Jul 29, 2002 - Keywords: Quantum information processing, quantum optical ... of quantum gates is the phase gateâa transformation which attaches a minus ...
INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF OPTICS B: QUANTUM AND SEMICLASSICAL OPTICS
J. Opt. B: Quantum Semiclass. Opt. 4 (2002) S430–S436
PII: S1464-4266(02)37816-9
Controlling dynamical phases in quantum optics T Calarco1,2 , D Jaksch1 , J I Cirac1,3 and P Zoller1 1
Institute for Theoretical Physics, University of Innsbruck, Technikerstrasse 25/2, A-6020 Innsbruck, Austria 2 European Centre for Theoretical Studies in Nuclear Physics and Related Areas, Strada delle Tabarelle 286, 38050 Villazzano (TN), Italy 3 Max-Planck Institut f¨ur Quantenoptik,Hans-Kopfermann Str. 1, D-85748 Garching,Germany
Received 1 January 2001 Published 29 July 2002 Online at stacks.iop.org/JOptB/4/S430 Abstract We review and compare several schemes for inducing precisely controlled quantum phases in quantum optical systems. We focus in particular on conditional dynamical phases, i.e. phases obtained via state- and time-dependent interactions between trapped two-level atoms and ions. We describe different possibilities for the kind of interaction to be exploited, including cold controlled collisions, electrostatic forces, and dipole–dipole interactions. Keywords: Quantum information processing, quantum optical implementations
1. Quantum gates based on controlled two-particle phases The high degree of precision which can be obtained in quantum optics in the manipulation of isolated quantum systems offers several advantages in view of the creation and control of entanglement, having several applications in the field of quantum information and computation [1]. In particular, for the realization of a general unitary transformation on an arbitrary number of quantum bits (i.e. a quantum computation), only arbitrary single-qubit rotations and a suitably chosen twoqubit entangling operation are required. An example of a two-qubit operation that allows us to build a universal set of quantum gates is the phase gate—a transformation which attaches a minus sign to just one component of logical states: |0|0 −→ |0|0 |0|1 −→ |0|1 |1|0 −→ |1|0
αβ
|α|β|ψ αβ −→ eiϕ |α|β|ψ αβ ,
(1)
(2)
αβ
|1|1 −→ −|1|1. Quantum optical systems in periodic microscopic potentials can be used to achieve such a transformation, by combining the good isolation from the environment and precise control by laser fields, achievable in quantum optics, with the 1464-4266/02/040430+07$30.00 © 2002 IOP Publishing Ltd
ability—usually associated with semiconductor technology— of manufacturing periodic structures to generate modulated fields on a microscopic scale. The general concept is to encode the logical states of each qubit into two internal states of a particle (neutral atom or ion). Single-qubit operations are obtained as Rabi rotations by applying resonant laser fields. Two-qubit gates are performed by inducing a state-dependent interaction over a certain time, making the particles acquire a conditional phase shift depending on their logical states. However, it is not always straightforward to realize in practice an interaction between two particles which couples only their internal states—other degrees of freedom, for instance the motional ones, are possibly affected. Therefore, our goal is to approximate the ideal transformation (1) by means of a stateselective evolution for two particles, making them acquire a phase π if and only if they are both in the internal state |1, and leaving eventually unaffected the external degrees of freedom. More generally, we aim at achieving a mapping of the kind
where |ψ describes the state of the external (non-logical) degrees of freedom of the two particles. Single-particle contributions to the phase can be undone by single-bit operations [2], leaving us with the gate phase ϕ = ϕ 11 − ϕ 01 − ϕ 10 + ϕ 00 , plus possibly an irrelevant two-particle global phase. For ϕ = π, the transformation (1) is recovered. The system is, in general, initially prepared in a superposition
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S430
Controlling dynamical phases in quantum optics
of the two-qubit computational basis states, |χ ≡
1
cαβ |α1 |β2 .
