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Quantum Photonics
创建日期：2024-03-30 10:17:30
所属分类：物理Physical
标签： Photonics
Quantum Photonics

Module Title: Quantum Photonics

This is an open book assessment. You may consult any of your own notes. You must provide an explanation for all your answers in your own words. This will vary from afew words to two to three sentences depending on the material. This will be to demonstrate your understanding of the course.

Do not just repeat answers from your notes without this explanation. Make sure your method of calculation is clearly shown.

If you make use of websites or textbooks to answer specific questions, you must list them at the end of the relevant answer.

Assessment information:  

  • This assessment is made up of 3 pages.
  • You must upload your answers via GradeScope to Minerva within 48 hours of the as- sessment being released. You are advised to allow up to four hours to photograph youranswers, and upload either as a PDF or series of images to GradeScope. The uploadlink will be found in the Assessment section for each module on Minerva and will be available throughout the period of the assessment.
  • Although the upload is open for the full period of the assessment, you are advised that the assessment should only require 1 hour to complete.
  • Youmust answer all of the questions in this assessment.  
  • Youshould cross out any work you do not want to be marked.
  • Youshould indicate by underlining the final answer to each question.
  • You must use black or dark blue ink.
  • You must write your answers on white or lightly coloured A4 plain, or lightly ruled, paper. You must leave a margin of at least 1cm.
  • As part of the process of submitted through GradeScope you must identify which ques- tions are answered on which uploaded pages. You must also check that you have uploaded all the work you wish to be marked as part of this assessment and that the answers uploaded are clearly legible.

1. Measurements and probabilities. 

Suppose the quantum state of two two-level atoms a and b with ground states |0) and

|1) equals

|ψ) = α |00) + β |01) + γ |10) + δ |11) ,

where α, β, γ and δ are complex coefficients with |α|2 +|β|2+|γ|2 +|δ|2 = 1. Moreover,|ij ) denotes the state with atom a prepared in |i ) and atom b prepared in |j).  

(a)What is the probability of finding both atoms in|1)?

(b)Whatis the probability of finding atom a in |1)?  

(c)Whatis the probability of finding atom a in a state |φ) = (|0) + |1))/ 2?

(d)Whatis the state |ψj) of the atoms after atom a has been found in |φ)?

(e)Whydoes the measurement change the state of the atoms into |ψ’)?[10 marks]

2.Operators and observables. 

Suppose an observable A of a two-level quantum system with energy eigenstates |0) and|1) equals

A = (a + b) I + (a − b) σ+ + σ− ,

where a and b are real numbers, I is the identity operator and σ+ and σ− are the Pauli operators σ+ = |1)(0| and σ− = |0)(1|. In addition, we assume in the following that the quantum system has been prepared in the state |ψ) = α |0) + β |1).

(a)Calculate the eigenvalues and eigenvectors ofA?

(b)Whatisits expectation value for a measurement of A given |ψ)?

(c)Whatis its expectation value for a measurement of A given |ψ)?[10 marks]

3.The dynamics of expectation 

Next we consider an observable A of a quantum system with Hamiltonian H which evolves according to the Schr¨odinger equation.

(a)Derivean equation for the time derivative (A˙ ) of the expectation value (A).

(b)SupposeA = |1)(1| and H  = kΩ (|0)(1| + |1)(0|). Calculate (A˙ ) for this

(c)Supposethe quantum state is initially prepared in |ψ(0)) = |0). Calculate (A) at a time t = 2π/Ω?[10 marks]

4.Trapping single atoms and ions. 

Explain your answers.

(a)Describeone technique which allows to trap individual ions in free space.

(b)Describeone technique which allows to trap individual atoms in free space.

[4 marks]

5. Manipulating the electronic states of single atoms and ions.

(a)Describetwo different methods which can be used to transfer an atom from an initial ground state 0 into another ground state 1 , while minimising spontaneous photon emission from excited atomic states.

(b)Nameone advantage and one disadvantage for each technique. [6 marks]

                                                                                                    

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