The time-dependent $CP$ asymmetry in $B_s^0 to J/psi K^+K^-$ decays is measured using $pp$ collision data, corresponding to an integrated luminosity of $3.0$fb$^{-1}$, collected with the LHCb detector at centre-of-mass energies of $7$ and $8$TeV. In a sample of 96 000 $B_s^0 to J/psi K^+K^-$ decays, the $CP$-violating phase $phi_s$ is measured, as well as the decay widths $Gamma_{L}$ and $Gamma_{H}$ of the light and heavy mass eigenstates of the $B_s^0-bar{B}_s^0$ system. The values obtained are $phi_s = -0.058 pm 0.049 pm 0.006$ rad, $Gamma_s equiv (Gamma_{L}+Gamma_{H})/2 = 0.6603 pm 0.0027 pm 0.0015$ps$^{-1}$, and$DeltaGamma_s equiv Gamma_{L} - Gamma_{H} = 0.0805 pm 0.0091 pm 0.0032$ps$^{-1}$, where the first uncertainty is statistical and the second systematic. These are the most precise single measurements of those quantities to date. A combined analysis with $B_s^{0} to J/psi pi^+pi^-$ decays gives $phi_s = -0.010 pm 0.039 $rad. All measurements are in agreement with the Standard Model predictions. For the first time the phase $phi_s$ is measured independently for each polarisation state of the $K^+K^- $ system and shows no evidence for polarisation dependence.
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