Download Introductory Quantum Optics Section 1. Single photon physics

Introductory Quantum Optics
Section 1. Single photon physics
Almut Beige
(Dated: November 9, 2007)
Summary. In the first lecture, we discuss the quantum character of light. That there are light
beams that cannot be split into weaker beams was first noticed by Planck, when he tried to explain
the black body radiation. The concept of single photons was then introduced by Einstein in his
seminal paper in 1905. Here, we won’t look at the work of these two great physicists. Instead,
we discuss recent linear optics experiments with single photons, which illustrate some surprising
features of quantum mechanical particles.
1.0 Introduction
This lecture is the second half of the course Photonics and Introductory Quantum Optics. Up to now,
the course focussed on classical light. We now continue with a summary of the basic features of quantum
mechanics and illustrate them through single photon experiments. A photon is the smallest quantity
in which light can occur. On this level, light has to be described by a quantum mechanical formalism.
Depending on the experimental setup, photons behave like particles and their energy and momentum are
given by
E = h̄ω and p = h̄k .
(1)
Here ω denotes the frequency and k denotes the wave vector of the photon.
In the following, we discuss the generation of entanglement and look at recent linear optics experiments
with single photons. These experiments were designed to test the foundations of quantum mechanics
or aim at finding implementations of new quantum technological devices. Recently, for example, it
became feasible to generate single photons on demand. Such an experiment requires an atom-cavity
setup, like the one shown in Figure 1. The presence of exactly one atom trapped inside the cavity
makes it possible to guarantee the emission of at most one photon, when a laser trigger pulse transfers one quant of excitation from the atom into the cavity field. The outcoupling mirror of the cavity
fixes the direction of the emitted photon which can the be used as the input for single photon experiments.
Another central subject of quantum optics is the interaction between matter and light. A big part of
this course will therefore be devoted to the emission of single photons from a laser driven two-level atom.
FIG. 1: Experimental setup for the generation of a single photon on demand. The pictures are from the webpage of
Prof. Rempe’s group in Garching by Munich, which has an operating photon pistol in the laboratory. Alternatively,
one can experiment with single photons from parametric down conversion sources or approximate them through
very weak laser pulses. More recently, relatively reliable single photon sources based on quantum dot systems
became available.
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Such a system can be described by rate equations, by the quantum jump approach and by the master
equations. For finding out more details about the course material, I recommend:
• C. C. Gerry and P. L. Knight, Introductory Quantum Optics, Cambridge University Press (Cambridge, 2005).
For further reading see for example:
• R. Loudon, The Quantum Theory of Light, Oxford University Press (Oxford, 2000).
• M. O. Scully and M. S. Zubairy, Quantum Optics, Cambridge University Press (Cambridge, 1997).
• S. M. Barnett and P. M. Radmore, Methods in Theoretical Quantum Optics, Oxford University
Press (Oxford, 1997).
Enjoy the course!
1.1 The Dirac notation
In the following we use the Dirac notation to describe the state of a single photon or atom. This notation
is always used, when describing the features of quantum mechanical systems other than their position and
momentum distribution. If you are not familiar with this notation, please look it up in an introductory
quantum mechanics book. Here is a short introduction into the Dirac notation.
• In the following, we denote the state of a quantum mechanical system by |ψi. In general, |ψi is a
superposition of normalised and pairwise orthogonal basis states |ii and can be written as
|ψi =
N
X
ci |ii .
(2)
i=1
• Suppose, we have two wave functions |ψi and |ϕi with |ψi as above and with |ϕi given by
|ϕi =
N
X
di |ii .
(3)
i=1
Then the complex scalar product between both state vectors is defined as
hψ|ϕi =
N
X
d∗i ci .
(4)
i=1
If we therefore define
hϕ| =
N
X
d∗i hi| ,
(5)
i=1
then the scalar product (4) becomes simply the product of the right hand side of Eq. (5) with the
right hand side of Eq. (2), since
1 if i = j ,
hi|ji =
= δij .
(6)
0 if i 6= j .
This applies, since the states |ii are basis vectors and implies

! N
N
N
X
X
X
hϕ|ψi =
d∗i hi| 
cj |ji =
d∗i ci ,
i=1
j=1
(7)
i=1
as in Eq. (4). The vector hψ| is in general called a bra-vector and the vector |ϕi is called a ketvector. The Dirac notation is therefore also called the bra-ket notation.
3
• Using the Dirac notation the norm squared of a vector can be written as
k |ψi k2 = hψ|ψi .
(8)
A state vector |ψi is therefore normalised, if
hψ|ψi =
N
X
|ci |2 = 1 .
(9)
i=1
Keep in mind that the Dirac notation is nothing other than writing vectors in a different way. It has
been developed to make quantum mechanical calculations more convenient. Its expressions contain only
the most necessary information that is needed to perform quantum mechanical calculations.
