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גוריון בנגב-אוניברסיטת בן Course: Atoms and Molecules By Ron Folman Lecture 10 Walks on the Bloch sphere: Coherent manipulations of an atomic two-state system Rabi Oscillations and Ramsey fringes Thanks to Menachem Givon for preparing the lecture Walks on the Bloch sphere 1. Introduction Since the dawn of quantum mechanics in the 1920s, the atomic two-state system has served as an important model of radiation-matter interaction. With the introduction (in the 1960s) of lasers as a source of monochromatic and coherent radiation, detailed studies of the coherent matter-radiation interactions became a major part of the research activities and technological development. Magnetic Resonance Imaging (MRI), atomic clocks, the Global Positioning System (GPS, based on cesium atomic clocks), are just few examples of the many applications of the two-state system. Much of this work (SHO 1990 and references therein) was based on atomic two-state models, sometimes modified to include effects of additional states. The appearance of quantum computing concepts in the 1980s (EIS 2004 and references therein) brought in a new possible role for the atomic two-state system: that of the qubit, the quantum equivalent of the bit. The qubit is the basic information storage and manipulation unit of the quantum computer, and realization of a qubit is an important step towards the construction of a quantum computer. The state Ψ of an atomic two-state system (and any other two-state system) can be represented as a superposition of the two eigenstates of the Hamiltonian H0 describing the system: 0 Ψ = C 0 0 + C1 1 Where C0,1 are complex numbers, ∑C C i (1.1) * i Z = 1, i = 0,1 θ Ψ i and H 0 0 = E 0 0 , H 0 1 = E1 1 (1.2) ϕ X Y Alternatively, we can write (1.1) as Ψ = cos θ θ 0 + e iϕ sin 1 2 2 (1.3) 1 Figure 1.1 – The Bloch sphere where we neglected the global phase, and 0 ≤ θ ≤ π , 0 ≤ ϕ ≤ 2π . Thus, the two angles θ, ϕ define the normalized state Ψ up to a global phase. As long as our two-state system does not interact coherently with the external world, this global phase does not affect any observable, and can therefore be neglected. We can geometrically demonstrate any such state as a point on a three-dimensional unit sphere, called the “Bloch Sphere” (Fig 1.1). The “north pole” represents the pure state 0 while the “south pole” represents the pure state 1 . Any other point represent a superposition state per (1.3), where θ, ϕ are the spherical polar coordinates of that point. Page 2 of 31 28/11/06 Walks on the Bloch sphere Once we established the correspondence between states and the Bloch sphere, we can use the Bloch sphere to visualize and demonstrate the system and its evolution (both θ and ϕ may vary with time). Side note: Much attention is being devoted today towards developing two-state systems as qubits of the future quantum computer. If time allows we will discuss the uniqueness of quantum computing and the application of two-state systems for the quantum computer. Several two-state systems are currently utilized, or studied, as qubits, including: Superconducting Quantum Interference Devices (SQUID), where the direction of the current in the superconducting loop is the binary observable (ORL 2004); NMRbased devices, that were used to build a quantum computer capable of factoring the number 15 (VAN 2001); systems based on the electron spin, that can be manufactured utilizing existing chip fabrication technologies (CAR 2005); cold trapped ions (KIN 1999), (PEA 2006); and cold trapped neutral atoms (SCH 2004), (KHU 2005), (YAV 2006), (TRE 2006), (LAC 2006), (CHA 2006). In 2000, DiVincenzo (DIV 2000) defined several requirements for the implementation of quantum computation: 1.1 Scalable physical system with well characterized qubits. Scalability is important, since any practical computer needs a large number of qubits. The meaning of the term “well characterized” is harder to define exactly. It includes: • • • • Stability, or at least meta-stability of two states that serve the qubit; Accuracy of the physical parameters and the internal Hamiltonian; Weak coupling to external fields (to avoid dephasing), but good coupling to specific external fields which will be used to manipulate the qubit; Weak coupling to other internal states and to other qubits (to prevent cross-talk). 1.2 Ability to initialize the qubits’ state. The main factor here is the initialization time – if it is not considerably shorter then the quantum gate time (the actual “computation”), one will need to replace the qubits after every gate operation. 1.3 Very long decoherence time (compared to gate time). This is necessary for the implementation of “error correction” procedures. 1.4 A “universal” set of quantum gates. This is the heart of the quantum computer. In principle, it is just a set of unitary transformations, applied sequentially, each to a finite set of qubits. This implies that we will be able to turn some interaction “ON”, direct it to a specific set of qubits, and then turn if “OFF”, without disturbing any other qubits. In practice, no such system have bean realized yet, but even partly realized system may be good enough for quantum computation. The details vary considerably with the choice of qubit, but we may translate this requirement to the following: the ability to address a specific qubit, to manipulate its state with Page 3 of 31 28/11/06 Walks on the Bloch sphere “enough” precision, to control its interaction with other qubits, and to do so during a time interval “much shorter” than the decoherence time. 1.5 Qubit-specific measurement capability. Theoretically, we should be able to measure the state of each qubit independently of any other parameters of the system, including the state of the nearby qubits, without disturbing any other qubit, and to do so in a time interval “much shorter” then the decoherence time. In practice, we may be able to relax this requirement by repeating the measurement several times and averaging the results. This can be done either by repeating the computation several times or by performing parallel computations simultaneously. Based on these criteria, we can compare specific systems and methods and assess their relative value as “components” of the future quantum computer. In this lecture, we will focus on experimental procedures for rotating qubits (on the Bloch sphere) that are based on ultracold neutral atoms. As an exercise, if time allows, we