(3)
αβ=0
Let |χ be the state obtained from |χ via the transformation (1). The state of the external degrees of freedom is, in general, mixed. The total state of the system can be described in terms of the density matrix σ (t0 ) ≡ ρ1 (t0 ) ⊗ ρ2 (t0 ) ⊗ |χχ|,
(4)
where ρ j (t0 ) is the external state of particle j at the initial time t0 . After the gate operation time τ , the system will be in some state σ (t0 + τ ). To evaluate the performance of a particular scheme, we have to compare the ideal gate transformation, as given by equation (1), with the one that can be achieved via the physical process actually realized in the laboratory. The figure of merit is the minimum fidelity F, given by
F = min tr ext χ |σ (t0 + τ )|χ , χ
(5)
where tr ext denotes the trace over the external degrees of freedom. Ideally, to achieve the optimal fidelity F = 1, we would like the external degrees of freedom to factorize after the gate operation, and the evolution to only have the effect of inducing a nontrivial two-particle phase, independent of the external state. Of course, in a real situation, not all these conditions can be perfectly satisfied. On one hand, the total density matrix after gate operation will not, in general, have a factorized form like σ (t0 ), bearing a certain amount of undesired entanglement between the internal and the external degrees of freedom. On the other hand, the starting state will correspond to a thermal distribution over the external energy eigenstates: ρ j (t0 ) ∝ e−H (t0 )/k B T . (6) In other words, at nonzero temperatures there will be a finite probability that each particle starts in an excited motional state, leading, in general, to different phases, which cannot be experimentally controlled and easily undone by single-qubit rotations. In this case the fidelity is expected to decrease with temperature, at a rate which depends on the different implementation schemes.
(8)
t0
In a real situation, as stated in the previous section, one has to also take into account external degrees of freedom. This is described by a Hamiltonian of the form 1 α,β=0
1
αβ
HT (t, x1 , x2 ) |α1 α| ⊗ |β2 β|,
α,β=0
HI (t, x1 , x2 ) =
1
αβ
HI (t, x1 , x2 ) |α1 α| ⊗ |β2 β|,
α,β=0
(10) and αβ
HT (t, x1 , x2 ) =
2 p2i + V α (t, x1 )|α1 α| 2m i=1
+ V β (t, x2 )|β2 β|.
(11)
αβ
The form of HI depends on the particular physical system chosen for the implementation, and will be specified below in each case. The state-dependent phase shift ϕ αβ , which is obtained after the evolution dictated by the Hamiltonian (9), is the sum of two kinds of contribution—the single-particle kinematic phases ϕ α + ϕ β due to the kinetic energy of the particles, and an interaction phase ϕ αβ due to coherent interactions between the two particles. The kinematic phases can be calculated by ignoring the effects of interactions as ϕ α + ϕ β = arg[ αβ |UT | αβ ],
(12)
t +τ t +τ where U ≡ exp{− h¯i t00 H (t ) dt }, UT ≡ exp{− h¯i t00 HT (t ) dt }, and | αβ ≡ |α|β|ψ αβ is the initial twoparticle state. The interaction phase is responsible for the entanglement, and is given by ϕ αβ = arg[ αβ |U † UT | αβ ].
(13)
The latter can also be evaluated in terms of the conditional energy shift arising from the interaction (14)
If the condition
In this section we want to describe the physical processes that lead to the transformation (1). The latter could be accomplished by means of a state-dependent interaction of the form Hint = E(t)|11 1| ⊗ |12 1|, (7)
H (t, x1 , x2 ) =
HT (t, x1 , x2 ) =
E αβ (t) ≡ αβ |UT† HI UT | αβ .
2. Dynamical quantum gates based on two-body interactions
acting over a time τ such that t0 +τ
E(t ) dt = ϕ.
