Open quantum systems, effective Hamiltonians, and device characterization

High fidelity models, which are able to both support accurate device characterization and correctly account for environmental effects, are crucial to the engineering of scalable quantum technologies. As it ensures positivity of the density matrix, one preferred model of open systems describes the dynamics with a master equation in Lindblad form. In practice, Linblad operators are rarely derived from first principles, and often a particular form of annihilator is assumed. This results in dynamical models that miss those additional terms which must generally be added for the master equation to assume the Lindblad form, together with the other concomitant terms that must be assimilated into an effective Hamiltonian to produce the correct free evolution. In first principles derivations, such additional terms are often canceled (or countered), frequently in a somewhat ad hoc manner, leading to a number of competing models. Whilst the implications of this paper are quite general, to illustrate the point we focus here on an example anharmonic system; specifically that of a superconducting quantum interference device (SQUID) coupled to an Ohmic bath. The resulting master equation implies that the environment has a significant impact on the system's energy; we discuss the prospect of keeping or canceling this impact and note that, for the SQUID, monitoring the magnetic susceptibility under control of the capacitive coupling strength and the externally applied flux results in experimentally measurable differences between a number of these models. In particular, one should be able to determine whether a squeezing term of the form $\widehat{X}\widehat{P}+\widehat{P}\widehat{X}$ should be present in the effective Hamiltonian or not. If model generation is not performed correctly, device characterization will be prone to systemic errors.

Figure 1

The energy eigenvalues of the effective system Hamiltonian ${\widehat{\mathcal{H}}}_{\text{eff}}$, including all anticommutator contributions from the environment, as a function of external flux ${\mathrm{\Phi }}_x$, for various coupling ratios $g$. Results are obtained with parameter values of Josephson energy $\hslash \nu =6.693\times {10}^{-22}\mathrm{J}=7.74\hslash {\omega }_0$, capacitance $C=5\times {10}^{-15}\mathrm{F}$, inductance $L=3\times {10}^{-10}\mathrm{H}$, damping rate $\gamma =0.05{\omega }_0$, and bath cut-off frequency $\mathrm{\Omega }=10{\omega }_0$. Increasing the coupling ratio has a noticeable effect on the energy level structure.

Figure 2

Spiderweb Diagram. The solid line shows the lowest eigenvalue for the SQUID Hamiltonian plus various combinations of anticommutator terms that arise from coupling to the environment, as labeled. Dashed and dotted lines show, for comparison, the lowest eigenvalues of ${\widehat{\mathcal{H}}}_0$ (the isolated SQUID) and ${\widehat{\mathcal{H}}}_{\text{eff}}$ (corrected for environmental contributions, but with ${\widehat{\mathcal{H}}}_{LS}={\widehat{\mathcal{H}}}_{\delta }$), respectively. Note that the contributions of ${\widehat{\mathcal{H}}}_{XP}, {\widehat{\mathcal{H}}}_{XS}$, and ${\widehat{\mathcal{H}}}_{PS}$ are not independent, and that the combination ${\widehat{\mathcal{H}}}_{XP}+{\widehat{\mathcal{H}}}_{XS}$ accounts for the majority of the alteration. The origin corresponds to $E=0$, the outer radius to $E=6.0\hslash {\omega }_0$. Here, ${\mathrm{\Phi }}_x=0.5{\mathrm{\Phi }}_0, g=1.8$.

Figure 3

Ground state magnetic susceptibility as a function of the external flux ${\mathrm{\Phi }}_x$, and capacitive coupling strength $g$, with all anticommutator terms included in ${\widehat{\mathcal{H}}}_{\text{eff}}$. As the coupling strength is increased, the external flux dependence of the magnetic susceptibility flattens out, showing measurable differences that could form the basis of experiments to certify specific candidate open-system models.

Figure 4

The two nonzero eigenvalues of the coefficent matrix $a_{AB}$ [Eq. (11) ], which determine the strength of the Lindblad operators, as a function of coupling ratio $g$. Making the minimum addition to ensure Lindblad form constrains $g$, for $\xi =0.25$, as here, $g\in (0.227,4.4)$. An alternative, more invasive change to $a_{AB}$, which affects both $a_{\text{PP}}$ and $a_{\text{SS}}$, produces the dot-dashed eigenvalues. Markers indicate values at $g=0.25$, used in decoherence calculations.

Figure 5

The real and imaginary coefficients of the component operators of the dominant Lindblad, eigenvalue ${\lambda }_1$, as a function of coupling ratio $g$. In addition to the expected generalized annihilator contributions, there are also significant contributions from $\widehat{S}$. This shows that the contribution from the Josephson junction term to the Lindblads are significant, and depend nontrivially on the coupling strength $g$. It is also noteworthy that this naturally leads to an external flux dependence in the Lindblad.

Figure 6

As in Fig. 5 for the second Lindblad, eigenvalue ${\lambda }_2$.

Figure 7

The steady state purity of a SQUID coupled inductively and capacitively to an Ohmic bath as a function of external flux for various capacitive coupling strengths at a bath cut-off frequency of $\mathrm{\Omega }=2{\omega }_0$. Note that the well created about ${\mathrm{\Phi }}_x=0.5{\mathrm{\Phi }}_0$, and flattens and widens as capacitive coupling strength is increased.

Figure 8

The steady state purity of a SQUID coupled inductively and capacitively to an Ohmic bath as a function of external flux for various bath cutoff frequencies at a coupling ratio of $g=0.55$. A drop in purity below a value of a half suggests leakage to higher energy levels than simply the ground state.