Homework
#2  (due 1/27/09)

2.1.  Calculate
the number of photons.

(a) Calculate the number of photons hitting an area of 1cm2  per sec  at a distance of 1m from  a 60 W light bulb. Assume that the light
is emitted at the wavelength of 500nm.

(b) Answer the same question for the number of photons per cm2
per second from the sunlight on the surface of Earth, assuming the surface is
normal to the direction of the sunlight and that the wavelength is also 500nm.

(c) For a linearly polarized plane wave, if the electric field is one
atomic unit, what is average power (per cycle) in units of Watts/cm2 ?

2.2.   Identifying the Hilbert space.

(a) For a
particle in an infinite 2D square well potential, i.e.,  V(x,y)=0 for    and V(x,y) is  infinite otherwise, define the Hilbert
space, the base state vectors and examples of observables.

(b) For two particles in an infinite 1D square well potential,
i.e., V(x)=0 for   and infinite otherwise, define the Hilbert
space if the two particles are distinguishable. If the two particles are
identical Fermions, what is the Hilbert space?  ( We have not talked about identical
particles yet but you have heard them before).

2. 3.  Measurement of polarized light.

(a) For a linearly polarized
light with the polarization axis at 45o from the x-axis on the
xy-plane, if you use a right-circularly polarized analyzer, what is the fraction
of the intensity that can be transmitted? What is the polarization of the light
after the measurement?

(b)   Following  the first measurement, you use another
linear polarizer ( 45o with respect to the x-axis) to analyze, what
will be the fraction of the intensity that will be transmitted with respect to
the original linearly polarized light?

Study
Material:

Adopted from:        http://en.wikipedia.org/wiki/Quantum_cryptography

Quantum key exchange—use quantum measurement
concept; use entanglement means

Will use photon polarization-- BB84 protocol - Charles H. Bennett and
Gilles Brassard (1984)

The sender (traditionally referred to as Alice)
and the receiver (Bob) are connected by a quantum communication channel which
allows quantum states to be transmitted. In the case of photons
this channel is generally either an optical fibre or simply free space.
In addition they communicate via a public classical channel, for example using
broadcast radio or the internet. Neither of these channels need to be secure;
the protocol is designed with the assumption that an eavesdropper
(referred to as Eve) can interfere in any way with both.

The security of the protocol comes from encoding
the information in non-orthogonal states. Quantum indeterminacy means that these states
cannot in general be measured without disturbing the original state (see No cloning theorem). BB84 uses two pairs of states,
with each pair conjugate to the other pair, and the two states
within a pair orthogonal to each other. Pairs of orthogonal states are referred
to as a basis. The usual polarization state pairs
used are either the rectilinear basis of vertical (0°) and
horizontal (90°), the diagonal basis of 45° and 135° or the circular basis of left- and right-handedness.
Any two of these bases are conjugate to each other, and so any two can be used
in the protocol. Below the rectilinear and diagonal bases are used.

Basis

0

1

The first step in BB84 is quantum transmission.
Alice creates a random bit
(0 or 1) and then randomly selects one of her two bases (rectilinear or diagonal
in this case) to transmit it in. She then prepares a photon polarization state
depending both on the bit value and basis, as shown in the table to the left.
So for example a 0 is encoded in the rectilinear basis (+) as a vertical
polarization state, and a 1 is encoded in the diagonal basis (x) as a 135°
state. Alice then transmits a single photon in the state specified to Bob,
using the quantum channel. This process is then repeated from the random bit
stage, with Alice recording the state, basis and time of each photon sent.

Quantum mechanics (particularly quantum indeterminacy) says there is no
possible measurement that will distinguish between the 4 different polarization
states, as they are not all orthogonal. The only measurement possible is
between any two orthogonal states (a basis), so for example measuring in the
rectilinear basis will give a result of horizontal or vertical. If the photon
was created as horizontal or vertical (as a rectilinear eigenstate)
then this will measure the correct state, but if it was created as 45° or
135° (diagonal eigenstates) then the rectilinear measurement will instead
return either horizontal or vertical at random. Furthermore, after this
measurement the photon will be polarized in the state it was measured in
(horizontal or vertical), with all information about its initial polarization
lost.

As Bob does not know the basis the photons were
encoded in, all he can do is select a basis at random to measure in, either
rectilinear or diagonal. He does this for each photon he receives, recording
the time, measurement basis used and measurement result. After Bob has measured
all the photons, he communicates with Alice over the public classical channel.
Alice broadcasts the basis each photon was sent in, and Bob the basis each was
measured in. They both discard photon measurements (bits) where Bob used a
different basis, which will be half on average, leaving half the bits as a
shared key.

Alice's random bit

0

1

1

0

1

0

0

1

Alice's random sending basis

Photon polarization Alice sends

Bob's random measuring basis

Photon polarization Bob measures

PUBLIC DISCUSSION OF BASIS

Shared secret key

0

1

0

1

To check for the presence of
eavesdropping Alice and Bob now compare a certain subset of their remaining bit
strings. If a third party (usually referred to as Eve, for 'eavesdropper') has
gained any information about the photons polarization, this will have
introduced errors in Bobs' measurements. If more than p bits differ they
abort the key and try again, possibly with a different quantum channel, as the
security of the key cannot be guaranteed. p is chosen so that if the
number of bits known to Eve is less than this, privacy amplification can be
used to reduce Eve's knowledge of the key to an arbitrarily small amount, by
reducing the length of the key.

The simplest type of possible attack is the
intercept-resend attack, where Eve measures the quantum states (photons) sent
by Alice and then sends replacement states to Bob, prepared in the state she
measures.   The probability Eve
chooses the incorrect basis is 50% (assuming Alice chooses her basis randomly),
and if Bob measures this intercepted photon in the basis Alice sent he will get
a random result, i.e. an incorrect result with probability of 50%. The probability
an intercepted photon generates an error in the key string is then 50% x 50% =
25%. If Alice and Bob publicly compare n of their key bits (thus
discarding them as key bits, as they are no longer secret) the probability they
find disagreement and identify the presence of Eve is

So to detect an eavesdropper with
probability Pd = 0.999999999 Alice and Bob need to compare n
= 72 key bits.