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physics laboratory代写 Assignment代写 beam splitter代写

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Assignment 2

physics laboratory代写 This assignment is due on Friday, October 18th by 11:59pm (A minute before midnight). There will be no late assignments

Due  October 21th

This assignment is due on Friday, October 18th by 11:59pm (A minute before midnight). There will be no late assignments accepted and that includes any kind of technical problems with Crowdmark, so I recommend handing in the day before to avoid this kind of problem!!! Please do not email the TAs or the Professor asking for a late assignment hand-in.physics laboratory代写

Some Photon BS (marked by Erickson) physics laboratory代写

One of the most basic experiments in a physics laboratory is interferometry. It typically consists of an array of mirrors arranged in such a way that to light beams will interfere, causing an interference pattern. The interference pattern contains information that can be used to deduce physics of the phenomenon under study. It is a very basic but versatile experimental setup that countless variations (different mirror setups or using electrons, neutrons, atoms, or even big molecules1 as the ‘source’) exists for equally many purposes. In this problem we will consider one of the simplest setup involving only five main ingredients: a light source, a beam splitter, mirrors,  phase  control and photodetectors. This is shown in Figure 1. The light source will be producing pulses with two possible polarizations states — horizontal and vertical polarization states |H) , |V ), analogous to the classical linear polarization in optics.

(a)A beam splitter (BS) is classically

a material that splits a light beam into two parts with smaller amplitudes along two different paths (cf. Figure 1). If it is a 50/50 BS, the light is split into two equal amplitudes. Quantum mechanically, this will be a material that takes in an input light state and gives a superposition of polarization statesand 50/50 BS means the superposition has equal weight, e.

physics laboratory代写
physics laboratory代写

1People are ‘this’ close to do interferometry and double-slit experiments with small viruses!’physics laboratory代写

Figure 1: setup of the interferometer.

We can then label path 1 as the path where only vertically polarized light is allowed, and path 2 otherwise (i.e. the BS acts as a polarization filter2).

Find  the  matrix  representation  of  the  beam  splitter  operator3,  denoted  Bˆ,  in  the polarization basis {|V ) , |H)}. Show your working.physics laboratory代写

(b)In interferometry there is equipment that can adjust the phase of a light beam on one path. We call this equipment a phase control4(PC) which can add a phase φ

to a state. If the PC adds a phase φ to |H) and does nothing to |V ), find the matrix representation for the phase control operator C in this basis. Show your working.

(c)At the end we have two photodetectors D1, D2 physics laboratory代写

which are positioned in such a way that D1 will register a click if a photon arrives via path 1 and D2 if the photon arrives via path  Let us assume as an approximation that those detectors carry out projective measurements. Find the matrix representation for the projection that D1

and D2 implement on |ψ).

(d)If the input state for the experiment is |V ), compute the output state right before the measurement is performed. What is the probability that D1and D2 registers a

click? Does the phase φ matter?

2Want to buy one? You can do it , e.g., herehttps://www.thorlabs.com/navigation.cfm?guide_id=2318.I am not paid to say this.

3This gives a nice heuristic interpretation that in experiments we can actually think of lab equipment as operators that acts on the state.

4You can find them online too!https://www.thorlabs.com/NewGroupPage9.cfm?ObjectGroup_ID=711

(e)Nowwe consider a variation physics laboratory代写

where just before the photodetectors, we insert another identical beam splitter (call it BS2) as in part (a), so the setup looks like Figure 2 below. Current technology allows us to insert this faster than the light-crossing time of the interferometer arms (before light reaches the end of the interferometer), so we can insert BS2 after we have sent the light pulse through BS15.

physics laboratory代写
physics laboratory代写

Figure 2: variation of the first interferometer which includes the second beam splitter BS2.

Compute the output state after BS2 and compute the probability that D1 registers a click. What happens to the detection probability when φ = 0, π/2, π? Is the outcome of the experiment different from part (d)?

Moral of the story: 

The light pulse was originated in the source, and only when it has already gone through the beam splitter, the extra beam splitter is introduced in part (e). That means the light pulse (or the ‘photon’, if you insist on calling it that) could not have made the decision of which path to go in advance. In view of part (d) and part (e), it seems that the fact that light behaves as a particle (no interference, as in part (e), which means that only one detector clicks, the photon goes only on one path) or as a wavelike entity (interference pattern as in part (d),

which means physics laboratory代写that both detectors click and the ‘photon’ is in a superposition of going through both paths) is experimentdependent and completely independent of any decisions made at the source. The source does not influence if the light behaves as a classical particle, or as a quantum entity that can go through both paths at the same time. It is indeed the experimental setup after the light is emitted that conditions how light behaves, and more concretely, whether we obtain information of which path the ‘photon’ goes

5This prevents the ‘photons’ from, for instance, know in advance whether they should make an interfer- ence pattern or not!physics laboratory代写

in our experiment. More on that on Block 3!

