Double well potentials and quantum gates
A system of particles in a double well potential is a widely studied and useful example for understanding quantum mechanics. This simple system has recently been used in theoretical proposals and related experiments as a way to make quantum logic gates for ultracold atoms confined in optical lattices. Such quantum gates are the fundamental building blocks for quantum information processing; in these proposals a regular array of cold atoms in an optical lattice serves as the quantum register. We explain how this research can be understood in terms of well-known principles for systems of identical particles.
Current address: ]National Institute of Standards and Technology, Gaithersburg, MD 20899, USA
A system with two particles in a double well potential has been studied in recent experimentsAnderlini ; Trotzky with ultracold atoms. The atoms were confined in optical lattice potentials, which are periodic potentials created by standing waves of laser light. These experiments are remarkable because many copies of the double well system can be prepared simultaneously, each with a well-defined and almost identical quantum state, which is then manipulated. Recently, there has been a related theoretical proposal to make a quantum logic gateHayes with neutral atoms held in double well potentials for the purpose of quantum information processing.
In this paper we explain how the basic principles underlying this current research can be understood using simple quantum mechanics. To aid accessibility, we explicitly express the states of the system in terms of multi-particle wave functions, rather than using the more compact formulation of creation and annihilation operations as in most of the research papers. To establish the notation and background, we first consider a system of two particles in a single well. The reader who is familiar with this background material can skip to Sec. IV.
Ii Two particles in a potential well
The well-known example of two particles in a potential well illustrates the important point that the energy levels of quantum systems do not depend on whether or not the particles are identical. The occupation of these energy levels is influenced by quantum statistics if, and only if, the particles are identical.
The Schrödinger equation for a system of two particles of mass in one dimension is
where the potential is the same for both particles. If the particles do not interact, can be written as the product of the wave functions for the individual particles
or linear combinations of such functions. The lowest configuration of the two-particle system has energy , and wave function , where is the wave function of the th particle in the ground state. The first excited level of the two-particle system has energy , and has two eigenfunctions (which are a product of single particle states):
The degeneracy of the first excited state (and higher states) arises from the symmetry of the system under the exchange of particle labels, that is, the energy of two particles of the same mass in the same potential well does not depend on which particle is in the excited state. To determine the effect of an interaction we need to use degenerate perturbation theory.
ii.1 The effect of an interaction between the particles
We now consider the effect of an interaction between the particles so that the Hamiltonian becomes
The potential represents the interaction between the two particles, which we consider as a perturbation. We only consider potentials that are a function of the distance between the two particles , for example, the Coulomb potential between charged particles.
At this stage those who have encountered similar situations, for example, in solving the helium atom, are inclined to anticipate the result and exploit the freedom arising from the degeneracy to rewrite the two-particle wave functions as linear combinations of the original wave functions that are symmetric or antisymmetric under exchange of particle labels. This approach may be prompted by the knowledge that both spatial and spin wavefunctions are either symmetric or antisymmetric for systems of identical particles. However, for now we consider only distinguishable particles to see the natural emergence of symmetric and antisymmetric states, without explicitly invoking the exchange symmetry operator.
The effect of on the non-degenerate ground state is given by the first-order perturbation:
To evaluate the effect of the perturbation on the degenerate states of Eq. (3) we write the general wave function in the basis of the two states as
This projection of the wave function onto two states is a good approximation when the perturbation is sufficiently small that it does not mix in significant amplitudes of other states. It is convenient to represent these two degenerate basis states as vectors:
In this truncated basis the Schrödinger equation reduces to a matrix equation (with eigenvalue ):
The eigenvalues are determined by the roots of the equation
Here is the ‘direct’ integral given by
and is the ‘exchange’ integral. (It might seem strange to call the exchange integral when the particles are distinguishable, but it has the same form for identical particles.) The exchange integral is given by
is real (the symmetry of the integral for interchange of variables implies that ). Unlike the direct integral, the exchange integral cannot be interpreted in terms of a simple classical picture; attempts to explain the exchange integral classically lead to nonsensical concepts such as ‘exchange forces’. (For example, the gross structure of the helium atom is determined by electrostatic interactions and nothing else.)
