Fidelity of a Bose–Einstein condensates

We investigate fidelity for the quantum evolution of a Bose–Einstein condensate and reveal its general property with a simple model. We find the fidelity decay with time in various ways depending on the form of initial states as well as on mean-field dynamics. When the initial state is a coherent state, the fidelity decays with time in the ways of exponential, Gaussian, and power-law, having a close relation to the classical meanfield dynamics. With the initial state prepared as a maximally entangled state, we find the behavior of fidelity has no classical correspondence and observe a novel behavior of the fidelity: periodic revival, where the period is inversely proportional to the number of bosons and the perturbation strength. An experimental observation of the fidelity decay is suggested. © 2005 Elsevier B.V. All rights reserved. PACS: 03.75.-b; 05.45.-a; 03.75.Kk; 42.50.Vk Instability issue of Bose–Einstein condensation (BEC) has been constantly addressed for its crucial role in control, manipulation and even future’s application of this newly formed matter. Dynamical instability [1], Landau or superfluid instability [2], modulation instability [3] and quantum fluctuation instability [4] have been discussed thoroughly. It is found that instability will break the coherence among the atoms and therefore lead to the collapse of BEC [5]. However, an important issue is still missing, namely, the sensitivity of the quantum evolution of a BEC with respect to a perturbation from outer environment. This instability is distinguished from the instability mentioned above in that the perturbation here is from outer environment rather than from the inner of system. It can be depicted by the so-called fidelity, or the Loschmidt echo, defined as the overlap of two states obtained * Corresponding author. E-mail address: [email protected] (J. Liu). 0375-9601/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.12.070 by evolving the same initial state under two slightly different (perturbed and unperturbed) Hamiltonians [6–9]. This issue is very essential for coherent manipulation of BEC as well as for future’s application of BEC to quantum information and quantum computation [10,11]. In this Letter, we discuss this issue by considering a twocomponent BEC trapped in a harmonic potential [12], subject to a periodic coupling (successive kicks) between the two components. This is a rather general model containing rich dynamical behavior as we show later, with a constant coupling it is a BEC system proposed recently to generate entangled state for quantum computation [11]. Taking this simple model for example, we investigate the new instability of BEC and reveal its general property. We also show that the fidelity instability (fast decay) may lead to a fadeaway of the inference pattern of recent experiment [13,14]. The two internal states of the BEC are coupled by a near resonant pulsed radiation field [12]. Total density and mean phase remain constant during the condensate evolution, then the J. Liu et al. / Physics Letters A 353 (2006) 216–220 217 Hamiltonian describing the transition between the two internal states reads [15] (1) Ĥ = μ(↠1 â1 − â 2 â2)+ g(↠1 â1 − â 2 â2)2 +KδT (t) ( â † 1 â2 + â 2 â1 ) , where K is the coupling strength between the two internal states, g is the interaction, and μ is the difference between the chemical potentials of two components. â1, â † 1 , â2 and â † 2 are boson annihilation and creation operators for the two components, respectively. δT (t) = ∑n δ(t − nT ) means that the radiation field is only turned on at certain discrete moments, i.e., integral multiples of the period T . The above model is a very simple model used to demonstrate our theory. In realistic experiments, the decoherence effects always exist. Generally, decoherence originates in the coupling to a bath of unobserved degree of freedom, or the interparticle entanglement process [16,17]. The main source of decoherence in a BEC is the thermal cloud of particles surrounding the condensate. Thermal particles scattering off the condensate will cause phase diffusion at a rate Γ proportional to the thermal cloud temperature. For internal states not entangled with the condensate spatial state, Γ maybe as low as 10−5 Hz under the coldest experimental condition [18]. Writing the above Hamiltonian in terms of the angular momentum operators [18], L̂x = â † 1 â2 + â 2 â1 2 , L̂y = â † 1 â2 − â 2 â1 2i , L̂z = â † 1 â1 − â 2 â2 2 , we have Ĥ = μL̂z + gL̂z + KδT (t)L̂x . The Floquet operator depicting the quantum evolution in one period takes the following form: (2) Û = exp[−i(μL̂z + gL̂2z)T ] exp(−iKL̂x). The Hilbert space is spanned by the eigenstates of L̂z, |l〉, with l = −L,−L + 1, . . . ,L, where L = N/2 and N is the total number of atoms. In the above expression and henceforth, the Planck constant is set to unit. The outer perturbation is mimicked by a small change on the interaction parameter or on the coupling strength, etc. Without losing generality, here we suppose that perturbation is on coupling, i.e., a small perturbation leads to the change of the coupling strength like K → K + ε, the corresponding evolution operator is denoted by Û . To investigate the influence of the perturbation on the quantum evolution, we need trace the temporal evolution of the fidelity function M(t)= |m(t)|2, here m(t) is the Loschmidt echo defined as (3) m(t = nT )= 〈Φ0| ( Û† )n ◦ (Û)|Φ0〉, where |Φ0〉 is the initial state and the fast decay of the fidelity means the rapid lose of the information during quantum evolution in the presence of perturbation. We first set the initial state as a coherent state, |Φ0〉 = e L+−αL−|−L〉, with α = π−θ e−iφ . The system parameters 2 Fig. 1. Fidelity decay in the mixed system whose classical phase space structure is shown in left panel of Fig. 2. L= 500 and = 6 × 10−4. The non-decaying solid line is the fidelity of an initial coherent state lying within the largest regular region. The other two solid curves correspond to fidelity of two initial coherent states lying in the chaotic region of the classical system. One of them has an exponential decay with Γ = 0.03. Unexpectedly, the other one first has a fast Lyapunov decay e−λt , with λ being the Lyapunov exponent, then follows the exponential decay as the first one. Detail refer to the text. are set as μ = T = 1. With three different coherent states of the parameters (φ, cos θ) = (0.25,0.04), (4.97,−0.2), and (4.91,−0.9), respectively, we then calculate the temporal behavior of the fidelity numerically by solving the operator equations (2), (3) with the standard FFT method. The results are shown in Fig. 1. We find fidelity decay depends strongly on the initial parameters (θ,φ): it may have no decay up to t = 200 as in the first case, or decays much faster in an exponential way as in the latter two cases. To understand the above phenomena, we need retrospect the classical limit of the above quantum system. The effective Planck constant of the system h̄eff = 1/L. In the limit N → ∞, it describes a classical spin on a Bloch sphere with Si = 1 L 〈L̂i〉 (i = x, y, z). The classical Hamiltonian takes the form, H = μSz + gcS z + KδT (t)Sx , and the equations Ṡi = [Si,H ]cl (i = x, y, z), where gc = gL. This classical equations is just the mean field Gross–Pitaevskii equation for the BEC system. They can be solved analytically: the free evolution between two consecutive kicks corresponds to a rotation around Sz axis with the angle (μ + 2gcSz)T , and the periodic kicks added at times nT give rotation around the Sx axis with the angle K . We then plot the classical orbits in left panel of Fig. 2. It shows one big island and four small islands. Inside the islands motions are stable or quasi-periodic, outside the islands are the chaotic points indicating the unstable motions. The chaotic motion is characterized by an exponential magnification of initial deviation having a positive Lyapunov exponent,