23.3.1.1 Optical Potentials

The Hamiltonian of an atom of mass m is given by H_A  = p ² /2m +  . Here p is the center of mass momentum operator and |e_j 〉 denotes the internal atomic states with energies ω_j (setting ℏ ≡ 1). We assume the atom to initially occupy a metastable internal state |e ₀〉 ≡ |a〉 that defines the point of zero energy. The atom is subject to a classical laser field with electric field E(x ,t) = E(x , t)ɛ exp(−iωt), where ω is the frequency and ɛ the polarization vector of the laser. The amplitude of the electric field E(x , t) is varying slowly in time t compared to 1/ω and slowly in space x compared to the size of the atom. In this situation, the interaction between atom and laser is adequately described in dipole approximation by the Hamiltonian H _dip = −μ E(x , t) + h .c ., where μ is the dipole operator of the atom. We assume the laser to be far detuned from any optical transition so that no significant population is transferred from |a〉 to any of the other internal atomic states via H _dip. We can thus treat the additional atomic levels in perturbation theory and eliminate them from the dynamics. In doing so, we find the AC‐Stark shift of the internal state |a〉 in the form of a conservative potential V(x) whose strength is determined by the atomic dipole operator and the properties of the laser light at the center of mass position x of the atom. In particular, V(x) is proportional to the laser intensity |E(x , t)| ². Under these conditions, the motion of the atom is governed by the Hamiltonian H = p ²/2m + V(x).

Let us now specialize the situation to the case where the dominant contribution to the optical potential arises from one excited atomic level |e〉 only. In a frame rotating with the laser frequency the Hamiltonian of the atom is approximately given by H_A  = p ²/2m + δ |e〉 〈e|, where δ = ω_e − ω is the detuning of the laser from the atomic transition |e〉 ↔ |a〉. The dominant contribution to the atom–laser interaction neglecting all quickly oscillating terms (i.e., in the rotating wave approximation) is given by H _dip = Ω(x) |e〉 〈a|/2 + h .c. Here Ω = −2E(x , t)〈e|μɛ|a〉 is the so‐called Rabi frequency driving the transitions between the two atomic levels. For large detuning δ ≫ Ω adiabatically eliminating the level |e〉 yields the explicit expression V(x) = |Ω(x)| ²/4δ for the optical potential. The population transferred to the excited level |e〉 by the laser is given by |Ω(x)| ²/4δ ², and this is the reason why we require Ω(x) ≪ δ for our adiabatic elimination to be valid.

23.3.1.2 Periodic Lattices

For creating an optical lattice potential, we start by superimposing two counter‐propagating running‐wave laser beams with E_± (x, t) = E ₀ exp(±ikx) propagating in the x‐direction with amplitude E ₀, wave number k and wavelength λ = 2π/k. They create an optical potential V(x) ∝ cos²(kx) in one dimension with periodicity a = λ/2. Using two further pairs of laser beams propagating in y‐ and z‐directions, respectively, a full three‐dimensional periodic trapping potential of the form

23.18

is realized. The depth of this lattice in each direction is determined by the intensity of the corresponding pair of laser beams that is easily controlled in an experiment.

23.3.1.3 Bloch Bands and Wannier Functions

For simplicity, we only consider one spatial dimension in Eq. 23.18 and write down the Bloch functions with q the quasi momentum and n the band index. The corresponding eigenenergies for different depths of the lattice V ₀ /E_R in units of the recoil energy E_R  = k ² /2m are shown in Figure 23.7. Already for a moderate lattice depth of a few recoil the energy separation between the lowest lying bands is much larger than their width. In this case, a good approximation for the gap between these bands is given by the oscillation frequency ω_T of a particle trapped close to one of the minima x_j (≡lattice site) of the optical potential. Approximating the lattice around a minimum by a harmonic oscillator we find ω_T = (29).

The dynamics of particles moving in the lowest‐lying, well‐separated bands will be described using the Wannier functions. These are complete sets of orthogonal normalized real‐mode functions for each band n. For properly chosen phases of the (x) the Wannier functions optimally localized at lattice site x_j are defined by (30)

23.19

where Θ is a normalization constant. Note that for V ₀ → ∞ and fixed k the Wannier function w_n (x) tends toward the wavefunction of the nth excited state of a harmonic oscillator with the ground state size a ₀ = . We will use the Wannier functions to describe particles trapped in the lattice since they allow (i) to attribute a mean position x_j to the particles in a given mode and (ii) to easily account for local interactions between particles because the dominant contribution to the interaction energy arises from particles occupying the same lattice site x_j .

