Consider a hot cup of coffee, that indispensable fuel for many a physicist. We know that if we leave it alone for long, that hot, steaming cup of coffee cools down, making it easier to drink but less delectable. To delay this, some of us use special insulating cups, or cover the open top of the cup. Also, consider that one can enjoy a steaming cup of coffee only for a short period of time if one drinks it at an outdoor Paris cafe in the dead of winter, as the cold, biting air will cause the coffee to cool down rapidly and freeze. On the other hand, there is more time to enjoy this same hot cup of coffee in this same outdoor cafe on a warm summer day, due to the air’s temperature being closer to that of the coffee. We can explain these phenomena by saying that the coffee interacts with the surrounding environment in such a way that its state changes depending on the environment surrounding it. Changing the environment also changes the rate at which the coffee’s state changes, and minimizing the interaction between the coffee and the environment allows it to change its state much more slowly, giving us more time to enjoy that hot cup of coffee.
This everyday system that I just described is what is known as an open system, wherein a physical system interacts with the surrounding environment. The interaction changes the system over time, while any change to the environment is minimal if the environment is big enough. Open systems may either be classical or quantum. For an open quantum system, we have a quantum system, such as a quantum harmonic oscillator, interacting with its surrounding environment, with this interaction between the system and the environment driving the time evolution of the quantum system, just like how the interaction between the coffee and its surrounding environment drives the change in temperature of the coffee.
An open quantum system is much more realistic than a closed quantum system, which is isolated from the environment, because it is almost impossible to completely isolate a quantum system from its surrounding environment. Furthermore, open quantum systems are of paramount importance in quantum information, quantum computing, and cavity optomechanics, among other fields, because interaction between a quantum system and the surrounding environment causes decoherence in the quantum system, and a consequent loss of the system’s quantum properties, thus rendering the system unusable in quantum computing or quantum information. Understanding the dynamics of an open quantum system will enable one to control the system appropriately in order to delay the emergence of decoherence in the system, prolonging the time in which the system behaves quantum mechanically.
There are three main elements to an open quantum system, namely the system itself, the environment, and the interaction between the system and the environment, each of which correspond to a separate Hamiltonian. The Hamiltonian gives us information about how each element of the open quantum system evolves over time. However, the bigger the environment, the more information is available about the open quantum system, most of which is of no importance in describing the system’s dynamics. As a result, with all the information available, the more difficult it is to describe the dynamics of the system. Thus, there is a need to reduce the amount of information available about the open quantum system up to that which are absolutely necessary to describe the dynamics of the system, which is done by treating the system in a probabilistic manner. In doing so, we can eliminate the information which is irrelevant in describing the dynamics of the system, and focus only on those few relevant events.
An analogy to the probabilistic approach can be seen in the treatment of a system undergoing a random walk. Here, if the system is located at a given point at a given instant of time, there is a 1 in N chance that the system will move in one of N possible directions. In particular, if the system can only move on a line, then if it is located at a given point x at a given instant of time t, then there is a 1 in 2 chance that it will move to the point x-1 or x+1. Eventually, if one plots the frequency of the system being at certain points x as a function of time, then one finds that the resulting plot, also known as the probability distribution of the system, resolves itself into a a familiar shape, namely the Gaussian distribution, also known as the bell curve, which has a pronounced peak at a given point then falls off gradually on either side of the peak. By looking at this probability distribution, we can then infer that the system will most likely be located at the point corresponding to the peak of the distribution. So in describing the dynamics of this system, we can then focus only on the points close to the peak of the distribution, and neglect all other points for which the probability is very low.
In treating an open quantum system using probability, we have to make certain approximations which will enable us to discard as much of the irrelevant information as possible to describe the dynamics of the system. One approximation in particular that has long been used for open quantum systems is the Markov approximation, wherein it is assumed that only events happening in the present can affect events happening immediately after it; equivalently, the system is said to have no memory of past events. An illustration of such a process can be done if we consider a simplified description of the price of a share in a company. Share prices rise or fall based on current information about the company available, such as data available about profits made or loss, or whether or not the company is about to merge with another company, and not information about the company in the past, such as profit statements from the past year or the ownership history of the company. Treating an open quantum system using the Markov approximation will minimize the amount of information that can be used to describe the dynamics of the system, but at the same time ensure that the information available provides as accurate a description of the system’s dynamics as possible at the present time and the immediate past, which will enable us to predict and analyze its dynamical behavior in the future.
However, this approximation may not be useful all of the time, since certain events in the past may affect how the system will behave; in other words, it is possible that the system will have a memory that will affect how it will evolve over time. To see how this happens, let us again go back to our example of the share price of the company. Suppose that, in the past, a sudden spike in the price of commodities in which the company trades in, say oil, caused its profits to rise. Then that increases the valuation of the company, causing its share prices to rise as well. Because of that event, which may have occurred at any time in the company’s past, we then know that any future spikes in oil prices will cause the company’s share price to rise, no matter what other event may be happening in the immediate past or in the present that affects the company. As such, if the open quantum system has a memory that affects how it evolves over time, other approximation methods must be used.
The physical application of open quantum systems are varied, and we will consider only a few examples here. One application of interest is in quantum computation and quantum state preparation. For both applications, a quantum system evolves over time in such a way that it acquires certain physical properties. This evolution process corresponds to the implementation of quantum gates in quantum computation, which are analogous to the logic gates used in classical computation. In quantum computation, the physical state that evolves according to the evolution process set by the series of quantum gates corresponds to a qubit, the fundamental unit of information in quantum computing. If we use an open quantum system to prepare a quantum state or to implement quantum gates, we have what is known as dissipative quantum state preparation or dissipative quantum computation, wherein the interaction between the system and the environment drives the dynamics of the system, causing it to evolve in time in such a way that it acquires the properties of the desired state.
Dissipative quantum state preparation and dissipative quantum computation minimizes the interaction between the experimenter and the system, since the interaction between the system and the environment as it evolves is what drives the time evolution and prepares the state. Also, another advantage for particular forms of these processes is that no matter what the initial form of the system may be, the interaction between the system and the environment causes the system to evolve towards a unique final state, thus making such processes almost universal in the sense that one can use any initial state to obtain the desired final state. One can also control the preparation of the state or the quantum computational process by adjusting the interaction between the system and the environment, thus allowing an efficient degree of control over the process.
Roughly speaking, this manner of quantum state preparation or quantum computation is analogous to roasting a chicken in an oven, wherein after being dressed, marinated and prepared for roasting, the chicken is put inside the oven, the temperature and timer are set, and the chicken is allowed to roast on its own in the oven, with the cook only having to check on the chicken after the preset time for cooking is up. In later posts we will be describing examples of such dissipative quantum computing and quantum state preparation schemes, which is currently one of the most active fields of research in quantum information.
In sum, open quantum systems are of great interest in quantum mechanics. First of all, this is because they allow a more realistic description of quantum systems, by taking into account their interaction with the environment. Second, this is because they have varied applications, which are of great interest now that quantum technologies are gaining in importance in everyday life. As such, greater attention must be paid to open quantum systems, which promises to be a rich and promising area of research in physics for decades to come.