where x j denotes the external degrees of freedom of particle j , and the explicit time dependence indicates that we can switch on and off a suitable interaction in order to obtain the desired effect. The Hamiltonian is the sum of a trapping part HT plus an interaction part HI , where
H αβ (t, x1 , x2 ) |α1 α| ⊗ |β2 β|, (9)
E αβ (t) h¯ ω
(15)
is satisfied, with h¯ ω the first excitation energy of the system, then a perturbative expression for the interaction phase can be obtained: 1 t0 +τ ϕ αβ ≈ dt E αβ (t ). (16) h¯ t0 In order for the interaction phase to have a nontrivial dependence on α and β, state selectivity is required either in the trapping or in the interaction Hamiltonian, and not necessarily in both. We will first examine schemes based on state-dependent trapping potentials, with different kinds of position-dependent interactions, and move later on to a scheme based on a state-selective interaction, with possible realizations in different physical systems. S431
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3. State-selective trapping potentials Let us assume, for the moment, that the interaction Hamiltonian only depends on the positions of the two particles, αβ i.e. HI (t, x1 , x2 ) ≡ HI (x1 , x2 ), while the external force couples in a different way to different internal states of the particles to be trapped. Initially, at t = t0 , the two particles are in the ground states of two separate trapping potential wells, whose centres are sufficiently far apart so that the particles’ wavefunctions do not overlap. Then the form of the potential wells is changed such that the wavefunctions of the two atoms are displaced in a different way according to their internal state; the particles interact with each other, and then (after a certain time τ ) the potential is restored to the original situation. The goal is to minimize the perturbation in the motional state due to the time evolution within the modified trap. We will analyse two opposing regimes in which this can be achieved, namely the cases of very slow (adiabatic) and of very fast (sudden switching) changing potentials. 3.1. Moving potentials One way of controlling the interaction between the particles is to move the centre position of the potentials towards each other in a state-dependent way while leaving the shape of the potential unchanged. Let us assume that we can design the trapping potentials such that the particle 1 (2) experiences a potential which is initially centred at position −x0 (x0 ). We assume that we can move the centres of each well, according to the displacement trajectory δ x(t), only if the particle sitting there is in internal state |1: 2 mω2 x1 + x0 − αδ x(t) ; 2 (17) 2 mω2 β V (t, x2 ) = x2 − x0 − βδ x(t) . 2 The trajectory δ x(t) is chosen along the direction x0 of the separation between the two traps, and in such a way that δ x(t0 ) = δ x(t0 + τ ) = 0 (see figure 1). The maximum displacement ≡ maxt |δ x(t)| needed for a gate operation will be d ≡ 2|x0 | in the case of the contact interaction represented by a collision, where the atoms have to overlap on the same site in order for an useful energy shift to be obtained. A much smaller displacement will be needed in the case of the (long-range) electrostatic interaction between ions. The fact that x0 does not depend on the internal atomic state and the shape of the two potentials is the same at times t = t0 , t0 + τ allows one to easily change the internal state at those times by applying laser pulses. If the motion of the potential is slow enough such that the atoms always remain in the ground state, the kinetic phase can be easily calculated [4] as 2 mα t0 +τ d ϕα = dt (18) δx(t) , 2¯h t0 dt V α (t, x1 ) =
while the interaction phase takes the form 1 t0 +τ αβ dt dx HI (x1 , x2 ) ϕ =− h¯ t0 × |ψ0 (x1 + x0 − αδ x(t))|2 |ψ0 (x2 − x0 − βδ x(t))|2 . (19) S432
Figure 1. State-selective moving potentials needed for realizing a quantum gate via collisions between neutral atoms (left) and via electrostatic interaction between ions. Curves depict schematically the variation with time of the potential minima for the two-qubit states. (This figure is in colour only in the electronic version)
3.1.1. Cold collisions between neutral atoms in optical lattices. The interaction between two neutral atoms undergoing an elastic s-wave collision can be described by the contact pseudopotential αβ
HI (t, x1 , x2 ) =
αβ
4πas h¯ 2 3 δ (x1 − x2 ), m
(20)
αβ