23.3.1.4 Lattice Geometry and Site Offset

The lattice site positions x _j determine the lattice geometry. For instance, the above arrangement of three pairs of orthogonal laser beams leads to a simple cubic lattice (shown in Figure 23.8a). Since the laser setup is very versatile different lattice geometries can be achieved easily. As an example, consider three laser beams propagating at angles 2π/3 with respect to each other in the xy‐plane and all of them being polarized in the z direction. The resulting lattice potential is given by V(x) ∝ 3 + 4 cos(3kx/2) cos( ky/2) + 2 cos( ky), which is a triangular lattice in two dimensions. An additional pair of lasers in the z direction can be used to create localized lattice sites (cf. Figure 23.8b). Furthermore, as will be discussed later in Section 23.3.2, the motion of atoms can be restricted to two or even one spatial dimension by large laser intensities. As will be shown, it is thus possible to create truly one‐ and two‐dimensional lattice models as indicated in Figure 23.8b and c respectively. The offset of the lattice sites can be manipulated through a superimposed magnetic trapping field or Stark shifts introduced by additional lasers. In particular, regular patterns are useful for quantum computing as they exploit the inherent scalability of optical lattice setups.

23.3.1.5 State‐Dependent Lattices

As already mentioned above, the strength of the optical potential crucially depends on the atomic dipole moment between the internal states involved. Thus, we can exploit selection rules for optical transitions to create differing traps for different internal states of the atom (31–33). We will illustrate this by an example that is particularly relevant in what follows. We consider an atom with the fine structure shown in Figure 23.9a, like for example, ²³Na or ⁸⁷Rb, interacting with two circularly polarized laser beams. The right circularly polarized laser σ ⁺ couples the level S _1/2 with m_s  = −1/2 to two excited levels P _1/2 and P _3/2 with m_s  = 1/2 and detunings of opposite sign. The respective optical potentials add up. The strength of the resulting AC‐Stark shift is shown in Figure 23.9b as a function of the laser frequency ω. For ω = ω_L, the two contributions cancel. The same can be achieved for the σ⁻ laser acting on the S _1/2 level with m_s  = 1/2. Therefore, at ω = ω_L the AC‐Stark shift of the levels S _1/2 with m_s  = ±1/2 are purely due to σ_± polarized light that we denote by V_± (x). The corresponding level shifts of the hyperfine states in the S _1/2 manifold (shown in Figure 23.9a) are related to V_± (x) by the Clebsch–Gordan coefficients, for example, (x) = V ₊(x), (x) = 3V ₊(x)/4 + V₋ (x)/4, and (x) = V ₊(x)/4 + 3V₋ (x)/4.

23.3.1.6 State Selectively Moving the Lattice

The two standing waves σ_± can be produced out of two running counter‐propagating waves with the same intensity as shown in Figure 23.10. Moreover, it is possible to move nodes of the resulting standing waves by changing the angle of the polarization between the two running waves (31). Let {e ₁, e ₂, e ₃} be three unit vectors in space pointing along the {x, y, z} direction, respectively. The position‐dependent part of the electric field of the two running waves E _1,2 is given by E ₁ ∝ e^ikx (cos(ϕ)e ₃ + sin(ϕ)e ₂), E ₂ ∝ e^−ikx (cos(ϕ)e ₃ − sin(ϕ)e ₂). The sum of the two electric fields is thus E ₁ + E ₂ ∝ cos(kx − ϕ)σ₋ − cos(kx + ϕ)σ ₊, where σ_±  = e ₂ ± ie ₃ and the resulting optical potentials are given by

23.20

By changing the angle ϕ, it is therefore possible to move the nodes of the two standing waves in the opposite directions. Since these two standing waves act as internal state‐dependent potentials for the hyperfine states the optical lattice can be moved in the opposite directions for different internal hyperfine states.

