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Two-dimensional lattice gauge theories with superconducting quantum circuits.

Marcos D, Widmer P, Rico E, Hafezi M, Rabl P, Wiese UJ, Zoller P - Ann Phys (N Y) (2014)

Bottom Line: Fractionalized confining strings and the real-time dynamics of quantum phase transitions are accessible as well.We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories.The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.

View Article: PubMed Central - PubMed

Affiliation: Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria.

ABSTRACT

A quantum simulator of [Formula: see text] lattice gauge theories can be implemented with superconducting circuits. This allows the investigation of confined and deconfined phases in quantum link models, and of valence bond solid and spin liquid phases in quantum dimer models. Fractionalized confining strings and the real-time dynamics of quantum phase transitions are accessible as well. Here we show how state-of-the-art superconducting technology allows us to simulate these phenomena in relatively small circuit lattices. By exploiting the strong non-linear couplings between quantized excitations emerging when superconducting qubits are coupled, we show how to engineer gauge invariant Hamiltonians, including ring-exchange and four-body Ising interactions. We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories. The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.

No MeSH data available.


Basic building block for the lattice gauge theory architecture shown in Fig. 5. Two superconducting qubits (transmons) are coupled through a Josephson junction in parallel with a capacitor. This enables hopping and Kerr interactions between quantized photonic excitations at nodes 1 and 2. The value of the capacitor can be chosen appropriately in order to control the hopping of excitations.
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f000030: Basic building block for the lattice gauge theory architecture shown in Fig. 5. Two superconducting qubits (transmons) are coupled through a Josephson junction in parallel with a capacitor. This enables hopping and Kerr interactions between quantized photonic excitations at nodes 1 and 2. The value of the capacitor can be chosen appropriately in order to control the hopping of excitations.

Mentions: To implement interactions between neighboring qubits, we now consider the basic building block shown in Fig. 6. Here two transmons are connected via an additional Josephson junction with energy in parallel with a capacitor . The associated Hamiltonian is  [41,54,55](21)H=12Q→C−1Q→T−∑ℓ=1,2EJ(ℓ)cos(ϕℓϕ0)−EJ(Q)cos(ϕ1−ϕ2ϕ0), where and are the charge and flux operators at a node , , and (22)C=(C1+CQ−CQ−CQC2+CQ), is the capacitance matrix. As above, for small phase fluctuations we can expand the cosine functions and write the resulting Hamiltonian as (23)H=∑ℓ=1,2Hℓ+Hint. Here, (24)Hℓ=Qℓ22C¯ℓ+(EJ(ℓ)+EJ(Q))ϕℓ22ϕ02−(EJ(ℓ)+EJ(Q))ϕℓ424ϕ04, are the modified Hamiltonians for each qubit, where (25)C¯1=C1+C2CQC2+CQ,C¯2=C2+C1CQC1+CQ. By assuming that and , the coupling junction does not qualitatively change the single-qubit Hamiltonians, , with slightly modified frequencies . The remaining interaction Hamiltonian is given by(26)Hint≈CQC1C2Q1Q2−EJ(Q)ϕ02ϕ1ϕ2−EJ(Q)4ϕ04ϕ12ϕ22+EJ(Q)6ϕ04(ϕ1ϕ23+ϕ13ϕ2), and when projected onto the spin subspace of interest, we obtain (27)Hint≈−Ω2(S1z+S2z)−μ(S1+S2−+S1−S2+)−ΩS1zS2z. We notice that here the subindexes 1 and 2 refer to respective circuit nodes of Fig. 6, which are located on the links of the two dimensional lattice of Fig. 5. The first term in this Hamiltonian is a small frequency shift, which can be absorbed into a redefinition of the qubit frequency, . The other two contributions represent a spin flip-flop and an Ising-type spin–spin interaction with coupling strengths (28)μ=ε2(EJ(Q)EJ−CQC)−Ω,Ω=U2EJ(Q)EJ, where we have assumed . Still under the assumption that the capacitance and the Josephson energy are sufficiently small, the coupling between different neighboring transmons on the lattice of Fig. 5 can simply be added up. Considering different coupling constants around plaquettes () and across lattice sites (), and taking (29)μ□=μ,Ω□=Ω′,μ+=0,Ω+=Ω, we obtain the model (13), from which we then derive the effective Hamiltonian (15), with parameters and as defined in Eq. (16).


Two-dimensional lattice gauge theories with superconducting quantum circuits.

