Integrability of the Rabi Model

Braak, Daniel

doi:10.1103/physrevlett.107.100401

Cited by 796 publications

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“…The ultrastrong coupling brings about fundamentally different physics deeply connected to the high degree of entanglement between the "matter" and the photon [15][16][17][18][19]. However, the effect of ultrastrong coupling on the lowenergy excitations of an array of coupled cavity-QED systems remains unclear, and is our main concern in this work.…”

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confidence: 99%

Large-scale maximal entanglement and Majorana bound states in coupled circuit quantum electrodynamic systems

Hwang

Choi

2013

Phys. Rev. B

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We have studied the low-lying excitations of a chain of coupled circuit-QED systems in the ultrastrong coupling regime, and report several intriguing properties of its two nearly degenerate ground states. The ground states are Schrödinger cat states at a truly large scale, involving maximal entanglement between the resonators and the qubits, and are mathematically equivalent to Majorana bound states. With a suitable design of physical qubits, they are protected against local fluctuations and constitute a non-local qubit. Further, they can be probed and manipulated coherently by attaching an empty resonator to one end of the circuit-QED chain.Confronted with formidable difficulties in solving strongly interacting many-body systems, it has been desired to find good quantum simulators. It may seem natural to simulate a many-body system with another tunable system of massive particles such as ultracold atomic gases [1]. In fact, any controllable quantum system, notably quantum computer if ever practical, can simulate efficiently many-body systems [2]. Indeed it has been recognized that photons confined in coupledcavities simulate closely the quantum behaviors of stronglycorrelated many-body systems [3,4]. Subsequent studies have revealed that Bose-Hubbard model [5], interacting spin models [4,6], and other exotic quantum phases [7] can be simulated efficiently using the coupled-cavities. Further, recent advances in solid-state devices such as circuit-QED systems [8,9] and micro-cavities [10,11] and ongoing efforts to fabricate large-scale cavity arrays [12] make the array of coupled cavities a promising candidate for an efficient quantum simulator.Meanwhile, the ultrastrong coupling regime of the cavity-QED system, where the light-matter coupling energy is comparable to or even higher than the energy of the cavity field, has been envisioned [13] and experimentally demonstrated [14]. The ultrastrong coupling brings about fundamentally different physics deeply connected to the high degree of entanglement between the "matter" and the photon [15][16][17][18][19]. However, the effect of ultrastrong coupling on the lowenergy excitations of an array of coupled cavity-QED systems remains unclear, and is our main concern in this work.In this paper, we investigate the low-lying excitations of a one-dimensional (1D) array of circuit-QED systems (cQEDs), with each cQED being in the ultrastrong coupling regime; see Fig. 1. It turns out that the array permits two nearly degenerate ground states separated by a finite energy gap from the continuum of higher-energy states. We find several intriguing properties of the two ground states: (i) They are Schrödinger cat states at a truly large scale, and involve maximal entanglement between the resonators and the qubits. (ii) With a suitable design of physical qubits, the two ground states are protected against local fluctuations and constitute a non-local qubit [20]. (iii) They are mathematically equivalent to the long-searched Majorana bound states [21]. (iv) They can be probed and mani...

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confidence: 99%

Large-scale maximal entanglement and Majorana bound states in coupled circuit quantum electrodynamic systems

Hwang

Choi

2013

Phys. Rev. B

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“…Another advantage is that a lumped-element LC oscillator has only one resonant mode. Together with the strong anharmonicity of the flux qubit, we can expect that our circuit will realize the Rabi model [19][20][21][22] , which is one of the simplest possible quantum models of qubitoscillator systems, with no additional energy levels in the range of interest.The coupling Hamiltonian can be written as 9 H c = g σ z (â +â † ), where g = MI p I zpf is the coupling energy and M( L c ) is the mutual inductance between the qubit and the LC oscillator. Importantly, a Josephson-junction circuit is used as a large inductive coupler 23 (Fig.…”

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Superconducting qubit–oscillator circuit beyond the ultrastrong-coupling regime