23.3.1.7 Validity

In all of the above calculations, we have only considered the coherent interactions of an atom with laser light. Any incoherent scattering processes that lead to spontaneous emission were neglected. We will now establish the validity of this approximation by estimating the mean rate Γ_eff of spontaneous photon emission from an atom trapped in the lowest vibrational state of the optical lattice. This rate of spontaneous emission is given by the product of the life time Γ of the excited state and the probability of the atom occupying this state. In the case of a blue detuned optical lattice (dark optical lattice) δ < 0, the potential minima coincide with the points of no light intensity and we find the effective spontaneous emission rate Γ_eff ≈ −Γω_T /4δ. If the lattice is red detuned (bright optical lattice) δ > 0, the potential minima match the points of maximum light intensity and we find Γ_eff ≈ ΓV ₀/δ. Since V ₀ > ω_T the spontaneous emission in a red detuned optical lattice will always be more significant than in a blue lattice, however, as long as V ₀ ≪ δ spontaneous emission does not play a significant role.

23.3.1.8 Typical Numerical Values

In a typical blue detuned optical lattice, with λ = 514 nm for ²³Na atoms (S _1/2 − P _3/2 transition at λ ₂ = 589 nm and Γ = 2π × 10 MHz), the recoil energy is E_R ≈ 2π × 33 kHz and the detuning from the atomic resonance is δ ≈ −2 .3 × 10⁹ E_R . A lattice with depth V ₀ = 25E_R leads to a trapping frequency of ω_T  = 10E_R yielding a spontaneous emission rate of Γ_eff ≈ 10 ⁻² /s while experiments are typically carried out in times shorter than one second. Therefore, spontaneous emission does not play a significant role in such experiments.

23.3.2.1 The (Bose) Hubbard Model

The Hamiltonian of a weakly interacting gas in an optical lattice is

23.21

with the bosonic field operator for atoms in a given internal atomic state |b〉 and V_T (x) a (slowly varying compared to the optical lattice V ₀ (x)) external trapping potential, for example, a magnetic trap or a superlattice potential. The parameter g is the interaction strength between two atomic particles. Only if atoms interact via s‐wave scattering, it is given by g = 4πa_s/m with a_s the s‐wave scattering length. We assume all particles to be in the lowest band of the optical lattice and expand the field operator in terms of the Wannier functions , where is the destruction operator for a particle in site x _i . We find

where

and

The numerical values for the offsite interaction matrix elements U_ijkl involving Wannier functions centered at different lattice sites as well as tunneling matrix elements J_ij to sites other than nearest neighbors (note that diagonal tunneling is not allowed in a cubic lattice since the Wannier functions are orthogonal) are small compared to onsite interactions U ₀₀₀₀ ≡ U and nearest neighbor tunneling J ₀₁ ≡ J for reasonably deep lattices V ₀ ≳ 5E_R . We can therefore neglect them and for an isotropic cubic optical lattice arrive at the standard Bose–Hubbard Hamiltonian

23.22

Here 〈i, j〉 denotes the sum over nearest neighbors and the terms ɛ_j  = V _T(x _j ) arise from the additional trapping potential. The physics described by H _BH is schematically shown in Figure 23.11a. Particles gain an energy of J by hopping from one site to the next while two particles occupying the same lattice site provide an interaction energy U. An increase in the lattice depth V ₀ leads to higher barriers between the lattice sites decreasing the hopping energy J as shown in Figure 23.11b. At the same time, two particles occupying the same lattice site become more compressed, which increases their repulsive energy U (cf. Figure 23.11b).

23.3.2.2 Tunneling Term J

In the case of an ideal gas, where U = 0, the eigenstates of H _BH are easily found for ɛ_i  = 0 and periodic boundary conditions. From the eigenvalue equation  = −2J cos(qa), we find that 4J is the height of the lowest Bloch band. Furthermore, we see that the energy is minimized for q = 0 and therefore particles in the ground state are delocalized over the whole lattice, that is, the ground state of N particles in the lattice is |Ψ _SF 〉 ∝  |vac〉 with |vac〉 the vacuum state. In this limit, the system is SF and possesses first‐order, long‐range, off‐diagonal correlations (29,34).

23.3.2.3 Onsite Interaction U

In the opposite limit, where the interaction U dominates the hopping term J, the situation changes completely. As discussed in detail in (29,34), a quantum phase transition takes place at about U ≈ 5 .8zJ, where z is the number of nearest neighbors of each lattice site. The long‐range correlations cease to exist in the ground state and instead the system becomes Mott insulating (MI). For commensurate filling of one particle per lattice site, this MI state can be written as |Ψ _MI 〉 ∝ ∏ _j |vac〉. Using two internal long‐lived states of atoms in this MI state realizes a scalable quantum register with coherence times of the order of seconds.

