3. Exact Numerical Simulation of the spin star Model
3.4 Results
The simulation algorithm described in the previous section was run using 106 reali- sations and again with 107 realisations to give some idea as to the tradeoff between accuracy and computing cost. It can be clearly seen that in the interval λt ∈ [0,1]
the results of the simulation converge well to the exact solution with a reasonable cost of computing time.
In the construction of the algorithm for simulating the dynamics of this model, it was shown that for each of the two independently evolving states, one component did not grow in magnitude and just included a term of the form (i)n while the other, the bath’s component, grew exponentially as a term of the form exp(Γt). It is clear that this second component will grow exponentially if the exponent is greater than one, which is usually the case as λt runs past one. This algorithm is designed in such a way that as the number of realisations approaches infinity, the expectation of this exponential term approaches one. This means that whilst an increase in λt will introduce greater errors, this can be offset by increasing the number of realisations used in computing the expectation to the point that the numerical plots approach the analytical solution exactly as the number of realisations increase. This tradeoff between computation time and accuracy is exponential, this means that this algorithm used in this study is reasonable to use for the region 0≤λt≤1 and as is evident in the figures, errors soon play a dominant role thereafter. For each of the plots the time was run fromλtbetween zero and one, and was run for both 106 and 107 realisations to illustrate the error reduction when more realisations are used.
From these results it can be seen that using the algorithm above is very efficient
Fig. 3.2: Numerical plot of density matrix elementρ11where the circles represent 106realisations and the diamonds 107 realisations with the initial condition |ψ(0)i=|12,−12i. The analytical solution is represented by the solid line.
if the behaviour one wishes to model lies within the aforementioned bounds and approaches the exact solution. Additionally, the algorithm presented here is designed to calculate the dynamics of every density matrix element resulting in a complete statistical description of the open system.
Fig. 3.3: Numerical plot of density matrix element ρ12 where the circles represent 106 realisations and the diamonds 107 realisations with the initial condition|ψ(0)i= 12 |12,12i+|12,−12i
. The analytical solution is represented by the solid line.
Fig. 3.4: Numerical plot of density matrix elementρ22where the circles represent 106realisations and the diamonds 107 realisations with the initial condition |ψ(0)i=|12,−12i. The analytical solution is represented by the solid line.
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.
4.2 Algorithm for Spin 1case
This section details the construction of the algorithm used to simulate the trajectory followed by the density matrix of the central spin’s state for the three level system.
The first part of the algorithm involves setting up parameters, exactly as in the previous chapter, such as the number of bath spins, the final time until which the simulation will run, and the coupling constant. Additionally, this parameter setup phase is where the probability distribution function for states |j, mi is set up. The case where the central system is a three level system rather than a simple qubit requires some important changes, however the bath remains unchanged. A cumulative probability distribution function is set up for all possible j values running from 1 to half the number of bath spins, N2. Identically to the previous chapter, the probability of the bath existing in state |j, mi, given by (2.18), is dependant on j but not on m. As a result, the quantum number j is picked according to this probability but the corresponding quantum number mis chosen according to a uniform distribution.
This is done because each m is equally likely for any given value of j.
Once the parameters are set up, the two independently evolving states|ψ1(t)iand
|ψ2(t)i are simulated. One realisation of this simulation involves stepping through time from t = 0 until the final time t = tf in steps of the random waiting time
τ = −log (η)Γ where η is a uniformly distributed number between 0 and 1, and as a result the jumps constitute a Poisson process. Every time a new τ is found and added to the existing time, this indicates that a jump occurs. Ensuring that the time does not run past tf, a record is kept of when these jumps occur and to which new state the system has jumped. In this case, the system exists in either |1,−1i,
|1,0i or |1,1i and so care must be taken since there may be more than one possi- ble state to jump into. As well as recording which state is occupied at each time interval, a record must also be kept of what factors the jumps have contributed to the state’s evolution. Recall that every jump contributes a factor (−i) to |ψi and between jumps there is no drift. For |χi, the deterministic drift is the factor eΓt, but one should note that in this case Γ(j, m) is dependant on the currently occupied state.
