6.4 One-photon interference signal

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Из курса от партнера École Polytechnique

Quantum Optics 1 : Single Photons

63 оценки

École Polytechnique

Quantum Optics 1 : Single Photons

63 оценки

This course gives you access to basic tools and concepts to understand research articles and books on modern quantum optics. You will learn about quantization of light, formalism to describe quantum states of light without any classical analogue, and observables allowing one to demonstrate typical quantum properties of these states. These tools will be applied to the emblematic case of a one-photon wave packet, which behaves both as a particle and a wave. Wave-particle duality is a great quantum mystery in the words of Richard Feynman. You will be able to fully appreciate real experiments demonstrating wave-particle duality for a single photon, and applications to quantum technologies based on single photon sources, which are now commercially available. The tools presented in this course will be widely used in our second quantum optics course, which will present more advanced topics such as entanglement, interaction of quantized light with matter, squeezed light, etc... So if you have a good knowledge in basic quantum mechanics and classical electromagnetism, but always wanted to know: • how to go from classical electromagnetism to quantized radiation, • how the concept of photon emerges, • how a unified formalism is able to describe apparently contradictory behaviors observed in quantum optics labs, • how creative physicists and engineers have invented totally new technologies based on quantum properties of light, then this course is for you.

Из урока

Wave-particle duality for a single photon in the real world

You are now ready to develop the description of a real experiment , which was the first one to reveal directly the dual nature -- wave and particle, of a real single photon wave-packet. You will not only be able to describe, with the formalism you have learned, both the particle-like and the wave-like behaviors, but you will also see how to take into account the features of a real experiment, which is never perfect. Last and not least, we will have the opportunity to think about the notions of wave-particle duality and complementarity, which should be not confused, and about thethe statement of Feynman, who named wave-particle duality “a great quantum mystery”. I will try to convince you that when one identifies a mysterious behavior, one should not complain, but rather explore the possibility that something new and interesting can emerge from that mystery.

6.0 Introduction to Lesson 63:21

6.1 Anti-correlation for a one-photon wave-packet on a beam-splitter13:16

6.2 Anti-correlation experiments: fully quantum behavior9:23

6.3 Anti-correlation with supplementary photons10:04

6.4 One-photon interference signal14:08

6.5 One photon interference experiment14:21

6.6 Wave particle duality and complementarity9:00

6.7 A fruitful mystery7:16

Познакомьтесь с преподавателями

Alain Aspect
Professor
Physics
Michel Brune
Professor
Physics

In a previous lesson, we already encountered a Mach-Zehnder interferometer,

the favorite interferometer of theorists.

Classically, a wave enters in input 1, is split into reflected and

transmitted components 3 and 4, which are recombined on a second beam splitter.

The intensities in the two outputs 5 and 6, depend on the difference in

the optical path 3 and 4, between the two beam splitters.

This means that when we move mirror M3,

which is mounted on a piezo transducer,

one observes a modulation of the signals at detectors D5 and D6.

You remember that we calculated the interference signal for the toy model, of

a one photon wave packet consisting of a monochromatic wave with a finite duration.

Today, we are speaking of real experiments

done with one photon wave packets emitted by spontaneous emission.

They are composed of many one photon states with different frequencies.

We are going to do the calculations in the Heisenberg formalism where

the field operators depend not only on position, but also on time.

In contrast, the state does not depend on time, and

the coefficients in its expansion keep their value at time t0.

The real positive coefficient, K, is chosen to normalize the state.

We have already encountered that state:

it corresponds to a wave packet starting at t0 and decaying exponentially.

To be more precise, this means that the probability of a photo detection at O

in the input channel 1, is a wave packet

rising suddenly at t0, and decaying exponentially.

The radiation state in input 2 is the vacuum,

so the input state vector should write as the product

of one in input 1 times the vacuum in input 2.

In fact, we will not keep that vacuum term in

the calculation since here it plays no role.

Note, however, that there are other calculations where the vacuum

in one input channel of an interferometer does play a role.

So be careful, and if you have any doubt, do not hesitate to take into

account the vacuum in the second input, and to check that it does not play a role.

It would be a good idea to do it when you study this lesson:

you could repeat the calculation I'm going to do now,

keeping the term associated with the vacuum in channel 2,

and check by yourself that it does not change the result.

Let us then develop our calculation of the probabilities

of single detections at detectors D5 and D6.

We know that what we have to do is to express the output fields E+_5 and

E+_6 as a function of the input fields E+_1 and

E +_2 and apply them to the input state.

We are going to use a useful feature of the Heisenberg formalism:

we can directly obtain the expressions of the field operators

by taking the classical expressions obtained in the same situation.

So let us recall the classical optics calculation

which we have already encountered in a previous lesson.

I recall here the expressions describing the classical fields

transformations on the two beam splitters.

The amplitude reflection and transmission coefficients for the complex amplitude

E+ are taken equal to the square roots of capital R and capital T.