23.3.3.1 Defect Suppressed Optical Lattices

One method to further decrease the number of defects in the lattice was recently described in (38). The proposed setup utilizes two optical lattices for internal states |a〉 and |b〉 with substantially different interaction strengths U_b  ≠ U_a and identical lattice site positions x _i . Initially, atoms in |a〉 are adiabatically loaded into the lattice and brought into a MI state, where the number of particles per site may vary between n = 1,…, n _max because of defects. Then, a Raman laser with a detuning varying slowly in time between δ_i and δ_f is used to adiabatically transfer exactly one particle from |a〉 to |b〉 as shown in Figure 23.12a and b. This is done by going through exactly one avoided crossing. During the whole of this process, transfer of further atoms is blocked by interactions. This scheme allows a significant suppression of defects and – with additional site offsets ɛ_i – can also be used for patterned loading (37) of the |b〉 lattice.

23.3.3.2 Irreversible Loading Schemes

Further improvement could be achieved via irreversible loading schemes. An optical lattice is immersed in an ultracold degenerate gas from which atoms are transferred into the first Bloch band of the lattice. By spontaneous emission of a phonon (39), the atom then decays into the lowest Bloch band as schematically shown in Figure 23.12. By atom–atom repulsion, further atoms are blocked from being loaded into the lattice. This scheme can also be extended to cool atomic patterns in a lattice. In contrast to adiabatic loading schemes, it has the advantage of being repeatable without removing atoms already stored in the lattice.

23.3.4.1 Basic Building Blocks of a Quantum Computer

The SF to MI transition is used to initialize the quantum register. In the MI state, each atom has a fixed position and the particle number fluctuations are very small. Using two internal states, each atom can thus be used to realize a qubit. Single‐qubit operations can be realized by laser interactions of atoms and light and controlled entanglement generation is achieved via interactions between atoms trapped in the lattice (31,32,40,41).

23.3.4.2 Single‐Qubit Gates

In principle, it is straightforward to induce single‐qubit gates by using Raman transitions between the two internal states |a〉 and |b〉. Raman transitions with Rabi frequency Ω _R and detuning δ are described by the Hamiltonian

23.23

which induce rotations of the qubit state on the Bloch sphere. The axis and angle of this rotation depend on the choice of laser parameters and can be chosen freely. The major problem in inducing single qubit operations is addressing of a single atom as it is difficult to focus a laser to spots of order of an optical wave length that is the typical separation between atoms in the lattice. Possible solutions to this difficulty are using schemes for pattern loading (29,42), where only specific lattice sites are filled with atoms or using additional marker atoms that specify the atom the laser is supposed to interact with.

MI atoms in an optical lattice have already experimentally been used as qubits, and it has been shown that they support multiparticle entangled states (43). The robustness of these qubits is, however, limited by stray magnetic fields and spin echo techniques need to be used to perform the experiments successfully. A different approach to obtaining robust quantum memory uses a more sophisticated encoding of the qubits (44). A 1D chain with an even number of atoms encodes a single qubit in the states |0〉 = |abab ⋯ ab〉 and |1〉 = |baba ⋯ ba〉. These states both contain the same number of atoms in internal states |a〉 and |b〉 and therefore interact with magnetic fields in exactly the same way avoiding dephasing of the quantum information stored in the chain. This and the fact that all the atoms have to flip their internal state to get from one logical state to the other make these qubits very robust against the most‐dominant experimental sources of decoherence. However, at the same time, this robustness makes it more difficult to manipulate them. A laser pulse no longer corresponds to a simple rotation on the Bloch sphere spanned by the logical states |0〉 and |1〉 that makes the realization of a single‐qubit gate difficult. We will describe how these problems can be circumvented later and further details can be found in (44).

23.3.4.3 Two‐Qubit Gates

Implementing a two‐qubit gate is more challenging than the single‐qubit gates. The different schemes for two‐qubit gates can be classified in two categories. The first version relies on the concept of a quantum data bus; the qubits are coupled to a collective auxiliary quantum mode, for example, a phonon mode in an ion trap, and entanglement is achieved by swapping the qubits to excitations of the collective mode. The second concept that is the basis for two‐qubit gates between atoms in optical lattices deploys controllable internal‐state dependent two‐body interactions. Examples for different interactions are coherent cold collisions of atoms, optical dipole–dipole interactions (31,32) and the “fast” two‐qubit gate based on large permanent dipole interactions between laser excited Rydberg atoms in static electric fields (40). Besides these dynamical schemes for entanglement creation, it is also possible to generate entanglement by purely geometrical means (45). We will now discuss the different ways to achieve two‐qubit operations in optical lattices.

