The feasibility of using fermionic Strontium to simulate the SSH model

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The feasibility of using fermionic Strontium to simulate the

SSH model

Joey van de Leijgraaf August 17, 2020

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1 Abstract

We simulate the evolution of fermionic Strontium that has been coupled such that it can be approximated as an SSH model. We examine the effects that error sources inherent to Raman coupling have on the coherence and fidelity of the quantum oscillations, and to what degree this is affected by the topology of the SSH model. We find that while with our assumptions it is possible to approximate the SSH model with fermionic strontium, the laser intensities and detunings required to combat decoherence caused by natural decay of the excited state and the laser linewidth are impractical. If this decoherence is not present, the system is well-protected against Rabi frequency noise of up to 5%

2 Introduction

Topology refers to the study of objects with properties called topological invariants, which are preserved under continuous deformation. An example of a topological invariant is the amount of twists in a loop[1]. The Mobius strip is topologically different from a regular loop because it has a twist[1]. Topological insulators are directly linked to the band theory of electric conduction, a theory that explains why some crystalline materials are insulators or conductors[2]. The theory states that insulators have fully occupied electron bands, while conductors have partially filled bands[2]. However, the discovery of the Quantum Hall[3] effect showed that we could not consider simply a conductor or insulator; topological insulators were discovered, which are insulators in the bulk but have conducting edge states that are the result of nontrivial topology of the occupied bands [2].

Just like regular topological objects, topological insulators have topological invariants. These invariants arise when one constrains the Hamiltonian to have additional symmetries [1]. An artifical SSH model, a model for a 1D topological insulator, has been successfully created before using interacting bosons that have been excited into Rydberg atoms[4]. The goal of this paper

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Figure 1: An N = 10 SSH lattice with the n = 6th unit cell dotted. It has an intracellular hopping amplitude v and intercellular hopping amplitude w. Grey cells belong on sublattice A and white cells belong on sublattice B. This figure is from A Short Course on Topological Insulators[2]

is to simulate the SSH model with a fermionic Strontium atom to examine the efficacy of these topological invariants in protecting its edge oscillations against the errors that appear because of Raman coupling. If these protections prove effective they could allow fermionic Strontium to be used in situations where high-fidelity quantum oscillations are required, such as qubits in quantum computing

2.1 What is the SSH model and why is it important

The SSH model is one of the simplest forms of a topological insulator. The SSH model describes a 1D-lattice with an even number of sites and alternating hopping amplitudes v and w[2], also referred to as J and J0[4][1]. If we imagine an SSH lattice of 2N lattice sites we can create N adjacent pairs of lattice sites, as seen in figure 1, that we will henceforth call unit cells. Because our hopping amplitudes alternate they now describe an intra- and an intercellular hopping amplitude. We take v as the intracellular hopping amplitude and w as the intercellular hopping amplitude.

Because there are no onsite potentials the Hamiltonian of the SSH model is now simply given by [2] ˆ H =v N X m=1 (|m, Bi hm, A| + h.c.)+ w N −1 X m=1 |m + 1, Ai hm, B| + h.c.) (1)

with A and B denoting the location on the unit cell sublattice. We can also represent it in its matrix form H =          0 v 0 · · · 0 0 v 0 w · · · 0 0 0 w 0 · · · 0 0 .. . ... ... . .. ... ... 0 0 0 · · · 0 v 0 0 0 · · · v 0          (2)

As mentioned, the SSH model is a topological insulator. The model is in its topological regime for w >> v[2]

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Figure 2: A three-level lambda system as represented in the Schrödinger picture(a) and the Rotating frame(b). This figure is from Adiabatic Elimination in a Lambda System[8]

2.2 Why Strontium

Strontium is an alkaline-earth element with four stable isotopes, three of which are bosonic and one of which is fermionic[5]. The fermionic isotype,87Sr, has a nuclear spin of I = 9₂[5][6]. While the energy levels of these different spin states are usually degenerate, this degeneracy is lifted when we apply a magnetic field and examine the hyper fine structure of Strontium[7]. While the effect this has on the ground state is negligible, it causes large shifts in the energy levels of the excited states.