Below you find a few simple examples of quantum mechanical operators in Dirac notation. An operator
applied to a vector yields another vector.
• An especially simple operator is the identity operator I. Applied to a vector, it returns the same
vector. Applying the identity operator to a state vector is therefore the same as multiplying the
vector with one. This is sometimes a useful operation. Suppose the states |ii form a complete basis.
Then I can be written as
I=
N
X
|iihi| .
(10)
i=1
Let us now apply this operator to the state vector in Eq. (2). Doing so and using Eq. (6), we obtain

! N
N
N
X
X
X
ci |ii = |ψi .
(11)
cj |ji =
I |ψi =
|iihi| 
i=1
j=1
i=1
This operator has indeed no effect on the state vector.
• If you are more familiar with matrices, you might find it helpful to note, for example, that the
linear operator
A11 A12
A =
(12)
A21 A22
c1
has the same effect on the vector ψ =
as the operator
c2
A = A11 |1ih1| + A12 |1ih2| + A21 |2ih1| + A22 |2ih2|
(13)
has on the vector |ψi = c1 |1i + c2 |2i.
• Finally, we introduce the terms eigenstates and eigenvectors of a quantum mechanical operator A.
In general, A |ψi is very different from |ψi. However, one can always find states |λi i with
A |λi i = λi |λi i .
(14)
These states |λi i are the eigenstates of A and the numbers λi are the corresponding eigenvalues.
Note that the states |λi i form a complete set of basis states with hλi |λj i = δij . Consequently, one
can write the operator A as
A =
N
X
λi |λi ihλi | ,
i=1
when all its N eigenstates |λi i and eigenvalues λi are known.
(15)
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1.2 The basic postulates of quantum mechanics
Before talking about concrete single photon experiments, we now summarise the three basic postulates of
quantum mechanics:
1. The state of a quantum mechanical system can always be described by a vector |ψi of length one.
|ψi is called the state vector and hψ|ψi = 1.
2. Suppose a measurement is performed on a system prepared in |ψi, whether it is prepared in a
certain state |ϕi. Then the probability to find the system in |ψi equals
pϕ = |hϕ|ψi|2 .
(16)
Note that the state of the quantum mechanical system becomes |ϕi, when found in this state.
Otherwise, repeating the same measurement would not give the same measurement result with unit
probability.
3. Quantum mechanical observables are represented by operators of the form
A=
N
X
λi |λi ihλi |
(17)
i=1
with hλi |λj i = δij . The numbers λi are the possible measurement outcomes, when measuring the
observable A. Moreover, the state |λi i is a state, which yields the measurement outcome λi with
probability one. As we show below, the expectation value for a measurement of the observable A
on an ensemble of systems prepared in |ψi equals
hAiψ = hψ| A |ψi .
(18)
Using only Eq. (16), one would expect that the expectation value for a measurement of an observable A
on an ensemble of systems prepared in |ψi equals
hAiψ =
N
X
λi |hλi |ψi|2 .
(19)
i=1
However, the right hand side of this equation is identical to
N
X
λi hψ|λi i hλi |ψi = hψ|
i=1
N
X
!
λi |λi ihλi |
|ψi = hψ| A |ψi .
(20)
i=1
And this is the same as the right hand side of Eq. (18). The main application of this equation is that
it allows for the calculation of expectation values without having to know all the eigenstates |λi i and
eigenvalues λi of an observable A.
1.3 The basic linear optics elements
To familiarise ourselves more with the quantum nature of single photons we now have a closer look at some
recent linear optics experiments with single photons in which single photons travel through a linear optics
network. Notice that photons do not interact with each other. They travel for example many light years
through space from one star to another without changing their frequency and polarisation. Photons also
do not interact with each other in a linear optics network. Even when several photons enter a linear optics
network simultaneously, each of them is redirected independently of the others to one of the output ports.
Below is a list of the four basic building blocks of linear optics networks. It is described how each of
them affects an incoming photon. See Figure 2 for the notation used for the different input and output
ports:
5
2’
1
2’
1’
1
1’
2
2
BS
PBS
1
1’
1
HWP
ϕ
1’
PP
FIG. 2: Symbols for the presentation of linear optics elements in schematic experimental setups.
• Beam splitter (BS) or half transparent mirror: splits a photon wave packet into two parts of equal
size such that
photon in input port 1 and none in 2 : |ψi1 −→ √12 |ψi10 + i |ψi20
photon in input port 2 and none in 1 : |ψi2 −→ √12 |ψi20 + i |ψi10 .
(21)
The phase factor i always has to be added, when a photon changes its direction by 900 . During the
reflection of a photon on a mirror by 1800 , the state vector of the photon accumulate a factor −1.