The two eigenvectors and corresponding spatial wave functions are
We see that the eigenstates of the Hamiltonian with an interaction between the distinguishable particles are the antisymmetric and symmetric combinations of the original product wave functions. Note that the form of these eigenstates has been derived without reference to the intrinsic exchange symmetry of indistinguishable particles. Instead, this behavior may be understood as a direct consequence of the symmetry of the Hamiltonian with respect to the exchange of the particle labels and .
For the corresponding problem with indistinguishable particles, that is, particles that possess an intrinsic exchange symmetry, the only difference is in the occupation of the two states of Eqs. (13) and (14). Particles with a certain exchange symmetry and a spin wavefunction of a certain exchange parity can occupy only one of the two states of Eqs. (13) and (14); the other is forbidden. Nevertheless, the wave functions and energies of the two states remain the same as for distinguishable particles.
ii.2 A delta-function interaction
The interaction between two ultracold neutral atoms can be well approximated by a delta-function potential
where is a constant, and for a repulsive interaction. For this so-called contact interaction, the direct and exchange integrals are the same:
Because , the interaction does not change the energy of the antisymmetric wave function in Eq. (13), which is an consequence of the function being equal to zero when . A similar argument is often used to explain why the triplet terms of the helium atom are lower in energy than the singlets, namely that the expectation value of the electrostatic repulsion between the two electrons is lower for the spatial wave function in which the two parts have opposite sign.
Iii Identical particles and spin statistics
If the two particles are identical, their spin has an influence on the occupancy of the energy levels. In particular, the requirements of overall exchange antisymmetry for fermions and symmetry for bosons introduces a connection between the spin and spatial wave functions. Consequently states with different spin have different energies, for example, the singlet and triplet terms in helium. It is tempting to think that the energies themselves depend on spin, but they do not. The wave functions for two fermions are linear combinations of the terms
where (A) denotes antisymmetric with respect to exchange of the particle labels , and (S) denotes symmetric. This subset of all the possible two-particle wave functions has overall antisymmetry with respect to the exchange of particle labels as required for identical fermions. The spin wave functions are given in Table 1. There is a singlet for and a triplet for making a total of four functions.
A well-known example of these spin states occurs in the helium atom: for the ground configuration (1s) there is only a symmetric spatial wave function associated with a spin singlet. The first excited configuration (1s2s) is split into two terms: a singlet () and a triplet () whose energy separation is twice the exchange integral given by
where is the Coulomb repulsion between the electron at and the electron at .Foot For two bosons the wave function is symmetric with respect to exchange of the particle labels, so that the wave functions are linear combinations of the terms
This brief review of the physics of two particles in a single well serves as an introduction for the treatment of a double well system in the following.
Iv Trapping atoms using light
Many experiments in ultracold atomic physics use laser light to trap atoms. The laser light interacts with an atomic resonance to form a spatially varying potential, which can confine the atoms. Consider a laser beam incident on a two-level atom, which has unperturbed eigenenergies of in the ground state and in the excited state. These energies refer to electronic (internal) states of the atoms rather than the motional states of the atomic center of mass. The laser beam has an oscillating electric field , which introduces a time-dependent coupling between the ground and excited states of the atom. The coupled Schrödinger equation for the ground and excited electronic states of the atom in the presence of the oscillating electric field of the laser beam can be shown to beFoot
where is a time-independent parameter proportional to .