23.3.4.4 Entanglement via Coherent Ground State Collisions

The interaction terms describing s‐wave collisions between ultracold atoms in one lattice site are analogous to Kerr nonlinearities between photons in quantum optics. For atoms stored in optical lattices these nonlinear atom–atom interactions can be large (31), even for interactions between individual pairs of atoms, thus providing the necessary ingredients to implement two‐qubit gates.

We consider a situation where two atoms in a superposition of internal states |a〉 and |b〉 are trapped in the ground states of two optical lattice sites (see Figure 23.13a). Initially, at time t = −τ, these wells are centered at positions sufficiently far apart so that the particles do not interact. The optical lattice potential is then moved state selectively and for simplicity we assume that only the potential for a particle in internal state |a〉 moves to the right and drags along an atom in state |a〉 while a particle in state |b〉 remains at rest. Thus, the wavefunction of each atom splits up in space according to the internal superposition of states |a〉 and |b〉. When the wavefunction of the left atom in state |a〉 reaches the second atom in state |b〉 as shown in Figure 23.13b, they will interact with each other. However, any other combination of internal states will not interact and therefore this collision is conditional on the internal state. A specific laser configuration achieving this state‐dependent atom transport has been analyzed in Ref. (31) for alkali atoms, based on tuning the laser between the fine structure excited p‐states. The trapping potentials can be moved by changing the laser parameters. Such trapping potentials could also be realized with magnetic and electric microtraps (46).

We therefore only need to consider the situation where atom 1 is in state |a〉 and particle 2 is in state |b〉 to analyze the interactions between the two atoms. The positions of the potentials are moved along the trajectories and  = const ., so that the wavepackets of atoms overlap for a certain time, until they are finally restored to the initial position at the final time t = τ. This situation is described by the Hamiltonian

23.24

Here, and are the position and momentum operators, V^a,b describe the displaced trap potentials and u^ab is the atom–atom interaction term (which leads to the interaction terms in the BHM). Ideally, we want to implement the transformation from before to after the collision,

23.25

where each atom remains in the ground state of its trapping potential and preserves its internal state. The phase φ = φ^a  + φ^b  + φ^ab will contain a contribution φ^ab from the interaction (collision) and (trivial) single‐particle kinematic phases φ^a and φ^b . The transformation Eq. 23.25 can be realized in the adiabatic limit, whereby we move the potentials slowly on the scale given by the trap frequency, so that atoms remain in their motional ground state. In this case, the collisional phase shift is given by φ^ab  =  ΔE(t)/ℏ, where ΔE(t) is the energy shift induced by the atom–atom interactions

23.26

with a_s the s‐wave scattering length. In addition, we assume that |ΔE(t)| ≪ ℏω_T so that no sloshing motion is excited.

The interaction phase thus only applies when atoms are in internal state |a, b〉 but not otherwise. Carrying out the above state‐selective collision with a phase φ^ab  = π, we obtain (up to trivial phases) the mapping (as above we identify |a〉 ≡ |0〉 and |b〉 ≡ |1〉)

23.27

which realizes a two‐qubit phase gate that is universal in combination with single‐qubit rotations.

23.3.4.5 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. An example is given by the interaction between state‐selectively switched electrical dipoles (40).

In each qubit, the hyperfine ground states |a〉 ≡ |0〉 is coupled by a laser to a given Stark eigenstate |r〉 that does not correspond to a logical state. The internal dynamics is described by a model Hamiltonian

23.28

with Ω _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. We have neglected any loss from the excited states |r〉 _j . We discuss two possible realizations of two‐qubit gates with 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 (δ ₁ = δ ₂ = 0) laser fields of the same intensity, corresponding to a Rabi frequency Ω₁ = Ω₂ ≫ 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 Ω₁(t) ≠ Ω₂(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|r1〉, the second pulse will be detuned from the state |rr〉 by the interaction strength u, and thus accumulate a small phase ≈ πΩ₂/2u ≪ π. The third pulse returns the system to the state , which realizes a phase gate with ϕ = π −   ≈ π (up to trivial single‐qubit phases). The time needed to perform the gate operation is of the order τ ≈ 2π/Ω₁ + 2π/Ω₂. Loss from the excited states |r〉 _j is small provided γΔt ≪ 1, that is, Ω _j  ≫ γ. A further improvement is possible by adopting chirped laser pulses with detunings δ _1,2(t) ≡ δ(t) and adiabatic pulses Ω_1,2(t) ≡ Ω(t), that is, with a time variation slow on the time scale given by Ω and δ (but still larger than the trap oscillation frequency), so that the system adiabatically follows the dressed states of the Hamiltonian H_I . As found in (40), in this adiabatic scheme the gate phase is