where as is the s-wave scattering length for the internal states α and β. In an external potential moving as described in equation (17), this interaction gives rise to a two-particle phase 8π¯h ω 01 t0 +τ 2 αβ dt e−2(mω/¯h )[δx(t)−d/2] . (21) ϕ ≈− as m t0 To evaluate the performance of a quantum phase gate based on this effect, we focus now on a specific implementation. We consider first that the atoms are trapped in an optical lattice (one can similarly consider trapping in other optical fields: see, for example, [5]). In order to perform gate operations we have to be able to selectively fill the lattice sites with exactly one particle. This can be achieved in principle by making use of the phase transition from a superfluid BEC phase to a Mott insulator (MI) phase at low temperatures, which has been predicted in the context of the Bose–Hubbard model [6] and recently observed experimentally [7]. Now we show how to move the optical potentials [4, 8]. We consider the example of alkali atoms with a nuclear spin equal to 3/2 (87 Rb, 23 Na) trapped by standing waves in three dimensions. The internal states of interest are hyperfine levels corresponding to the ground state S1/2 . Along the z axis, the standing waves are in the lin lin configuration (two linearly polarized counter-propagating travelling waves with the electric fields E 1 and E 2 forming an angle 2θ ). The total electric field is a superposition of right and left circularly polarized standing waves (σ ± ) which can be shifted with respect to each other by changing θ . The lasers are tuned in such a way that the polarizabilities α±± due to σ ± are identical (α++ = α−− ≡ α) [4]. We choose for the states |0 and |1 the hyperfine structure states |0 ≡ |F = 1, m f = 1 and |1 ≡ |F = 2, m f = 2 of 87 Rb. Due to angular momentum conservation, these states are stable under collisions. More αβ precisely, for such states always as ≈ 5.1 nm independently
Controlling dynamical phases in quantum optics
1-F
F 5
5
Figure 2. Fidelity F against temperature kT /¯h ω for gate operation in a moving optical lattice with 87 Rb, having as = 5.1 nm. Here ω = 2π × 100 kHz, d = 390 nm, and the gate operation is performed in about 1 ms.
of the combination of internal states α and β. Thus, even though the collisional interaction does in principle depend upon the atomic internal state, for the particular species we chose here this is actually not the case, as anticipated at the beginning of this section. By varying the angle θ , the corresponding potentials are displaced with respect to each other, and a dynamics of the kind described by equation (17) is obtained. The optical lattice potential has actually a sinusoidal shape, which moreover changes as it moves. These features have been taken into account in the calculations made in [4], resulting in the fidelity plotted in figure 2. 3.1.2. Electrostatic force between ions in microscopic traps. In this section we consider the conditional dynamics for two charged particles, trapped in separate harmonic wells [9], interacting via electrostatic repulsion and under the influence of an external state-dependent force, which can be generated, for example, by an off-resonant laser standing wave [2], or alternatively by the same method as described in the previous section. The interaction Hamiltonian is again state independent: 1 q2 αβ . HI (t, x1 , x2 ) ≡ e 4πε0 x1 − x2
(22)
√ We assume the trapping ground-state width a0 ≡ h¯ /(mω) to be much smaller than the trap separation d. As anticipated above, due to the strength and the long-range nature of the Coulomb interaction, the maximum trap displacement here can be much smaller than in the collisional case, i.e. d. Under these assumptions, the quantum evolution can be calculated analytically in a full three-dimensional treatment. If the gate operation takes place adiabatically, i.e. for |δ x˙ | a0 ω, the obtained gate phase is [3] ϕ≈√
qe2 τ 2 8π h¯ ε0 d 3
.
(23)
To second order in a0 /d, ϕ is even independent of the motional state of the ions in the trap. This means that the fidelity does not depend on the ions’ temperature. A weak temperature dependence shows up when higher-order contributions in a
10
15
20
Figure 3. Fidelity for a phase gate operating over τ ≈ 40 µs with ions in microtraps, without (solid curve) and with (dashed line) the symmetrization procedure described in the text. We chose to work with Ca+ ions and chose the parameters ω = 2π × 1 MHz, d = 20 µm. Inset: detail of the departure from unity of the same quantities, on a logarithmic scale.
multipole expansion of the interaction term are taken into account. However, the odd contributions can be compensated for via a simple symmetrization procedure [3]. Indeed, to all orders in a0 /d, the interaction phase vanishes when both ions are in the same internal state, since in this case their mutual distance is not varying. On the other hand, odd multipoles correspond to odd powers of the interparticle distance, and therefore give an opposite contribution to ϕ 01 (i.e. in the case when the particles get farther apart during gate operation) than they do to ϕ 10 (when the particles get closer). Thus, as is shown in detail in [3], a simple qubit swap in the middle of the gate operation has the effect of undoing odd-multipole contributions, leaving us with fourth-order corrections in a0 /d to the fidelity:
F(T ) ≈ 1 − 6
3
πk B T h¯ ω
2
a0 d
4 .
