Time-optimal processes for interacting spin systems

Reversible adiabatic processes connecting thermal equilibrium states are usually considered to be infinitely slow. Recently, fast reversible adiabatic processes for quantum systems have been discussed. Here we present time-optimal processes for a paradigmatic ensemble of two interacting spin1 2 systems in an external magnetic field, which previously had been employed as working fluid in a quantum refrigerator. These processes are realized by appropriate bang-bang or quasi–bang-bang controls of the external magnetic field. Explicit control protocols including the necessary times for a transition connecting thermal equilibrium states depending on the limiting conditions on the magnetic field strength are presented. Copyright c © EPLA, 2012 Optimal control of quantum systems has attracted considerable interest in recent years [1–7]. Sources of interest in the area are manifold and range from quantum computing and control of chemical reactions to manipulating Bose-Einstein condensates. Our present interest in such control stems from the quest to attain lower and lower temperatures with more and more complicated systems [8]. Heat pumps that reach such extreme temperatures perforce require quantum treatment. Lossless control of the working fluid can be achieved in the adiabatic limit which requires slow operation. Due to heat leaks from the surroundings, the objective for heat pumping from extremely low temperatures is not the coefficient of performance but rather the rate of heat removal from the system being cooled. That is not to say that irreversibilities do not play an important role for such engines. Quite the contrary, any irreversibility in the branch cooling the working fluid will limit the temperature that the working fluid can attain as well as the subsequent rate of heat transfer out of the cooled system. Thus, the branch of the cooling cycle during which the working fluid is cooled is the natural focus for optimal control. Fast cooling of working fluids runs into the problem of exciting internal degrees of freedom, a phenomenon also termed quantum friction [9,10]. Surprisingly, however, we have found [11] that there exist very fast effectively (a)E-mail: [email protected] adiabatic processes, i.e., processes that begin and end at states that the adiabatic process would connect thereby eliminating the quantum frictional irreversibilities. In a previous effort [8,11], we have exhibited such processes for cooling a working fluid made of harmonic oscillators. In the present work, we show that such fast effectively adiabatic cooling can also be achieved for a working fluid composed of two coupled spins – the original system for studying quantum friction. Whether such fast effectively adiabatic processes exist for other quantum systems is an important open problem. This paper presents a second example. Optimal control of spin systems has a long history [4,12,13]; the resultant controls have a rich and well– worked-out structure, which are important for NMR [1,4]. In general, the structure of the achievable controls depends strongly on the type of control one assumes possible. For example, the controls applied in [12] had more degrees of freedom than those used in the present paper. The present effort also deviates somewhat from the standard problem in the field in its view of the initial and final states as well as the natural constraints on the allowed controls. Rather than considering arbitrary transfer of populations between excited and ground states, we focus on controls that carry our system from one thermal state to another. This makes it natural to consider the problem of minimum time transfer using external fields constrained to operate between initial and final frequencies. We carry