Quantum Optimal Control Theory - A Brief IntroductionJoel Gardner<Text-field style="Heading 1" layout="Heading 1"></Text-field>IntroductionChemistry has always been a practical science, and new insights into chemical theory often yield new methods and tools to apply to practical chemistry. Quantum Chemistry is no exception, and the theoretical advances made over the past century in this field have yielded new experimental techniques for synthesis and characterization. Quantum Optimal Control Theory (QOCT) applies the mathematical and computational methods of optimization to quantum mechanical systems in order to determine the most effective external controls to achieve a given objective. To provide an illustrative example (which will be examined more thoroughly later in the worksheet), consider a simple diatomic gas with a number of rotational, vibrational, and electronic excited states and the task of exciting as much of the gas as possible into a specific electronic excited state while expending a set quantity of energy. From a classical standpoint, this would be very difficult to do in a targeted way - dumping heat indiscriminately into the system would probably excite some of the molecules into the correct state, but much of the energy would be wasted in other molecular degrees of freedom. However, the insights given by quantum mechanics - in this case, the identification of specific frequencies and transition dipoles with different excitations - allow energy to be applied in a more efficient and targeted way. In this case, the system of interest is the molecule's wavefunction, the objective is the excitation of the wavefunction into the chosen electronic state, and the external controls applied is an external electric field that will be iteratively modified to push the system as close to the objective as is possible.<Text-field style="Heading 1" layout="Heading 1"></Text-field>Selection of a Cost FunctionalAt the core of QOCT is the construction of a mathematical expression known as a cost functional. The general form of a cost functional is as follows: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, the cost functional is a measure of how good or how bad an external control is. The person defining the cost functional has broad latitude is determining how to evaluate the control. However, the three most important considerations that are evaluated in the cost functional are typically:Objective Considerations. How good is the control at actually accomplishing the quantum objective? There are several different ways of evaluating whether the quantum objective has been accomplished or not. The metric that will be used in this worksheet is state control, which minimizes the difference between the final quantum state and a desired quantum state. However, there are other metrics, such as evolution-operator control (minimizing the difference between the evolution operator of the system and a chosen unitary operator) and observable control (targeting the value of an observable property of the system) that can be used in other situations. Practical Considerations. If the control is to be used in a laboratory setting, then it is important for the control to be one that can be realistically created in laboratory conditions. For instance, there may be a circumstance where a quantum objective can be maximized by using arbitrarily strong fields. In this case, it can be valuable for the cost functional to constrain the strength of the fields to realistic levels.Physical Considerations. The transformation of the quantum state must be physically possible. If we can choose an expression for the cost functional that incorporates all of these costs in some fashion and minimizes the cost with respect to all considerations, then we can find the best possible control to accomplish our objective.