2.3 Raman coupling and adiabatic elimination

We make use of Raman coupling to excite the fermionic Strontium. Consider a 3-level Lambda system, as shown in figure 2, consisting of two ground states |ai, |bi and one excited state |ei. While figure 2 shows red detuning to be positive, in this paper we consider blue detuning positive δ_i = ω_i(L)− ω_i. Our Hamiltonian in the Rotating wave approximation with the basis |ψi =   a e b 

 is then given by[8]

H = ~    δ 2 Ω∗ a 2 0 Ωa 2 ∆ Ωb 2 0 Ω ∗ b 2 − δ 2    (3)

with ground detuning δ ≡ δa− δb, intermediate detuning ∆ ≡ δa+δ₂ b and rabi frequencies Ωa

and Ω_b which couple |ai respectively |bi to |ei,

If we consider only σ− and π-polarized lasers for simplicity, we can imagine each excited mf state from −9₂ to 7₂ as part of their own 3-level Lambda system, coupling to their respective

ground states and the ground states with 1 higher spin. We assume that a large enough B-field has been applied to lift the degeneracy of the excited m_f states such that each laser addresses a single transition. We set δ = 0 and choose Ω_a and Ω_b for each Lambda system such that

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Figure 3: The coupling between the ground states of Strontium and excited states of adjacent nuclear spin via σ−and σ+_{coupling. There is a slight error in the series as represented, both 20 and 35 should} be 1 higher. Figure from Detection and manipulation of nuclear spin states in fermionic strontium [6]

the complete Hamiltonian has alternating real rabi frequencies

H = ~                 0 Ω1 2 0 0 0 · · · 0 0 0 Ω1 2 ∆ Ω1 2 0 0 · · · 0 0 0 0 Ω1 2 0 Ω2 2 0 · · · 0 0 0 0 0 Ω2 2 ∆ Ω2 2 · · · 0 0 0 0 0 0 Ω2 2 0 · · · 0 0 0 .. . ... ... ... ... . .. ... ... ... 0 0 0 0 0 · · · 0 Ω1 2 0 0 0 0 0 0 · · · Ω1 2 ∆ Ω1 2 0 0 0 0 0 · · · 0 Ω1 2 0                 (4)

Do note that to achieve this we must account for the Clebsch-Gordan coefficients in our σ− coupling, as seen in figure 3.

Consider once again our three-level Lambda system from equation 3. It’s possible to adiabatically eliminate the excited state from the Hamiltonian by assuming there is no change in the population of the excited state ˙γ(t) = 0[8]. This is true in the regime |∆| >> |Ωa| ,

|∆| >> |Ω_b| and |∆| >> |δ| . This adiabatic elimination gives us for a 3-level Lambda system [8] H = ~ δ 2 + Ω2 a 4∆ Ω∗aΩb 4∆ ΩaΩ∗b 4∆ − δ 2 + Ω2 b 4∆ ! (5) If we take δ = 0 and choose our rabi frequencies to be real and equal we find

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H = ~ Ω2 4∆ Ω2 4∆ Ω2 4∆ Ω2 4∆ ! (6) If we consider this for all our Lambda systems and reduce the global energy level by subtracting Ω21+Ω22

4∆ , we arrive at a matrice similiar to that of the SSH Hamiltonian 2.

H = ~          −Ωw Ωv 0 · · · 0 0 Ωv 0 Ωw · · · 0 0 0 Ωw 0 · · · 0 0 .. . ... ... . .. ... ... 0 0 0 · · · 0 Ωv 0 0 0 · · · Ωv −Ωw          (7) with Ωv = Ω 2 1 4∆ and Ωw = Ω2 2

4∆. Which is close to the SSH Hamiltonian, but has a detuning on

the edge states of −Ω22

4∆. It’s possible to correct for this edge-detuning, but this will result in

the same correction in our original Raman coupling Hamiltonian. The full Raman coupling Hamiltonian considered in this paper will thus be

H = ~                  −Ω22 4∆ Ω1 2 0 0 0 · · · 0 0 0 Ω1 2 ∆ Ω1 2 0 0 · · · 0 0 0 0 Ω1 2 0 Ω2 2 0 · · · 0 0 0 0 0 Ω2 2 ∆ Ω2 2 · · · 0 0 0 0 0 0 Ω2 2 0 · · · 0 0 0 .. . ... ... ... ... . .. ... ... ... 0 0 0 0 0 · · · 0 Ω1 2 0 0 0 0 0 0 · · · Ω1 2 ∆ Ω1 2 0 0 0 0 0 · · · 0 Ω1 2 − Ω2 2 4∆                  (8)

2.3.1 Resulting error sources

In order to test the resiliency of the topological protections we consider the following error sources

Error source Predicted Effect Laser Intensity uncertainty in Ω₁, Ω2

Laser Linewidth uncertainty in detuning Natural decay Decoherence Approximations not valid Decoherence Where we note that, since Ωw

Ωv =

Ω2 2

Ω2 1

, any noise in Ω1 and Ω2 likewise gets squared when

considering the topology of the system. This means that small errors in laser intensities could cause large changes in the topological regime.

