Steady state in the regime of ultra strong system-bath coupling and high temperature

Understanding better the dynamics and steady states of systems strongly coupled to their environment (idealized as bosonic thermal baths) is a great theoretical challenge with promising applications in several fields of quantum technologies. Among several strategies to gain access to the steady state, one consists in obtaining approximate expressions of the mean force Gibbs state – the reduced state of the global system-bath thermal state – largely credited to be the steady state. Here, we present analytical expressions of corrective terms to the ultrastrong coupling limit of the mean force Gibbs state [1], which has been recently derived [2]. We find that the first order term coincides precisely with the first order correction obtained from a dynamical approach [3] – master equation in the strong-decoherence regime. This strengthens the identification of the reduced global thermal state with the mean force Gibbs state. Additionally, we also compare our expressions with an other recent result obtained from a high temperature expansion of the mean force Gibbs state [4]. We observe good numerical agreements for ultra strong coupling as well as for high temperatures. This confirms the validity of all these results. In particular, we show that, in term of coherences, all three results allow one to sketch the transition from ultrastrong coupling to weak coupling. Finally, we briefly sketch a method to obtain higher order terms, and provide an upper bound of the error between the approximated expression containing terms up to order n, and the exact expression. We present additional numerical simulations containing terms up to fifth order with the corresponding error bound. This allows one to evaluate the range of validity of the above techniques.