Quantum Tweezer for Atoms: Extracting Neutral Atoms from a BEC using a Quantum Dot, Lab Reports of Health sciences

Download Quantum Tweezer for Atoms: Extracting Neutral Atoms from a BEC using a Quantum Dot and more Lab Reports Health sciences in PDF only on Docsity! VOLUME 89, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 AUGUST 2002Quantum Tweezer for Atoms Roberto B. Diener,1 Biao Wu,1,3 Mark G. Raizen,1,2 and Qian Niu1 1Department of Physics, The University of Texas, Austin, Texas 78712-1081 2Center for Nonlinear Dynamics, The University of Texas, Austin, Texas 78712-1081 3Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6032 (Received 31 January 2002; published 26 July 2002)070401-1We propose a quantum tweezer for extracting a desired number of neutral atoms from a reservoir. A trapped Bose-Einstein condensate is used as the reservoir, taking advantage of its coherent nature, which can guarantee a constant outcome. The tweezer is an attractive quantum dot, which may be generated by red-detuned laser light. By moving at certain speeds, the dot can extract a desired number of atoms from the condensate through Landau-Zener tunneling. The feasibility of our quantum tweezer is demonstrated through realistic and extensive model calculations. DOI: 10.1103/PhysRevLett.89.070401 PACS numbers: 03.75.–bPosition E ne rg y Trap BEC dot µ v v FIG. 1. A quantum dot (tweezer) moves out of a trapped BEC (reservoir) with the speed of v. The inset illustrates that a resonance occurs as the dot moves further away from the trap center such that the energy of the atoms matches the chemical potential  of the condensate. If one of the atoms is tunneled into the BEC, the energy level of the dot is lowered, due to the absence of repulsion from the lost atom. Thus, nocompared to the self-interaction; the eigenstates of the sys- tem are then Fock states in which the dot contains a definite other atom has a chance of leaking back to the condensate at this position.The manipulation and control of isolated single neutral atoms has been a long term goal with important applica- tions in quantum computing [1,2] and fundamental physics. Trapping and cooling of single neutral atoms was first achieved in magneto-optical traps and more re- cently in a dipole trap [3–6]. Despite these impressive successes, all existing methods share a common weakness: The trapping process itself is random and not deterministic. In this Letter, we propose a quantum tweezer that can extract a definite number of atoms from a reservoir at will, with the atoms in the ground state of the tweezer. A trapped Bose-Einstein condensate (BEC) is used as a res- ervoir, and its coherent nature makes the constancy of the output possible. An attractive quantum dot, created by a focused beam of red-detuned laser light, serves as a quan- tum tweezer to extract a desired number of atoms from the BEC reservoir. In a typical operation of the quantum tweezer, a quantum dot is turned on adiabatically inside the bulk of the BEC and moves out of the BEC at a certain speed so that a desired number of atoms is extracted (see Fig. 1). In the initial stage of this operation, it is important that the system remains in the ground state of the trap
dot potential. The superfluidity of the BEC helps to suppress the excitations which might otherwise be induced by the turning on and movement of the quantum dot. The speed of the dot just needs to be slower than the speed of sound, and the rate of turning on of the dot potential should be smaller than the frequency of phonons whose wavelength is comparable to the size of the dot. The crucial part of the tweezer operation is when the dot moves out of the BEC. Inside the BEC, when the coupling between the trap and the dot is still stronger than the atom self-interaction within the dot, the system is in a coherent state in which the number of atoms in the dot strongly fluctuates. Outside the BEC, the coupling drops exponen- tially with distance and eventually becomes negligible0031-9007=02=89(7)=070401(4)$20.00 number of atoms. In the general case, the dot exits the condensate in a superposition of eigenstates. However, under certain circumstances, we can steer the state into a prescribed final state, with a definite number of particles in the dot. Starting from the ground state of the system with the dot at a certain position inside the BEC, we start moving the dot outwards. At an infinitesimally slow speed, the system always stays in the lowest energy state and no atoms are extracted, simply because moving out of the BEC costs potential energy of the atoms. At some finite speed, the system may get stuck in a nonzero number state of the dot, and become decoupled from the BEC before the atoms in the dot have a chance to leak back. In the following, we2002 The American Physical Society 070401-1 VOLUME 89, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 AUGUST 2002will give a detailed account of this phenomenon through a realistic model calculation. In Fig. 