Preparing states in the degenerate subspaces
4.2 Preparing given states in the degenerate subspaces
In what follows we assume that the mean number of photons hni (which will simply be referred to as n in the following) is large. In 4.2 we will calculate expectation values of the creation and annihilation operators, scaling with√
n. These expectation values are affected by a fluctuationδnin the number of photons as
√n+δn−√
where in the last step a Taylor approximation was applied.
For large n, however, the width δn in the distribution of the number of photons is small compared ton. For example for a coherent state holds δn=√
n[33], yielding
This results in the system being subjected to the same field intensity during the experiment.
4.2 Preparing given states in the degenerate subspaces
We first disregard the coupling expressed byHI. The state of the composite system consisting of the qubits and the electromagnetic field can be written as|ψ, ni, where the labels in the ket are|qubits,fieldi, i.e. qubits in the state|ψi, andn being the number of photons.
For propagating the system from the ground state to one of the excited two-fold degenerate subspaces, we choose the frequency of the mode to be resonant with the transition frequency from the ground state to the excited level (the indices ’e’ and ’g’ stand for the ground state and the excited states, respectively),
~ω=Ee−Eg . (4.16)
Consider the three states
|g, ni , |e1, n−1i , |e2, n−1i , (4.17) where|e1i and |e2i are two arbitrary states in the degenerate subspace.
The three states in (4.17) are energetically degenerate eigenstates of theuncoupled Hamilto-nian¡ We can now introduce the couplings between the field and the qubits. The couplings cor-respond to absorption (|g, ni → |e1, n−1i,|g, ni → |e2, n−1i) and stimulated emission (|e1, n−1i → |g, ni,|e2, n−1i → |g, ni) processes.
34 4 Preparing states in the degenerate subspaces since the number states are orthogonal,hn|mi=δnm.
The elements between the ground state and the excited states, however, are non-zero, hg, n|gIVˆ0(a†+a)|e1, n−1i = gI√
nhg, n|Vˆ0|e1, ni, (4.20) hg, n|gIVˆ0(a†+a)|e2, n−1i = gI√
nhg, n|Vˆ0|e2, ni. (4.21) We write down the matrix representing the reduced perturbation in the tree-fold degenerate subspace, spanned by the states in (4.17). The basis states are numbered as
|g, ni= With this choice for the basis, the reduced perturbation takes the form
Vˆ0red=gI√
4.2.1 Driving the low-energy subspace
For convenience purposes, we calculate ˆV0red in the coupled basis introduced in appendix A, Vˆ0red=gI√
As pointed out in chapter 3.3.2 and explicitly written down in appendix B, the lower energy subspace is spanned by|ψ˜1Liand|ψ˜2Li, which are superpositions of|˜v5iand|˜v7i(|˜v6iand|˜v8i, respectively), whereas the ground state|E˜4iis a superposition of{ |˜v1i, . . . ,|˜v4i}. Moreover, the state vectors have only real entries (the Hamiltonian is purely real). This enables us –without further knowledge about the structure of the states– to write
|˜gi=|E˜4i=
4.2Preparing given states in the degenerate subspaces 35 We obtain
Vˆ0red = − r2
3gI √
n(e g2+p
1−e2g3)
0 √
3κ1 κ1+ 2κ2
√3κ1 0 0
κ1+ 2κ2 0 0
=
= ~
0 ω1 ω2 ω1 0 0 ω2 0 0
. (4.26)
Here, all the constants1 have been substituted by the Rabi frequenciesω1 and ω2.
When we take into account this coupling between|g, ni and |e1, n−1i, as well as between
|g, ni and |e2, n−1i, as expressed by ˆV0red, we obtain three perturbed states |1(n)i, |2(n)i,
|3(n)i(the eigenstates of ˆV0red), two of which are shifted up and down, respectively, in energy by ~Ω with Ω := p
ω12+ω22. These states are called dressed states. In the dressed state language, this configuration can be understood as two two-state systems, the first one con-sisting of|g, niand|e1, n−1i, the second one of|g, niand|e2, n−1i. The missing coupling between|e1, n−1i and|e2, n−1ion the other hand, results in an energy shift of zero (state
|2(n)i) with respect to the original energies.
