Rising time of entanglement between scattering spins
We investigate the time evolution of entanglement in a process where a mobile particle is scattered by static spins. We show that entanglement increases monotonically during a transient and then saturates to a steady-state value. For a quasi-monochromatic mobile particle, the transient time depends only on the group-velocity and width of the incoming wavepacket and is insensitive to the interaction strength and spin-number of the scattering particles. These features do not depend on the interaction model and can be seen in various physical settings.
pacs:03.65.Ud, 03.67.Bg, 03.67.Hk
Scattering, an almost ubiquitous mechanism in physics and a rather well-studied topic, has very recently gained renewed interest within the community working on quantum mechanics in virtue of the potential that it has in many respects dechiara (); imps (); ciccarello (); ciccarello1 (). Scattering processes between two subsystems are effective in order to probe correlation properties of quantum many-body systems dechiara (). Moreover, under proper conditions, scattering can be exploited to prepare non-classical states of unaccessible systems imps (); ciccarello (); ciccarello1 (). Interesting proposals have been put forward for the generation of quantum correlated light-matter states via off-resonant coherent scattering kupryianov (). Frequently, a stationary approach to scattering processes is chosen, especially for the sake of entanglement generation: the system under scrutiny is observed at long-time scales, when it should have reached steady conditions. This cuts the (often complicated) time-evolution from the effective description of scattering dynamics. While this approach is computationally convenient and frequently useful, it is not fully satisfactory since it leaves some interesting questions, related to the details of the dynamical evolution, unanswered vanhove (). In particular, it does not give information on the time needed by entanglement to reach its stationary value, which is a pivotal point for the aims of coherent quantum information processing (QIP). In fact, an estimate of such transient time will help us to anticipate, quantify and eventually counteract the effects that decoherence might have in a given process.
In this paper, we focus on a prototypal setting involving both mobile and static spin particles imps (); ciccarello (); ciccarello1 () to shed some light onto these issues. We clearly identify the physical parameters that determine the duration of the scattering process. Independently of the Hamiltonian chosen in order to model the spin-spin coupling, we show that when a quasi-monochromatic incoming wavepacket of the mobile spin is prepared, with a given average momentum , stationary conditions are reached in a time dictated only by the wavepacket width in momentum space . This parameter can thus be used to tune the duration of the scattering event and the rate of entanglement generation in the system. Counter-intuitively, the interaction strength of a given spin-spin Hamiltonian model coupling mobile and static particles does not affect the scattering transient time but only determines the maximum entanglement attainable in the process. Although, as anticipated, our conclusions do not depend on the details of the model being considered, most of our quantitative results are presented for spin-spin interactions induced by the Heisenberg exchange coupling. This model arises naturally in the spin-spin interaction of magnetic impurities embedded in a one-dimensional electron nanowire as well as in other situations, including the interaction between a free and a bound electron in a carbon nanotube imps (). However, in the second part of our work we quantitatively address the results for an anisotropic XYZ model. Tuning the ratio of its parameters, we are able to span a wide range of significant spin-spin Hamiltonians. Remarkably, our results can be extended to an arbitrary number of static particles and intrinsic spin numbers, which makes them valid in a heterogeneous set of physical settings (ranging from spintronics to cavity-quantum electrodynamics) imps (); ciccarello (); ciccarello1 (); kupryianov (); photonmodel (). In passing, we also reveal an unexpected and general monotonic time-behavior of entanglement, observable under easily-matched conditions. Although wiggling of the reflected wavefunction of the mobile particle is observed due to interference at the scattering site(s), entanglement can only grow in time. First, we empirically observe this behaviour for relevant coupling models and then give a clear physical explanation for the case of the Heisenberg exchange coupling and one scattering center. The features described above make it clear that a scattering-based mechanism holds the promises for a genuinely control-limited distribution of entanglement in a partially addressable quantum register. In contrast to procedures based on the temporal gating of spin-spin interactions the process addressed in this paper allows for a dramatic relaxation of the time control, which is a major advantage in light of a potential experimental implementation.
The remainder of the paper is organized as follows. In Sec. I we introduce the single-scatterer version of the system addressed in our work and study the entanglement rising-time. An important bench-mark is set by comparing the results of this study with those corresponding to the scattering-less case of two exchange-coupled static particles. Sec. II extends the analysis to the two-scatterer case. In Sec. III we quantitatively prove the insensitivity of the entanglement rising-time to the spin-spins coupling strength. Finally, in Sec. IV we summarize our findings and remark their most important implications.
