4. Substituting a Spin 1 Qutrit for the Central Spin
4.1 PDP for Spin 1 case
Instead of a single spin one-half particle as the central system, the case where the central system consists of a single spin one particle surrounded by N spin one-half particles is investigated. This model is illustrated by Figure 4.1, the most notable change from the previous chapter is the central spin. In this model, the central spin exists as a three level system. In contrast to the two level system of a spin half particle, a spin one particle can be treated as a three level system with probabilities for transition between these three states which shall be explored. The bath is unchanged from the previous chapter and retains the symmetry that enables this model to be solved analytically. Nevertheless, it is interesting to study the numerical techniques presented in this study and compare their results to the analytical results given in [23].
The bath exists initially in the state|j, mi, wherej is the total collective spin and m is thez-component of the total spin. The probability for the bath to exist in state
|j, mi is given again, as in (2.18), by P(j, m) = 1
2N
N
N 2 +j
−
N
N
2 +j+ 1
. (4.1)
The Hamiltonian of this system is given by H = λ
√N (σ+J−+σ−J+). (4.2) The initial condition of the system is the state
|Φ(0)i=|S, Szi ⊗ |j, mi. (4.3)
Fig. 4.1: A diagram representing a spin star model where the central system is a three level system, or qutrit, interacting with a bath of spin half particles.
Writing the Hamiltonian (4.2) in the form
H =X
α
Aα⊗Bα,
where A ∈ Hs and B ∈ HB, leads to the following equations describing the jump
processes for the central spin and bath states;
|S, Szi → −i|| |S, Szi ||
||σ±|S, Szi ||σ±|S, Szi
|j, mi → || |j, mi ||
||J∓|j, mi ||J∓|j, mi. (4.4) In contrast to the previous chapter, this model describes a three level system having four distinct jumps; |1,−1i → |1,0i, |1,0i → |1,−1i, |1,1i → |1,0i and |1,0i →
|1,1i. The jump rates for this process can again be calculated according to (3.25), using
Γαν = ||Aαψν||.||Bαχν||
||ψν||.||χν|| .
This system will randomly jump between the three states |1,−1i, |1,0i and |1,1i, with each jump contributing a factor (−i). Complications arise from the fact that an even number of jumps no longer guarantees with certainty the final state and it should also be observed that the four jump rates are not equal unlike in the previous chapter. If the system makes an even number of jumps after initially starting in the state|1,0i, it will with certainty end in the same state. If, however, the system makes an odd number of jumps from the initial state|1,0i, it will end up in one of the two remaining states with a some probability. Similarly, if the system starts in either of the states|1,−1ior|1,1i, an odd number of jumps will result in the final state|1,0i with absolute certainty. On the other hand, an even number of jumps from the initial states |1,−1i or |1,1i will result in the final state being either |1,−1i or |1,1i with the same aforementioned probability. This probability is given by the two jump rates out of the state |1,0i. The probability to jump from |1,0i to |1,1i is given by the equation
π0→1 = Γ0→1
Γ0→1+ Γ0→−1
, (4.5)
where Γ0→1 and Γ0→−1 represent the probabilities of jumping from|1,0i to|1,1i and
|1,0i to |1,−1i respectively. In the same way, the probability to jump from |1,0i to
|1,−1i is given by the equation
π0→−1 = Γ0→−1
Γ0→1 + Γ0→−1
. (4.6)
More explicitly, if the system starts in state |1,1i, then after an even number of
jumps
|S, Szi = (−i)nj|1,1i
|j, mi = |j, miexp
nj
X
i
Γi(j, m)ti
!
, (4.7)
with probability π0→1, or
|S, Szi = (−i)nj|1,−1i
|j, mi = |j, m+ 2iexp
nj
X
i
Γi(j, m)ti
!
, (4.8)
with probability π0→−1 where nj is the number of jumps that have occurred, Γi is the jump rate after i jumps and ti is the random waiting time before the ith jump.
This is an important change to the previous chapter since the jump rate is no longer constant but rather depends on the state of the system at a given time. After an odd number of jumps
|S, Szi = (−i)nj|1,0i
|j, mi = |j, m+ 1iexp
nj
X
i
Γi(j, m)ti
!
