Source: https://www.groundai.com/project/interference-of-spin-orbit-coupled-bose-einstein-condensates/
Timestamp: 2019-04-21 22:12:49+00:00

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Interference of atomic Bose-Einstein condensates, observed in free expansion experiments, is a basic characteristic of their quantum nature. The ability to produce synthetic spin-orbit coupling in Bose-Einstein condensates has recently opened a new research field. Here we theoretically describe interference of two noninteracting spin-orbit coupled Bose-Einstein condensates in an external synthetic magnetic field. We demonstrate that the spin-orbit and the Zeeman couplings strongly influence the interference pattern determined by the angle between the spins of the condensates, as can be seen in time-of-flight experiments. We show that a quantum backflow, being a subtle feature of the interference, is, nevertheless, robust against the spin-orbit coupling and applied synthetic magnetic field.
Interference of matter waves is one of the most interesting effects in quantum physics. The interference of two expanding Bose-Einstein condensates is a clear manifestation of quantumness in macroscopic systems [2, 3, 4] . It can be observed by preparing two condensates in spatially separated harmonic traps, that are released afterwards. Then, the condensates can expand freely and eventually overlap, producing an interference pattern.
The quantum dynamics becomes much richer for spin-orbit coupled Bose-Einstein condensates, where optically produced pseudospin is coupled to the atomic momentum and to a synthetic, also optically produced, magnetic field [5, 6, 7] . These effects, which open a venue to the simultaneous control of orbital and spin degrees of freedom and to experimental observation of new phases and dynamic processes have been discussed for a variety of ultracold atomic systems [8, 9, 10, 11, 12, 13, 14, 15, 17, 16, 18] including recently produced and studied Fermi gases with synthetic spin-orbit coupling [19, 20] . In quantum information technologies, spin-orbit coupled Bose-Einstein condensates can serve as a realization of macroscopic qubits, as proposed in  . State-of-the-art reviews can be found in [21, 22] .
Figure 1: Two condensates with mean momenta per particle p1, p2 and spin-orbit coupling constant α and spins precessing in a synthetic magnetic field characterized by Zeeman splitting Δ. Dashed ellipses show the time-dependent spins of condensates, and vectors n1 and n2 defined in Eq.(7) mark corresponding precession axes.
Here we consider time-of-flight control of interference of two spin-orbit- and Zeeman-coupled one-dimensional condensates (as shown in Fig.1) producing their entangled state, which might be required for quantum information purposes  . The condensates, that move freely in a waveguide realized by tight confinement in the transverse directions, give rise to an interference pattern that strongly depends on the relative orientation of their pseudospins. We study the role of the synthetic magnetic field on the interference and show that it can be fully controlled by changing the synthetic Zeeman coupling. In addition, we show that the quantum backflow [23, 24, 25, 26, 27, 28, 29, 30] , being a subtle effect of the interference, is rather robust against mutual orientation of spins of the condensates.
where ⟨p⟩ is the mean momentum, w is the initial width, and xin is the initial position.
where Hs=α⟨p⟩ˆσz/ℏ+Δσx/2 is the mean value of a spin contribution to the Hamiltonian (1). We use Eq.(5) below for a qualitative analysis of the packets’ interference.
where ℏΩ(n⋅σ)=2Hs. Figure 1 shows rotating spins in the presence of Zeeman splitting and directions of the vectors nj in (6), where index j=1,2 labels the condensate, and we use pj for corresponding mean values.
where x±≡x±αt/ℏ. The α-determined phase shift between ψ↑(x,t) and ψ↓(x,t) in (8) leads to a coordinate-dependent spin rotation. The same results for spin motion can be obtained by gauging out the spin-orbit coupling in Eq.(1) by a coordinate-dependent spin rotation exp[iσzx/Lso] (Lso≡ℏ2/Mα is the spin rotation length), calculating the resulting dynamics, and then making the inverse transformation to obtain the observables [31, 32] . However, in the presence of a Zeeman field, which is of our interest, gauging out the spin-orbit coupling leads to a coordinate-dependent effective magnetic field. Although transport effects can be obtained with Eq.(5) (see, e.g  ), general dynamics is difficult to treat beyond perturbation theory  . For this reason we use the direct calculation rather than the spin rotation approach.
where Ψx≡∂Ψ/∂x. The experimentally measured density ρ(x,t)=Ψ†(x,t)Ψ(x,t) is related to J(x,t) by the continuity equation. For the weak coupling considered below we neglect the α-related terms in Eqs.(9)-(11).
Here the amplitudes A1 and A2 are normalized as A21+A22=1, Bj(0)=(βj1(0),βj2(0))T is the corresponding spinor, and gj(p) is defined by (4). The average velocities of the packets vj are determined by (10) for the corresponding momentum pj and spin state, and from now on we omit ⟨…⟩ in the notation of averages. Using (3) and (12) we obtain the exact evolution of two initial wave packets with spin-orbit coupling.
where B1≡B1(t) and B2≡B2(t). Equation (15) shows that the interference, seen here as the fast oscillations in the coordinate or time-dependence of the current, is controlled by the spin states through the product B†1B2.
Equation (16) shows how the mutual orientation of the spins of the condensates and, in turn, their interference, depends on the time of flight in the presence of spin-orbit or Zeeman coupling. In particular, if at t=t0 the spin states are orthogonal, the interference disappears, which shows that it can be controlled by manipulating the condensate spin.