Stern-Gerlach again and again… (Marked by Richard) physics laboratory代写

In 1922 Otto Stern and Walter Gerlach experimentally verified the prediction of Peter Debye and Arnold Sommerfeld that electrons have an intrinsic6 angular momentum that is quantized. In Fig. 3 the experiment is shown. We consider a source that provides silver ions

physics laboratory代写
physics laboratory代写

Figure 3: Experimental setup for the Stern-Gerlach experiment.

which are spin-1/2 particles, and assume that half of the atoms are prepared in the state corresponding to the positive eigenvalue of the spin operator in the z  direction sˆz  =  1 σˆz, and the other half prepared in the state corresponding to the negative eigenvalue. Denote the eigenstates by z). These atoms fly then through an inhomogeneous magnet where the magnetic field is along the z direction and the gradient of the field is parallel to that.physics laboratory代写

The interaction potential of the atom and the field is Vˆ  = µˆ · Bˆ, where the magnetic moment of the atom is µˆ  = gµBsˆ, g  is the Land´e factor, µB  is the Bohr magneton and sˆis the spin operator of the atom sˆ = (sˆx, sˆy, sˆz).

(a)If the force experienced by the atoms of the beam is proportional to the gradientof the  interaction  potential  Fˆ  =  −∇Vˆ ,  show  that  the  interaction  results  in  a  spatial splitting of the two different preparations of the atoms.

(b)Assume now that the atoms

with negative spin in the z-component after the first magnet are discarded. This is equivalent to doing a projective measurement on the z-directionofspin on the remaining beam, with its corresponding update  Now let the remaining beam of atoms go through a second magnet which is aligned in the x-direction. How many beams of atoms will appear on the glass plate gathering the

6Different from the orbital angular momentum Xˆ  × Pˆ.physics laboratory代写

results of the experiment? Relate your answer with the fact that the atoms cannot be simultaneously in an x and z eigenstates of the corresponding spin operators.

(c)What is  the  expectation  value  and  variance  of  sˆx  for  the  ensemble  of  atoms  in  the state |z) after the first magnet?

(d)Now assume again that after the second magnet the atoms with negative spin in sˆx are      The  other  atoms  go  through  a  third  magnet  aligned  along  an arbitrary direction given by the unit spatial vector n which forms an angle θ with the z axis and an angle φ with the x axis (the usual co-latutide and azimuthal angles in spherical coordinates).  What is the spin operator sˆn that this third splitter would be measuring (represented in the eigenbasis of sˆz)?

(e)Whatare the eigenstates of sˆn in the eigenbasis of sˆz?

(f)Calculate the probabilities that an atom after the second magnet is found in the eigenstate corresponding  to  the  positive  and  negative  eigenvalues  of  sˆn (spin  up  or down with respect to the n direction). Particularize your result for the special cases:

(a)θ = 0, φ = π,

(b) φ = 0, θ =π/2,

(c) θ, φ = π/2.

Quantum Guess Who? (Marked by Pipo)physics laboratory代写

Consider that you are given a qubit, that is, a two-level quantum system i.e the Hilbert space is two-dimensional. Let {|0) , |1)} be a basis of the qubit Hilbert space. The person who has given you the qubit tells you that they do not remember the state of the system,but they recall that it was either

  • |ψ)1= |0) or
  • |ψ)2= (|0) |1))/2

Since you go around claiming that you are a physicist, the person asks for your help.physics laboratory代写

(a)First,you wonder whether to perform a projective measurement in the {|0) , |1)} basis of the qubit would do the job. Compute the probabilities of getting each result, 0 or1, and explain if you can distinguish unambiguously the state you were given, eitherψ)1 or ψ)2 by doing this kind of projective measurement. Notice that you can only distinguish the state unambiguously if for both options there is an outcome of your measurement such that you can be sure that the state was either one or the other.

(b)Now consider the following POVMelements:

Discuss if with this set of measurements you can distinguish the state. Take into account that some outcomes from the measurement can be inconclusive, but never leading to a mistake.

Further ranting on PVMs (marked by Pipo)

In this exercise you will analyze in detail one of the numerous problems related to measuring schemes involving projector valued measures (PVM). More concretely, we will analyze the interplay between Heisenberg’s uncertainty principle and the fairy tale called “wavefunction collapse7”.