will use the above criteria to estimate the suitability of the procedure as part of a future quantum computer. In section 2 we will present some of the relevant theory. We will start with the Schrödinger equation for a two-state atom and its interaction with external excitation. Then we proceed to describe the density matrix, the Bloch sphere and the Bloch vector. We will describe Rabi oscillations and manipulating two-state systems with π pulses, chirped pulses, Raman pulses and Stimulated Raman Adiabatic Passage (STIRAP). In section 3 we will review the state of the art in the field of manipulating two-state systems based on ultracold atoms. Page 4 of 31 28/11/06 Walks on the Bloch sphere 2. Theory In this section we review some of the theory relevant to the atomic two-state system. The purpose here is to review the relevant theoretical background and establish notation. Based on this background we will review the state of the art, and define the intended research work included in this proposal. In this section we will mainly follow Shore (SHO 1990), unless otherwise noted. 2.1 The Schrödinger equation for a two-state atom. Let’s look at a bound atom which can be described by a stationary Hamiltonian H0 and a complete set of orthonormal eigenfunctions ψ n such that: and H0 ψn = En ψn (2.1) ψ n ψ m = δ n,m (2.2) Any state Ψ(t) of the system can then be described as a linear combination of the Ψ(t) = ∑ C n (t) ψ n (2.3) C n (t) = ψ n Ψ(t) ≡ n Ψ (2.4) ψ n s: n where The Cn(t)s are complex numbers representing probability amplitudes. Pn(t), the probability of finding the system in the state ψ n at the time t is: 2 Pn (t) = C n (t) = n Ψ and, since ψ n is a complete set, ∑ P (t) = 1 n 2 (2.5) (2.6) n Obviously, when there are no interactions between the bound atom and its environment the system is stationary and all the Pn(t)s are constants. It follows that only the phase of the Cn(t)s change with time for an isolated system: C n (t) = e i − En t h C n (0) (2.7) We are interested in situations where we manipulate the atom with controllable, timedependent external interactions. Typically, those interactions are small relative to the internal forces that define the stationary states ψ n . (If this is not the case, the external forces will completely alter the internal atomic structure, the ψ n s will become meaningless, and the perturbative approach we are following will collapse). Under this Page 5 of 31 28/11/06 Walks on the Bloch sphere assumption we can define the Hamiltonian H, describing both the bound atom and its interaction with the applied external forces as Then the Schrödinger equation H(t) = H 0 + V(t) . (2.8) ∂ Ψ(t) = H(t)Ψ(t) ∂t (2.9) ih describes the time evolution of the atomic state Ψ(t) under the influence of the external interaction V(t), that may also be represented by its matrix elements Vnm(t): Vnm (t) = ψ n V(t) ψ m (2.10) Vnm=Vmn* (2.11) with, since V(t) is Hermitian, Let us now focus our attention at an atomic two-state system. Such a system is, by definition, fully described by the two states: 0 ≡ ψ 0 and 1 ≡ ψ1 . Using (2.3) we can expand any wave function Ψ(t) in 0 and 1 . Substituting this expansion in the Schrödinger equation (2.9) yield the following equation for the expansion coefficients: ih 0 V01 (t) ⎤ ⎡C 0 (t)⎤ d ⎡C 0 (t)⎤ ⎡E 0 + V 00 (t) = ⎢ ⎥⎢ ⎢ ⎥ ⎥ 0 dt ⎣ C1 (t) ⎦ ⎣ V10 (t) E1 + V11 (t)⎦ ⎣ C1 (t) ⎦ (2.12) Once V(t) and the initial conditions are specified, Eq. (2.12) above provides the time evolution of the two-state system subject to the external interaction. 2.2 Coherent interaction – Majorana spin flips A bound two-state atom with no interaction with its environment will remain indefinitely in its initial condition, as far as observables are considered. To illustrate some of the phenomena that occur when such an interaction is present, let us subject a bound twostate atom to a very simple external time-dependent potential: zero at t < 0 and constant afterwards. t ≤ 0 : Vij (t) = 0, ( i, j = 0,1) h (2ϖ − ω 0 ) − E 00 , V11 (t) = h (2ϖ + ω 0 ) − E10 , 2 2 1 V01 (t) = V10 (t) = const ≡ 2 hΩ t > 0 : V00 (t) = (2.13) where hω 0 is the energy difference between the two states, and hϖ is the average of the energies of the two states. Substituting (2.13) into (2.12) and setting ϖ = 0 (the zero point of the energy is arbitrary) we get: Page 6 of 31 28/11/06 Walks on the Bloch sphere i which leads to d ⎡C 0 (t)⎤ 1 ⎡− ω 0 = dt ⎢⎣ C1 (t) ⎥⎦ 2 ⎢⎣ Ω Ω ⎤ ⎡C 0 (t)⎤ + ω 0 ⎥⎦ ⎢⎣ C1 (t) ⎥⎦ (2.14) ~ ~ && (t) = − 1 Ω C Ci (t), with Ω = Ω 2 + ω02 , i = 0,1 i 4 (2.15) i i Solving (2.15) with initial conditions: C0(0)=1, C1(0)=0 C& 0 (0) = ω 0 , C&1 (0) = − Ω and 2 2 calculating the probability P1(t) of finding the system in the upper state we get: ( )( ( )) ~ 2 ~ P1 (t) = 12 Ω Ω 1 − cos Ωt (2.16) So we see that by adding this simple interaction, we caused periodic oscillations in the two-state probability. When the interaction is weak ( V01 << hω 0 , or Ω << ω 0 ) the oscillation amplitude is small, and its frequency is close to ω 0 . When the interaction is strong ( Ω >> ω 0 ) the entire population moves periodically between each of the two states, and its oscillation frequency is close to Ω . Similar oscillations of the two-state atom are produced by almost any external excitation. 2.3 Pure and mixed states, coherence, density matrix, Bloch sphere and Bloch vector The periodic probability oscillations described in the previous section can never be observed by one measurement of a single two-state atom. To notice these oscillations we need an ensemble of such atoms (and several other conditions that we will summarize in the end of this section). In this section we will describe the basic tools to deal with ensemble of quantum states, following Hideo Mabuchi (MAB 2001). (Note that many observations of a single system constitute an ensemble) Lets assume that at time t = 0 we can define an ensemble of two-state atoms through a set of wave functions Ψ i (0) , each representing a fraction pi of the atoms in that ensemble. We can then define the density matrix, or the density operator, as: ρ̂(0) = ∑ p i Ψ i (0) Ψ i (0) , with i ∑p i =1 (2.17) i It can be shown that the equation of motion of ρ̂(t) is given by: ih ∂ ρ̂(t) = [H, ρ̂(t)] ∂t (2.18) and that the expectation value of any operator  is given by  = Tr(ρ̂Â) Page 7 of 31 (2.19) 28/11/06 Walks on the Bloch sphere If one of the p i equals 1, say p1 = 1, then Ψ1 represents all the atoms in the ensemble, and we have what is usually