Let’s now see what happens if we send one photon in |ϕi into input port 1 and one photon in |ψi
into input port 2:
|ϕi1 |ψi2 −→ √12 |ϕi10 + i |ϕi20 √12 |ψi20 + i |ψi10
= 12 |ϕi10 |ψi20 + i |ϕi10 |ψi10 + i |ϕi20 |ψi20 − |ψi10 |ϕi20 .
(22)
Task: How likely is it to find one photon in each output port?
• Polarising beam splitter (PBS): redirects vertically polarised photons (in state |Vi) without affecting
horizontally polarised photons (in state |Hi) such that
photon in input port 1 and none in 2 : α |Hi1 + β |Vi1 −→ α |Hi10 + β |Vi20
photon in input port 2 and none in 1 : α |Hi2 + β |Vi2 −→ α |Hi20 + β |Vi10
(23)
Task: What would happen to the initial state α |Hi1 |Hi2 + β |Vi1 |Vi2 ?
Task: How likely is it to find a photon in each output port in this case?
• Half wave plate (HWP): changes the polarisation of the incoming photon according to
photon in input port 1 : |+i1 ≡ √12 |Hi1 + |Vi1 −→ |Vi10 ,
|−i1 ≡ √12 |Hi1 − |Vi1 −→ |Hi10 .
• Phase plate (PP):
(24)
adds an overall phase factor to the state of the incoming photon such that
photon in input port 1 : |ψi1 −→ eiϕ |ψi10 .
(25)
1.4 A maximally entangled photon pair
An entangled state is a state whose state vector |ψi cannot be written as a product of the individual
states of its subsystems. For example, any state describing two photons in two separate output ports
that is not of the form
|ψi = α |Hi10 + β |Vi10 · γ |Hi20 + δ |Vi20
(26)
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Alice
Bob
H
V
V
+
H
−
−
+
FIG. 3: Possible measurement outcomes of a polarisation measurement performed on the maximally entangled
photon pair prepared in |ψent i.
is entangled. An example of a maximally entangled two-photon state is the state obtained when sending
one photon prepared in |Hi and one photon photon prepared in |Vi simultaneously through a beamsplitter
(c.f. Figure 2). From Eq. (21) we know that the state of the two photons changes according to
|Hi1 |Vi2 −→ 12 |Hi10 + i|Hi20 · |Vi20 + i|Vi10
= 12 |Hi10 |Vi20 − |Hi20 |Vi10 + i |Hi10 |Vi10 + i |Hi20 |Vi20 .
(27)
Under the condition of the detection of one photon per output port, which happens with probability 12 ,
the state of the system becomes therefore
(28)
|ψent i = √12 |Hi10 |Vi20 − |Vi10 |Hi20 .
This is a maximally entangled state. To analyse some features of |ψent i we first note that it can alternatively be written as
|ψent i = √12 − |+i10 |−i20 + |+i20 |−i10
(29)
with
|+i =
√1
2
|Hi + |Vi
and |−i =
√1
2
|Hi − |Vi .
(30)
Suppose an Alice collects the photons in output port 1’ and measures the polarisation of each incoming
photon and Bob measures the polarisation of each photon in 2’. If both receive a photon, they
compare their results. The possible measurement outcomes of Alice’s and Bob’s measurements, if both
distinguish either between |Hi and |Vi or between |+i and |−i, are shown in the table in Figure 3.
Remarkably, whatever Alice measures, Bob’s measurement outcome is different. Note that this happens
independently of whether they measure the polarisation states |Hi and |Vi or the basis states |+i and |−i.
Task: Use Eq. (30) to show that the states (28) and (29) are indeed the same.
1.5 Interference of a single photon with itself
You are probably all very familiar with the interference of classical light but also single photons can
interfere. A single photon always interferes only with itself. The interpretation of quantum mechanical
interference experiments is therefore different from the interpretation of classical interference experiments
although there are also many similarities. To illustrate this, let us have a look at the setup shown in
Figure 4 and assume that a photon prepared in |ψi enters input port 1. Using Eqs. (21)-(25), we find
that the initial state transforms according to
|ψi1 −→ √12 i |ψi10 + i eiϕ |ψi20 .
(31)
Here we assumed that a reflection on the mirror adds a phase factor i to the photon state. Taking the
second beam splitter into account, the transformation (31) becomes
|ψi1 −→ 12 i |ψi100 + i |ψi200 + i eiϕ |ψi200 − eiϕ |ψi100
= 12 i 1 − eiϕ |ψi100 + i 1 + eiϕ |ψi200 .
(32)
7
1’’
ϕ
2’
2’’
photon
detector
1’
incoming 1
photon
mirror
2
input:
vacuum
FIG. 4: Experimental setup for a one-photon interference experiment.