Figure 4.1: Level diagram of the qubit+field system showing the dressed states. The bare states are perturbed by the coupling encountered via absorption and induced emission, resulting in new eigenstates |1(n)i,|2(n)i, and |3(n)i (dressed states). The frequency of the field ω is resonant with the qubits’ level splitting. ~Ω is the energy separation induced by the coupling.
In the following, we aim for exploring the dynamical behavior of the states, that is to say we derive the Rabi formula by means of our dressed state approach. We expect the probability to find the system in the state|e1, n−1i(|e2, n−1i, respectively) after a timetif we started in the ground state|g, ni at timet= 0 to be a sinusoidal function of time, oscillating at the
1The eigenstates of the system are supposed not to change during the short duration of the pulses. Moreover, κ1 andκ2 shall remain fixed during the driving process.
36 4 Preparing states in the degenerate subspaces
Bohr frequency Ω associated with the perturbed levels [33]. For resonant coupling as in our case, the Rabi frequency equals the Bohr frequency.
In the interaction picture, the time evolution of the system is governed by the perturbation and the Schr¨odinger equation reads
i~∂
∂t|ψ(t)i= ˆV0red|ψ(t)i. (4.27) Vˆ0red is time independent (as a result of the used dressed state approach), and we can solve (4.27) by
|ψ(t)i= ˆU(t, t0)|ψ(t0)i=e−~iVˆ0red(t−t0)|ψ(t0)i. (4.28) Our objective is to calculate the propagator ˆU(t, t0). The dressed states together with the corresponding eigenvalues read
We get (in the following we sett0 = 0 without loss of generality) Uˆ(t) = ˆT
The explicit form of ˆU(t) can be found in appendix E.
Consider we start with a fully occupied ground state without any population in the excited levels. The effect of the propagator onto this initial state then looks like
|ψ(t)i= ˆU(t)
We obtain the expected sinusoidal behavior mentioned above. Complete depopulation of the ground state can be achieved by applying aπ-pulse of length
tπ = π
2Ω , (4.33)
yielding the final state (disregarding a global phase)
|ψ(tπ)i= 1
By choosing the amplitudes of the sources κ1 and κ2 (and thereby the Rabi frequencies ω1 and ω2) appropriately, we can completely depopulate the ground state and prepare states with arbitrary amplitude ratio (in a given basis) in the degenerate subspace. Note that this
4.2Preparing given states in the degenerate subspaces 37 could not be achieved by a symmetric driving in ² (²1(t) = ²2(t) = ²3(t)); the Hamiltonian H0 in (3.1) has no transition matrix elements between the ground state (living in the upper 4×4 block) and the degenerate subspaces (living in the lower 2×2 blocks).
We now want to enhance this scheme by additionally introducing a relative phase, that is preparing a target state
4.2.2 Introducing a relative phase
We assume the two microwave sources, so far just characterized by their amplitudesκ1 and κ2, to be independent not only in amplitude but also in their relative phase. However, both sources shall be kept on resonance, as expressed by the condition (4.16), oscillating on the same frequency,
Quantization of the field introduces the creation and annihilation operators, whereas the exponentialse±iωt disappear for a single-mode field (in the interaction representation) [34],
B~1 = ~² B1(a+a†),
B~2 = ~² B2(eiθa+e−iθa†). (4.37) The interaction Hamiltonian then takes the form
HI=gIκ
and the reduced perturbation operator reads (cp. (4.26))
Vˆ0red = −
The operator can still be written in terms of the two frequencies ω1 and ω2 and in addition a phase ϕ; however, note that in general, ω2 as in (4.39) is different from ω2 as in (4.26).
38 4 Preparing states in the degenerate subspaces
Determining the propagator ˆU(t) =e−~iVˆ0redtin the same way as above and applying it to the initial ground state gives
|ψ(t)i= ˆU(t)
1 0 0
=
cos Ωt
−iω1sin ΩtΩ
−iω2sin ΩtΩ eiϕ
. (4.40)
The explicit form of ˆU(t) can again be found in appendix E.
For aπ-pulse of the same length as in (4.33) and disregarding a global phase we get a final state
|ψ(tπ)i= 1 Ω
0 ω1 ω2eiϕ
. (4.41)
We obtain the delighting result that amplitude as well as phase can be controlled by amplitude and phase of the applied pulses. This enables us to prepare arbitrary states in the subspace.
By an optimal choice of ω1, ω2 and ϕ, states with maximized entanglement can be created (see appendix D.2). We will concentrate on these states in the following.