called a “pure state”. Otherwise, we have a “mixed state”. (“Pure ensemble” and “mixed ensemble” seem more suitable names, but we will stick with tradition). r We now introduce the spin operator S r hr h S = σ = (σ̂ x , σ̂ y , σ̂ z ) = 2 2 h ⎛⎛0 1⎞ ⎛0 − i⎞ ⎛1 0 ⎞⎞ ⎜⎜ ⎟, ⎜ ⎟, ⎜ ⎟⎟ 2 ⎜⎝ ⎜⎝ 1 0 ⎟⎠ ⎜⎝ i 0 ⎟⎠ ⎜⎝ 0 − 1⎟⎠ ⎟⎠ (2.20) r and the Bloch vector ν , (which is the unit vector connecting the origin to a point θ, ϕ on r r the Bloch sphere). S and ν will enable us to link together Bloch sphere, the density operator and the pure and mixed states concepts. The Bloch vector components, in Cartesian coordinates, are: r ν = (ν x , ν y , ν x ) = (sinθ cosϕ , sinθ sinϕ , cosθ ) (2.21) r Calculating the expectation value S for the pure state represented by θ, ϕ (Eq. 1.3) we r h get (see annex 1 for details): S = (sinθ cosϕ , sinθ sinϕ , cosθ ) (2.22) 2 So by combining (2.19), (2.21) and (2.22) we see that for a pure state we have: r r hr r S = Tr(ρ̂S) = ν, with ν = 1 2 (2.23) Eq. (2.23) motivates the definition of the generalized Bloch vector (both for pure and for mixed states) as: r hr ν B = Tr(ρ̂S) 2 (2.23a) Studying the properties of the density matrix and the generalized Bloch vector, we can learn the following: (1 + νr B ⋅ σr ) • The density matrix can be written as ρ̂ = • The diagonal matrix elements of ρ̂ are real; they give the relative populations of the two basis states 0 and 1 for the ensemble represented by ρ̂ . Therefore, their sum 1 2 (2.24) is 1, or Tr (ρ̂ ) = 1 . Page 8 of 31 28/11/06 Walks on the Bloch sphere • • • The off-diagonal elements of ρ̂ , ρ12 = ρ ∗21 , represent the coherence between the atoms in the ensemble. For example, if we have a completely incoherent ensemble of r two-state atoms, the off-diagonal elements, (as well as ν B ), vanish. r r r dν B 1 r = Tr[σ[H, (1 + ν B ⋅ σ )]] (2.25) dt 2 Example: when a static magnetic field Bz is applied, the solution of this equation shows that the Bloch vector precesses about the z axis at the Larmor frequency, given by ω L = γB z , where γ is the gyromagnetic ratio of the atom. For simplicity, the r dynamics of ν B may be calculated in a reference frame that rotates about the z axis at the Larmor frequency. r The equation of motion for ν B is: ih Finally, the Bloch vector can represent dissipation and dephasing. For that purpose, we define 3 additional parameters: longitudinal relaxation time T1, transverse relaxation time T2 and ν 0z , which is the z component of the Bloch vector at thermal r equilibrium. Adding these parameters to the equation of motion for ν B , and solving in the rotating frame with no field except Bz, we obtain the following results for the components of the Bloch vector: ν x (t) = ν x (0)e ν y (t) = ν y (0)e ( − t T2 − t T2 (2.26) ) ν z (t) = ν z (0) − ν e 0 z − t T1 + ν 0z This result is valid only in a very simple case, but it does present the general picture. We see that if we start at t=0 at some pure state on the surface of the Bloch sphere, the projection of the Bloch vector on the XY plane will start to shrink to zero with the T2 time constant, representing dephasing, or decoherence process. In parallel, but at a different rate defined by T1, the Z component of the Bloch vector will decay towards its thermal equilibrium value ν 0z . Let us briefly return to the Majorana spin flips described in section 2.2. We now see that there are several requirements if we wish to actually observe the probability oscillations: • We need an ensemble of measurements. It can be realized by repeated measurement on a single two-state atom, or by a single measurement of a group of such atoms, or some combination of these two methods. • We have to prepare the ensemble so that its initial state is as close as possible to a pure state. Otherwise, the atoms in the ensemble will have random phases relative to Page 9 of 31 28/11/06 Walks on the Bloch sphere each other. This will mask any observation of the probability oscillations. Typically, the ensemble is prepared by some type of optical pumping (YUA 2004). • All the operations, including final measurements, must be completed well before the smaller of the two relaxation times T1 and T2. Failing to do this will again lead to averaging out the probability oscillations. • As we will see in section (2.5), there is one more condition: the initial state of the system at t=0 should not be an eigenstate of the full Hamiltonian. If it is, the system will remain in that state indefinitely. All of these conditions are required if we wish to observe the Majorana spin flips, or any other coherent phenomena. 2.4 Rabi oscillations, Rabi frequency, Rabi flopping pulses, Ramsey fringes The two-state atom is described by the Hamiltonian H0, the eigenvalues E0, E1 and the eigenstates 0 , 1 . When we subject the atom to external periodic excitation (such as laser light), the atom is now described by the full Hamiltonian Ĥ(t) = H 0 + V̂(t) . For circular polarized light we may have (SHO 1990, §3.7 – annex 2) V00 (t) = V11 (t) = 0 V01 (t) = V10* (t) = 12 hΩ e-i (ωt + φ ) (2.27) where ω is the laser light frequency and Ω is the Rabi frequency. For laser beam of intensity I (W/cm2), the Rabi frequency is given by: Ω= r r d 10 ⋅ ε ea 0 8π(ea 0 ) I h 2c 2 2 (2.28) r r r d10 is the dipole matrix element 1 d 0 and ε is the unit vector in the direction of the electric field of the laser light. (Note: this dipole moment, and Eq. (2.28), are valid only if 0 and 1 are non-degenerate states) Substituting (2.27) into the time dependent Schrödinger equation, transforming to the rotating wave picture (reference frame that rotates at ω) and manipulating we get the following equations for the probability amplitudes: i d ⎡C 0 (t)⎤ 1 ⎡− (ω 0 − ω) = ⎢ Ω dt ⎢⎣ C1 (t) ⎥⎦ 2 ⎣ Ω ⎤ ⎡C 0 (t)⎤ (ω 0 − ω)⎥⎦ ⎢⎣ C1 (t) ⎥⎦ (2.29) where hω 0 is the energy difference between the two states. Solving this equation with the initial condition C0(0)=1, and calculating the probabilities we get: Page 10 of 31 28/11/06 Walks on the Bloch sphere 2 [ ( )] Ω 1 ~ ~ P1 (t) = 1 − P0 (t) = 1 − cos Ωt , Ω = 2 2 2 (ω 0 − ω) + Ω Ω + (ω 0 − ω ) 2 2 (2.30) We see that the probability of finding the atom in the state 1 oscillates indefinitely (Rabi oscillations) with a frequency that is the RMS value of the Rabi frequency and the detuning δ = ω 0 − ω of the laser light from the frequency of the atomic transition. This behavior is similar to the Majorana spin flips presented in section 2.2. However while the amplitude of the Majorana spin flips depend mainly on Ω , the amplitude of the Rabi oscillations depends only on the detuning. At δ = 0 the amplitude is maximum and it drops as the excitation field frequency moves away from this resonance point. Figure 2.1 below present the Rabi probability oscillations for Ω = 2rad/sec, and several values of the detuning. 