I Scattering by a single static particle
In order to provide a significant mile-stone for our main results, we first review the well-known case of two static spin-1/2 particles, and , interacting via the Heisenberg coupling . Here, () is the spin operator of particle () and is the coupling rate (we use units such that throughout the paper). We assume the initial state
where , are the spin states of each particle and are steady states of with energies and , respectively. Upon evolution, the state reads
As a measure of the entanglement between the particles in , we use the logarithmic negativity logneg (). A straightforward calculation gives , which oscillates with characteristic time .
is maximized at (), which shows that for a set coupling strength a fine control on is required. This is not the case when the spin-spin interaction takes place during scattering.
To show it, let us address the case of a mobile particle , interacting with the static spin during scattering. To fix the ideas, we consider a one-dimensional wire along the -axis where spin is embedded at position . Spin-spin interaction occurs, via a Heisenberg model, when the mobile spin reaches and is scattered by . Assuming a quadratic dispersion law for , the Hamiltonian reads
where () is the momentum operator (effective mass) of and is the coupling strength (notice that, unlike , has the dimensions of a frequency times length). The incoming mobile spin is supposed to have wavevector (, ) and be prepared in the spin state while the static spin is initially in . We call () the set of spin quantum numbers of mobile and static particles before (after) the scattering event. Correspondingly, () is the probability amplitude that is reflected (transmitted) in state , while the spin-state of is changed into . These probability amplitudes depend on () and are computed by solving the time-independent Schrödinger equation (SE) associated with Eq. (3) and imposing proper boundary contitions at marsiglio (). The steady state of the system has wavefunction of the form
where we have omitted a factor and is the Heaviside step function. As the kinetic energy of is the only free-energy term in Eq. (3) the system’s spectrum is and thus coincides with that of a free-propagating . In order to investigate the dynamics of entanglement during scattering, we consider as prepared in a Gaussian wavepacket such that
with the average position and momentum of . As for the uncertainties, we have . The overall initial state is taken as , whose time-evolved version is found by solving the respective time-dependent SE.
In Fig. 1, we study the evolution of spin-spin entanglement between and when the former is prepared in a quasi-monochromatic wavepacket. Clearly, in the processes at hand, the entanglement associated with the full quantum state involves both motional and spin degrees of freedom. The study of this hybrid form of entanglement, which is in general a difficult task requiring ad hoc tools for his approach, is beyond the scopes of our work. A possible way to tackle it would be the investigation of the non-locality content of the state of particles and and involving both internal and external degrees of freedom, along the lines as in Ref. wodkiewicz (). In Fig. 1(a) we set , and plot for increasing values of up to 10. The differences with respect two static spins are striking. First, there is no oscillatory behavior. At small , no entanglement is found as far as has not yet reached . At large times, is far from spin 1 and takes a steady value. At intermediate times, the entanglement shows a monotonic increase before eventually saturating to a steady value, which does not depend on . Second, unlike the case of static spins, the characteristic time of the entanglement evolution is now independent of the spin-spin coupling strength . To illustrate this, in Fig. 1(b) we set and study for various ’s. Remarkably, although the steady value of depends on , the rising time does not, as it is revealed by the top-most curve in Fig. 1(b). There, for an assigned value of , we have rescaled each to the respective stationary value and found that the curves are identical. This marks a profound difference with the static-particle case: the introduction of motional degrees of freedom in our problem of interacting spins does not result in a mere spoiling effect of the system spin coherences, as it might be naively expected, but deeply affects the dynamics of the particles involved in the process. To identify the parameters upon which depends in the dynamical-scattering process, we first observe that in Fig. 1(a) an increase in of one order of magnitude makes ten times smaller, suggesting an inverse proportionality between these quantities. Then, in Fig. 1(a) (dashed lines) we plot the probability density of finding particle at for , i.e. the free-propagation case. It can be clearly seen that the time needed by to reach its steady value coincides with the time during which is found at with non-negligible probability in the absence of any scattering. Thus, by using the free-particle time-energy uncertainty principle cohen () we obtain , which explains nota-time-energy () the aforementioned inverse proportionality between and for a given . In other words, in the scattering case and for a quasi-monochromatic mobile particle the characteristic time over which entanglement changes is decided only by the kinetic parameters specifying the incoming wavepacket . As for a given the steady value of is insensitive to [cfr. Fig. 1(a)] we conclude that the rising time can be tuned simply by adjusting , with no effect on the stationary value of entanglement (for a set ). This result is key to the study of the feasibility of scattering-based QIP protocols imps (); ciccarello (); ciccarello1 () and the quest for effective ways to relax the control on a system. We have checked that these features hold even when particle has spin number . We point out that although Fig. 1 might at first glance suggest that entanglement rises linearly with time in general this is not the case. Rather, our findings show quite clearly that such functional behavior is dictated by the shape of the incoming wavepacket of particle (here assumed to be Gaussian).