2, we show a result for the probability of extracting a single atom as a function of the speed of the dot. The plateau extends several orders of magnitude of the speed, demonstrating the robustness of our quantum tweezer. Our focus is on the crucial stage of the quantum tweezer operation—when the dot is leaving the BEC cloud. In this case, the density is low and the interaction between atoms in the dot and atoms in the condensate is weak. The state of the system can then be expressed as a combination of atoms in the dot (with wave function d) and atoms in the BEC trap (with wave function B, properly orthogon- alized to d [7]). These two wave functions are chosen as the adiabatic ground state of the system when the dot is motionless and the coupling between these two sets of atoms is negligible. The Hamiltonian of the system is Ĥ  R dx ̂yx   h 2 2Mr 2
Vtx
Vdx; t
g 2 ̂ yx̂x ̂x. We can write ̂x  Bxĉ
dxâ in the weak coupling limit, where ĉ annihilates an atom in the trap and â annihilates an atom in the dot. We shall denote the state with n atoms in the dot (and N  n atoms in the BEC) by jni; an atom jumping from the dot to the BEC corresponds to the transition jni ! jn 1i. Given that dx is much more localized than Bx, the repulsion felt by the atoms in the dot is stronger than the one felt by the ones in the BEC. This asymmetry between the two potentials yields n much smaller than N, in general, and sets our system apart from two-state condensates discussed elsewhere [8].0.0001 0.001 0.01 0.1 1 10 100 Speed 0 0.2 0.4 0.6 0.8 1 Pr ob ab ili ty FIG. 2. The probability of extracting a single atom as a function of the speed of the dot. The calculation was per- formed for a one-dimensional BEC with N  10 000 atoms in a harmonic trap with frequency !  0:005. The dot is a square well with depth U0  8 and width a  1, and the effective coupling constant is g  8. The units for all these parameters are defined in the text. For sodium, speed is measured in units of 2:75 mm=s, so that for many speeds shown it takes a fraction of a second to extract one atom. The plateau exhibited extends several orders of magnitude. 070401-2The nonvanishing matrix elements of the Hamiltonian are (for n  N) hnjĤjni  En  nE1
nn 1 2  ; (1) hnjĤjn
1i  hn
1jĤjni   n
1 p 
nG ; (2) hnjĤjn
2i  hn
2jĤjni   n
1n
2 p A: (3) The parameters depend on the position xd of the dot and can be explicitly calculated. E1  d
Vtxd 
4A accounts for the energy difference between the ground state in the dot and the chemical potential , while   gJ0;4 represents the repulsion an atom in the dot feels from another atom there. We have defined the generalized over- lap integrals as Jm;n  R dxBmdn. Notice that E1 increases as the dot moves away from the center of the BEC. The off-diagonal terms are the couplings that allow an atom to tunnel from the dot to the BEC (or vice versa) either by itself or in pairs. The two terms in   N p hBjVtjdi
gNJ3;1 correspond to quantum tunnel- ing over a barrier and the interaction of a particle in the dot with three atoms in the BEC trap, respectively: this last term dominates when the dot is inside the BEC cloud. Equivalently, G  g  N p J1;3 is due to the interaction of three atoms in the dot with one atom in the trap. Finally, A  gNJ2;2=2 is due to the interaction of two atoms in the trap with two in the dot. Outside of the BEC, the off- diagonal terms vanish exponentially, since the overlap integrals do so. Although our scheme works in any dimensionality, we concentrate in what follows on a dilute, one-dimensional condensate [9] as an example. Such a system can be obtained by tightly confining the cloud in the transverse directions, in which the atomic dynamics are frozen out. The coupling constant is g  4ash=MatomL2, where as is the s-wave scattering length and L is the length of the perpendicular confinement. We shall express our results in the following units: length in units of L0  1 m, time in units of MatomL20=h, and energy in units of h 2=MatomL20. We plot in the bottom panel of Fig. 3 our calculation of the energy levels as a function of the position for a har- monic trap with frequency !  0:005 and N  10 000 atoms. The dot used is a square potential with depth U0  8 and width a  1; the coupling constant is g  8. The edge of the condensate cloud is marked by the dotted line [10]. For comparison, the top panel shows the curves Enx, corresponding to the energies of states with n atoms in the dot in the absence of the tunneling terms. The wave function for the BEC was calculated by numerical solution of the Schrödinger equation in imaginary time [11]. Let us consider the evolution of the number of atoms in the dot as the dot moves out of the BEC, with the help of Fig. 3. It is possible for an atom to tunnel out of the dot when there is no extra energy required to do so, i.e., when the energy for n atoms in the dot is equal to the energy of n 1 atoms in the dot. This is shown in the top panel of Fig. 3 as the locations where the curves for En and En1070401-2