1 δ=0 P1(t) 0.75 δ=Ω 0.5 0.25 0 0 1 2 3 4 5 6 δ=2Ω time Figure 2.1 – Rabi oscillations 2.4.1 Rabi pulses Focusing our attention on the resonance case, and following Kuhr (KUH 2003), we can r represent the effect of the Rabi pulse of duration time t on the Bloch vector ν B (0) (section 2.3) by the following 3x3 matrix: 0 0 ⎞ ⎛1 ⎜ ⎟ r r ˆ ν B (t ) = ⎜ 0 cos φ (t ) sin φ (t ) ⎟ ⋅ν B (0) = Θ φ ( t ) ⋅ν B (0) ⎜ 0 − sin φ (t ) cos φ (t ) ⎟ ⎝ ⎠ r (2.31) t with φ (t ) = ∫ Ω(t ' )dt ' (2.32) 0 In other words, a Rabi pulse rotates the Bloch vector about the x axis. Page 11 of 31 28/11/06 Walks on the Bloch sphere For φ (t) = π and φ (t) = π/2 and the matrix becomes very simple: Θπ 2 ⎛1 0 0⎞ ⎛1 0 0 ⎞ ⎜ ⎟ ⎜ ⎟ = ⎜ 0 0 1 ⎟ , Θπ = ⎜ 0 − 1 0 ⎟ ⎜ 0 −1 0⎟ ⎜ 0 0 − 1⎟ ⎝ ⎠ ⎝ ⎠ (2.33) Θ π is called a π pulse, or Rabi flopping pulse, and Θ π / 2 is called a π 2 pulse. The matrix representation for the free precession is: ⎛ cos φ (t ) sin φ (t ) 0 ⎞ t ⎜ ⎟ Θ FREE (t ) = ⎜ − sin φ (t ) cos φ (t ) 0 ⎟, φ (t) = ∫ δ (t' )dt' 0 ⎜ 0 0 1 ⎟⎠ ⎝ (2.34) Note that the frequency of the free precession of the Bloch vector is δ = ω 0 − ω , since our results were obtained in a rotating wave frame that rotates at ω . a. b. 0 0 c. 0 φ 1 0 d. 1 1 e. 0 1 1 f. 0 1 Figure 2.2: some examples of Rabi pulses. is the Bloch vector before the pulse, is the vector after the pulse. Figure 2.2 above presents some examples of Rabi pulses: (a) presents a general Rabi pulse. The x component of the Bloch vector is saved, while its projection on the yz plane Page 12 of 31 28/11/06 Walks on the Bloch sphere is rotated by φ . (b) Presents the operation of a π pulse on 0 . It flips the population from 0 to 1 , which explains its name - Rabi flopping pulse. In (c) a π/2 pulse takes the population from 0 to a point on the equator representing coherent equal mixture of the two base states with no phase difference between them (ϕ = 0) . In (d) another π/2 pulse applied immediately takes the population from that point to 1 . (e) shows free precession of the Bloch vector in the xy plane. Finally, (f) shows how a π/2 pulse effect the Bloch vector after it was allowed to precess for some time. The Rabi pulses may be utilized directly to manipulate our two-state atom system. However, the “two-state atom” is usually just an approximate model, where we pick two out of its many states and set them to be the states that will realize a qubit or any other useful device. As it turns out, typically those two states are separated by microwave scale energies (∼10GHz), so that the Rabi pulses will need to have microwave frequency. Microwave pulses are much harder to manipulate than optical frequency laser beams, and their spatial resolution is measured in centimeters, compared with less than a micrometer for the optical range. Moreover, in some cases the direct transition between the two states may be forbidden or limited by selection rules, making direct Rabi flopping impossible. Therefore, Rabi pulses are rarely utilized for direct manipulation of two-state ensembles. 2.4.2 Ramsey fringes Consider the following sequence of operations, performed on an ensemble of two-state atoms: • Prepare the ensemble in the pure state Ψ (0) = 0 . • Apply a π/2 Rabi pulse by a coherent, monochromatic laser beam at ω ≈ ω 0 • • Block the laser beam for time T Apply a second π/2 Rabi pulse by same laser • Measure the population p1 (t) of level 1 In terms of the Bloch vectors and the matrices defined in (2.33) and (2.34), we can r summarize the sequence described above as measuring the z component of ν RAMSEY (T ) defined below: r r ν RAMSEY (T ) = Θπ 2 ⋅ Θ FREE (T ) ⋅ Θ π 2 ⋅ν B0 (2.35) Based on the examples presented in Figure 2.2, it is clear that p1 (t) is a periodic function of the accumulated phase φ (see 2.34). Actually, we have: T ν Z , RAMSEY (T ) = − cos φ = − cos( ∫ δ (t )dt ) (2.36) 0 Page 13 of 31 28/11/06 Walks on the Bloch sphere To observe the oscillatory behavior of (2.36), we need δ to be small compared to the Rabi frequency and the spectral pulse width, such that the pulse can be approximated as near resonant one, and complete population transfer can occur. The “Ramsey fringes” created by (2.36) are a very delicate interferometeric tool. The interferometeric nature can be revealed if we look at a π/2 pulse as a 50/50 optical beamsplitter. The first pulse splits the atomic wavefunction into a superposition of the two states. During the free propagation time, the relative phase of the two states evolves at a rate proportional to the energy difference between the states, and the coupling driving field accumulates a phase ωT . The second pulse recombines the two states, interferometrically comparing the accumulated relative phase. Below are some examples to demonstrate how this tool works. To simplify, let’s define the following set of units: ~ δ =δ Ω (2.36a) ~ T = TRAMSEY /2πΩ (2.36b) where Ω is given by (2.28). In the following example, we will set the Rabi frequency to a convenient number: Ω =1000*2π. First let review the effect that T has when the detuning is small and independent of time: ~ δ = 0 ÷ 0.5 . For very small detuning, 2.36 is a good approximation. However, for larger detuning we need to replace Θ π 2 in (2.35) with (2.31). Based on that we can calculate the ~ ~ population of 1 : P1 = (1 − ν Z,RAMSEY (T))/2 . (Note that T measures time in units of Rabi cycles). We can see below that as the detuning goes up, the cycling of the population is more rapid, while the maximum population decreases. 1 0.8 Detuning =0 P1 0.6 0.4 Detuning =0.1 Detuning =0.5 ` 0.2 0 0 5 10 T Page 14 of 31 28/11/06 Walks on the Bloch sphere In the graph in this page we see the effect of the detuning on the population of 1 , for 1 0.8 two values of T. We can see that as T is larger, the population cycling is more rapid. Thus, using the Ramsey fringes with large T, enables us to determine the Rabi frequency with higher accuracy. 