Ii Scattering by two static particles
We now consider the situation where the mobile spin is scattered by two static spin-1/2 particles, and , placed at and , respectively. The Hamiltonian is the same as in Eq. (3) with the inclusion of the term . Each steady state () has energy and differs from Eq. (4) for the replacement and the addition of . The coefficients () depend implicitly on and . As shown in ciccarello (); ciccarello1 (), once boundary conditions at are imposed, the steady states can be determined. This configuration has recently been proposed as a potential way to set entanglement between remote spins imps (); ciccarello (); ciccarello1 () via mediation of and only mild time-control: It is enough to wait for a time , so that a steady state is reached. Clearly, the quantification of in the case of an incoming wavepacket (instead of a plane wave as in Refs. imps (); ciccarello (); ciccarello1 ()) is key to estimate the influence of dissipation and decoherence affecting the scattering particles. This would be a pivotal point in view of potential experimental implementations. We take the initial state , whose time-evolution is computed through the time-dependent SE. We focus on the dynamics of entanglement between static particles and . Here
is the state obtained by tracing over the spatial and spin degrees of freedom of . We call
the probability to find within the interaction region at time .
In Fig. 2(a) we set , , and study the behavior of and free-propagating-particle for increasing values of up to 10. On the other hand, Fig. 2(b) considers the effect of different choices of over the entanglement dynamics, for set values of and . Features similar to those revealed for a single scattering center are found [see Figs. 1(a) and (b)]. Entanglement increases with time to reach a steady value that depends on and . As in the single-scatterer case, is determined only by the time needed by the free-propagating to cross . In Fig. 2 our focusing on the regime is due to the fact that such setting allows for efficient entanglement distribution schemes ciccarello (); ciccarello1 (). However, we have checked that other values lead to similar conclusions.
In reaching our conclusions, we have performed a few technical steps which is instructive addressing here. In each of the studied configurations, the time evolution of (with the total initial spin state nota ()) can be expanded over the basis of steady states with coefficients given by . Provided that , i.e. at spin is out of the domain , these scalar products are well-approximated by taking and neglecting contributions from other -functions in Eq. (4), so that
with the Fourier transform of (an analogous expression holds in the many-scatterer case). Eq. (8) allows us to evaluate
which can be solved analytically via a power-series expansion of each spin-dependent amplitude () around the carrier wavevector . The replacement of the -th order expansion of these coefficients in results in integrals of the form (, and ), which can be computed in terms of exponential and error functions. The behavior of is then found by tracing the overall state over the spin degrees of freedom. As expected cohen (), we find that during scattering wiggles at due to interference between incoming and reflected waves. The spin state is found by tracing over the spatial and, in the many-scatterer case, spin degrees of freedom of mesh ().
Iii Model-independence of entanglement rising-time
We are now in a position to explain the insensitivity of to the values of . Key to this task is the observation that as long as the wavepacket is narrow enough around the term proportional to (with ) in Eq. (8) can be neglected. Thus meaning that, within the limits of our study, the spectral decomposition of is the one corresponding to . On the other hand, the spectrum does not depend on [see discussion after Eq. (4)] so that the energy and time uncertainties nota-time-energy () and , are the same as in the free-particle case, i.e. . These features hold for a quasi-monochromatic incoming wavepacket. Most importantly, it is clear that our proof does not rely on the specific form of the interaction Hamiltonian, the number of scattering particles as well as their intrinsic spin numbers. While all these parameters affect the shape of ’s and ’s, we have just shown that, for quasi-monochromatic wavepackets, they cannot influence the entanglement rising-time.