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3 Method

For the purpose of this paper we make use of Qutip [9] to simulate the evolution of states. While a closed system is governed in its entirety by the Schrödinger equation, this is not the case for open systems. To simulate these open system the Lindblad Master Equation Solver[9] makes use of the Von Neumann equation, the analog of the Schrödinger equation in the density matrix formalism, given by[9]

˙ ρ = −i

~[Htot, ρtot(t)]

(9) with Htot= Hsys+ Henv+ Hintconsisting of the system, environment and interaction

Hamil-tonian. The evolution of our system’s density matrix ρ = T renv[ρtot] can then be represented

as[9] ˙ ρ(t) = −i ~[H(t), ρ(t)] + X n 1 2[2Cnρ(t)C + n − ρ(t)Cn+Cn− Cn+Cnρ(t)] (10) where Cn= √

γnAn are collapse operators, An the operators through which the environment

couples to the sytem in H_intand γ_n the rates.

We use the natural decay rates of our transitions3P1 and3P2 with γ_3P1

2π = 7.4 kHz[6] and γ_3P2

2π = 6.8 mHz[10] as the rates. The decay from our excited states to adhere the selection

rules ∆m = 0, ±1.

In other words,the interaction operators Anare given by |gi he| where e can hold all odd

num-bers and g differs from e by -3, -1 or 1 with the condition that 0 <= g <= 18.

The linewidth of our laser can also be simulated through the collapse operators. As the laser linewidth interacts with the detuning we consider interaction operators An = 0.5(|ni hn| −

|n + 1i hn + 1| with rate γ_llw simply the linewidth in angular frequency.

Recall that variations in the intensity of the laser result in variations of the Rabi frequency. In other words, the ideal SSH Hamiltonian given by equation 1 would be constructed with an additional δ term ˆ H = N X m=1 (v + δi,v)(|m, Bi hm, A| + h.c.)+ N −1 X m=1 (w + δi,w)(|m + 1, Ai hm, B| + h.c.)

where δ_i depends on the laser powering that transition. For the purposes of this paper we consider a uniformly distributed δ with a maximum error δi,v,max

v = δi,w,max w and a minimum error δi,v,min v = δi,w,min w = − δi,v,max

v . We average the oscillations over 100 iterations.

The oscillation between edge states we expect for the SSH model will break down when we consider randomization of the couplings. For now we will simply guess that this breakdown behaves like exponential decay of the oscillation

f (t) = A cos(2π t P) ∗ e

−t

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with A the amplitude, P the period, τ the 1_e lifetime of the oscillations and C the midpoint of the oscillations.

We will justify the use of this function for the SSH Hamiltonian in the results.

For other graphs we make use of a running mean to remove smaller oscillations that are experimentally difficult to measure. We divide our data into 100 subsets, with the timescale of our simulations such that an ideal SSH Hamiltonian with only the v and w data would have roughly one oscillation.

4 Results & Discussion

We first make some preliminary remarks about these results.

1. Multiplying the Hamiltonian by some factor N simply decreases the timescale by a factor

1

N. This follows directly from the Schrödinger equation.

2. While the Hamiltonian is denoted in angular frequencies, we denote the values in this paper in linear frequencies.

3. Unless mentioned otherwise we take Ω₁ = 0.3 MHz and ∆ = −10 MHz

4. Our Ω_v and Ω_wscale with the square of our physical rabi frequencies Ω₁ and Ω₂, as seen in equation 8

5. The "ideal Raman coupling system" means a Raman coupling system with an ideal laser of linewidth 0 and no natural decay from the excited states.

4.1 Topological behaviour

We consider the behaviour of the SSH model in figure 4. For a set v we increase w untill we first note topological behaviour around w_v = 3, as seen in subplot B. We note that while deeper topologies have longer oscillations, there is also less bulk occupancy.

The Pi-fidelity in the context of the SSH model is the expectation value of the opposite edge state after half a period. It is an important measure of the coherence of oscillations which we will use to examine the effectiveness of our protections. An alternative would be the

π

2-Fidelity, which would be less affected by oscillation decay but is still affected the same by

population of the bulk.