(24)
In figure 3 we show F(T ) as evaluated in a classical model [3], which reproduces the behaviour of equation (24) but allows as well for an analytic calculation in the case of no symmetrization, which is also shown for comparison. 3.2. Switching potentials The interaction between the particles can also be controlled in other ways: for example by changing with time the shape of the potentials depending on the particles’ internal states. We will consider raising and lowering a potential barrier between the two particles, in such a way as to leave them free to interact for a definite time if they are in certain internal states. This model is best suited for application with the collisional interaction (20). The potential is initially composed of two separated wells. Ideally, the atoms have been cooled to the vibrational ground states of the two wells. At t = t0 , the barrier between the wells is suddenly removed in a selective way for atoms in internal state |0: an atom in state |1 feels no change, whereas one in state |0 finds itself in a new harmonic potential, centred on x = 0 with frequency ω < ω0 . The atoms S433
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are allowed to oscillate for some time, and then at t = t0 + τ — where τ is a multiple of Tosc , the oscillation period in the well of unperturbed frequency ω—the barrier is suddenly raised again to trap them at the original positions. We can model the situation with the following potential: V α (t, x) = v α (t, x) + v⊥ (y) + v⊥ (z),
(25)
with mω02 (x)(x − x0 )2 + (−x)(x + x0 )2 ; (26) 2 1 t < t0 , t > t 0 + τ ; v (t, x) v 0 (t, x) = mω2 (27) x2 t 0 t t0 + τ . 2
v 1 (t, x) =
v⊥ (y) =
2 mω⊥
y 2. (28) 2 Here, (x) denotes the step function and the confinement is assumed to be much stronger in the directions perpendicular to the trap separation than along this one. The collisional phase can be evaluated perturbatively under the assumption (15). If, moreover, the maximum velocity that atoms reach at the centre of the well v 0 (t0 < t < t0 + τ ) is large with respect to the analogous quantity for the ground-state motion in the wells v 1 , i.e. if x0 ω a0 ω0 /4, then the velocity of each particle and the shape of its wavefunction do not vary during the interaction. In this case the time and space integrations in equation (19) can be interchanged [10], and the latter factors out, yielding simply the normalization of the motional eigenfunctions. As long as the above conditions are satisfied, the following simple expression for the gate phase can be found: ϕ(t0 + 2nπ/ω) ≈
4nas11 ω⊥ x02 ω2 − h¯ ω/(4m)
,
(29)
which is independent of the motional eigenstate of the particle inside the trap. Therefore, the fidelity is expected to decrease with temperature at a smaller rate in this switching scheme than it does in the moving scheme, where no such independence of the phase from the motional state can be found, even for the lowest eigenstates. 3.2.1. Neutral atoms in magnetic microtraps. We now consider the implementation of a switching potential by means of static electric and magnetic trapping forces. The interplay between the state-dependent magnetic term Vmagn ∝ m F |B | (with |F, m F being the internal state of an atom in a magnetic field B ) and the state-independent electric one Vel ∝ |E |2 can be exploited in order to obtain a trapping potential whose shape depends on the internal state of the atoms [10]. As an example, we consider an atomic mirror in an external bias field, generating a field whose modulus has minima along x and z, where atoms can be trapped. The spacing between two nearest minima along x is of the order of the mirror’s magnetization period, which can be as small as 1 µm with the system studied in [11], or even close to 100 nm using existing magnetic storage technologies. With present-day technology, trapping frequencies can range from a few tens of kHz up to S434
Figure 4. Fidelity for gate operation with neutral atoms colliding in magnetic microtraps. We choose ω = 2π 17.23 kHz and ω⊥ = 2π 150 kHz, corresponding to ground-state widths a0 ≈ 82 nm, a⊥ ≈ 28 nm, with the initial wells having frequency ω0 = 2ω and displaced by d = 10a0 . The gate operation time is τ ≈ 0.7 ms.