0.6 P1 T=5 T=15 0.4 0.2 0 -1.5 -1 -0.5 0 0.5 1 1.5 Detuning 2.5 Dressed states, adiabaticity, chirped pulses, adiabatic population transfer Our two-state atom is described by the Hamiltonian H0, the eigenvalues E0, E1 and the eigenstates 0 , 1 . When we add external excitation, the atom is now described by the full Hamiltonian Ĥ(t) . It is of interest to study the full Hamiltonian eigenstates. Let the interaction with the external excitation (such as monochromatic laser beam) be: h⎡ 0 V̂(t) = ⎢ − iωt 2 ⎣Ωe Ω*eiωt ⎤ ⎥ 0 ⎦ (2.37) and the rotating wave frame state: Ψ(t) = C0 (t) 0 eiωt 2 + C1 (t) 1 e-iωt 2 (2.38) (The energy zero is midway between the states). Substituting (2.37) and (2.38) in the time dependent Schrödinger equation lead to: ih ∂ ⎛ C 0 (t) ⎞ ⎜ ⎟= ∂t ⎜⎝ C1 (t) ⎟⎠ h ⎛ - δ Ω * ⎞⎛ C 0 (t) ⎞ ⎜ ⎟⎜ ⎟ 2 ⎜⎝ Ω δ ⎟⎠⎜⎝ C1 (t) ⎟⎠ (2.39) where δ = ω 0 − ω, and hω 0 = E1 − E 0 . As we can see, in the rotating wave frame the matrix representation of Ĥ(t) is time independent, and so are its eigenvalues, given by: ~ h hΩ 2 2 E± = ± Ω +δ =± 2 2 Page 15 of 31 (2.40) 28/11/06 Walks on the Bloch sphere Calculating the eigenvectors we get (up to a global phase): Φ + (t) = e iωt 2 Φ − (t) = e i ωt 2 1 δ 1 δ (1 + ~ ) 0 + e -iωt 2 (1 − ~ ) 1 2 2 Ω Ω 1 δ 1 δ (1 − ~ ) 0 + e -iωt 2 (1 + ~ ) 1 2 2 Ω Ω (2.41) the states Φ ± (t) are called “dressed states”, as opposed to the “bare states 0 , 1 . It is as if the full Hamiltonian “dresses” the bare states. We can utilize the dressed states to manipulate our system of two-state atoms. • First, from (2.41) we can see that at far red detuning, Φ − (t) is mainly 0 , while at far blue detuning, Φ - (t) is mainly 1 . • Second, we note that if a system is represented at some time by either one of the dressed states, it will remain in this state indefinitely, as this state is an eigenstates of the full Hamiltonian. • Third, it turns out that if we change the frequency of the exciting field “slowly”, a system that is in a pure dressed state will remain in this dressed state. Such a change is an adiabatic change. The adiabaticity can be measured by (RUB 1981) γ= Ω 2 dδ dt (2.42) The probability that an atom will make a transition to another dressed state is P = e −2πγ so that with large enough γ (see 2.24), we can assume that no atom will make a transition to another dressed state. There are several procedures that utilize adiabaticity to manipulate atomic ensembles. Chirped pulses To demonstrate this procedure, let’s start with an ensemble that has all the atoms in the ground state 0 . (This may be the thermal equilibrium at the ensemble’s temperature, or a result of some other procedure). Then we slowly ramp up the intensity of a far-red detuned laser field. Since the Φ − (t) dressed state is then mainly 0 , all the atoms will be in this state. Then we sweep the frequency adiabatically up to a far-blue detuned region. There, the Φ − (t) dressed state is then mainly 1 . At this point we adiabatically ramp the intensity down to zero, leaving almost all the atoms in 1 . Thus we have a “chirped pulse” method to manipulate the state of an ensemble of atoms. Page 16 of 31 28/11/06 Walks on the Bloch sphere 2.5 Adding other states: Raman transitions, STIRAP Typically, our two-state atom has many other states that we ignore. Sometimes, however, it is advantageous to bring more states into the game. It adds additional parameters that may be used to manipulate our two-state system, and may facilitate otherwise forbidden transitions. We will review two schemes here: Raman transitions and STIRAP. 2.5.1 Raman transitions Figure 2.3 presents a scheme to excite Rabi probability oscillation between our two states through a third state. This is known as a Raman transition. The scheme shown in figure 2.3 is typically called a “Λ configuration”. Two coherent light beams are applied simultaneously to the system. (By “coherent” we mean that the phase of the beat-note of the two light fields is independent of time.) We define here two Rabi frequencies per (2.28): Ω P for the transition, and Ω S for the 1 ⇔ 2 ω0 2 ⇔ 3 transition. The energy zero is at Figure 2.3: Raman transition scheme level 1 . Following Dotsenko (DOT 2002), this system can be described by the state: Ψ (t ) = C1 (t ) 1 + C 2 (t ) 2 + C3 (t ) 3 (2.43) and its Hamiltonian Ĥ is given by (using the rotating wave approximation) : ⎛ 0 ⎜ h Hˆ = ⎜ Ω P 2⎜ ⎝ 0 ΩP 2Δ ΩS 0 ⎞ ⎟ ΩS ⎟ 2δ ⎟⎠ (2.44) Substituting (2.43) and (2.44) into the time dependent Schrödinger equation, we get equations for the Ci(t)s. For C2(t) we have: iC& 2 (t ) = 12 (Ω P C1 (t ) + Ω S C3 (t )) + ΔC 2 (t ) (2.45) As typically Δ >> Ω P , Ω S , C2(t) oscillates at high frequency. Therefore we may replace its time derivative in (2.45) with its average over large number of cycles, namely zero (see DOT 2002, pp5). Using the modified (2.45) to calculate C2(t) as a function of the other Ci(t)s, we get an effective Hamiltonian: Page 17 of 31 28/11/06 Walks on the Bloch sphere 2 Ω P ΩS Δ ⎞ 1⎛ Ω Δ ⎟ Ĥ EFF = ⎜⎜ P 4 ⎝ Ω P Ω S Δ Ω S2 Δ − 4δ ⎟⎠ (2.46) The off diagonal elements give the coupling of the state 1 and 3 by the Raman beams, while the diagonal elements represent the light shift of the energy levels due to the interaction with the light fields. By solving the time dependent Schrödinger equation with the effective Hamiltonian, and calculating the population probabilities, we obtain: P3 (t) = 1 − P1 (t) = Ω 2R sin 2 (Ω 0 t 2) Ω 02 (2.47) where at t=0 all the population is in 1 , and we have defined Ω 0 = Ω 2R + δ 2 and Ω R = Ω P Ω S 2Δ . So operating with two light fields we may to induce Rabi oscillations between our two states as if we were coupling them with a single δ-detuned field. 2.5.2 STIRAP - Stimulated Raman Adiabatic Passage STIRAP is a method for the complete transfer of an ensemble of atoms from one state to another, with the indirect “help” of a third state. In this subsection we will follow Bergmann (BER 1998) and use his notation. In Fig 2.4 we see the three levels: Level 1 is the ground state while level 3 is the excited state. The idea is to transfer the population from 1 to 3 , without populating level 2 . (Atoms excited to level 2 will undergo spontaneous emission to levels other than 1 and 3 and would be lost). Fig 2.4: STIRAP scheme The STIRAP method utilize two laser fields: the Pump beam, linking 1 and 2 , with detuning Δ P , and the Stokes beam, linking 2 and 3 , with detuning Δ S . Clearly, this system is very similar to the Raman scheme (Figure 2.3), and we will use a similar Hamiltonian as a starting point: 0 Ω P (t ) ⎞ ⎛ 0 ⎟ ⎜ h Hˆ (t ) = ⎜ Ω P (t ) 2Δ P Ω S (t ) ⎟ 2⎜ Ω S (t ) 2(Δ P − Δ S ) ⎟⎠ ⎝ 0 Page 18 of 31 (2.48) 28/11/06 Walks on the Bloch sphere Here, however, we intend to modify the