. (4.9)
The jump rates for each of the four possible jumps, Γ−1→0(j, m), Γ1→0(j, m), Γ0→−1(j, m) and Γ0→1(j, m), are now calculated according to (3.25).
Γ−1→0(j, m) = λ
√ N
||σ+|1,−1i ||.||J−|j, mi ||
|| |1,−1i ||.|| |j, mi ||
= λ
√N
||[12(1(1 + 1) + 1(−1 + 1))]12|1,0i ||.||J−|j, mi ||
|| |1,−1i ||.|| |j, mi ||
= λ
√N
|| |1,0i ||.||J−|j, mi ||
|| |1,−1i ||.|| |j, mi ||
= λ
√N||J−|j, mi ||
= λ
√N||[j(j+ 1)−(m−1)m]12 |j, m−1i ||
= λ
√
N[j(j+ 1)−(m−1)m]12. (4.10)
In exactly the same way the remaining jump rates are calculated as Γ0→1(j, m) = λ
√ N
||σ+|1,0i ||.||J−|j, m−1i ||
|| |1,0i ||.|| |j, m−1i ||
= λ
√N
||[12(1(1 + 1) + 0(0 + 1))]12 |1,1i ||.||J−|j, m−1i ||
|| |1,0i ||.|| |j, m−1i ||
= λ
√N
|| |1,1i ||.||J−|j, m−1i ||
|| |1,0i ||.|| |j, m−1i ||
= λ
√N||J−|j, m−1i ||
= λ
√N||[j(j+ 1)−(m−1)(m−2)]12 |j, m−2i ||
= λ
√
N[j(j+ 1)−(m−1)(m−2)]12, (4.11)
Γ1→0(j, m) = λ
√N
||σ−|1,1i ||.||J+|j, m−2i ||
|| |1,1i ||.|| |j, m−2i ||
= λ
√ N
||[12(1(1 + 1)−1(1−1))]12 |1,0i ||.||J+|j, m−2i ||
|| |1,1i ||.|| |j, m−2i ||
= λ
√N
|| |1,0i ||.||J+|j, m−2i ||
|| |1,1i ||.|| |j, m−2i ||
= λ
√N||J+|j, m−2i ||
= λ
√N||[j(j+ 1)−(m−1)(m−2)]12 |j, m−1i ||
= λ
√N[j(j + 1)−(m−1)(m−2)]12 (4.12) and
Γ0→−1(j, m) = λ
√N
||σ−|1,0i ||.||J+|j, m−1i ||
|| |1,0i ||.|| |j, m−1i ||
= λ
√N
||[12(1(1 + 1) + 0(0 + 1))]12 |1,−1i ||.||J+|j, m−1i ||
|| |1,0i ||.|| |j, m−1i ||
= λ
√N
|| |1,−1i ||.||J+|j, m−1i ||
|| |1,0i ||.|| |j, m−1i ||
= λ
√N||J+|j, m−1i ||
= λ
√N||[j(j + 1)−(m−1)m]12 |j, mi ||
= λ
√N[j(j+ 1)−(m−1)m]12. (4.13) The total jump rate Γ(j, m) is the sum of all allowed jump rates,
Γ(j, m) =X
α
Γα(j, m).
Whether or not a particular jump is allowed is dependant on the state of the central spin. This three level system has a total of four possible jumps. If the central system is in either of the states|1,1ior|1,−1i, only one jump is allowed and care should be taken to note that they are generally unequal. If the central system is in the state
|1,0i, two jumps are allowed and the total jump rate is their sum, and the jump direction is chosen randomly with the probabilities π0→−1 and π0→1.
After calculating the four jump rates, and using (4.7) along with the definition for the density matrix in the form of an expectation (3.1), the element of the density matrix of the central spinsρ11 can be expressed as
ρ11(t) = h1,−1|ρS(t)|1,−1i
= E[h1,−1|ψ1(t)i hψ2(t)|1,−1i hχ2(t)|χ1(t)i]. (4.14) It is assumed for the equations above that the initial condition of the system is such that the central spin begins in state |ψ(0)i =|1,−1i. It is not sufficient in this case that the number of jumps that each of the two states |ψν(t)i is even in order for the density matrix element to be considered non-zero, it is also a requirement that the two states|ψ1iand|ψ2iare both|1,−1i, since there three rather than just two states possible.