<Text-field style="Heading 1" layout="Heading 1"></Text-field>Defining a Quantum System & ObjectiveConstructing a cost functional is heavily dependent on the quantum system we select and the objective we choose to maximize. For the sake of mathematical simplicity, we will use an easy example - the Hydrogen molecule: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We will ignore the vibrational and rotation excited states, and consider only the ground and first electronic excited state. The excitation of the wavefunction by an external electric field is driven by the interaction between the external field and the electronic dipole associated with a transition between the electronic states: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We will also need the energies of the two states: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The time-independent Schrodinger equation gives the evolution of the wavefunction in time: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 can decompose the Hamiltonian in terms of the unperturbed Hamiltonian and the coupling with the control field: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The diagonal elements represent the ground-state Hamiltonian, while the off-diagonal elements represent the coupling. As previously mentioned, the coupling is produced by the interaction between the control field and the transition dipole, and can be represented as the product of the field strength at a point in time and the transition dipole at that same point in time. The transition dipole moment oscillates with the natural frequency of the transition: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One of our objectives is to excite the molecule from the ground excited state into the first excited state to the degree that it is possible: the coefficient array of the wavefunction will have value (1, 0) at the initial time and the success of the control field will be evaluated on the basis of how different it is from (0, 1) at the final time. We will also attempt to minimize the energy we expend doing so. At this time, we can also solve the time-independent Schrodinger equation for the derivative of the wavefunction, although until we have chosen a control field, it will be purely symbolic: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<Text-field style="Heading 1" layout="Heading 1"></Text-field>Selection of a Cost Functional RevisitedNow that we have defined the quantum system and objective we are working with, we can develop a functional to minimize.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This functional functional considers all the constraints that we enumerated above. The absolute value difference measures the distance between the pure excited state wavefunction and the system wavefunction at the end time - minimizing it maximizes the similarity to the desired final state. The first integral takes into account the practical consideration of minimizing the fleunce of the field. The second integral with LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEidEYnL0YwUSV0cnVlRidGMi9GNlEnaXRhbGljRidGNUYyRjVGNQ==is used to measures the difference between the actual evolution of LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JlEmJnBzaTtGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy1JKG1mZW5jZWRHRiQ2JS1GIzYkLUYsNiZRInRGJy9GMFEldHJ1ZUYnRjIvRjZRJ2l0YWxpY0YnRjVGMkY1LUYsNiNRIUYnRjU=and the physically expected evolution as given by the Schrodinger equation. The other wavefunction in this integral LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEoJnRoZXRhO0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvJSxtYXRodmFyaWFudEdRJ25vcm1hbEYnLUkobWZlbmNlZEdGJDYlLUYjNiQtRiw2JlEidEYnL0YwUSV0cnVlRidGMi9GNlEnaXRhbGljRidGNUYyRjVGNQ== is a Lagrange Multiplier - a technique from optimization which allows us to incorporate the physical constraint as we search for an optimal control field.In order to find a minimum of J, the derivative must be taken with respect to the Lagrange Multiplier, the Wavefunction, and the control field. The details of this derivation can be found in the original paper from which this worksheet was partially adapted, which is the