Rabi frequencies during the process, so we explicitly indicated their time dependence. The next step is to define the following 3 eigenstates through the mixing angle Θ: TanΘ = Ω P (t) Ω S (t) a + = sin Θ sin Φ 1 + cos Φ 2 + cos Θ sin Φ 3 a 0 (2.49) = cos Θ 1 − sin Θ 3 a − = sin Θ cos Φ 1 − sin Φ 2 + cos Θ cos Φ 3 (The angle Φ is a function of the Rabi frequencies and detuning and is not relevant here.) The three eigenstates above have the following eigenvalues (dressed states): ω + = Δ P + Δ 2P + Ω 2P + Ω S2 ω0 = 0 (2.50) ω − = Δ P − Δ 2P + Ω 2P + Ω S2 We will now describe the population transfer procedure, with the help of figure 2.5. At t=0, both laser intensities and the mixing angle are zero, all the population is in state 1 , and the three dressed states are degenerate. In the first step we ramp the Stokes laser up adiabatically (a). We notice (c) a split of the dressed states to three distinct levels. The population (d) is still at 1 since the Stokes laser couples two empty states. However, as the mixing angle (b) is still zero all the population is at a 0 also, as a 0 = 1 at this point. At the next step we ramp up (a) the Pump laser, while ramping down the Stokes laser. This is done in order to smoothly change the mixing angle (b) from zero to 90°. During this step, all the population remains in a 0 , since it is an Figure 2.5: STIPAP procedure eigenstate of the full Hamiltonian. Page 19 of 31 28/11/06 Walks on the Bloch sphere However, as Θ changes smoothly from 0 to 90°, a 0 , which is a superposition of 1 and 3 , changes smoothly from 1 to 3 (see (2.49)). We now ramp the Pump laser intensity down smoothly to finish the procedure. Using this sequence, we are able to transfer all the population from state 1 to 3 without populating 2 , from which population may be lost via spontaneous emission. This method can be modified to include more than one additional level, and may be used for arbitrary manipulation of a two-state atom. Page 20 of 31 28/11/06 Walks on the Bloch sphere 3. State of the art During the last decade, a large amount of work was aimed at the manipulation of twostate atoms, also known as qubit rotations. Several methods were used, such as single pulses, chirped pulses, several variations of the STIRAP methods and Raman beams. The two-state neutral atoms manipulated thereby were held in magnetic traps on atomchips, in optical traps (such as optical lattice) and in magneto-optical traps. In other experiments the atoms were part of a moving beam, or confined to a vapor cell. To picture the state of the art in this field, we present below several examples of these efforts that have been performed in the last three years. 3.1 STIRAP based method We will start our review with a method based on STIRAP, performed in Kaiserslautern by Vewinger et al (VEW 2003). They used a STIRAP procedure to achieve a coherent superposition of two degenerate states in a moving beam of 20Ne atoms. The relevant level diagram is presented in Figure 3.1. At the start of the sequence, the atoms are at the 3P0 state. The target is to transfer them to the states 3P2(m = +1), 3P2(m = -1), and control the relative phase. Two laser beams are used: The linearly polarized Pump laser P, coupling 3P0 with the intermediate level 3P1(m = 0), and another linearly polarized laser S, which can be viewed as a superposition of two σ+ and σ− beams. These beams, marked S+, S-, couple the intermediate level Figure 3.1 – Level diagram (see text) with the target states. The Hamiltonian of the combined system (atom and radiation) has four adiabatic states, two of which are “dark” states, which do not have components of the 3P1 state. These states are used for the adiabatic transfer process: Φ1 (t) = cosϑ (t) 0,0 − sinϑ (t)cosϕ (t)e −iχ 2,−1 − sinϑ (t)sinϕ (t)e + iχ 2,+1 (3.1) Φ 2 (t) = sinϕ (t)e −iχ 2,−1 − cosϕ (t)e +iχ 2,+1 where the states are marked by J, m notation, the mixing angles ϑ and ϕ given by: tanϑ (t) = Ω P (t) Ω S (t) Ω (t) tanϕ (t) = s+ , Ω s (t) = Ω S2+ + Ω S2− Ω S− (t) (3.2) and the angle χ measures the inclination of the polarization plane of the S beam, as seen in figure 3.2 below. Page 21 of 31 28/11/06 Walks on the Bloch sphere Generally, Φ1 and Φ 2 are coupled. In our case they are not coupled, since ϕ is constant. Therefore, in the adiabatic limit the system reaction to detuning changes is given by Φ1 , as Φ 2 does not include the initial state. The 20Ne atoms interact first with the Stokes beam (see Figure 3.2 below). As they move on, they interact with both the stokes and the pump beams, with the stokes beam intensity fading and that of the pump beam rising. Then pump beam fade out and the atoms move out of the interaction region. (see section 2.5.2 for details). With this sequence, we have (see 3.2) ϑ (−∞) = 0 at the beginning of the sequence and ϑ (+∞) = π / 2 at its end. Since we know that the system will evolve per Φ1 (in the adiabatic limit) we can substitute these values in Eq. 3.1, and see that the system will evolve from the initial state 0,0 to the desired final state: Ψ = [ 1 2,−1 e −i ( χ +φ ) + 2,+1 e +i ( χ +φ ) 2 ] (3.3) where the relative phase can be controlled by rotating the polarization plane of the S Beam. The paper then describes the method to measure the actual resulting superposition, which we will not follow here. Figure 3.2: Geometry of the experiment. The direction of polarization (indicated by the small arrows) for the pump laser (λ = 616nm) is chosen to be the z-axis, its direction of propagation defines the x-axis, while the neon beam propagates in y-direction. The two circularly polarized Stokes lasers are generated by a linearly polarized laser (λ = 588 nm) propagating in zdirection, whose direction of polarization forms an angle χ with the x-axis. The propagation of the so called filter laser (λ = 588 nm) is parallel to the Stokes laser and its direction of polarization forms an angle α with the x-axis. The detection laser (λ= 633 nm) is unpolarized. Page 22 of 31 28/11/06 Walks on the Bloch sphere 3.2 Spin flips with π pulses and chirped pulses In 2004, Schrader et al. (SCH 2004), (Bonn University, Meschede’s group) developed a quantum register based on trapping a string of single cesium atoms in an optical dipole trap (Figure 3.3). Cesium atoms were transferred from a MOT to a standing wave dipole trap created by a far-red detuned laser beam (λ=1064nm). The light from the cesium atoms was collected by an Avalanche PhotoDiode (APD) and Intensified ChargeFigure 