We can now calculate the probability of finding a photon in output port 200 and obtain
2
2
2
P200 = 12 1 + eiϕ |ψi200 = 41 1 + eiϕ = 14 e−iϕ/2 + eiϕ/2 = cos2 12 ϕ ,
(33)
which oscillates between 0 and 1 as a function of ϕ.
Quantum mechanical interference effects occur, when there are different trajectories that contribute
to the same event. Note that the detection of a photon in the detector 2” can be caused by the
photon traveling through the lower part of the interferometer. However, the photon can also travel
through or the upper part of the interferometer in Figure 4 and exit the setup via 2”. Both respective amplitudes of the state vector contribute to the probability P200 and thereby interfere with each other.
The same interference pattern occurs, when one sends many photons at once through the setup in
Figure 4. In an analogous classical experiment, a light beam consisting of many photons would enter
input port 1. Each photon would travel independently from the others through the setup. The signal at
output port
2” would be a light beam whose intensity equals the initial beam intensity multiplied with
cos2 12 ϕ . The reason isthat each incoming photon leaves the network through output port 2” only
with probability cos2 12 ϕ . What we discussed above is thus in good agreement with predictions from
classical optics.
1.6 A two-qubit quantum gate operations
Quantum gates are like the gate operations in a classical computer. They process information. Now
the information is not encoded in the bits of a classical computer but in the bits of a quantum mechanical
systems, the so-called qubits. An example is the controlled phase gate which implements the operation
Ugate = |00ih00| + |01ih01| + |10ih10| − |11ih11| .
(34)
This means, it transforms a two-qubit system in the general state
|ψi = α |00i + β |01i + γ |10i + δ |11i
(35)
Ugate |ψi = α |00i + β |01i + γ |10i − δ |11i .
(36)
into the state
The final state still contains the same amount of information as the initial state of the qubits. However,
the state of the first qubit has changed from |1i to −|1i conditional on the second qubit being in |1i.
Here we are interested in a possible implementation of such a gate operation. Suppose the qubits are
photons and information is encoded in the polarisation of their states. In the following we assume that
8
output 2 output 4
PBS
input 1
output 1
input 2
output 2
input 3 input 4
FIG. 5: Linear optics network with two polarising beam splitters that can be used for the implementation of a
controlled two-qubit phase gate.
the state |Hi encodes a 0 and the state |Vi encodes a 1. We now have a look at the linear optics network
shown in Figure 5. Suppose the two photonic qubits, whose state is of the general form
|ψin i = α |Hi1 |Hi2 + β |Hi1 |Vi2 + γ |Vi1 |Hi2 + δ |Vi1 |Vi2 ,
(37)
enter the network through input ports 1 and 2. After passing through the network, their state equals
|ψout i = α |Hi1 |Hi2 + β |Hi1 |Vi4 + γ |Vi3 |Hi2 + δ |Vi3 |Vi4 .
(38)
This means, depending on their state, the photons get distributed to different output ports.
In order to perform the above described gate operation, we need to send two auxiliary photons, prepared
in the maximally entangled state
(39)
|ϕin i = 21 |Hi3 |Hi4 + |Hi3 |Vi4 + |Vi3 |Hi4 − |Vi3 |Vi4 ,
simultaneously through the input ports 3 and 4. These leave the network in the state
|ϕout i = 21 |Hi3 |Hi4 + |Hi3 |Vi2 + |Vi1 |Hi4 − |Vi1 |Vi2 .
(40)
Let us now calculate the state of the four photons under the condition of one photon per output port.
After normalisation, this state equals
|ψcond i = α |Hi1 |Hi2 |Hi3 |Hi4 + β |Hi1 |Vi2 |Hi3 |Vi4 + γ |Vi1 |Hi2 |Vi3 |Hi4 − δ |Vi1 |Vi2 |Vi3 |Vi4 . (41)
If we now measure whether or no there is a photon in the output ports 1 and 2, without detecting their
state, we postselect the state
|ψcond i = α |Hi3 |Hi4 + β |Hi3 |Vi4 + γ |Vi3 |Hi4 − δ |Vi3 |Vi4 .
(42)
This is exactly the expected output state for a conditional two-qubit phase gate.
Unfortunately, this output cannot be obtained with probability one. Can you carefully go through the
above calculations and show that the success rate of the above described operation equals 14 ?
1.7 Final remarks
In this section, we discussed the quantum mechanical features which dominate the behaviour of light on
the level of single photons. Single photons have to be described by a state vector |ψi and are characterised
by properties like their frequency, polarisation, path and arrival time. We saw that a single photon can
interfere with itself inside a linear optics network which is a nice demonstration of the wave particle
duality of quantum mechanical systems. Sending two photons through a beam splitter can result in the
generation of a maximally entangled state.