To solve for the steady state of Eq. (20) we make a time-dependent substitution for the wavefunction
When the frequency detuning of the laser light from the transition frequency and the laser coupling parameter are much less than the transition frequency , Eq. (22) can be simplified. The rapidly oscillating terms vary with a much shorter period than all other time scales in the problem. Therefore the integral of a term with a rapidly varying phase factor such as is much less than the integral of the term without this factor . Hence we can discard terms from the coupled equations with a rapidly varying phase. This is called the rotating wave approximation; it turns out to be a very good approximation for laser-atom interactions. The resulting effective time-independent Hamiltonian in matrix form is
Diagonalizing this Hamiltonian gives the energy shift of the atomic states due to the laser beam. The frequency detuning of the laser light used in experiments is typically much greater than the coupling parameter in order to minimize the spontaneous scattering of the laser light from the trapped atoms. In this limit the energy shifts for the ground and excited state energy levels due to the laser light are
This energy shift is often called a dipole energy shift or an AC Stark shift. As is proportional to the electric field , the energy shifts are proportional to the intensity of the laser beam. The intensity of laser beams can vary spatially so that the dipole energy shift can be used to trap atoms within a potential. For example, when the frequency of the laser beam is less than the frequency of the atomic transition (), ground state atoms are trapped at laser intensity maxima, such as the focus of a single laser beam.
The simple relation between the atomic potential and laser intensity is the basis for the trapping of ultracold atoms using lasers. All manner of light patterns can be created to form potential landscapes for the ultracold atoms. For example, a standing wave arising from the interference of two counter-propagating laser beams creates a periodic potential of hills and valleys for the atoms called an optical lattice. These periodic potentials can be extended to two and three dimensions using additional pairs of laser beams. A double well trap, such as discussed in this paper, may be formed using two independently focused laser beams. The laser beams, and hence the potential that the atoms experience, can be altered in real-time, making laser dipole traps a highly flexible experimental tool with which to manipulate ultracold atoms.
V Double well potential
v.1 A single particle in a double well
We consider double wells that have symmetry with respect to inversion (), so that the eigenfunctions of particles in this potential have definite parity. (The Hamiltonian of the system commutes with the parity operator , that is, , and hence these operators have simultaneous eigenfunctions.) The even and odd parity wave functions for a single particle in a double square well potential with a thin barrier are illustrated in Fig. 1.
The energy of the state whose wave function has even parity is less than that of the wave function with odd parity ; the barrier has less effect on the odd wave function because has a node at no matter the height of the barrier, whereas the energy of the even parity wave function decreases as the barrier height decreases. For the simple case of square well potentials, when the barrier is reduced from to zero, so that the well becomes twice its original length, the ground state energy reduces to a quarter of its original value. A qualitatively similar argument holds for any double well with reflection symmetry: the solution of the Schrödinger equation with no nodes has the lowest eigenenergy.
We can form linear superpositions of the energy eigenstates
which correspond to wave functions in which the particle is located predominantly in the left or right well respectively, as illustrated in Fig. 1. These are not eigenstates of the system, and thus the wave function of a particle initially prepared in the left well at evolves according to
where is the tunneling rate between the wells. If for integer , the wave function is proportional to , and if , the wave function is proportional to , corresponding to the particle having tunneled into the other well. For negligible tunneling between the wells (very high barrier) it is possible to describe the particle as being either in the left or right well. This is clearly the case when the wells are very far apart so that the particle is localized in one of the wells, and is the situation at the start of the experiments we shall describe.
v.2 Two interacting particles in a double well
The previous sections have prepared all the tools we need to understand the behavior of two particles in a double well. We choose a potential with a high barrier so that initially the system is close to the limit of two separate wells. There is an antisymmetric spatial wave function which can be expressed in terms of the left/right wave functions of the individual particles as
Equation (27) can be rewritten using the ket notation to represent
Similarly, the symmetric wave functions () are linear combinations of the three terms
Writing the four linearly independent spatial wave functions in this way highlights their similarity with the form of the four spin wave functions for two spin one-half particles; both have one and three . We combine space and spin functions with the restriction of overall antisymmetry, as in Eq. (17), and find by inspection of Table 1 that there are six two-particle wave functions for identical spin one-half particles.
Now consider the effect of a repulsive contact interaction between the particles. This interaction has little effect on the energy when the two particles are localized in different wells (with small overlap of their wave functions), that is, for and . However, for the wave functions containing () and (), the interaction raises the energy by an amount which we denote as . The integral for resembles Eq. (16):
In the following we shall mainly consider the subset of states with the particles in different wells. These are separated by the energy gap from the higher lying states and .