4.2 protections & breakdowns

As we justify the similiarity of the Raman coupling Hamiltonian to the SSH Hamiltonian by its adiabetic elimination, it is vital that we choose detunings such that we are in the adiabetic regime.Recall that our Lambda system is in an adiabatic regime for |∆| >> |Ω|. In figure 5, we can see that the behaviour of the ideal Raman coupling system is very similiar to the SSH model at our generic detuning −10 MHz. In addition, when we examine the Pi-fidelity of the ideal Raman coupling system and vary the intermediate detuning as seen in figure 6, we see no loss of fidelity near our considered regime. If we inspect the graphs near where there is breakdown, in figure 7, we can see that the loss in Pi-fidelity is indeed due to population of

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Figure 4: Different ratio’sw_v in the SSH Hamiltonian for v = 1. We consider the regimes w_v = 1, 3, 5, 10 in subplots A, B, C, D. We start seeing topological behaviour for w

v = 3. Note that more topological regimes have far longer oscillations and lower average bulk occupancy

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Figure 5: A comparison between a simulation of the ideal Raman coupling system(left) and the SSH system(right). Simulation performed for w_v =Ωw

Ωv of 3

excited states.

4.2.1 Average Hamiltonian

We motivate our use of the decaying cosine to extract data from the offset by considering sev-eral simulations in figure 8. The decaying cosine behaves well for smaller offsets, around 10%, but is not reliable for anything but an approximation of the first oscillation at very high offsets. We first consider how the Pi-fidelity changes with random v and w in figure 9. While the more topological regimes have a higher Pi-fidelity, the fidelity seems to drop in the same man-ner. We examine this behaviour further by normalizing the Pi-fidelities to the non-randomized coupling Pi-fidelity in figure 10. The drop in fidelity is indeed similiar for small offsets, but this behaviour starts becoming more divergent for larger offsets.

4.2.2 Dissipative Hamiltonian

We now consider the full Light coupling Hamiltonian, and in particular the consequences of natural decay and a non-zero laser linewidth.

When examining the effects of laser linewidth we consider different transitions and topo-logical regimes, as seen in figure 11. We note that while the 3P2 transition dominates at 0

linewidth, the topological regime proves more important when we increase the linewidth: The more topological regime loses fidelity far more quickly than the less topological regime. As we saw in figure 4, topological regimes have longer oscillations. This means that they will be subject the decoherence caused by the linewidth for longer before their first oscillation. We inspect the validity of this claim in figure 12. While the bulk states in the more topological regime w_v = 5 (subplot B) are less populated than that of w_v = 3(subplot D) at the same point in time, the oscillation of the more topological regime takes roughly seven times longer than the w_v = 3 regime. We also note that subplot D is the only system that shows significant population in the excited states.

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Figure 6: Pi-fidelity of the Raman coupling Hamiltonian with no collapse operators and varied inter-mediate detuning

Figure 7: A comparison between raman coupling systems with ideal laser, no excited state decay and in the topological regime Ωw

Ωv = 3. Subplot A has an intermediate detuning of -1 MHz, subplot B -3 MHz. We see that a lower intermediate detuning results in a shorter period and can cause excited state population

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Figure 8: SSH Hamiltonian with randomized coupling amplitudes, averaged over 100 iterations. The left column contains offsets of 10%, the right 50 %. The top row is for the topological regime w

v = 3, the bottom for w

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Figure 9: Pi-fidelity: offsets. The Pi-fidelity of SSH oscillations with a random offset in coupling strength for various topological regimes(v=1), averaged over 100 iterations. SSH case

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Figure 10: Pi-fidelity: offsets. A graph of how the Pi-fidelity of the SSH model is affected by an in-creasing random offset in coupling strength for various topological regimes. Each regime is normalized to its 0-offset Pi-fidelity.

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We noted earlier that oscillations on a longer timescale are more affected by decoherence. A low intermediate detuning can increase decoherence, as we saw in figure 7, but we must remember that Ω_w and Ω_v are inversely proportional to this intermediate detuning. By de-creasing the intermediate detuning we thus avoid decoherence from the laser linewidth at the cost of going less far into the adiabatic regime.

Let us first consider an ideal laser of linewidth 0, as shown in figure 13. Our only source of decoherence is from natural decay and low intermediate detuning. While the3P2 transition

is already fully coherent, it seems the3P1 transition enjoys an increased fidelity further into

the adiabatic regime. Because the decoherence of 3P1 was mainly caused by decay from the

excited states, going further into the adiabatic regimem, which limits excited state population, helped avoid this decoherence.