3.3: setup for quantum register (see text) Coupled Device camera (ICCD). They chose the levels: F = 4, m F = 4 ≡ 0 , F = 3, m F = 3 ≡ 1 as the qubit states, separated by a ~9.2GHz transition. When they operated the system, cesium atoms filled some of the optical dipole traps created by the standing wave light radiation. (Figure 3.4, a). To address each atom separately, they added a magnetic field with a gradient of 15 G/cm along the trap axis. Due to the Zeeman shift of the energy levels, this magnetic field created a shift of ~9kHz in the resonance frequency for atoms separated by 2.5 μm. Each atom has its own resonance frequency as a function of its position. The position of each cesium atoms is located by the ICCD camera, and its resonant frequency is calculated and store in the system’s computer (Fig. 3.4 – a). The next step is initialization (by optical pumping) of all atoms (5 in our case) to the state 00000 (Figure 3.4 b). Then two microwave π pulses, one with the resonanse frequency of the second atom and one with that of the fourth atom, are sent to the register via open ended wave guide. Now our register reads 01010 (Figure 3.4 c). To verify that this is really the case, a push-out laser beam removes atoms in the 0 state from the trap. (Figure 3.4 d). This is confirmed by the camera picture (e). Figure 3.4: Cesium atoms register (see text) Page 23 of 31 28/11/06 Walks on the Bloch sphere Schrader et al measured decoherence time of 0.6ms, and crosstalk (effect on neighboring atoms) of about 1%. Using the same system, Khudaverdyan et al (KHU 2005) manipulate the state of the cesium atoms with chirped pulses (otherwise known as adiabatic passage). They used two microwave signal generators: one provides a 10.2 GHz signal with fast AM capabilities, and the other provides 1 GHz signal with FM capabilities. Using a mixer to subtract the signals, they got the required 9.2 pulse with computer controlled AM and FM modulation. Thus, they were able to produce pulses with arbitrary shape. The sequence they used was: Trapping cesium atoms in the optical dipole trap, initializing them to 0 with optical pumping, then transmitting the microwave pulse and finally measuring the resulting population. They showed a robust transition efficiency of up to 90%- 95% for variation of up to ±40 kHz in the signal’s central frequency. In addition, they demonstrated an ability to use a fixed frequency microwave radiation to flip the atoms while the atoms were in motion. They moved the atoms by an optical conveyor belt across a region where their resonance frequency was a function of position (due to magnetic field gradient). As the atoms reached the point where their resonance frequency matches the fixed microwave field, their state was flipped. 3.3 Raman transitions in optical tweezers M. P. A. Jones et al, (Jon 2006) in Cedex, France (Grangier’s Group) used a tightly focused far-off resonant (λ=810nm) laser beam to create a dipole trap for 87Rb atoms. Their trap had the property that the number of atoms in the trap to be either zero or one, and was loaded with cold rubidium atoms (90μK) from optical molasses. As seen in Figure 3.5 below, they chose as qubit states and F = 1, m F = 0 ≡ 0 F = 2, mF = 0 ≡ 1 . The also utilized the laser beam that created the trap as one of the Raman beams. The other beam was produced by injection locking a sideband of a microwave modulated bridge laser to the master (FORT) laser, and using the other sideband to lock a slave laser. Thus the slave laser’s beam frequency is 6.8 GHz away from the master’s, while maintaining the coherence between the beams. The Raman beams have orthogonal linear polarization in the y-z plane, to drive ΔmF = 0 transitions. The experiment sequence was as follows: The system monitors the fluorescence from the molasses cooling light via the avalanche photodiode (APD). As an atom populate the trap, the fluorescence goes drastically up, triggering the shut down of the molasses light and the experiment sequence start. First the atom is optically pumped to 0 . Then a Raman pulse is send to the atom, followed by a “push out” laser beam that remove atoms in the 1 state from the trap. Finally, the molasses light are turned on again and the status of the trap is detected. Page 24 of 31 28/11/06 Walks on the Bloch sphere Rabi frequency of 2π x 6.7MHz was demonstrated, providing π/2 rotation time of 37ns. Coherence time was 370μs. Using spin echo method, coherence time of 34 ms was reached, 6 orders of magnitude higher then the qubit rotation time. Figure 3.5: System setup and level diagram (see text) 3.4 Raman transitions on atom chip Treutlein at el (TRE 2006) review the work in Max-Planck institute for quantum optics. They realized single qubit rotation based on microwave and radio frequency Raman transition. They chose the following levels F = 1, m F = −1 ≡ 0 , F = 2, m F = +1 ≡ 1 as the qubit states. The motivations for this choice are: • Both states can be trapped by magnetic potential (“low field seekers”) • The magnetic moments and the corresponding static Zeeman shifts of the two states are approximately equal, leading to a strong common mode suppression of magnetic field induced decoherence • Both states have nearly identical trapping potentials in magnetic traps. This will reduce undesirable entanglement between the internal and external states. In addition, at a magnetic field of 3.23G, both states experience the same first-order Zeeman shift. Therefore, the atoms were held in magnetic traps with B0=3.32G, leading to Zeeman shift of few MHz. Figure 3.6 (top) presents the level diagram of the 87Rb atom, and the two Raman beams used in this experiment. The microwave Raman beam was produced by a microwave signal generator phase locked to a 10 MHz reference oscillator, and detuned 1.2 MHz above the intermediate 2,0 state. An RF signal generator, phase locked to the same reference oscillator, generate the second (RF) Raman beam. Page 25 of 31 28/11/06 Walks on the Bloch sphere An ensemble of about 104 0 atoms, at 0.6μK, trapped in a Ioffe-type microtrap was prepared, and then Raman pulse was applied. After The Raman pulse, the number N1 of atoms transferred to 1 was measured. Figure 3.6 (bottom) is a plot of N1 as a function of the Raman pulse length, Showing 320Hz Rabi oscillations. The reason for this low Rabi frequency is the low radiation power. Both beams are transmitted form outside the vacuum chamber. Decoherence time, measurement by Ramsey spectroscopy, is 