More about qubits

27 Jan 2021

As Nielsen and Chuang write in their classic book on quantum computing,

A classical bit is like a coin: either heads or tails up. By contrast, a qubit can exist in a continuum of states between \(|0 \rangle\) and \(|1 \rangle\) – until it is observed.

Despite this rather strange behaviour, qubits are physical entities that can be realised in laboratories. Some of the ways in which this realisation can occur are:

1. As two different polarisations of a photon
2. As the alignment of a nuclear spin in a uniform magnetic field
3. As the two states of an electron in an atom.

In the atom model, the ground state and the excited states of the electron can be considered as the basis states \(|0 \rangle\) and \(|1 \rangle\). An electron in ground state \(\left(|0 \rangle \right)\) can be moved to the excited state \(\left(|0 \rangle \right)\) by shining light on the atom for a sufficiently long time. Surprisingly, by shining light for a shorter duration of time, we can move the electron to a state that is ‘in between’ \(|0 \rangle\) and \(|1 \rangle\).¹ This state is a superposition of ground and excited states, and an example for the state \(|+ \rangle\) that we considered here.

Reference:

1. Nielsen and Chuang. Quantum Computation and Quantum Information

1. I have two questions here:

   1. An electron can occupy a ‘in between’ state only if the ground state and excited states are separated by \(\Delta n \ge 2\), where \(\Delta n\) is the difference in principle quantum numbers corresponding to ground and excited states. For example, if the ground state corresponds to \(n = 1\), and the excited state corresponds to \(n = 2\), then \(\Delta n = 1\), and there is no question of an ‘in between’ state. If the excited state corresponds to \(n = 3\), then the energy level corresponding to \(n = 2\) might be considered as an ‘in between’ state. So, the existance of an ‘in between’ state is depends on how we define the excited state. Is this understanding correct?

   2. When we shine light on an atom, the photons transfer their enrgy to the electron, and the electron jumps from ground state to an excited state. The energy received by the electron is a function of frequency alone and specifically, not a function of duration of illumination. Then, how can we make the electron move to an ‘in between’ state by shining light for a shorter duration of time?

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