23.29

with  = δ − Ω² /(4δ + 2u) the detuning including a Stark shift. For a specific choice of pulse duration and shape Ω(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 Ω _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 atoms, since the distance‐dependent part of the entanglement phase is  ≪ 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.

23.3.4.6 Universal Quantum Simulators

Building a general purpose quantum computer, which is able to run, for example, Shor's algorithm, requires quantum resources that will only be available in the long term future. Thus, it is important to identify nontrivial applications for quantum computers with limited resources, which are available in the lab at present. Such an example is provided by Feynman's universal quantum simulator (UQS) (47). A UQS is a controlled device that, operating itself on the quantum level, efficiently reproduces the dynamics of any other many‐particle system that evolves according to short‐range interactions. Consequently, a UQS could be used to efficiently simulate the dynamics of a generic many‐body system, and in this way function as a fundamental tool for research in many‐body physics, for example, to simulate spin systems.

According to Jane et al. (48), the very nature of the Hamiltonian available in quantum optical systems makes them best suited for simulating the evolution of systems whose building blocks are also two‐level atoms, and having a Hamiltonian H_N  = ∑ _a H ^(a) + ∑ _a≠b H^(ab) that decomposes into one‐qubit terms H ^(a) and two‐qubit terms H ^(ab). A starting observation concerning the simulation of quantum dynamics is that if a Hamiltonian decomposes into terms K_j acting in a small constant subspace, then by the Trotter formula e^−iKτ  = lim _m→∞ , we can approximate an evolution according to K by a series of short evolutions according to the pieces K_j . Therefore, we can simulate the evolution of an N‐qubit system according to the Hamiltonian H_N by composing short one‐qubit and two‐qubit evolutions generated, respectively, by H ^(a) and H ^(ab). In quantum optics, an evolution according to one‐qubit Hamiltonians H ^(a) can be obtained directly by properly shining a laser beam on atoms or ions that host the qubits. Instead, two‐qubit Hamiltonians are achieved by processing some given interaction (see the example below) that is externally enforced in the following way. Let us consider two of the N qubits, that we denote by a and b. By alternating evolutions according to some available, switchable two‐qubit interaction for some time with local unitary transformations, one can achieve an evolution

where with u_j and v_j being one‐qubit unitaries. For a small time interval U(t) ≃ 1 − it with p_j  = t_j /t, so that by concatenating several short gates U(t), U(t) = exp  + O(t ²),we can simulate the Hamiltonian

for larger times. Note that the systems can be classified according to the availability of homogeneous manipulation, u_j  = v_j , or the availability of local individual addressing of the qubits, u_j  ≠ v_j .

Cold atoms in optical lattices provide an example where single atoms can be loaded with high fidelity into each lattice site, and where cold‐controlled collisions provide a way of entangling these atoms in a highly parallel way. This assumes that atoms have two internal (ground) states |0〉 ≡ |↓〉 and |1〉 ≡ |↑〉 representing a qubit, and that we have two spin‐dependent lattices, one trapping the |0〉 state, and the second supporting the |1〉 state. An interaction between adjacent qubits is achieved by displacing one of the lattices with respect to the other as discussed in the previous subsection. In this way, the |0〉 component of the atom a approaches in space the |1〉 component of the atom a + 1, and these collide in a controlled way. Then, the two components of each atom are brought back together. This provides an example of implementing an Ising ∑ _a≠b  = ∑ _a  ⊗ σ_z ^(a+1) interaction between the qubits, where the σ ^(a)′s denote Pauli matrices. By a sufficiently large, relative displacement of the two lattices, also interactions between more distant qubits could be achieved. A local unitary transformation can be enforced by shining a laser on the atoms, inducing an arbitrary rotation between |0〉 and |1〉. On the time scale of the collisions requiring a displacement of the lattice (the entanglement operation), these local operations can be assumed instantaneous. In the optical lattice example, it is difficult to achieve an individual addressing of the qubits. Such an addressing would be available in an ion‐trap array as discussed in Ref. (49). These operations provide us with the building blocks to obtain an effective Hamiltonian evolution by time averaging as outlined above.