some MHz. Microscopic electrodes can be nanofabricated on the mirror’s surface [12] below each pair. Charging them produces an attractive potential for atoms, giving confinement along the y direction. The barrier between two nearest wells can be lowered by increasing the charge on the corresponding electrode. Note that |1 has an interaction with the magnetic field twice as high as |0. Therefore, the charge can be adjusted in such a way that the barrier is removed just for atoms in internal state |0, but still remains in place for atoms in |1. In this way it is possible to design a potential with the characteristics described above. Furthermore, we assume that the potential minimum for state |1 is displaced along the transverse direction from the one for state |0 by means of an additional electrostatic field [11, 12], so that the atoms interact if and only if they are both in state |0. This is meant to avoid energy exchange via collisions between atoms in different motional states [10].
4. State-selective interaction potential An alternative possibility, for a nontrivial logical phase to be obtained, is to rely on a state-independent trapping potential, while defining a procedure where different logical states couple to each other with different energies. A good example is given by the interaction between state-selectively switched electrical dipoles [13]. 4.1. Dipole–dipole interactions In this scheme we store qubits in two internal ground states: for example, atomic hyperfine levels or spin states of an excess electron in a quantum dot (QD). In each system, the ground state |1 is coupled by a laser to a given Stark eigenstate |r . The internal dynamics is described by a model effective nonHermitian Hamiltonian HI (t, x1 , x2 ) = (δ j (t) − iγ )|r j r | j =1,2
j (t, x j ) − (|1 j r | + h.c.) + u|r 1 r | ⊗ |r 2 r |, 2
(30)
Controlling dynamical phases in quantum optics
with j (t, x j ) Rabi frequencies, and δ j (t) detunings of the exciting lasers. Here, u is the dipole–dipole interaction energy between the two particles and γ accounts for loss from the excited states |r j . We will discuss two possible physical realizations of this dynamics. The most straightforward way to implement a two-qubit gate is to just switch on the dipole– dipole interaction by exciting each qubit to the auxiliary state |r , conditioned on the initial logical state. This can be obtained by two resonant (δ1 = δ2 = 0) laser fields of the same intensity, corresponding to a Rabi frequency 1 = 2 u. After a time τ = ϕ/u, the gate phase ϕ is accumulated and the particles can be taken again to the initial internal state. However, besides ϕ being sensitive to the atomic distance via the energy shift u, during the gate operation (i.e. when the state |rr is occupied) there are large mechanical effects, due to the dipole–dipole force, which create unwanted entanglement between the internal and the external degrees of freedom. These problems can be overcome by assuming single-qubit addressability and by moving to the opposite regime of small Rabi frequencies 1 (t) = 2 (t) u. The gate operation is then performed in three steps, by applying (i) a π-pulse to the first atom, (ii) a 2π-pulse (in terms of the unperturbed states) to the second atom, and, finally, (iii) a π-pulse to the first atom. The state |00 is not affected by the laser pulses. If the system is initially in one of the states |01 or |10 the pulse sequence (i)–(iii) will cause a sign change in the wavefunction. If the system is initially in the state |11 the first pulse will bring the system to the state i|r 1, the second pulse will be detuned from the state |rr by the interaction strength u, and thus accumulate a small phase ϕ˜ ≈ π2 /2u π. The third pulse returns the ˜ |11, which realizes a phase gate system to the state ei(π −ϕ) with ϕ = π − ϕ˜ ≈ π (up to trivial single-qubit phases). The time needed to perform the gate operation is of the order of τ ≈ 2π/ 1 +2π/ 2 . Loss from the excited states |r j is small provided γ t 1, i.e. j γ . A further improvement is possible by adopting chirped laser pulses with detunings δ1,2 (t) ≡ δ(t) and adiabatic pulses 1,2 (t) ≡ (t), i.e. with a time variation slow on the timescale given by and δ (but still larger than the trap oscillation frequency), so that the system adiabatically follows the dressed states of the Hamiltonian HI . As found in [13], in this adiabatic scheme the gate phase is t0 +τ ˜ − δ˜2 + 22 |δ| ˜ dt sgn(δ) ϕ(τ ) = 2 t0 √ (31) − sgn(δ) |δ| − δ 2 + 2 with δ˜ = δ − 2 /(4δ + 2u) the detuning including a Stark shift. For a specific choice of pulse duration and shape (t) and δ(t) we achieve ϕ(τ ) = π. To satisfy the adiabatic condition, the gate operation time τ has to be approximately one order of magnitude longer than in the other scheme discussed above. In the ideal limit j u, the dipole–dipole interaction energy shifts the doubly excited state |rr away from resonance. In such a ‘dipole-blockade’ regime, this state is therefore never populated during gate operation. Hence, the mechanical effects due to atom–atom interaction are greatly suppressed.