Consider a free quantum-mechanical particle in one dimensional space. The Hilbert space associated with this physical system is L2(R). Consider also the following family of projectors

physics laboratory代写
physics laboratory代写

(4.1)

where Ω is a subset8 of R.  Note that the projector on the whole real line is identity:  ΠˆR = 1 and that if A and B  are two disjoint subsets of R, i.e.  A  B , then ΠˆAB  = ΠˆA + ΠˆB . Clearly, these projectors define a probability distribution in the sense that you were

told in previous courses, that is, one can always define the probability of a particle in a state |ψ) to be in a region Ω with Born’s rule

p(particle in Ω) = (ψ|ΠˆΩ|ψ) = ∫ |ψ(x)| dx. (4.2)physics laboratory代写

Since this is a projective measurement, we now the update rule: if you find the particle somewhere it will remain in that somewhere right? Or in other words, the state after the measurement should then be given by

                             (4.3)

7One of the less cruel German fairy tales, unlike H¨ansel and Gretel or the Sleeping Beauty (for reference check the original stories)

8e.g., an interval.

but this leads to some problems, as we will see.

4.1 Part I: Oops, there is a problem with this. physics laboratory代写

(a)Show that the new wave function in the position representation, i.e. (x|ψj) = ψj(x), is of the form

                         (4.4)

where

physics laboratory代写
physics laboratory代写

(b)Calculatethe variance (uncertainty squared) of the momentum operator Pˆ,i. e.  ∆2 ,in the new state |ψj). Consider states not initially moving on average for simplicity, that  is,  (ψ|Pˆ|ψ) =  0.   Also,  consider  Ω  to  be  an  arbitrary  interval  (a, b).   Will  the new state |ψj) always be in the domain of the momentum operator provided that |ψ) was?

4.2 Part II: Wait, maybe we can fix it?

One could think that the problem appearing in previous part is somehow related to the sharpness of the measurement, that is, that the projectors defined by equation (4.1) con- strain too much the information about that system to have a smooth notion of “state update”. Therefore, one could perhaps expect to be able to find a set of better behaved projectors.physics laboratory代写

Indeed, consider a set of functions defined by taking a smooth, compactly supported9 function ψ0(x) and modulating with a phase and translating it along the real line, i.e.

(x|ψm,n) = ψm,n(x) = ei2πmxψ0(x  n),          (4.5)

where m, n Z. Perhaps these functions span all possible states of the system by linear

combination (i.e., ny wavefunction may be written as a combination of them). Further, theoverlap of an arbitrary state with this set of functions (their inner product) should give anapproximate localization of the particle in the real line, provided that the function ψ0(x) issupported around zero. Pictorially You can imagine ψ0(x) as some sort of smooth window function that decays away from the origin, adding the translation displaces that window, so we have a set of window functions covering the whole space.physics laboratory代写

So! do we have a nice basis to project on that is localize in space? As we will see, we are not required to give an explicit expression for ψ0(x) to check that, even in this case, the problem persists.

9Compactly supported:

The support if the function is compact. In other words, for a real function, the function is closed and bounded (is zero everywhere except it is finite for a finite set of intervals).

For the set of ψn,m) to give a reasonable measuring scheme, i.e. to assign probabilities, we require it to be an orthogonal basis of the Hilbert space, in such a way that

               (4.6)

Moreover, in order to avoid the problem of part I we also require that each |ψn,m) belongs to the domain of both the momentum and the position operators, in such a way that we

can measure the position and the momentum of the particle consecutively according to the update rule

physics laboratory代写
physics laboratory代写

(4.7)

where

Πˆm,n  |ψn,m)(ψn,m| . (4.8)

You are asked10 to prove that these two requirements are incompatible by answering the following questions.The way to proceed is to assume that ψ0) belongs to the domainsof both position and momentum, and that the set of functions ψm,n) generated by it forms a basis. This will lead to a contradiction.

(a)Consider the  vectors  Xˆ |ψ0) and  Pˆ |ψ0),  by  assumption  they  belong  to  the  Hilbert Show that

(ψ0|Xˆ|ψm,n) (ψm,n|Xˆ|ψ0) (4.9)

and that

(ψ0|Pˆ|ψm,n) (ψm,n|Pˆ|ψ0) (4.10)

(b)By assuming the completeness relation of the set |ψm,n), i.e. equation (4.6), show that

(ψ0|XˆPˆ|ψ0) (ψ0|PˆXˆ|ψ0) . (4.11)

(c)Finally, reach a contradiction (it’s an obvious..).

10In case there is any doubt after so much text, you need to answer only a), b) and c). Anything before that is for you to read and understand.

physics laboratory代写
physics laboratory代写

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