You remember that the minus sign on one of the reflection terms

is associated with the unitarity of the transformation.

It ensures energy conservation in classical optics and

the conservation of the commutation relations in quantum optics.

The different choices for the position of the minus

signs associated with the two different beam splitters correspond to

the fact that the second beam splitter is reversed, as shown on the figure.

We assume that the coefficients R and

T do not depend on the frequency of the field components,

otherwise, we should use the Fourier decomposition of the fields.

Note also, that I ignore the polarization of the fields for simplification.

You can think of polarization perpendicular to the figure and

consider the values on the axis perpendicular to the figure.

As just explained, we will ignore the field E_2

corresponding to an empty input and erase the corresponding term.

I have not yet written explicitly the dependence on time,

since all values are taken at the same point O,

but we must do it as soon as fields propagate between O and O'.

In the vacuum or in a non-dispersive medium,

a simple retardation term is enough to take into account the propagation.

This allows us to express the classical field complex amplitudes in output 5 and

6 as a function of the input field amplitude in channel 1.

We can now substitute the classical complex field amplitudes

with the corresponding E+ operators to obtain

the output fields as a function of E1+_hat.

Let us then calculate the photo-detection probability at D5.

We must express E5+_hat at D5 as a function of

E1+_hat applied to the input state and take the squared modulus.

The propagation factor from O' to the detector D5 is

an exponential with a complex exponent, and its squared modulus is 1.

So it is enough to calculate the expression involving E5+_hat, at O'.

Note that if we had kept the E2+_hat terms, they would give null contributions.

Be sure to understand this point.

In order to use the expression of E5+_hat,

as a function of E1+_hat,

we must calculate E1+_hat at time t-L3/c

applied to the one photon wave packet, and similarly, for t-L4/c.

We have already done that calculation in section one.

The result depends only on the retarded times tau_3 or tau_4.

In the case of a spontaneous photon wave packet,

we have found the result written here within a normalization constant

that I ignore, to keep only the most important factors.

The operator E1+ applied to the one photon state

yields the vacuum state times a function f(tau).

That function f is a decaying exponential starting at tau equals 0.

Remember that we have two different values of tau for channels 3 and 4.

H(tau) is the step function, null for negative taus,

We thus have a wave packet starting at tau equals 0, and decaying

exponentially with an inverse time constant gamma over 2 for the amplitude.

The phase factor evolves at the central frequency omega_0.

Now, adding the two terms associated with L3 and L4,

we find that the vacuum state factorizes,

and we obtain the squared modulus of the sum of two complex

amplitudes as would be written with a classical wave packet.

So the probability of a single detection in the output 5 will

exhibit the usual interference term.

Let us write the expansion showing the interference term,

in the case of balanced beam splitters,

that is to say with R = T = one-half.

To complete the calculation it is convenient to introduce

the average L_bar of the two paths and the path difference delta_L.

We now assume that the path difference,

delta_L is small enough that the corresponding time shift is

small compared to the time constant, 1 over gamma.

This is the case for typical atomic transitions,

where gamma is larger than one nanosecond, while delta_L is a few wavelength,

that is to say, a few micrometers for visible light:

delta_L over C is then of the order of a few femtoseconds,

as you can check yourself.

It means that the rising edge and the decaying exponential can be taken

identical for the two terms, and we can use their expression as a function of tau bar

which depends on the average propagation time in the interferometer.

This simplification cannot be made for the phase factor which

changes by 2pi when the length L changes by the one wavelength.

This is why we have the different plus and minus delta L factors in f3 and f4.

Inserting these approximate values of f3 and

f4 in the expression of the w_1 of D5,

we obtain the familiar interference term: 1-cosine of the phase difference.

Remember that w_1 is the rate of detection per unit surface of the photodetector D5,

and to obtain the observed signal, we must integrate over the detector.

This is the result of this integration, yielding the rate of photodetection at D5.

In order to obtain this expression, I have adapted

the calculation of the previous lesson to obtain the exact prefactor.

Epsilon is the global quantum efficiency

including the collection efficiency and the detector efficiency.

Note that I have also replaced 1-cosine by

twice the squared sine of half the phase difference.

The result is the decaying exponential

characteristic of the spontaneous emission wave packet,

with the retardation associated with the propagation in the interferometer.

The rate of photodetection is the product of the decaying exponential

by the interference term.

Integrating over the gate, we obtain the probability of a photodetection per pulse,

that is to say, the interference term only

multiplied by the global detection efficiency epsilon.

It depends only on the path difference delta_L, and

it can be measured by repeating the experiment a large enough

number of times at each value of the position of the mirror.

A similar calculation would give the rate of photodetection at D6.

Its integral is the probability per pulse to detect

the photon at D6 for a given value of delta_L.

Note that it is complementary of P of D5, so

that the sum of P of D5 and P of D6 is constant and equal to epsilon.

Now that we have obtained these results,

we are ready to understand the results of the real experiments.

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