If we now consider a second source of decoherence, a laser linewidth of 100 Hz, we can see more divergent behaviour in the pi-fidelities shown in figure 14.

The 3P1 transition still mainly suffers decoherence from its natural decay, which means

increasing the intermediate detuning and thus going further into the adiabatic regime is more beneficial. The 3P2 transition, whose decoherence is mainly from the linewidth, enjoys an

increased fidelity by lowering the detuning and thus increasing the speed of its oscillations. Note that because the reason for the lower fidelity of the topological regime is the speed of its oscillations, we can once again enjoy the benefits of the topological regime by increasing the associated Rabi frequencies. In figure 14 we have shown this by increasing the Ω1 rabi

frequency from 0.3 MHz to 10 MHz. While this is a more favourable regime, we also have to note that besides the high intensity laser this requires, the intermediate detunings required might cause us to excite a different transition entirely.

5 Conclusion

Assuming all other parameters are ideal, fermionic Strontium coupled in a way as to create an artificial SSH model is resilient to variations in laser intensity.

While the simulation of the SSH model with fermionic Strontium is possible, the fidelities that result after considering the decoherence caused by the laser linewidth and natural decay of the atom are far too low to consider this use of SSH-like coupling in Strontium to create Qubits for any reasonable values of the laser intensity and intermediate detuning. Especially as the

3_P

2 transition, which has far more favourable fidelities, has a very narrow transition linewidth.

We cannot consider arbitrarily large detunings in part due to one of our assumptions. We assumed for our Raman coupling Hamiltonian, equation 8, that the differences between the energies of the excited m_f states was large enough that any laser pulse tuned for a specific transition could be disregarded for the other transitions. However, when we consider very large detunings we may start exciting these other transitions.

Aside from the technical limitations associated with a high-intensity laser, a high Rabi frequency might also result in off-resonant scattering.

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Figure 11: Pi-fidelity: LLW. A graph of how the Pi-fidelity is affected by laserlinewidths at various topological regimes for3_P

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Figure 12: An examination of laser linewidth decoherence for the 3P2 transition. figures A and C have LLW 0, B and D have LLW 100, A and B are regime w_v = 3. C and D are regime w

v = 5. We note that the oscillations of the more topological regime are much slower, which results in decoherence dominating the system

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Figure 13: Pi-fidelity: detuning at 0 Hz laser linewidth. A graph of how the Pi-fidelity is affected by the intermediate detunings at various topological regimes for3_P

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Figure 14: Pi-fidelity: detuning at 100 Hz laser linewidth. A graph of how the Pi-fidelity is affected by the intermediate detunings at various topological regimes for3_P

1 and3P2. 3P2transitions mainly suffer decoherence from the linewidth, and thus benefit from a lower detuning, while3_P

1 transitions mainly suffer decoherence from natural decay and benefit from a higher detuning. While we take Ω1 = 0.3 MHz for most transitions, a much faster oscillation Ω1 = 10 MHz would avoid most of the decoherence in the system.

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We have neglected how the feasibility of achieving certain rabi frequencies for σ− light is affected by the C.G. coefficients presented in figure 3. To combat this a combination of σ+ and σ− coupling could be used, but this has not been inspected in this paper.

We also note that our choices for the distribution and size of our errors has been generalized. Magnetic field noise would disproportionately affect the intermediate detuning on the higher-spin excited states. The laser linewidth, which is also an error source related to detuning, would instead cause errors in δa and δb.

Offsets in laser intensity would not only cause shifts in the laser coupling,but also a global detuning as shown in equation 5. This means that variations in laser intensity could not only affect coupling but also the detuning along the diagonal in a way similiar to the laser linewidth. We also note that the error calculation of the couplings was examined by considering the average of many Hamiltonians that did not vary with time. It would be more accurate instead to consider the evolution of a Hamiltonian whose errors are time-dependent.

Considering the alternating nature of the SSH model it might be possible to perform this simulation with only half the amount of lasers. If we split a laser and tune it such it is blue detuned for one transition and red for the transition next to it we might find a

Ωw Ωv = ∆vC 2 w ∆wG2v

where C denotes that Clebsch-Gordan coefficient on a scale of 0 to 1. We could manipulate this ratio such that Ωw

Ωv >> 1.

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The feasibility of using fermionic Strontium to simulate the SSH model