 50 lattice sites to the left. For the real-time evolution the harmonic confinement is again centered around the middle of the lattice such that the initial state is situated on the edge of the harmonic confinement.
In Fig. 6 the time evolution of CoM density (3) and single-particle density (4) is compared for two different ratios of interaction strength to tunneling strength: for panel (a) shows the initial state localized to the left of the delta-like barrier potential. The scattering at the barrier potential at about leads to the creation of a nonlocal two-particle NOON-state (32). The strength of the scattering potential is chosen to ensure 50%-50%-splitting. The created NOON-state is depicted in Fig. 6 (b) at , where two distinct peaks can be observed both for single-particle and CoM density. Removing the barrier potential gives rise to an interference of both parts of the wave function at as depicted in Fig. 6 (c). For the CoM density we obtain nearly perfect contrast (21) while it is considerably reduced for the single-particle density (cf. Gertjerenken and Weiss (2012)).
The same is depicted in Fig. 6 (d)-(f) for the lower interaction strength . While the single-particle density in Fig. 6 (e) still corresponds to a 50%-50%-probability to find the particles on either side of the potential, significant contributions of the state to the CoM density are observed, giving rise to the peak in the middle of the harmonic confinement. The contrast (21) in the resulting single-particle density interference pattern again is considerably reduced, but an interference pattern in the CoM density is still clearly visible with contrast , cf. Fig. 6 (f).
For a further analysis, in the left panels of Fig. 7 the time-evolution of the occupation probabilities is displayed for interaction strengths and . The worsening in contrast for lower interaction strength can be explained with the growing contribution . Removing it numerically can further improve the contrast of interference patterns in the CoM density, as displayed in Fig. 7 (d).
Experimentally, in principle, the suggested protocol could be investigated in lattices of double-wells, cf. experiments like Cheinet et al. (2008): preparing the atoms initially in one of the wells and then switching off the short-wavelength laser, the two particles would be initially prepared on the edge of the remaining approximately harmonic confinement, leading to an oscillation in the potential well. A possibility to realize delta-like barrier potentials could be single atoms as scattering potentials.
We have numerically investigated scattering bright solitons off barrier potentials with a focus on signatures of nonlocal high- and low-fidelity quantum superposition states. The chosen one-dimensional set-up with experimentally typical harmonic confinement gives rise to signatures that clearly distinguish quantum superposition states from statistical mixtures. The presented protocols naturally have the advantage that no opening of the harmonic trap is required, such that excitations due to trap opening Castin (2009) are avoided.
We use the mathematically rigorous Weiss and Castin (2012) effective potential approach Weiss and Castin (2009); Sacha et al. (2009) suitable for elastic scattering: switching off the barrier potential and recombining both parts of the NOON-state leads to high-contrast interference patterns in the CoM density both for narrow and broader barrier potentials. In combination with measurements confirming that all the particles are always clustered in a single lump with 50% probability to either side of the barrier potential (to exclude single-particle effects), these interference patterns can serve as a clear indication of quantum superposition states. Another protocol particularly suited for narrow barriers is scattering twice off the barrier potential, cf. also Gertjerenken et al. (2012). While in general excitations can be a severe problem we have shown that both protocols are remarkably robust in this respect: finite temperature effects have been modeled by taking into account excitations of the CoM wave function to higher lying oscillator states.
For two-particle solitons and delta-like barrier potentials clear interference patterns – again, particularly well-observable in the CoM density – have been demonstrated not only for high- but also for low-fidelity quantum superposition states. While the presented results have been obtained for two particles a deduction of qualitatively similar results to higher particle numbers seems reasonable. This extends former considerations to interesting target states, advantageous in view of decoherence and allowing for shorter time-scales of the presented protocols by choosing higher initial CoM kinetic energies.
Despite in a different experimental regime than required for the proposed protocols, scattering bright quantum solitons off potential barriers is currently investigated experimentally Pollack et al. (); Marchant et al. (2013). The presented signatures could be used in future experiments to distinguish quantum superposition states from statistical mixtures. Due to the large number of data points required it seems advantageous to use arrays of one-dimensional tubes Greiner et al. (2001).
I thank M. Holthaus, M. Oberthaler, A. Streltsov, W. Zurek and especially C. Weiss for discussions. I acknowledge funding by the Studienstiftung des deutschen Volkes and the Heinz Neumüller Stiftung.
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