first reference given at the bottom of this worksheet. Global minimum points for the cost functional can only be found where the derivatives with respect to all three are zero. This yields the following system of equations: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Respecting the boundary conditions that the wavefunction be purely in the ground state at the initial time, and that the Lagrangian multiplier have no ground state component at the final time (with an excited state component equal to the excited state component of the wavefunction at the final time). The final equation gives an expression for a new control field, where the energy E (defined as the integral of the field squared across the time interval) is held constant and O(t) represents the overlap function. The overlap function is given as follows: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 LUklbXN1Ykc2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlElJm11O0YnLyUnaXRhbGljR1EmZmFsc2VGJy8lNnNlbGVjdGlvbi1wbGFjZWhvbGRlckdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSSVtcm93R0YkNiQtSSNtbkdGJDYkUSMyMUYnRjVGNS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYn is the transition dipole operator. In more intuitive terms, the overlap function measures how similar the lagrangian and wavefunction are at every given time, and the equation to get the next control field is weighted to have a greater amplitude at times where the lagrangian and wavefunctions are more different. For a more detailed derivation of these equations, the first reference listed below is an excellent resource.There doesn't appear to be any easy way of analytically solving for the ideal control field, so we are going to use the following procedure:First, choose a trial control field that we think will do a good job of exciting the wavefunction. Determine the total energy associated with the field over the time interval to constrain the field in subsequent iterations.Use the expression for the evolution of the wavefunction and the boundary condition to numerically estimate the value of the wavefunction on the time interval.Use the expression for the evolution of the Lagrangian and the boundary condition to numerically estimate it's value as well (note: as the boundary condition for the Lagrangian is at the endpoint of the interval, this estimate will have to be done in the negative time direction).Use the calculated values of the Lagrangian and the wavefunction to calculate the overlap function, and from the overlap function, the next control field.Repeat the process from the next control field until the algorithm converges.We will do this process with multiple initial fields and see what the result is!<Text-field style="Heading 1" layout="Heading 1"></Text-field>A Note On ControllabilityThe controllability of a quantum system is a measure of whether control laws (under given constraints) can be devised which can achieve a chosen quantum objective. Not every system is controllable for every objective under every set of constraints, and determining systematic criteria under which quantum systems can be controlled is an ongoing area of research. Before we attempt to achieve our objective, we must stop and ask ourselves whether it is possible. Inspecting the derivative shows that in this system as we've defined it, fully exciting the wavefunction to to the excited state is impossible. In the limit of no control law, the diagonal elements dominate the evolution of the system which, unless the system is in a pure excited state, will cause the wavefunction to relax into the ground state. On the other hand, in the limit of an infinite control field, the off-diagonal elements dominate - and as the rate of change towards the ground and excited states in these elements depend on the proportion of the wavefunction in the other, this term will tends towards equal occupancy.Therefore, no matter what field we use, the long-term maximum of the projection of the wavefunction into the excited state will be below the projection into the ground state, at a value determined by the relative weights of the control field/transition dipole coupling and the natural relaxation of the wavefunction towards the ground state. With this in mind, the criteria for success should be the excitation of the wavefunction to a state that is nearly evenly split between ground and excited contributions, while at the same time minimizing the energy spent.