Once this procedure has been concluded twice, once for each independently evolv- ing |Φν(t)i, this constitutes one realisation. For each realisation, a new pair of quan- tum numbers j and m are picked for the bath state, and then for this bath state the composite system state is simulated twice, once for each of the two independent
|Φν(t)i’s.
If the initial condition is assumed to be |12,−12i, the density matrix elementρ11(t) for a specified time interval can be calculated by averaging over many realisations of the product
h1,−1|ψ1(t)i hψ2(t)|1,−1i hχ2(t)|χ1(t)i, (4.15) where h1,−1|ψ1(t)i is either 0 or (−i)nj1, and nj1 is the number of jumps|ψ1(t)ihas undergone. Similarly, hψ2(t)|1,−1i is either 0 or (+i)nj2, where nj2 is the number of jumps |ψ2(t)i has undergone. The contributions from hχ2(t)|χ1(t)i are each just eΓt but as previously mentioned, one must be careful to remember that Γ(j, m) depends on which state ψ(t) is in at that time. In the case of the qutrit as the central system, one must use the record of which states were occupied at each time interval to determine the function Γ for each time interval as it will change as the central state changes. Averaging the product of (4.15) with the probability distribution (2.18) over many realisations results in an approximation of the density matrix element which approaches the exact value given enough realisations.
4.3 Results
Presented here are the results of running the algorithm described in this study for a three level system. Shown also are the graphs of each analytical expression [23]
corresponding to the numerical plots of the density matrix elements in Figure 4.2.
Fig. 4.2: Plot of the analytical expression for the density matrix elements ρ11(t) (dashed), ρ22(t) (dotted) andρ33(t) (dot-dashed) with the initial conditionρ11(0) = 1.
It is clear that the numerical plots closely approximate the analytical results, but also evident is the exponential growth of the errors. These analytical results are ex- actly those for the situation where a pair of spin one-half qubits as the central system interacts with a bath of spin one-half qubits, since the state |0,0i remains forever decoupled from the interaction with the bath [23]. As a consequence, this numerical approximation technique is efficient up toλt= 1 and beyond this limit the computa- tional cost of reducing the errors becomes infeasible. In this particular example of the three level system interacting with a spin bath, there is some interesting behaviour that one may notice from the analytical plots. These points of interest, local minima, occur beyond the limit attainable numerically with a reasonable program running time. This is not to say that the results cannot be produced with this program, only that it would require a very large time.
The fact that these local minima are outside the scope of there particular numerical
Fig. 4.3: Numerical plot of the density matrix elements ρ11(t) (circles with dots), ρ11(t) (squares with lines) andρ11(t) (diamonds with dashes) with the initial conditionρ11(0) = 1, where the simulation was run for 106 realisations. The analytical solution is represented by the solid line.
plots does not in any way diminish the importance of these results. As previously mentioned, with an investment in significant computational power, this simulation could be run further in λt. The other important result of these numerical plots is the good agreement with the exact analytical plots which underscores the usefulness of this numerical algorithm which may be applied to situations where an analytical expression may not be possible.
The difference between the case where the central system consists of just one spin half particle, and the case here where the system consists of a spin one particle, or any three level system - a qutrit, is large in terms of the complexity involved in deriving analytical expressions for the dynamics of the central open system. In the case where even more complex systems are studied as the central open system, it becomes much more attractive to use numerical techniques in order to produce plots of the system’s time evolution.
A remarkable difference can be seen between Figure 4.3 and 4.4 in that the error bars are significantly smaller when the simulation is run for two orders of magnitude longer. The accuracy of this algorithm is limited only by available computational
Fig. 4.4: Numerical plot of the density matrix elements ρ11(t) (circles with dots), ρ11(t) (squares with lines) andρ11(t) (diamonds with dashes) with the initial conditionρ11(0) = 1, where the simulation was run for 108 realisations. The analytical solution is represented by the solid line.
time.
To summarise, these results shown here agree very well with the analytical results, and despite the restrictions introduced by the error growth rate on the timespan across which the simulation can be run, it is nevertheless useful in situations where it may not be possible to derive the equations of motion analytically.