As an example, let us consider the ferromagnetic [antiferromagnetic] Heisenberg Hamiltonian

23.30

where J > 0 [J < 0]. An evolution can be simulated by short gates with = γσ_z  ⊗ σ_z alternated with local unitary operations

without local addressing, as provided by the standard optical lattice setup. The ability to perform independent operations on each of the qubits would translate into the possibility to simulate any bipartite Hamiltonians.

An interesting aspect is the possibility to simulate effectively different lattice configurations: for example, in a 2D pattern a system with nearest neighbor interactions in a triangular configuration can be obtained from a rectangular array configuration. This is achieved by making the subsystems in the rectangular array interact not only with their nearest neighbor but also with two of their next‐to‐nearest neighbors in the same diagonal (see Figure 23.14).

One of the first and most interesting applications of quantum simulations is the study of quantum phase transitions (50). In this case, one would obtain the ground state of a system, adiabatically connecting ground states of systems in different regimes of coupling parameters, allowing to determine its properties.

23.3.4.7 Multiparticle Maximally Entangled States in Optical Lattices

Let us assume that the Rydberg interactions between neighboring atoms are turned on all the time and that atoms are arranged in a 1D chain. Furthermore, we allow each atom to either tunnel between two adjacent wells whose ground states are denoted by |a〉 and |b〉 or to have an additional laser driving Raman transitions between two internal states (again denoted by |a〉 and |b〉). This situation is schematically shown in Figure 23.15.

The dynamics of this setup can be understood as follows. Hopping between the two modes |a〉 _j and |b〉 _j of the jth atom is described by

23.31

with the Pauli x‐matrix for the two states (viewed as a spin) of the jth atom and B > 0 the energy associated with this process. The ground state of this Hamiltonian is obtained by putting each atom into an equal superposition of states written in spin notation as |↓_x 〉 _j  = (|↑_z 〉 − |↓_z 〉)/ , where |↑_z 〉 ≡ |a〉 and |↓_z 〉 ≡ |b〉, that is, we find

23.32

Note that this part of the Hamiltonian is identical to a spin chain in a magnetic field B along the x‐axis.

The interaction between two nearest neighbors due to Rydberg interactions is given by

23.33

accounting for the energy difference of having two adjacent particles in the same vs. different internal states. In this case, the ground state depends on the sign of the interaction parameter W. For repulsive interactions W > 0, the interaction will be minimized by arranging the particles as

23.34

while for attractive interactions the ground state is

23.35

Both of these states are maximally entangled multiparticle states, that is, extensions of the well‐known GHz states for three particles.

In the total Hamiltonian H = H_h  + H_i , the interaction energy and the hopping energy compete with each other resulting in a quantum phase transition. When the interaction energy is kept constant and the hopping term is switched off adiabatically, as shown in Figure 23.15, the state of the system will dynamically change from |Ψ _h 〉, which is a product state to one of the two states |Ψ _ir 〉 or |Ψ _ia 〉 depending on the sign of the interaction. The exact values of the parameters α and β depend on the details of the dynamics and are discussed in (44). These maximally entangled states can serve different purposes depending on the sign of W. Let us discuss the possible applications of these two kinds of maximally entangled states.

23.3.4.8 Repulsive Interactions

In this case, both parts of the superposition state have the same number of particles in each of the two internal states and thus external stray fields act identically on both, therefore not affecting the parameters α and β. Because of this stability, one can use them for storing quantum information in a robust way. Single‐qubit gates can be performed by dynamically going back and forth through the quantum phase transition in the whole chain changing the parameters α and β in a controlled way.

23.3.4.9 Attractive Interactions

For attractive interactions, the terms in the superposition of |ψ_ia 〉 will respond to external fields very differently. Therefore, the relative phase between the parameters α and β will be very susceptible to these fields, in fact for N particles in the chain, this phase will be N times larger than if there was just a single particle. The two parts of |ψ_ia 〉 can thus be used as two arms of an entanglement enhanced atomic interferometer.