<Text-field style="Heading 1" layout="Heading 1"></Text-field>FunctionsThis section defines the procedures that we will use to implement the algorithm. It is important to evaluate these blocks of code for the rest of the worksheet to work.Numerical Integration of coupled differential equations (forward time direction):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Evaluation of a function on a grid spanning a range: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 Integration of coupled differential equations (backwards time direction):LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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Calculation of the overlap term:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZicS1JI21pR0YkNiVRKG92ZXJsYXBGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSJ+RicvRjNRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y9LyUpc3RyZXRjaHlHRj0vJSpzeW1tZXRyaWNHRj0vJShsYXJnZW9wR0Y9LyUubW92YWJsZWxpbWl0c0dGPS8lJ2FjY2VudEdGPS8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRkwtRjY2LVEqJmNvbG9uZXE7RidGOUY7Rj5GQEZCRkRGRkZIL0ZLUSwwLjI3Nzc3NzhlbUYnL0ZORlNGNS1GNjYvUSVwcm9jRicvJSVib2xkR0YxL0YzUSVib2xkRicvJStmb250d2VpZ2h0R0ZlbkY7Rj5GQEZCRkRGRkZIRkpGTS1JKG1mZW5jZWRHRiQ2JC1GIzY2LUYsNiVRJWdyaWRGJ0YvRjItRjY2LVEiLEYnRjlGOy9GP0YxRkBGQkZERkZGSEZKL0ZOUSwwLjMzMzMzMzNlbUYnRjUtRiw2JVEqZ3JpZF9zaXplRidGL0YyRmBvRjUtRiw2JVEnYV92YWxzRidGL0YyRmBvRjUtRiw2JVEnYl92YWxzRidGL0YyRmBvRjUtRiw2JVErbGFnX2FfdmFsc0YnRi9GMkZgb0Y1LUYsNiVRK2xhZ19iX3ZhbHNGJ0YvRjJGYG9GNS1GLDYlUSdvX2xpc3RGJ0YvRjJGOUY5LUYsNiNRIUYnLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRkwvJSZkZXB0aEdGYHEvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRjY2L1EmbG9jYWxGJ0ZYRlpGZm5GO0Y+RkBGQkZERkZGSEZKRk1GNS1GLDYlUSRpZHhGJ0YvRjJGYG9GNS1GLDYlUSZ0X3ZhbEYnRi9GMkZgb0Y1LUYsNiVRI3J2RidGL0YyRmBvRjUtRiw2JVEmb192YWxGJ0YvRjJGNS1GXHE2JkZecUZhcUZjcS9GZnFRJWF1dG9GJ0ZbcUZnci1GNjYvUSxkZXNjcmlwdGlvbkYnRlhGWkZmbkY7Rj5GQEZCRkRGRkZIRkpGTUY1LUkjbXNHRiQ2I1FLQ29tcHV0ZX50aGV+b3ZlcmxhcH5mdW5jdGlvbn5hY3Jvc3N+YX5ncmlkRictRjY2LVEiO0YnRjlGO0Zjb0ZARkJGREZGRkhGSkZURltxRltxRmdyLUY2Ni5GOC8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RidGOUY7Rj5GQEZCRkRGRkZIRkpGTS1GNjYwUSRmb3JGJ0ZYRmdzRlpGZm5GO0Y+RkBGQkZERkZGSEZKRk1GZXMtRiw2JkZdckYvRmdzRjJGZXMtRjY2MFElZnJvbUYnRlhGZ3NGWkZmbkY7Rj5GQEZCRkRGRkZIRkpGTUZlcy1JI21uR0YkNiVRIjFGJ0Znc0Y5RmVzLUY2NjBRI2J5RidGWEZnc0ZaRmZuRjtGPkZARkJGREZGRkhGSkZNRmVzRmJ0RmVzLUY2NjBRI3RvRidGWEZnc0ZaRmZuRjtGPkZARkJGREZGRkhGSkZNRmVzLUYsNiZGaG9GL0Znc0YyRmVzLUY2NjBRI2RvRidGWEZnc0ZaRmZuRjtGPkZARkJGREZGRkhGSkZNRmVzRltxRltxLUYsNiZGYHJGL0Znc0YyRmVzLUY2Ni5GUUZnc0Y5RjtGPkZARkJGREZGRkhGUkZURmVzLUYsNiZGX29GL0Znc0YyLUZpbjYnLUYjNiRGXXRGOUZnc0Y5LyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRmVzLUYjNiVGaHAtRmluNigtRiM2Jy1GLDYlUSJzRidGL0YyLyUrZm9yZWdyb3VuZEdRKlswLDAsMjU1XUYnLyUscGxhY2Vob2xkZXJHRjEvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHRjFGOUY5L0krbXNlbWFudGljc0dGJFElVW5pdEYnL0ZcdlEoJmxvYnJrO0YnL0ZfdlEoJnJvYnJrO0YnRmF3RjktRjY2LlEiOkYnRmdzRjlGO0Y+RkBGQkZERkZGSEZSRlRGW3FGZXMtRiw2JkZmckYvRmdzRjJGZXNGY3UtRiw2JlEiSUYnRi9GZ3NGMi1GNjYuUScmc2RvdDtGJ0Znc0Y5RjtGPkZARkJGREZGRkhGSkZNLUYsNiZRKGNvbnZlcnRGJ0YvRmdzRjItRmluNiUtRiM2Ki1GLDYmUTlleGNpdGF0aW9uX2RpcG9sZV9tb21lbnRGJ0YvRmdzRjItRmluNiUtRiM2JEZhdUY5RmdzRjktRjY2LkZib0Znc0Y5RjtGY29GQEZCRkRGRkZIRkpGZG9GZXMtRjY2LlEiJ0YnRmdzRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4xMTExMTExZW1GJ0ZNLUYsNiZRKnVuaXRfZnJlZUYnRi9GZ3NGMkZjeUY5RmdzRjlGYHgtRmluNiUtRiM2NEZlcy1GLDYmRmRwRi9GZ3NGMkZndUZlc0ZgeC1GLDYmRltwRi9GZ3NGMkZndUZlcy1GNjYuUSomdW1pbnVzMDtGJ0Znc0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GZ3pGZXMtRiw2JkZhcEYvRmdzRjJGZ3VGZXNGYHhGZXMtRiw2JkZecEYvRmdzRjJGZ3VGOUZnc0Y5Rmh3RltxRltxRmVzLUYsNiZRJ0FwcGVuZEYnRi9GZ3NGMi1GaW42JS1GIzYnLUYsNiZGZ3BGL0Znc0YyRmF5RmVzRlt4RjlGZ3NGOUZod0ZnckZbcUZlcy1GNjYwUSRlbmRGJ0ZYRmdzRlpGZm5GO0Y+RkBGQkZERkZGSEZKRk1GZXNGXnUtRjY2LkZkc0Znc0Y5RjtGY29GQEZCRkRGRkZIRkpGVEZlc0ZbcUY1RmFyRjVGT0Y1RmVwLUY2Ni1GandGOUY7Rj5GQEZCRkRGRkZIRlJGVEZbcUZlc0ZmW2xGZXMtRjY2MEZXRlhGZ3NGWkZmbkY7Rj5GQEZCRkRGRkZIRkpGTUZpW2xGOQ==Calculation