Simulation of supersymmetric quantum mechanics in a Cooper-pair box shunted by a Josephson rhombus

Abstract

Supersymmetries in quantum mechanics offer a way to obtain degeneracies in the excitation spectrum which do not originate from selection rules. The mechanism behind the degeneracies is the same as the one that leads to the miraculous cancellations of divergences in supersymmetric field theories found in the high energy physics context. There is up to now no realistic proposal of nonintegrable systems that show level degeneracies due to a supersymmetric structure. Here, we propose an implementation of a quantum-mechanical supersymmetry in a Cooper-pair box shunted by a Josephson junction rhombus which is effectively π-periodic in the superconducting phase difference. For a characteristic ratio between the strength of the 2π-and the π-periodic junction, we find a two-fold degeneracy of all the energy levels all the way from the weak junction/charge qubit limit to the strong junction/transmon regime. We provide explicit experimental values for the parameters of the system and show that tuning in and out of the supersymmetric point is easily achieved by varying an external gate voltage. We furthermore discuss a microwave experiment to detect the supersymmetry and conclude that it can indeed be implemented with currently existing Josephson junction technology.PACS numbers: 03.67. Ac, 11.30.Pb, 85.25.Cp, 42.50.Pq The macroscopic quantum mechanics of superconducting circuits has allowed the experimental simulation of many complex quantum phenomena such as phase transitions, 1 quantum spins, 2 or dynamics in open systems.3 Theoretically, the quantum simulation of intricate subjects such as Hawking radiation 4 and lattice gauge theories 5-7 has been proposed. In the plethora of phenomena that can be simulated with the help of superconducting circuits, 8,9 degeneracies due to quantummechanical supersymmetries have notably been absent. Typically, degeneracies in the spectrum arise when the Hamiltonian commutes with all group elements of a nonAbelian symmetry which translates into selection rules demanding vanishing off-diagonal and equal diagonal matrix elements of the Hamiltonian within the same irreducible representation.10 The degeneracy of the states thus always follows from the dimension of the representation. Supersymmetry on the other hand does not simply forbid different states to couple but it makes sure that in each order of perturbation theory there is always a pair of terms canceling each other. 11,12It is intriguing that the degeneracies of supersymmetric quantum mechanics occur by the same mechanism 11 that leads to a miraculous cancellation of divergences in supersymmetric field theories and makes supersymmetries an important tool of particle physics.13 In trivial cases like the free particle 14 or the Jaynes-Cummings model, 15 however, the supersymmetric structure is irrelevant since the spectrum is exactly solvable. In order to deepen the connection to the ideas in the high-energy context, it is thus of vital importance to propose a non-integrable system where the level degeneracy can be solely understo...

Conditional quasi-exact solvability of the quantum planar pendulum and of its anti-isospectral hyperbolic counterpart

“…The same applies for the dimensions of the corresponding matrices N (Γ ) κ defined in equations (23) and (24). In summary, as implied by the equality of the matrix representations of the respective Schrödinger operators, equation (32), the hyperbolic Razavy system is, like the planar pendulum, a C-QES system, i.e., analytic solutions can only be found under the condition that the topological index κ be an integer. At the same time, the Razavy system is also QES, i.e., only a finite number of analytic solutions exist, and this number is given by the value of κ.…”

“…Internal rotation, molecular torsion [5][6][7][8][9][10] Uniaxial paramagnets or single-molecule magnets [11][12][13] Molecular alignment/orientation [14,15] Quasi-exactly solvable double well potential [16][17][18] Molecules in combined fields [3] Quantum field theory [19][20][21] Band structure of condensed matter systems [16,17,22] PT-symmetry [23] Nonlinear dynamics [24,25] Nonlinear coherent structures [26,27] Solitons [28,29] Quantum theory of instantons [29][30][31] Josephson Junction Rhombus [32] Model of a proton in a hydrogen bond [17,21,33] Optical lattice [34] each of them associated with one of the four irreducible representations of the C 2v point group. For each of the irreducible representations, the Hamiltonian of the planar pendulum is found to be an infinite tridiagonal matrix containing a finite-dimensional block characterised by a particular condition imposed on the pendulum's parameters and expressed in terms of an integer, termed the topological index.…”

Observation of Supersymmetry and its Spontaneous Breaking in a Trapped Ion Quantum Simulator

“…Although the SUSY theory has inspired wide applications in fields like optics 7 , condensed matter physics 8 , quantum chaos 8 or even outside physics 9 , whether it correctly describes the physical world has not yet been determined by the state-of-the-art high energy physics experiments 10,11 or other indirect experimental evidences 12,13 . However, many theoretical proposals already exist [14][15][16][17][18] to examine its effects by quantum simulation 19,20 . As one of the leading platforms for quantum information processing with long coherence time, convenient initialization and readout, as well as accurate laser or microwave control [21][22][23][24][25] , ion trap has demonstrated the quantum simulation of various phenomena such as quantum phase transitions 26,27 , manybody dynamics 26 , relativistic effects 28 and quantum field theories 29 .…”

Double-periodic Josephson junctions in a quantum dissipative environment

“…Moreover, there exist other means to realize different periodicities, including hybrid junctions involving superconducting and ferromagnetic layers which have been theoretically predicted to exhibit a controllable Josephson periodicity 37 . Another proposal is a specific arrangement of four Josephson junctions with a π periodicity called the Josephson rhombus and also appearing in certain Josephson arrays 27,[38][39][40][41] .…”