Furthermore, this version of the gate is only weakly sensitive to the exact distance between the atoms, since the distancedependent part of the entanglement phase ϕ˜ π. For the same reason, possible excitations in the particles’ motion do not alter significantly the gate phase, leading to a very weak temperature dependence of the fidelity. These features allow one to design robust quantum gates with atoms in lattices that are not filled regularly. 4.1.1. Rydberg-excited neutral atoms. Rydberg states [14] of a hydrogen atom within a given manifold of a fixed principal quantum number n are degenerate. This degeneracy is removed by applying a constant electric field E along the z-axis (linear Stark effect). Consider two atoms, initially prepared in Stark eigenstates, with a dipole moment along z and a given m, as selected by the polarization of the laser exciting the Rydberg states from the ground state. We are interested in the limit where the electric field is sufficiently large so that the energy splitting between two adjacent Stark states is much larger than the dipole–dipole interaction. We will assume that transitions between adjacent m manifolds are suppressed by an appropriate choice of the initial Stark eigenstate [13]. As an illustration we choose the hydrogen state |n, q = n − 1, m = 0 and find that the interaction energy at a fixed distance d of two atoms in such a state is u = −9[n(n − 1)]2 (a0 /d)3 (e2 /8π0 a0 ) ∝ n 4 (in alkali atoms we have to replace n by the effective quantum number ν [14]). By identifying this state with the auxiliary state |r of the model Hamiltonian (30), we can implement the dynamics described in the previous section. For instance (see [13]), the gate phase (1) can be realized in Na atoms for ν = 18, d = 300 nm with an applied electric field of 100 V cm−1 , and a laser intensity of approximately 40 kW cm−2 . A detailed analysis of various imperfections, including decoherence mechanisms, is given in [13]. 4.1.2. Atomic ensembles as qubits via dipole blockade. The dipole-blockade mechanism described above can be used for coherent manipulation and entanglement of collective excitations in mesoscopic ensembles of cold atoms [15]. This is accomplished by optically exciting the ensemble into states with a strong dipole–dipole interaction. Under certain conditions the level shifts associated with these interactions can be used to block the transitions into states with more than a single excitation. The resulting ‘dipole-blockade’ phenomenon can take place in an ensemble with a size that can exceed many optical wavelengths. This allows one to considerably alleviate many stringent requirements for the experimental implementation of various quantum processing protocols. 4.1.3. Excitons in charged quantum dots. We would like to mention here another possible realization of a quantum gate based on dipole–dipole interaction in a completely different context. We consider an excess conduction electron in a semiconductor QD, and take its spin as the qubit. Single-qubit operations can be done optically [16, 17] or on a picosecond timescale by employing g-factor modulated materials or timedependent magnetic fields [18]. The dynamics required S435
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to perform the gate operation exploits a Pauli-blocking mechanism [19]. We assume that the QDs can be individually addressed via laser excitation, using, for example, near-field techniques [16] or energy-selective addressing of QDs [20]. The control of the phase accumulated by Coulomb interactions is obtained by shining a σ + polarized laser pulse on the QD: due to the Pauli exclusion principle a |M Je = −1/2, M Jh = +3/2 electron–heavy hole pair is created only if the excess electron, already present in the QD, has a spin projection 1/2. This allows us on one hand to perform gate operations on the timescale (of the order of picoseconds) given by the biexcitonic interaction, while annihilating the excitons right after the gate operation and, on the other hand, to take advantage of the comparatively long (µs) coherence times of electron spins in semiconductors.
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