et al. 2016

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The interaction between an atom and the electromagnetic field inside a cavity 1-6 has played a crucial role in developing our understanding of light-matter interaction, and is central to various quantum technologies, including lasers and many quantum computing architectures. Superconducting qubits 7,8 have allowed the realization of strong 9,10 and ultrastrong 11-13 coupling between artificial atoms and cavities. If the coupling strength g becomes as large as the atomic and cavity frequencies (∆ and ω o , respectively), the energy eigenstates including the ground state are predicted to be highly entangled 14 . There has been an ongoing debate 15-17 over whether it is fundamentally possible to realize this regime in realistic physical systems. By inductively coupling a flux qubit and an LC oscillator via Josephson junctions, we have realized circuits with g/ω o ranging from 0.72 to 1.34 and g/∆ 1. Using spectroscopy measurements, we have observed unconventional transition spectra that are characteristic of this new regime. Our results provide a basis for ground-state-based entangled pair generation and open a new direction of research on strongly correlated light-matter states in circuit quantum electrodynamics.We begin by describing the Hamiltonian of each component in the qubit-oscillator circuit, which comprises a superconducting flux qubit and an LC oscillator inductively coupled to each other by sharing a tunable inductance L c , as shown in the circuit diagram in Fig. 1a.The Hamiltonian of the flux qubit can be written in the basis of two states with persistent currents flowing in opposite directions around the qubit loop 18 , |L q and |R q , as H q = − (∆σ x + εσ z )/2, where ∆ and ε = 2I p 0 (n φq − n φq0 ) are the tunnel splitting and the energy bias between |L q and |R q , I p is the maximum persistent current, and σ x, z are Pauli matrices. Here, n φq is the normalized flux bias through the qubit loop in units of the superconducting flux quantum, 0 = h/2e, and n φq0 = 0.5 + k q , where k q is the integer that minimizes |n φq − n φq0 |. The macroscopic nature of the persistent-current states enables strong coupling to other circuit elements. Another important feature of the flux qubit is its strong anharmonicity: the two lowest energy levels are well isolated from the higher levels.The Hamiltonian of the LC oscillator can be written asC is the resonance frequency, L 0 is the inductance of the superconducting lead, L qc ( L c ) is the inductance across the qubit and coupler (see Supplementary Section 2), C is the capacitance, andâ (â † ) is the oscillator's annihilation (creation) operator. Figure 1b shows a laser microscope image of the lumped-element LC oscillator, where L 0 is designed to be as small as possible to maximize the zeropoint fluctuations in the currentand hence achieve strong coupling to the flux qubit, while C is adjusted so as to achieve a desired value of ω o . The freedom of choosing L 0 for large I zpf is one of the advantages of lumped-element LC oscillators over coplanar-waveguide ...

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“…28 For higher couplings g/ω a,m ≳ 0.1 the RWA is no longer applicable and the excitation number conservation of the JC model is replaced by a conservation of excitation number parity. 10,29 In this regime, making the RWA would lead to a deviation in the energy spectrum of the system known as Bloch-Siegert shift χ BS , marking the entry into the USC regime. 30 Our samples, depicted in Fig.…”

Section: Resultsmentioning

confidence: 99%

“…This ultra-strong coupling (USC) regime, described by the quantum Rabi model, shows the breakdown of excitation number as conserved quantity, resulting in a significant theoretical challenge. 9,10 In the regime of g=ω a;m ' 1, known as deep-strong coupling (DSC), a symmetry breaking of the vacuum is predicted 11 (i.e., qualitative change of the ground state), similar to the Higgs mechanism or Jahn-Teller instability. To date, U/DSC with superconducting circuits has only been realized with flux qubits 6,12 or in the context of quantum simulations.…”

Section: Introductionmentioning

confidence: 99%

Multi-mode ultra-strong coupling in circuit quantum electrodynamics

et al. 2017

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With the introduction of superconducting circuits into the field of quantum optics, many experimental demonstrations of the quantum physics of an artificial atom coupled to a single-mode light field have been realized. Engineering such quantum systems offers the opportunity to explore extreme regimes of light-matter interaction that are inaccessible with natural systems. For instance the coupling strength g can be increased until it is comparable with the atomic or mode frequency ω a,m and the atom can be coupled to multiple modes which has always challenged our understanding of light-matter interaction. Here, we experimentally realize a transmon qubit in the ultra-strong coupling regime, reaching coupling ratios of g/ω m = 0.19 and we measure multi-mode interactions through a hybridization of the qubit up to the fifth mode of the resonator. This is enabled by a qubit with 88% of its capacitance formed by a vacuum-gap capacitance with the center conductor of a coplanar waveguide resonator. In addition to potential applications in quantum information technologies due to its small size, this architecture offers the potential to further explore the regime of multi-mode ultra-strong coupling. With strong coupling, 3 where the coupling is larger than the dissipation rates γ and κ of the atom and mode respectively, experiments such as photon-number resolution 4 or Schrödinger-cat revivals 5 have beautifully displayed the quantum physics of a single-atom coupled to the electromagnetic field of a single mode. As the field matures, circuits of larger complexity are explored, 6-8 opening the prospect of controllably studying systems that are theoretically and numerically difficult to understand.One example is the interaction between an (artificial) atom and an electromagnetic mode where the coupling rate becomes a considerable fraction of the atomic or mode eigen-frequency. This ultra-strong coupling (USC) regime, described by the quantum Rabi model, shows the breakdown of excitation number as conserved quantity, resulting in a significant theoretical challenge. 9,10 In the regime of g=ω a;m ' 1, known as deep-strong coupling (DSC), a symmetry breaking of the vacuum is predicted 11 (i.e., qualitative change of the ground state), similar to the Higgs mechanism or Jahn-Teller instability. To date, U/DSC with superconducting circuits has only been realized with flux qubits 6,12 or in the context of quantum simulations. 13,14 Using transmon qubits for USC is interesting due to their higher coherence rates 15 as well as their weakly-anharmonic nature, allowing the exploration of a different Hamiltonian than with flux qubits. 16 Since transmon qubits are currently a standard in quantum computing efforts, [17][18][19] implementing USC in a transmon architecture could also have technological applications by

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Integrability of the Rabi Model

Cited by 796 publications

References 42 publications

Large-scale maximal entanglement and Majorana bound states in coupled circuit quantum electrodynamic systems

Large-scale maximal entanglement and Majorana bound states in coupled circuit quantum electrodynamic systems

Superconducting qubit–oscillator circuit beyond the ultrastrong-coupling regime

Multi-mode ultra-strong coupling in circuit quantum electrodynamics

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