To further illustrate the discussed insensitivity of , in Fig. 3(a) we set and analyze the entanglement between and when these are coupled to with strengths . The case of spin- scattering centers and equal couplings is also reported. In Fig. 3(b) we set the same as in Fig. 3(a) and address the XYZ spin-spin model
We study the cases of (embodying an XXZ model), with (isotropic XY) and with (anisotropic XY). The ratios of the parameters are chosen so as to provide the best visibility of each plots. The insets in Figs. 3(a) and (b) report all the curves rescaled to their maximum value, showing that the entanglement rising-time is not affected by the specific quantum spin-number or the interaction model, which only affect the stationary value of entanglement. In particular, the applicability of our results to the XY model is remarkable since an effective XY model is indeed found ciccarello1 () considering the dispersive interaction of a single photon travelling across a 1D waveguide (a GaAs/GaN nanowire, for instance) and static atom-like systems (such as InAs/GaInN quantum dots or nitrogen-vacancy centers in diamond photonmodel ()). The (typical) case of linear dispersion law for a photon crossing a waveguide matches the requirements of our study, i.e. a narrow-bandwidth mobile particle. These considerations demonstrate the broad applicability of our investigation and results, which cover a wide range of experimental situations, from spintronics to quantum optics imps (); ciccarello (); ciccarello1 (); kupryianov (); photonmodel (). In the second scenario, in particular, the proven ability to experimentally engineer the temporal shape of photonic wavepackets harris () makes scattering-based techniques analogous to the one discussed here and in Refs. imps (); ciccarello (); ciccarello1 () quite advantageous against strategies using time-dependent “modulation” or “pulsing” of spin-spin interactions. In the former case, once the kinematic properties of the mobile-particle wavepackets are set, there is no necessity for time-control of the entanglement evolution. We remark that, in order for the steady-state entanglement not to depend on [as in Figs. 1 and 2], this has to be smaller than the inverse of the characteristic length associated with -amplitudes. In particular, for two static spins, it must be .
For the Heisenberg model, the monotonic rise of entanglement altroremark () can be interpreted as due to the progressive construction of a phase-difference between spin components of the overall wavefunction. Here, we discuss the steps required in order to prove this feature. For the sake of argument, we focus on the case of Eq. (3) and we assume that the incoming wavepacket is quasi-monochromatic. The Hamiltonian in Eq. (3) commutes with and , where is the total spin of and 1. It follows that for the initial spin state [see also Eq. (1)]
where are evolved wavepackets fulfilling the condition and . Upon trace over the spatial variable of we get with the spatial overlap between and . Now, for the spin-spin interaction in Eq. (3) reduces to an effective static potential with and marsiglio () with associated reflection and transmission probability amplitudes are
The steady states are similar to those in Eq. (4) with associated energies . Assuming , the projections of onto the stationary states are well-approximated by a form analogous to Eq. (4) (with due replacements). Provided that the width of the incoming wavepacket around is narrow enough, we find . Moreover, . Moreover, . Thus, by calling () the phase-difference between and ( and ), we get
where () are evolved free wavepackets with average positions at and Fourier transforms with . In our derivation we have neglected terms proportional to due to the assumption of dealing with an incoming wavepacket narrow enough around . As and are left- and right-propagating wavepackets, respectively, both and are increasing functions of time whose shapes are similar to those in Fig. 2(b) for a Gaussian wavepacket . Hence, the behavior of depends on . As (regardless of and ) the sine functions are always positive. Therefore, is an increasing function of time, proving the monotonic character of . Physically, it is now clear that the build-up of entanglement relies on the phase-difference acquired by the singlet and triplet components of the incoming wavepacket once is scattered off. Dynamically, as soon as the scattering process starts and the reflected and transmitted waves are progressively generated, the singlet and triplet components start to build-up the mentioned phase-difference and entanglement grows. For up to and up to , the largest relative increments of and in the range are of the order of , confirming the validity of our arguments. Clearly, the monotonic increase of entanglement does not hold for any initial spin state. For instance, starting from an entangled state of and , may decrease with time.
The study of wavepackets having , namely the regime opposed to quasi-monochromaticity, is under ongoing investigations.
Our work reveals that in a scattering process involving one mobile and many static spins, a situation that can be engineered in various physical settings imps (); ciccarello (); ciccarello1 (); kupryianov (); photonmodel () the time required for entanglement to reach its steady value can be tuned by preparing the mobile spin’s wavepacket, regardless of the specific spin-spin interaction model. If a wavepacket narrow enough in frequency is prepared, the strength of the interaction between mobile and static spins determines just the stationary value of the entanglement which, while scattering takes place, can only grow in time. Our findings show that the degree of control on the class of systems we have addressed can be significantly reduced down to the off-line engineering of a proper mobile spin’s wavepacket. The quantification of the scattering transient-time enables the analysis of noise effects, such as phase-damping affecting the static spins, on protocols for entanglement distribution via scattering imps (); ciccarello (); ciccarello1 ().
Acknowledgements.We thank A. Del Campo, M. S. Kim and D. Sokolovski for discussions. We acknowledge support from PRIN 2006 “Quantum noise in mesoscopic systems”, CORI 2006, EUROTECH, the British Council/MIUR through the British-Italian Partnership Programme 2007-2008. MP is supported by the UK EPSRC (EP/G004579/1).