Figure 2: Plot of current (a) vs time at x=x0 and (b) vs coordinate at t=t1 (t1=16), for the parameters in (18). Color lines correspond to values of spin-orbit coupling from (19) and (20), α1 - red dashed line, α2 - blue solid line.
corresponding to both condensates with the spin oriented along the x-axis. We take time-of-flight t0=20 and the wave packet “collision” point at x0=0. To make connection with possible experimental observations, we take 87Rb atom as an example. The resulting velocity unit ℏ/(MRb×10−4 cm) is 0.072 cm/s and, therefore, the unit of time is approximately 1.4×10−3 s. As a result, t0=20 corresponds to about 28 milliseconds and the initial distance between the packets (for p1=8 and p2=2) of 120 microns. Below we consider two realizations of the condensates, with equal and different widths.
Figure 3: Plot of current density for p1=5, p2=2.5, α=0.09, w1=0.5, w2=20 and other parameters from (18). For these parameters the number of no-interference points in (25) is Ncosθ=0≈1.
In Fig. 2 one can see that, for α=α1 there is no interference in the flux, while for α=α2 the flux is characterized by a strong interference pattern and by the presence of backflow, namely a negative current density, J(x0,t)<0, see Fig. 2 [23, 24, 25, 26, 27, 28, 29, 30] . For other values of spin-orbit coupling the flux interference is between these two limits.
where v1 and v2 are the velocities of the packets determined in (10), w1(t0) and w2(t0) are the widths of the packets at the meeting time t0 determined in (9), and we have taken into account that the traveling time of wave function is of the order of 2wj(t).
In this formula the factor 2 means that, in one period of rotation of the angle between spins, the interference is destroyed twice when spin states are orthogonal. In addition, if the packets are initially narrow, and, therefore, spread with a large rate of the order of ℏ/Mwj, one can see the effect of multiple interferences better. From Fig. 3 one can see that, during the interference time the spin states become orthogonal once, and the interference is destroyed at this instant.
Figure 4: Products of spinors for values in (18) for the spin-orbit coupling from (19) and (20): (a) - α1 and (b) - α2. Lines correspond to: Δ=0 - red dot line, Δ=0.1 - blue dashed line, and Δ=0.5 - black solid line.
Figure 5: The evaluation of interferences (a) and backflow (b) for values from (18), w1=w2=1 and spin-orbit coupling from (19) and (20) α1-dashed red line, α2-solid blue line. In (a) small circles mark appearance and disappearance of the backflow.
Below we consider the effects of the Zeeman term limiting ourselves to equal initial widths of the packets with all other initial parameters (18) unchanged. It is important here that if the condition Δ≫αpj/ℏ is satisfied, the vectors (6) are very close to each other and to the x-axis, and the spins be always parallel to each other with a high accuracy. Fig. 4 demonstrates the spin states (16) and shows that the expression (16) is zero only when Δ=0. As a result with synthetic magnetic field the interference cannot be completely destroyed by spin-orbit coupling.
Figure 6: The fluxes for values from (18), spin-orbit coupling is α2 from (20), and the plots correspond to the fields with Δ=0.26 (a) and Δ=0.63 (b). Lines correspond to functions (11) - blue solid line and (26) - with nmax=10 red dashed line.
The evaluation of interference and backflow, (29) and (30) dependent on Δ is plotted in Fig. 5 for given values of spin-orbit coupling (19) and (20). Fig. 5 and 6 show that it is possible to control the interference of two condensates using the spin-orbit coupling and synthetic magnetic field. For strong field spins of particles are frozen in one direction and interference is maximal. The zero value of the function Γ(Δ,α) corresponds to the absence of backflow, where the flux J(x0,t)>0 for any t. As one can see, the intervals of its zero values are relatively small, meaning that the backflow is robust against the spin-dependent interactions. Figure 5 shows that for the given system parameters the backflow disappears if the interference parameter F(Δ,α) is less than 0.5.
As for the role of the interactions, they do not influence the momentum of the packet, so they do not change its mean spin precession rate, affecting the spins only marginally. However they do influence the packet width and can prevent collision if they are strong enough. To avoid these effects in the regime p1w1≫ℏ and p2w2≫ℏ it is sufficient to satisfy the condition of small contribution of the interatomic repulsion into the packet width. Since in the absence of repulsion the packet spreads with the rate of the order of ℏ/Mwj, the interaction energy per atom should be less than ℏ2/Mw2j to satisfy this condition. A good candidate for a very weakly interacting BEC is 7Li ensemble, although, to the best of our knowledge, spin-orbit coupling effects have not been reported for this isotope.
We have shown that the superposition of two freely moving spin-orbit coupled condensates gives rise to interference effects strongly dependent on the spin state of the condensates at the collision time. The interference - characterizing both the density and the flux - is strong when the spins of the two condensates are parallel, and it disappears when the spin states are orthogonal. These effects can be clearly seen in time-of-flight experiments, and are at reach with the current technology for ultracold atoms. In addition, the system exhibits a spin-dependent quantum backflow behavior, which is relatively robust against synthetic spin-orbit coupling and magnetic field. The ability to control the interference by synthetic spin-orbit coupling and magnetic field can be useful for investigating the quantum properties of atomic condensates and for interference of macroscopic spin-orbit coupled BEC-based qubits for quantum information applications.
This work was supported by the University of Basque Country UPV/EHU under program UFI 11/55, Spanish MEC (FIS2012-36673-C03-01 and FIS2012-36673-C03-03), and Grupos Consolidados UPV/EHU del Gobierno Vasco (IT-472-10). S.M. acknowledges EU-funded Erasmus Mundus Action 2 eASTANA, ”evroAsian Starter for the Technical Academic Programme” (Agreement No. 2001-2571/001-001-EMA2). M.P. is supported by the Doctoral Scholarship of the University of Basque Country UPV/EHU.
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