Vi Quantum gate for two fermions
In quantum computing notation the state in which both particles are spin-up can be written as and conversely . The state of the system is encoded in terms of the four basis states as
where , , , and are complex coefficients. The state corresponds to a spin-up particle in each well for which the antisymmetrized wave function (unnormalized) is
Equation (32) can be expressed in terms of the energy eigenstates as
The state corresponds to spin-up in the left well and spin-down on the right for which the antisymmetrized (unnormalized) wave function is
Equation (34) can be expressed in terms of the energy eigenstates as
and similarly for and . Thus we write the quantum computational basis states in terms of the four eigenstates of two spin one-half particles (which are degenerate for zero tunneling) as
The barrier between the wells can be altered by changing the intensity or separation of the two laser beams used to form the double well potential. When the barrier is lowered slowly, and adiabatically evolve into and respectively, which are no longer degenerate. Further details are given in Ref. Hayes, (see also Table 2). To understand the operation of the gate all we need to know is that a phase difference accumulates between these states because they have different eigenenergies during the gate operation. Thus if the barrier is lowered for a certain interval and then raised again (to switch off the tunneling), the wave function that starts as becomes
Controlling the process so that the accrued phase shift causes this state to evolve into
The global phase factor is unimportant. All other combinations also pick up this phase, and therefore it does not affect the relative phase.
It can be seen that a system initialized in evolves into and vice versa, which implements the SWAP operation . When , the system cycles back to its initial state. More generally, the states and evolve into a coherent superposition, while the and states remain unaffected. The SWAP operation can be written as the matrix
for the basis states in Eq. (36). The SWAP gate cannot be used to entangle qubits. The operation on two unentangled qubits is
which (as the name implies) swaps the state of the qubits without entangling them.
A closely related gate called the “square-root-of-SWAP” gate can perform a useful entangling operation in quantum computation. This operation has the property
A matrix satisfying this condition is
To implement the gate of Eq. (42) the system undergoes half of the phase evolution of the SWAP gate. A square-root-of-SWAP gate cannot be broken down into a combination of single qubit gates, that is, gates operating on one qubit at a time. Furthermore, it entangles two unentangled qubits giving the resultant state [from Eq. (42)]
which cannot be written as a product of two single qubit states.
A square-root-of-SWAP gate, used alongside single-qubit operations, is sufficient to provide a universal set of gates for quantum computation.Nielsen Single qubit gates would be implemented in this system by the rotation of the atomic spin orientation of the atom in one of the wells, which may be accomplished simply by optical (Raman) or microwave pulses. The implementation of these powerful ideas with fermionic atoms in optical lattices is described in Ref. Hayes, . Similar principles can also be applied to a system of two bosons (selecting just two out of the possible spin states to form a suitable basis). More details of a gate for bosons are given in Refs. Vaucher, ; Anderlini, ; Trotzky, .
We have shown how recent experimental and theoretical research on ultracold atoms in double well potentials can be simply understood. The experiments have been carried out with bosonic atoms (the isotope Rb) for technical reasons, but for simplicity the fermionic case has been used to expound the principles. This system described in Ref. Hayes, is particularly intriguing for the way in which the influence of the nuclear spin on the spatial wave functions is exploited.
Second quantized notation tends to be used for a more general solution, as in most of the papers we have cited, because this notation becomes more efficient in keeping track of exchange symmetry as the number of particles in the system increases.
There is great current interest
in using cold atoms in optical lattices for quantum information processing, including the direct quantum
simulation of strongly correlated many particle systems (such as
those in condensed matter physics). The simple approach outlined in
this paper gives an
intuitive way of understanding aspects of these systems.
The authors would like to thank the referees and editors for helpful comments. M.S. would also like to thank Christ Church (Oxford), NIST (Gaithersburg) and the Lindemann Trust for financial assistance.
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|State||Spin states||Exchange symmetry||Parity||Eigenfunctions for a high barrier||Eigenfunctions for a low barrier|