- (1) J. Hald, J. L. Sorensen, J. L. Leick, and E. S. Polzik, Phys. Rev. Lett. 80, 3487 (1998); J. Stenger, S. Inouye, A. P. Chikkatur, D. M. Stamper-Kurn, D. E. Pritchard, and W. Ketterle, Phys. Rev. Lett. 82, 4569 (1999); I. B. Meckhov, C. Maschler, and H. Ritsch, Nature Phys. 3, 319 (2007); K. Eckert, O. Romero-Isart, M. Rodriguez, M. Lewenstein, E. S. Polzik, and A. Sanpera, Nature Phys. 4, 50 (2008); G. De Chiara, C. Brukner, G. M. Palma, and V. Vedral, New J. Phys, 8, 95 (2006).
- (2) A. T. Costa, Jr., S. Bose, and Y. Omar, Phys. Rev. Lett. 96, 230501 (2006); G. L. Giorgi and F. De Pasquale, Phys. Rev. B 74, 153308 (2006); L. Lamata and J. León, Phys. Rev. A 73, 052322 (2006); K. Yuasa and H. Nakazato, J. Phys. A: Math. Theor. 40, 297 (2007); M. Habgood, J. H. Jefferson, and G. A. Briggs, Phys. Rev. B 77, 195308 (2008); F. Buscemi, P. Bordone, and A. Bertoni, arXiv:0810.4093v1 [quant-ph]; H. Schomerus and J. P. Robinson, New J. Phys. 9, 67 (2007); D. Gunlycke, J. H. Jefferson, T. Rejec, A. Ramsak, D. G. Pettifor, and G. A. D. Briggs, J. Phys: Condens. Matter 18, S851 (2006).
- (3) F. Ciccarello, G. M. Palma, M. Zarcone, Y. Omar, and V. R. Vieira, New J. Phys. 8, 214 (2006); J. Phys. A: Math. Theor. 40, 7993 (2007); Las. Phys. 17, 889 (2007); F. Ciccarello, G. M. Palma, and M. Zarcone, Phys. Rev. B 75, 205415 (2007).
- (4) F. Ciccarello, M. Paternostro, M. S. Kim, and G. M. Palma, Phys. Rev. Lett. 100, 150501 (2008).
- (5) D. V. Kupriyanov, O. S. Mishina, I. M. Sokolov, B. Julsgaard, and E. S. Polzik, Phys. Rev. A71, 032348 (2005).
- (6) L. van Hove, Phys. Rev. 95, 1374 (1954).
- (7) M. Atatüre, J. Dreiser, A. Badolato, A. Högele, K. Karrai, and A. Imamoglu, Science 312, 551 (2006); K. Hennessy, A. Badolato, M. Winger, D. Gerace, M. Atatüre, S. Gulde, S. Fält, E. L. Hu, and A. Imamoglu, Nature (London) 445, 896 (2007); K.-M. C. Fu, C. Santori, P. E. Barclay, I. Aharonovich, S. Prawer, N. Meyer, A. M. Holm, and R. G. Beausoleil, arXiv:0811.0328v1 [quant-ph]; P. E. Barclay, K.-M. Fu, C. Santori, and R. G. Beausoleil, arXiv:0904.0500 [quant-ph].
- (8) P. Kolchin, C. Belthangady, S. Du, G. Y, Yin, and S. E. Harris, Phys. Rev. Lett. 101, 103601 (2008).
- (9) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. (in press, 2009); see also arXiv:quant-ph/0702225 [quant-ph].
- (10) W. Kim, R. K. Teshima and F. Marsiglio, Europhys. Lett. 69, 595 (2005).
- (11) K. Wódkiewicz, New. J. Phys. 2, 21 (2000).
- (12) C. Cohen-Tannoudji, B. Diu, and F. Laloe, Quantum Mechanics (John Wiley & Sons, New York, 1977).
- (13) We point out that in this work we make use of the heuristic time-energy uncertainty principle cohen () rather than the rigorous inequality by Mandelshtam and Tamm [L. I. Mandelshtam and I. E. Tamm, J. Phys. (USSR) 9, 249 (1945)]. Our goal here is indeed to merely estimate the characteristic time scale over which entanglement builds up. A more accurate quantification of such a time is under ongoing investigation.
- (14) We assume that belongs to the uncoupled-spin basis . The generalization to arbitrary (pure or mixed) initial spin states is straightforward.
- (15) We numerically trace over the spatial variable using a mesh of the region where is non-negligible and checking the stability of the outcomes vs. the number of mesh points.
- (16) The monotonic rise of entanglement observed in Figs. 1-3 clearly depends on the initial spin state (if the static spins are already entangled, a decrease may take place).