2.8 seconds. A possible improvement for this method is to include microwave wave guide and RF wires on the atom chip. This will increase dramatically the Raman power that reaches the atoms, and thus raise the Rabi frequency to 1.8 MHz. Figure 3.6: See text Page 26 of 31 28/11/06 Walks on the Bloch sphere 3.5 Proposal for realization of a qubit on atom chip As the last item in this chapter, we present the proposition of Charron et al (CHA 2006) regarding the realization of qubits on an atom chip. The central idea in this proposal is to use the external degrees of freedom of a trapped atom for quantum “calculations”, and then use Raman beams to transfer the “results” to the internal states for long term storage. In Figure 3.7 (top) we see the physical arrangement of the trap. The trapping potential is calculated for single ultracold 87Rb atoms, and for realistic currents and magnetic fields. The resulting double well potential of the trap is presented in figure 3.7 (bottom). It is presented along x` axis, tilted by 14.8° to the original x axis. The transverse (y`) potential is neglected, as it is assumed that the energy of the atom is well below the quanta of the transverse degree of freedom. The height of the internal barrier (marked ξ) can be controlled by a proper change of the currents. When the barrier is high (Figure 3.7, a) the atoms in the two wells do not interact. When the barrier is lowered (b), they do interact, and the combined wavefunction depend upon the states of each atom prior lowering of the barrier. Figure 3.7: Double wall microtrap (see text) The following interaction sequence is proposed: • • At t = 0 the barrier is high. During 0 < t < T0 the barrier is smoothly lowered. • During T0 < t < T0+T1 the barrier is kept at its low position. • During T0+T1 < t < 2T0+T1 the barrier is smoothly raised to its initial condition. Assuming that the atoms have low enough energy so that they can be either in the ground vibrational state g = 0 or in the first exited vibrational state e = 1 , we have a two qubit gate, encoded in the vibrational states of the two atoms. Page 27 of 31 28/11/06 Walks on the Bloch sphere Then Charron et al propose to use two-photon Raman transition to encode the result of the gate operation to the internal state of the atoms. Some 87Rb levels are presented in figure 3.8. Out of the 8 ground state sub-levels only 3 are low field seekers which can be trapped by static magnetic potential: 1,−1 2,1 and 2,2 . The motivation for the assignment of the qubit states 1,−1 ≡ 1 2,1 ≡ 0 was already discussed in section 3.4. To induce two-photon Raman transitions between those two levels, one of Raman beams has to be σ− polarized, and the other σ+. If we detune the Raman beams below the 5P1/2 level, we may disregard the 5P3/2 16 sublevel, as the Raman beams are more then 7 THz detuned from them. (Typically, Raman beams may be detuned up to 300 GHz from the relevant level). Out of the remaining 8 5P1/2 sublevels, selection rules dictate that only 4 will participate in the transition, marked A, B, C, D in Figure 3.8. Figure 3.8: 87 Rb levels (see text) Charron et al proceed to calculate the effective interaction Hamiltonian of the system for r two (σ±) Raman beams with ω± frequencies and k ± wavevector, and their result is: ~ ~ ~ ~ ⎞⎤ σ ⎞ ⎛σ σ hΩ − Ω + iη ⎡⎛ σ ~ H int = e ⎢⎜⎜ 01 − 01 ⎟⎟ + ⎜⎜ 01 − 01 ⎟⎟⎥ + Hermitian conjugate terms (3.4) 4 3 ⎣⎝ Δ A1 Δ C1 ⎠ ⎝ Δ A0 Δ C0 ⎠⎦ ( ) ( ) r r r r where η = k + ⋅ r − ω + t − k - ⋅ r − ω - t , Ω± are the Rabi frequencies of the σ± Raman beams, The detunings Δ are given by: Δ A0 ≡ ω A − ω 0 − ω - Δ A1 ≡ ω A − ω1 − ω + Δ C0 ≡ ω C − ω 0 − ω - Δ C1 ≡ ω C − ω1 − ω + (3.5) and the operators ~ σ are given by: ~ = i j ; i, j = 0,1, A, B, C, D σ ij (3.6) When we wish to set the Raman beam frequencies ω±, we need to detune them far enough from any 5S1/2⇔5P1/2 transition frequency, otherwise we will have non-negligible population in the 5P1/2 levels during the Raman transition, with unwanted spontaneous decay to the 5S1/2 levels. Typically, the Raman detuning is few orders of magnitude bigger then the excited level natural line width. However, in this case we may have Page 28 of 31 28/11/06 Walks on the Bloch sphere Δ A1 ≈ Δ C1 , Δ A0 ≈ Δ C0 and our effective interaction Hamiltonian (3.4) will vanish identically. The physical meaning is that that the two Raman processes through the A and C levels interfere destructively, so that the probability for a transition between the qubit states is zero. However, the very structure of the 5P1/2 sublevels offers a possible solution to this problem. While the line width of the 5P1/2 sublevel is 5.75MHz, the hyperfine splitting between its F=1 and F=2 levels (between A and C) is some 800MHz. So it may possible to detune the Raman beams far enough (relative to 5.75MHz) below A to prevent populating it during the Raman pulse, but close enough (relative to 800MHz), to prevent the Hamiltonian (3.4) from vanishing. 3.6 Summary Neutral atoms can behave, under favorable conditions, as a two-state system. The two hyperfine ground levels of alkali atoms seem like most favorable candidates to serve as qubits. As a result, considerable scientific effort is devoted to the theoretical and experimental study these levels and their manipulation. We have presented recent samples of these works. In this summary, we would like to note that it seems that the larger part of works deal with neutral ultracold atoms held by optical dipole traps (optical tweezers, optical lattice, etc.) Less work is devoted to atoms held by magnetic microtrap, typically realized on atomchips. As for the status of the magnetic microtraps on atomchips, we would like to quote Treutlein (TRE 2006): “Atom chips combine many important features of a scalable architecture for quantum information processing. The long coherence lifetimes of qubits based on hyperfine states of neutral atoms, accurate control of the coherent evolution of the atoms in tailored micropotentials, and scalability of the technology through microfabrication – which allows the integration of many qubits in parallel on the same device while maintaining individual addressability. Furthermore, atom chips offer the exciting perspective of creating interfaces between the atomic qubits and other QIP systems integrated on the same chip, such as photons in optical fiber cavities or solidstate QIP systems located on the chip surface. 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