5.1 The Border Between Classical and Quantum
The theory of quantum mechanics, and also quantum statistical mechanics, has pro- vided many new tools which may be used to describe phenomena which do not obey classical theories. Any new theory should of course agree with currently established theories in the classical limit - the limit as the Planck constant becomes zero. The point at which a line should be drawn between classical and quantum descriptions is very unclear[24]. There have been studies showing, for example, that the Joseph- son junction should be modeled as a quantum system[25], but there is evidence that it can also be treated as a classical non-linear oscillator[26]. Another example of a system with semiclassical behavior are the so-called Rydberg atoms[27]. Aside from classical behaviour emerging from quantum decoherence, one should remember the Bohr Correspondence Principle which simply argues that The systems consisting of a large number of particles and/or emerging in quantum states characterised by large quantum numbers behave classically[28]. Proposed here is a derivation of a witness which can decide whether a given system should be treated as quantum, or whether a classical probabilistic treatment is sufficient. Local hidden variable theories are for this purpose not considered, a system is taken to either be classical or quantum. This is not intended to make a general statement as to the quantum of classical nature of all possible systems, but is of interest, for example, in deciding whether for a given set of experimentally accessible states, with a set of experimentally accessible observables, any quantum behaviour is detectable.
5.2 An Example of a Quantum Witness for a Qubit
For some two-dimensional system, for example a qubit, one can express the state using the Bloch representation of its density matrix as
ρ= 12(11+vxσx+vyσy+vzσz), (5.1) where ~v is the so-called Bloch vector for the system. In matrix form, this can be written as
ρ= 12 1 +rcos(θ) eıφrsin(θ) e−ıφrsin(θ) 1−rcos(θ)
!
. (5.2)
In this representation, 0≤θ ≤π is the polar coordinate, 0≤φ≤2πis the azimuthal coordinate, and r is the radial coordinate for the Bloch vector. The total space spanned by all Bloch vectors is just the space of all two-dimensional density matrices and in the Bloch representation is a sphere of radius 1.
Equation (5.2) is the most general form of a density matrix for a qubit with 0 ≤ r ≤ 1. The pure states on the surface of the Bloch sphere are a special case where r = 1. A state in which r = 0 represents the maximally mixed state ρ = 1211. One may defineAandBto be operators acting on the density matrix of some system.
The operatorW =B2−A2 is a witness, where the operatorsAandB are normalised:
0< A < B <11. The normalised witness studied here is such that if
Tr{ρW}<0, (5.3)
then the statistics of experimental data obtained by measurements of A and B in a state ρ cannot be described by a classical model. Such a witness, as expected, describes a plane through the Bloch sphere. The conditions 0< A < B <11 requires that the eigenvalues of (B−A) be positive, and without loss of generality, B can be chosen diagonal and such that
B = 1 0 0 b
!
, (5.4)
where 0< b <1. The observable A is then written as A= a1 ξ
ξ∗ a2
!
. (5.5)
Under these conditions, one arrives at the equation
Tr{W ρ(r, θ, φ)} = 12 1−a21−a22+b2−2ξ2
(5.6) + 12 1−a21+a22−b2
rcos(θ)
(5.7)
− (a1+a2)rξcos(φ) sin(θ). (5.8) For any chosen state ρ(r, θ, φ), one can find a minimum value for this witness, taking this minimum over all experimentally accessible observables A and B. In Ref. [29], a minimum value for this witness was found, optimising over all possible observablesAandB using numerical techniques. These techniques combined random search and simulated annealing algorithms. The choice of parameters for the two observablesA and B giving this minimum are
A= 0.724 0.249 0.249 0.0854
!
, B = 1 0 0 0.309
!
, (5.9)
and the corresponding eigenstate for the witness W is Ψ = 0.391
0.920
!
. (5.10)
For these parameters, the witness gives the value
hWiΨ=−0.0590. (5.11)
Using these same parameters for the observables A and B, the Bloch sphere of all possible statesρ(r, θ, φ) along with the plane described by Tr{W ρ(r, θ, φ)} is plotted in Figure 5.1. This figure depicts the Bloch sphere; the darker shaded portion, for which Tr{W ρ} > 0, and the lighter portion, where Tr{W ρ} < 0, are separated by the plane for which Tr{W ρ}= 0.
5.3 Applications of This Test
The previous section gave an example of how this test can be used for a qubit, but it is also generally applicable in higher dimensions [24]. For the models presented in