of the next control field, given an energy constrained and values of the overlap function: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The control field generated from this method usually has a real component that is far larger than the imaginary component, so for the sake of simplicity, this function converts the control field into a real-valued array (by taking the real part of each complex number)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Finally, this function wraps everything up into a single procedure to run one iteration of the algorithm, essentially moving from the last control field to the next: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 function removes the units from an array (which can make thongs easier for plotting purposes)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 function gives the units to a control grid array (V/m) where needed (which is necessary, as units are often removed in intermediate stages to speed up calculations)
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 function estimates the integral of the energy (from a grid of values of the field)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Any other definitions we may need: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<Text-field style="Heading 1" layout="Heading 1"></Text-field>Case 1: Uniform Control FieldWe will begin with the simplest case: subjecting the system to a uniform electric field. As the timescales of quantum processes are short, we will only consider a timespan of a single femtosecond: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As we will often be estimating the values of integrals across this time range, we will also generate a grid of 100 evenly spaced points that we will use throughout the worksheet: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We will use a field with a potential difference density (the required unit of the control field from the differential equation):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The cost integral associated with this field is:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JVEiRUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIn5GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGTC1GNjYtUSomY29sb25lcTtGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjc3Nzc3OGVtRicvRk5GU0Y1LUkobXN1YnN1cEdGJDYnLUY2Ni1RKyZJbnRlZ3JhbDtGJ0Y5L0Y8USZ1bnNldEYnL0Y/RmZuL0ZBRjEvRkNGZm4vRkVGMS9GR0Zmbi9GSUZmbkZKRk0tRiM2Jy1JI21uR0YkNiRRIjBGJ0Y5Ri8vJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9sZGVyR0YxRjItRiM2Jy1GLDYlUSJURidGL0YyRi9GY29GZm9GMi8lMXN1cGVyc2NyaXB0c2hpZnRHRmJvLyUvc3Vic2NyaXB0c2hpZnRHRmJvLUYsNiZRJyYjOTQ5O0YnRi8vJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRjItSSVtc3VwR0YkNiUtSShtZmVuY2VkR0YkNiUtRiM2JS1GLDYmUSJ0RidGL0ZkcEYyLyUrZXhlY3V0YWJsZUdGPUY5RmRwRjktRiM2Ji1GYG82JVEiMkYnRmRwRjlGL0ZkcEYyRl1wLUY2Ni5GOEZkcEY5RjtGPkZARkJGREZGRkhGSkZNLUY2Ni1RMCZEaWZmZXJlbnRpYWxEO0YnRjlGZW5GZ24vRkFGZm5GaW4vRkVGZm5GW29GXG9GSkZNLUYsNiVGYXFGL0YyRmJxRjk=While this appears to be rather costly, keep in mind that the short timescale and tiny size needed for the laser shows that this isn't outrageous. We will need to generate the value of this field on our grid: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Use the procedure we defined to run one iteration of the algorithm: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 can plot the output of the algorithm to see what happened. Firstly, we can observe the relative (normalized) probabilities of the wavefunction being in each state, as a function of time:LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=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 allows us to see the competition between the coupling and relaxing forces. After the first few tenths of a femtosecond, the wavefunction appears to reach a sort of long-term equilibrium where the long-term tendency of the wavefunction to relax cancels out the coupling of the control field to the electronic transition dipole over the course of the period of the transition dipole.The next piece of our algorithm's output are the normalized probabilities of the Lagrangian. The boundary condition of the langrangian is that the wavefunction is completely in an excited state. The ideal control would yield a wavefunction that has the initial time condition of the main wavefunction and the final time condition of the Lagrangian, so in a sense one way to measure how close our system is to achieving its goal is by observing how different the two wavefunctions are.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 for us, the two don't look much alike at the endpoints. However, as our control improves, hopefully the two will converge. Finally, setting the stage for the next iteration, we can plot the old control field against the new one.LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYuLUkjbWlHRiQ2JlEwdWZfY29udHJvbF9ncmlkRicvJSdpdGFsaWNHUSV0cnVlRicvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi5RIn5GJ0YyL0Y2USdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGQC8lKXN0cmV0Y2h5R0ZALyUqc3ltbWV0cmljR0ZALyUobGFyZ2VvcEdGQC8lLm1vdmFibGVsaW1pdHNHRkAvJSdhY2NlbnRHRkAvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZPLUY5Ni5RKiZjb2xvbmVxO0YnRjJGPEY+RkFGQ0ZFRkdGSUZLL0ZOUSwwLjI3Nzc3NzhlbUYnL0ZRRlZGOC1GLDYmUSlVRl9hcnJheUYnRi9GMkY1LUkobWZlbmNlZEdGJDYlLUYjNictRiw2JlEtY29udHJvbF9ncmlkRidGL0YyRjUtRjk2LlEiLEYnRjJGPEY+L0ZCRjFGQ0ZFRkdGSUZLRk0vRlFRLDAuMzMzMzMzM2VtRidGOC1JI21uR0YkNiVRJDEwMEYnRjJGPEY8RjJGPC1GOTYuUSI6RidGMkY8Rj5GQUZDRkVGR0ZJRktGVUZXLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRk8vJSZkZXB0aEdGX3AvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRltwNiZGXXBGYHBGYnAvRmVwUSVhdXRvRictRiw2JlEpZGF0YXBsb3RGJ0YvRjJGNS1GZm42JS1GIzY1LUYsNiZRJWdyaWRGJ0YvRjJGNUZdb0Y4LUZmbjYnLUYjNidGK0Zdb0Y4LUYsNiZRMW5ld19jb250cm9sX2dyaWRGJ0YvRjJGNUY8RjJGPC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0Zdb0Y4LUYsNiZRKGNhcHRpb25GJ0YvRjJGNUY4LUY5Ni5RIj1GJ0YyRjxGPkZBRkNGRUZHRklGS0ZVRldGOC1JI21zR0YkNiNRO09sZH5hbmR+TmV3fkNvbnRyb2x+RmllbGRzRidGXW9GOC1GLDYmUSdsZWdlbmRGJ0YvRjJGNUY4RmVyRjgtRmZuNictRiM2KS1GLDYjUSFGJy1GaXI2I1EqT2xkfkZpZWxkRictRjk2LUZfb0Y8Rj5GYG9GQ0ZFRkdGSUZLRk1GYW8tRjk2LUY7RjxGPkZBRkNGRUZHRklGS0ZNRlAtRmlyNiNRKk5ld35GaWVsZEYnRmNzRjxGMkY8RlxyRl9yRjxGMkY8Rjw=The new field looks very different from the old field - appearing to incorporate some of the periodicity of the transition dipole moment. It also appears that it requires a lower total energy cost, and by calculating the integral of the square of the field strength across this range we can verify that the total energy cost of this control has dropped to just over 50% of the original cost.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At this point, if we wanted to focus on maximizing the objective, we could boost the amplitude of this field by a scalar factor to return to the original energy cost (and hopefully excite the wavefunction even more). However, we already know that there is a hard upper limit on the degree to which we can excite this wavefunction (and one we appear to have reached with the constant field). A more interesting route is to see if the new wavefunction can reach the same upper limit (or one that is nearly as high). We can store the control grids that we generate and the associated probability functions: 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 we will run the algorithm ten more times to see how the control laws change: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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=Plotting the first and the final controls against each other reveals that final control law fell slightly short of the original control law in terms of the degree of excitation of the wavefunction at the end of the time interval. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYlLUkjbWlHRiQ2JlEpZGF0YXBsb3RGJy8lJ2l0YWxpY0dRJXRydWVGJy8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYlLUYjNjMtRiw2JlElZ3JpZEYnRi9GMkY1LUkjbW9HRiQ2LlEiLEYnRjIvRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRkgvJSpzeW1tZXRyaWNHRkgvJShsYXJnZW9wR0ZILyUubW92YWJsZWxpbWl0c0dGSC8lJ2FjY2VudEdGSC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUY5NictRiM2KS1GLDYmUSZQX2FyckYnRi9GMkY1LUY5NictRiM2JC1JI21uR0YkNiVRIjFGJ0YyRkRGREYyRkQvJSVvcGVuR1EiW0YnLyUmY2xvc2VHUSJdRidGQC1GQTYuUSJ+RidGMkZERkYvRkpGSEZLRk1GT0ZRRlNGVS9GWUZXRmluLUY5NictRiM2JC1GYW82JVEjMTBGJ0YyRkRGREYyRkRGZG9GZ29GREYyRkRGZG9GZ29GQC1GLDYmUShjYXB0aW9uRidGL0YyRjVGam8tRkE2LlEiPUYnRjJGREZGRl1wRktGTUZPRlFGUy9GVlEsMC4yNzc3Nzc4ZW1GJy9GWUZdcUZqby1JI21zR0YkNiNRaG9Ob3JtYWxpemVkfkV4Y2l0ZWR+U3RhdGV+UHJvYmFiaWxpdHl+YW5kfnRoZX5CZWdpbm5pbmd+YW5kfkVuZH5vZn50aGV+UHJvY2Vzc0YnRkBGam8tRiw2JlEnbGVnZW5kRidGL0YyRjVGam9GaXBGam8tRjk2Jy1GIzYpLUYsNiNRIUYnLUZgcTYjUS5GaXJzdH5Db250cm9sRictRkE2LUZDRkRGRkZJRktGTUZPRlFGU0ZVRlgtRkE2LUZccEZERkZGXXBGS0ZNRk9GUUZTRlVGXnAtRmBxNiNRLkZpbmFsfkNvbnRyb2xGJ0ZqcUZERjJGREZkb0Znb0ZERjJGREZEHowever, the energy expended in the process has been substantially reduced.<Text-field style="Heading 1" layout="Heading 1"></Text-field>Case 2: Gaussian Laser PulseWe can try to take advantage of the periodic nature of the transition dipole moment by shaping our field to coincide with the maximum value of the transition dipole. The first way we will do this is using a laser pulse with a Gaussian shape. We will need to know the period of the transition dipole moment in order to do this: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We can center the wave after a single period has elapsed, so that the maximum of the wave and the maximum of the dipole coincide. We would like to use a relatively narrow pulse to maximize the coupling with the greatest amplitude of the dipole - so we will set the standard deviation to be a fraction of the period. 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The height of the pulse is related to the total energy that the pulse delivers. We will calculate the total energy first, and then will choose the height so that this field has the same energy flux as the previous.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(Despite the strange units, this is equivalent to the potential density used in the previous section). Thus, the pulse with the correct height is: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 again, let's try to use the algorithm once and see what happens.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\134Observe that the energy reduction, although still considerable, is less than that of the uniform field as the Gaussian field expends less energy out-of-phase with the electronic transition dipole moment.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 first iteration raises a good question - why did the Gaussian pulse, timed to coincide with the electronic transition dipole and avoid the wasted energy of the uniform field, in fact do a worse job of exciting the wavefunction at the endpoint? There are two answers. Firstly, part of it is an artifact of how the time interval is constructed: the peak probability of the Gaussian and uniform fields are very similar, but the uniform field happens to peak at the end of the time interval while the Gaussian peaks near the beginning. Secondly, the way we calculate energy use makes the uniform field more efficient. The derivative of the wavefunction is proportional to the strength of the control field, but we count energy in terms of its square. Therefore, the peak of the Gaussian is roughly 3.5 times more effective at exciting the wavefunction, but is over 10 times as costly!