23.3.4.10 A Single‐Atom Transistor in a 1D Optical Lattice

As a second example, we consider a spin‐1/2 atomic impurity that is used to switch the transport of either a 1D BEC or a 1D degenerate Fermi gas initially situated to one side of the impurity (51). In one spin state, the impurity is transparent to the probe atoms, while in the other, it acts as single atom mirror prohibiting transport. Observation of the atomic current passing the impurity can then be used as a quantum nondemolition measurement of its internal state, which can be seen to encode a qubit, |ψ_q 〉 = α|↑〉 + β|↓〉. If a macroscopic number of atoms pass the impurity, then the system will be in a macroscopic superposition, |ψ(t)〉 = α|↑〉|φ_↑ (t)〉 + β|↓〉|φ_↓ (t)〉, which can form the basis for a single‐shot readout of the qubit spin. Here, |φ_σ (t)〉 denotes the state of the probe atoms after evolution to time t, given that the qubit is in state σ (see Figure 23.16). In view of the analogy between state amplification via this type of blocking mechanism and readout with single‐electron transistors (SET) used in solid‐state systems, we refer to this setup as a single‐atom transistor (SAT).

We consider the implementation of a SAT using cold atoms in 1D optical lattices: probe atoms in the state |b〉 are loaded in the lattice to the left of a site containing the impurity atom |q〉, which is trapped by a separate (e.g., spin dependent) potential (cf. Figure 23.16). The passage of |b〉 atoms past the impurity q is then governed by the spin‐dependent effective collisional interaction . By making use of a quantum interference mechanism, we engineer complete blocking (effectively U _eff → ∞) for one spin state and complete transmission (U _eff → 0) for the other.

The quantum interference mechanism needed to engineer U _eff can be produced using an optical or a magnetic Feshbach resonance (52–54), and we use the present example to illustrate Hamiltonians for impurity interactions involving Feshbach resonances and molecular interactions. For the optical case, a Raman laser drives a transition on the impurity site, 0, from the atomic state via an off‐resonant excited molecular state to a bound molecular state back in the lowest electronic manifold (Figure 23.17a). We denote the effective two‐photon Rabi frequency and detuning by Ω _σ and Δ _σ , respectively. The Hamiltonian for our system is then given by Ĥ = Ĥ_b  + Ĥ ₀, with

23.36

23.3723.37

Here Ĥ _b is a familiar Hubbard Hamiltonian for atoms in the state |b〉; Ĥ ₀ describes the additional dynamics due to the impurity on site 0, where atoms in the state |b〉 and |q〉 are converted to a molecular state with effective Rabi frequency Ω _σ and detuning Δ _σ , and the last two terms describe background interactions, U_αβ,σ for two particles α, β ∈ {q_σ , b, m}, which are typically weak. This model is valid for U_αβ, J, Ω, Δ ≪ ω _T (where ω _T is the energy separation between Bloch bands). Because the dynamics for the two spin channels q_σ can be treated independently, in the following we will consider a single spin channel, and drop the subscript σ.

To understand the qualitative physics behind the above Hamiltonian, let us consider the molecular couplings and associated effective interactions between the |q〉 and |b〉 atoms for (i) off‐resonant (Ω ≪ |Δ|) and (ii) resonant (Δ = 0) laser driving. In the first case, the effective interaction between |b〉 and |q〉 atoms is U _eff = U_qb  + Ω² /Δ, where the second term is an AC‐Stark shift that plays the role of the resonant enhancement of the collisional interactions between |b〉 and |q〉 atoms due to the optical Feshbach resonance. For resonant driving (Δ = 0), the physical mechanism changes. On the impurity site, laser driving mixes the states and m ^† |vac〉, forming two dressed states with energies ε_±  = (U_qb )/2 ± ( /4 + Ω²)^1/2 (Figure 23.17b, II). Thus, we have two interfering quantum paths via the two dressed states for the transport of |b〉 atoms past the impurity. In the simple case of weak tunneling Ω ≫ J and U_qb  = 0 second‐order perturbation theory gives for the effective tunneling (|ε| ≪ Ω) that shows destructive quantum interference, analogous to the interference effect underlying electromagnetically induced transparency (EIT) (55), and is equivalent to having an effective interaction U _eff → ∞.

In Ref. (51) the exact dynamics for scattering of a single |b〉 atom from the impurity is solved exactly, confirming the above qualitative picture of EIT‐type quantum interference. Furthermore, in this reference, a detailed study of the time‐dependent many‐body dynamics based on the 1D Hamiltonian (23.38) is presented for interacting many‐particle systems including a 1D Tonks gas.