Marcos D, Widmer P, Rico E, Hafezi M, Rabl P, Wiese UJ, Zoller P - Ann Phys (N Y) (2014)

Basic building block for the lattice gauge theory architecture shown in Fig. 5. Two superconducting qubits (transmons) are coupled through a Josephson junction in parallel with a capacitor. This enables hopping and Kerr interactions between quantized photonic excitations at nodes 1 and 2. The value of the capacitor can be chosen appropriately in order to control the hopping of excitations.
© Copyright Policy - CC BY-NC-ND
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4263216&req=5

f000030: Basic building block for the lattice gauge theory architecture shown in Fig. 5. Two superconducting qubits (transmons) are coupled through a Josephson junction in parallel with a capacitor. This enables hopping and Kerr interactions between quantized photonic excitations at nodes 1 and 2. The value of the capacitor can be chosen appropriately in order to control the hopping of excitations.
Mentions: To implement interactions between neighboring qubits, we now consider the basic building block shown in Fig. 6. Here two transmons are connected via an additional Josephson junction with energy in parallel with a capacitor . The associated Hamiltonian is  [41,54,55](21)H=12Q→C−1Q→T−∑ℓ=1,2EJ(ℓ)cos(ϕℓϕ0)−EJ(Q)cos(ϕ1−ϕ2ϕ0), where and are the charge and flux operators at a node , , and (22)C=(C1+CQ−CQ−CQC2+CQ), is the capacitance matrix. As above, for small phase fluctuations we can expand the cosine functions and write the resulting Hamiltonian as (23)H=∑ℓ=1,2Hℓ+Hint. Here, (24)Hℓ=Qℓ22C¯ℓ+(EJ(ℓ)+EJ(Q))ϕℓ22ϕ02−(EJ(ℓ)+EJ(Q))ϕℓ424ϕ04, are the modified Hamiltonians for each qubit, where (25)C¯1=C1+C2CQC2+CQ,C¯2=C2+C1CQC1+CQ. By assuming that and , the coupling junction does not qualitatively change the single-qubit Hamiltonians, , with slightly modified frequencies . The remaining interaction Hamiltonian is given by(26)Hint≈CQC1C2Q1Q2−EJ(Q)ϕ02ϕ1ϕ2−EJ(Q)4ϕ04ϕ12ϕ22+EJ(Q)6ϕ04(ϕ1ϕ23+ϕ13ϕ2), and when projected onto the spin subspace of interest, we obtain (27)Hint≈−Ω2(S1z+S2z)−μ(S1+S2−+S1−S2+)−ΩS1zS2z. We notice that here the subindexes 1 and 2 refer to respective circuit nodes of Fig. 6, which are located on the links of the two dimensional lattice of Fig. 5. The first term in this Hamiltonian is a small frequency shift, which can be absorbed into a redefinition of the qubit frequency, . The other two contributions represent a spin flip-flop and an Ising-type spin–spin interaction with coupling strengths (28)μ=ε2(EJ(Q)EJ−CQC)−Ω,Ω=U2EJ(Q)EJ, where we have assumed . Still under the assumption that the capacitance and the Josephson energy are sufficiently small, the coupling between different neighboring transmons on the lattice of Fig. 5 can simply be added up. Considering different coupling constants around plaquettes () and across lattice sites (), and taking (29)μ□=μ,Ω□=Ω′,μ+=0,Ω+=Ω, we obtain the model (13), from which we then derive the effective Hamiltonian (15), with parameters and as defined in Eq. (16).

Bottom Line: Fractionalized confining strings and the real-time dynamics of quantum phase transitions are accessible as well.We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories.The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.

View Article: PubMed Central - PubMed

Affiliation: Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria.

ABSTRACT

A quantum simulator of [Formula: see text] lattice gauge theories can be implemented with superconducting circuits. This allows the investigation of confined and deconfined phases in quantum link models, and of valence bond solid and spin liquid phases in quantum dimer models. Fractionalized confining strings and the real-time dynamics of quantum phase transitions are accessible as well. Here we show how state-of-the-art superconducting technology allows us to simulate these phenomena in relatively small circuit lattices. By exploiting the strong non-linear couplings between quantized excitations emerging when superconducting qubits are coupled, we show how to engineer gauge invariant Hamiltonians, including ring-exchange and four-body Ising interactions. We demonstrate that, despite decoherence and disorder effects, minimal circuit instances allow us to investigate properties such as the dynamics of electric flux strings, signaling confinement in gauge invariant field theories. The experimental realization of these models in larger superconducting circuits could address open questions beyond current computational capability.

No MeSH data available.