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 we will run the algorithm ten more times to see how the control laws change:JSFHLUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZncC1JI21vR0YkNjBRJGZvckYnLyUlYm9sZEdRJXRydWVGJy8lK2V4ZWN1dGFibGVHRjEvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjYvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjsvJSlzdHJldGNoeUdGOy8lKnN5bW1ldHJpY0dGOy8lKGxhcmdlb3BHRjsvJS5tb3ZhYmxlbGltaXRzR0Y7LyUnYWNjZW50R0Y7LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGSi1GLDYuUSJ+RidGMi9GNVEnbm9ybWFsRidGOUY8Rj5GQEZCRkRGRkZIRkstSSNtaUdGJDYmUSRpZHhGJy8lJ2l0YWxpY0dGMUYyL0Y1USdpdGFsaWNGJ0ZNLUYsNjBRJWZyb21GJ0YvRjJGNEY3RjlGPEY+RkBGQkZERkZGSEZLRk0tSSNtbkdGJDYlUSIxRidGMkZQRk0tRiw2MFEjYnlGJ0YvRjJGNEY3RjlGPEY+RkBGQkZERkZGSEZLRk1GZ25GTS1GLDYwUSN0b0YnRi9GMkY0RjdGOUY8Rj5GQEZCRkRGRkZIRktGTS1GaG42JVEjMTBGJ0YyRlBGTS1GLDYwUSNkb0YnRi9GMkY0RjdGOUY8Rj5GQEZCRkRGRkZIRkstRiw2LlEiOkYnRjJGUEY5RjxGPkZARkJGREZGL0ZJUSwwLjI3Nzc3NzhlbUYnL0ZMRltwLUknbXNwYWNlR0YkNiYvJSdoZWlnaHRHUSYwLjBleEYnLyUmd2lkdGhHRkovJSZkZXB0aEdGYnAvJSpsaW5lYnJlYWtHUShuZXdsaW5lRictRlM2J1EjcnZGJ0ZWRjIvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRlgtRiw2L1EqJmNvbG9uZXE7RidGMkZdcUZQRjlGPEY+RkBGQkZERkZGam9GXHAtRiw2L0ZPRjJGXXFGUEY5RjxGPkZARkJGREZGRkhGSy1GUzYnUTJydW5fb25lX2l0ZXJhdGlvbkYnRlZGMkZdcUZYLUkobWZlbmNlZEdGJDYmLUYjNjQtRlM2J1EiRUYnRlZGMkZdcUZYLUYsNi9RIixGJ0YyRl1xRlBGOS9GPUYxRj5GQEZCRkRGRkZIL0ZMUSwwLjMzMzMzMzNlbUYnLUZTNidRJWdyaWRGJ0ZWRjJGXXFGWEZgckZjcS1GaG42JlEkMTAwRidGMkZdcUZQRmByRmNxLUYjNiYtRlM2J1EoVV9hcnJheUYnRlZGMkZdcUZYLUZpcTYmLUYjNigtRlM2J1ExbmV3X2NvbnRyb2xfZ3JpZEYnRlZGMkZdcUZYRmByRmNxRmlyRjJGUEYyRl1xRlBGMkZQRmByRmNxLUZTNidRI2RhRidGVkYyRl1xRlhGYHJGY3EtRlM2J1EjZGJGJ0ZWRjJGXXFGWEYyL0ZecVEoMkR+TWF0aEYnRlBGMkZdcUZQLUYsNi9GaW9GMkZdcUZQRjlGPEY+RkBGQkZERkZGam9GXHBGXXBGY3FGZXNGYHJGY3EtRlM2J1EsYV9wcm9iX3ZhbHNGJ0ZWRjJGXXFGWEZgckZjcS1GUzYnUSxiX3Byb2JfdmFsc0YnRlZGMkZdcUZYRmByRmNxLUZTNidRK2xhZ19hX3Byb2JGJ0ZWRjJGXXFGWEZgckZjcS1GUzYnUStsYWdfYl9wcm9iRidGVkYyRl1xRlhGY3FGYHFGY3FGanAtRmlxNigtRiM2Ji1GaG42JkZqbkYyRl1xRlBGMkZedEZQRjJGXXFGUC8lJW9wZW5HUSJbRicvJSZjbG9zZUdRIl1GJ0ZgckZqcC1GaXE2KC1GIzYmLUZobjYmUSIyRidGMkZdcUZQRjJGXnRGUEYyRl1xRlBGZHVGZ3VGYHJGY3FGanAtRmlxNigtRiM2Ji1GaG42JlEiM0YnRjJGXXFGUEYyRl50RlBGMkZdcUZQRmR1Rmd1RmByRmNxRmpwLUZpcTYoLUYjNiYtRmhuNiZRIjRGJ0YyRl1xRlBGMkZedEZQRjJGXXFGUEZkdUZndUZgckZjcUZqcC1GaXE2KC1GIzYmLUZobjYmUSI1RidGMkZdcUZQRjJGXnRGUEYyRl1xRlBGZHVGZ3VGYHRGXXBGY3EtRlM2J1EmRV9uZXdGJ0ZWRjJGXXFGWEZjcUZgcUZjcS1GUzYnUSxkZWZfbnVtX2ludEYnRlZGMkZdcUZYRmFzRmB0LUZTNiNRIUYnRl1wRl1wRmNxLUZTNidRJ0FwcGVuZEYnRlZGMkZdcUZYLUZpcTYmLUYjNigtRlM2J1EmUF9hcnJGJ0ZWRjJGXXFGWEZgckZjcUZldEYyRlBGMkZdcUZQRmB0Rl1wRmNxRl94LUZpcTYmLUYjNigtRlM2J1EmQ19hcnJGJ0ZWRjJGXXFGWEZgckZjcUZlc0YyRlBGMkZdcUZQRmB0RmNxRl1wRmNxLUYsNjFRJGVuZEYnRi9GMkZdcUY0RjdGOUY8Rj5GQEZCRkRGRkZIRktGY3EtRiw2MUZmb0YvRjJGXXFGNEY3RjlGPEY+RkBGQkZERkZGSEZLRmB0RjJGUA==JSFHJSFHPlotting the first and the final controls against each other reveals that in this case, the algorithm did improve the control law insofar as the results are being evaluated at the endpoint. While the peak is still diminished, the final control law still does a reasonable good job considering the energy efficiency.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. Kosloff et al, Chem. Phys. 139, 201-220 (1989). "Wavepacket Dancing: Achieving Chemical Selectivity by Shaping Light Pulses"J. P. Dowling and G. J. Milburn, Phil. Trans. R. Soc. Lond. A 361, 1655-1674 (2003). "Quantum Technology: the second quantum revolution"D. E. Kirk, Optimal Control Theory: An Introduction 1st Edition (Dover Publications, Inc. 2004).G. M. Huant and T. J. Tarn, J. Math. Phys. 24, 2608 (1983). "On the controllability of quantum-mechanical systems"A. Gilchrist, N. K. Langford, and M. A. Nielsen, Phys. Rev. A. 71 062310 (2005). "Distance measures to compare real and ideal quantum processes"A. J. Brizard, Introduction to Lagrangian and Hamiltonian Mechanics 2nd Edition (WSPC 2014)C. Brif et al, New J. Phys. 12 075008 (2010). "Control of quantum phenomena: past, present, and future"LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=