Liouvillian exceptional points in continuous variable system

The Liouvillian exceptional points for a quantum Markovian master equation of an oscillator in a generic environment are obtained. They occur at the points when the modified frequency of the oscillator vanishes, whereby the eigenvalues of the Liouvillian become real. In a generic system there are two parameters that modify the oscillator's natural frequency. One of the parameters can be the damping rate. The exceptional point then corresponds to critical damping of the oscillator. This situation is illustrated by the Caldeira--Leggett (CL) equation and the Markovian limit of the Hu--Paz--Zhang (HPZ) equation. The other parameter changes the oscillator's effective mass whereby the exceptional point is reached in the limit of extremely heavy oscillator. This situation is illustrated by a modified form of the Kossakowski--Lindblad (KL) equation. The eigenfunctions coalesce at the exceptional points and break into subspaces labelled by a natural number $N$. In each of the $N$-subspace, there is a $(N+1)$-fold degeneracy and the Liouvillian has a Jordan block structure of order-$(N+1)$. We obtain the explicit form of the generalized eigenvectors for a few Liouvillians. Because of the degeneracies, there is a freedom of choice in the generalized eigenfunctions. This freedom manifests itself as an invariance in the Jordan block structure under a similarity transformation whose form is obtained. We compare the relaxation of the first excited state of an oscillator in the underdamped region, critically damped region which corresponds to the exceptional point, and overdamped region using the generalized eigenvectors of the CL equation.Comment: To appear in Physica A (2023

General symmetry in the reduced dynamics of two-level system

We study general transformation on the density matrix of two-level system that keeps the expectation value of observable invariant. We introduce a set of generators that yields hermiticity and trace preserving general transformation which casts the transformation into simple form. The general transformation is in general not factorized and not completely positive. Consequently, either the parameter of transformation or the density matrix it acts on needs to be restricted. It can transform the system in the forward and backward direction with regard to its parameter, not as a semigroup in the time translation symmetry of dynamical maps. The general transformation can rotate the Bloch vector circularly or hyperbolically, dilate it or translate it. We apply the general transformation to study the general symmetry of amplitude damping and phase damping in two-level system. We generalize the generators to higher level systems.Comment: Accepted by European Physical Journal

Groundwater quality studies: A Case study of the Densu Basin, Ghana

Groundwater samples from 68 communities within the Densu basin were sampled and analysed over a period of 1 year for various physico-chemical water quality parameters using appropriate certified and acceptable international procedures, in order to assess the water types as well as the suitability of groundwater within the basin for drinking and other domestic uses. The study showed that most of the physico-chemical parameters were within the World Health Organization limits recommended for drinking water. However, a few of the boreholes were slightly acidic. Someboreholes showed high level of mineralization. Borehole (GaD 6) at Pokuasi recorded the highest conductivity value of 7780.0 ìS/cm. High levels of nitrates were also recorded in certain communities within the basin. These include Aponsahene (105.8 mg/l), Damang (66.0 mg/l), Adzen Kotoku (61.5 mg/l), Afabeng (50.8 mg/l), New Mangoase (48.3mg/l), Asuoatwene (41.3 mg/l), Potrase (33.6 mg/l) and Maase (33.3 mg/l). Correlations between major ions showed expected process-based relationship between Ca2+ and Cl- (r = 0.86); Mg2+ and Cl- (r = 0.84); Na+ and SO4 2- (r =0.77); Na+ and Cl- (r = 0.75); Mg2+ and SO42- (r = 0.74); Mg2+ and Ca2+ (r = 0.71); Ca2+ and SO4 2- (r = 0.58); and K+ and SO4 2- (r = 0.51), derived mainly from the geochemical and biochemical processes within the aquifer. Two major hydrochemical water types constituting 41% of groundwater sources within the basin have been delineated. These are Ca-Mg-HCO3 water (19%) and Na –Cl or Na –Cl –HCO3- Cl water (22%) types. Fifty-nine per cent of groundwatersources are mixed waters with no particular cation predominating, and having either HCO3-or SO4 2- ions as the main anion

Results at 24 months from the prospective, randomized, multicenter Investigational Device Exemption trial of ProDisc-C versus anterior cervical discectomy and fusion with 4-year follow-up and continued access patients.

BackgroundCervical total disk replacement (TDR) is intended to address pain and preserve motion between vertebral bodies in patients with symptomatic cervical disk disease. Two-year follow-up for the ProDisc-C (Synthes USA Products, LLC, West Chester, Pennsylvania) TDR clinical trial showed non-inferiority versus anterior cervical discectomy and fusion (ACDF), showing superiority in many clinical outcomes. We present the 4-year interim follow-up results.MethodsPatients were randomized (1:1) to ProDisc-C (PDC-R) or ACDF. Patients were assessed preoperatively, and postoperatively at 6 weeks and 3, 6, 12, 18, 24, 36, and 48 months. After the randomized portion, continued access (CA) patients also underwent ProDisc-C implantation, with follow-up visits up to 24 months. Evaluations included Neck Disability Index (NDI), Visual Analog Scale (VAS) for pain/satisfaction, and radiographic and physical/neurologic examinations.ResultsRandomized patients (103 PDC-R and 106 ACDF) and 136 CA patients were treated at 13 sites. VAS pain and NDI score improvements from baseline were significant for all patients (P < .0001) but did not differ among groups. VAS satisfaction was higher at all time points for PDC-R versus ACDF patients (P = .0499 at 48 months). The percentage of patients who responded yes to surgery again was 85.6% at 24 months and 88.9% at 48 months in the PDC-R group, 80.9% at 24 months and 81.0% at 48 months in the ACDF group, and 86.3% at 24 months in the CA group. Five PDC-R patients (48 months) and no CA patients (24 months) had index-level bridging bone. By 48 months, approximately 4-fold more ACDF patients required secondary surgery (3 of 103 PDC-R patients [2.9%] vs 12 of 106 ACDF patients [11.3%], P = .0292). Of these, 6 ACDF patients (5.6%) required procedures at adjacent levels. Three CA patients required secondary procedures (24 months).ConclusionsOur 4-year data support that ProDisc-C TDR and ACDF are viable surgical options for symptomatic cervical disk disease. Although ACDF patients may be at higher risk for additional surgical intervention, patients in both groups show good clinical results at longer-term follow-up