Download
1 / 42
441 likes | 455 Views
Coherent and squeezed states of the radiation field. By Amir Waxman. Introduction.
E N D
Coherent and squeezed states of the radiation field By Amir Waxman
Introduction Classically, an electric field consists of waves which are well defined both in amplitude and phase. That is not the case for the quantum radiation field. An electromagnetic field in the state |n> got a well defined amplitude, but completely uncertain phase. We can also describe the field in terms of 2 conjugates quadrature components If those components have minimum uncertainty relations, the state is said to be a coherent state. In general, it is possible to create a state with different uncertainties that steel has a minimum uncertainty relations. in that case, the fluctuations in one of the components will be reduced, on the expanse of greater fluctuations. In the conjugate component. That is a squeezed state. We will start with defining a the field operator for a single mode, and then on to definition of the number state. we will present the coherent state as a superposition of the number state, and then we will go to squeezed vacuum and then to squeezed coherent states, with all their properties.
Single mode field operator when we made the quantization of the radiation field, we defined the field operator: Where: Is a phase angle. By a convention we can get rid from the constants: And by the transformation to quadrature operators:
We get: From the commutation realations of the “a” operators we get: This commutation relation is only for the single mode while the complete field continue to commute. The uncertainty relations can be derived from the last equation: Where the variance of the electric field is defined by:
By this we can see that if the phase differs in л so we can measure the field precisely but then uncertainties of other pairs of angles will increase.for the vacuum state the mean is: And the variance is: The coherent signal S is defined as the expectation value of the field: From the matrix elements of destruction and creation operators we know that a coherent signal occurs only for a state that contains superposition of photons-number states differ in value by 1.
The uncertainty of the field can be refered to as noise: And the signal to noise ratio will thus be: SNR can be seen to depend on phase. The first order degree of coherence of the single mode state (g) can be defined as: And in second order:
We can define the degree of coherence with the mean square photon number (by using commutators and definition of the number operator): The photon number must be a an affirmative quantity: And than, the coherence from second order must satisfy: We thus see that in a single mode the second order coherence is independent in time and space.
Number states Number states are the basic states of the quantum theory of light. They form a complete set for the state of single mode. They are easy to manipulate in calculation of quantum optical properties. They are on the contrary less easy to generate experimentally. The eigen value equations for n states is : There for: The sec. degree of coherence then follows: For n<1 the limit is zero.
The energy eigenvalue relation can be written in some forms: Thus the number state has the quadrature-operator eigenvalue property: The quadrature- operator expectation values are: And the variance is: We can see that the number of state has got the same property for each one of the quadrature operators. For the vacuum state (n=0) the variance have the smallest values. We thus say that the vacuum state is a minimum uncertainty state.
By the expectation value and variance of the electric field we can now calculate the signal and the noise:
Coherent states The most commonly found single mode states correspond not to the number state, but to a superposition of number states. The coherent state is an important example for such a state as a single mode laser, operated well above threshold, generates a coherent state excitation. The coherent state is defined: When α is a complex number, and the normalization can be easily verified:
The coherent states are right eigen states of the destruction operator as can be seen form: The creation operator thus satisfies Remembering that: We can now write the coherent state using the vacuum state:
According to Baker-Hausdorf formula, any pair of operators that keep the relations: Also keep: In our case if A and B are: We get (with using the vacuum condition) : Which defines the displacement operator:
D is an equivalent to a creation operator for the complete state, analougus to the number state creation operator N D keeps the condition of unitarity: And the effect on the destruction operator is: With the hermitian conjugate relations:
Some properties of orthogonal states: Different coherent states are not orthogonal this can be shown from: Thus: The coherent-state expectation values for the number operator are: And for the second moment:
The photon number variance will then be: And the fractional uncertainty in the photon number: And this decreases while the coherent state amplitude increases. In a classical field the variance of intensity can vanish for a single wave. In quantum optics it cannot because of the particle-like aspects in quantum theory. The probability to find n photons in the mode is there for: Which is a Poisson distribution. For large values of n it approaches a gaussian distribution.
the poisson photon number distribution for various mean photon numbers
The coherent state expectation values of the quardature operators we defined earlier are the following: Where we have put: In a similar way we can calculate: The expectation values of the squares, can be now calculated, when we use the normal ordering procedure we used for the value of the square photon number n
The quadrature variances will there for be : From that we can conclude that the coherent state is a quadrature minimum uncertainties state for all photon mean number <α> ( unlike the situation in the n state). Using the expectations values for the quadrature components, we can easily calculate the coherent signal: And the variances help us to calculate the noise. The noise is there for phase independent and it has the minimum value allowed according to uncertainty principle. Next to be obtained is the signal to noise ratio: With maximum value at χ=θ
In the picture we present the coherent states properties. the arrow point on the mean field value. The length of the arrow is |α| and it is inclined in the χ-θ angle from the real field. Χ is determined by the evaluation in position and time of the field averages and there for is a property of the measurement and can be controlled by the experimentalist. Θ is a property of the field excitation on which the measurement is made and in principle is not controlled by the experimentalist. The black disc in the picture stands for the field uncertainty, and resolved to amplitude and phase
The amplitude result gives us the same uncertainty we got for the photon number uncertainty: The uncertainty in the phase may also be extracted, by geometric calculations: The product will there for be: This equation represents the trade off relation between amplitude and phase of the field in a coherent state. We can see that both the change in phase and fractional uncertainty are dependent in 1/α, which means that if we have a large number of photons the field is better defined both in amplitude and phase.
In the following graph we can see the phase dependence of the electric field ( <n>=4). The dashed lines stand for the noise band.
We use the formula for phase distribution : And: To get the phase distribution of coherent states: P is an even function of θ-φ and it follows that the mean phase equals:
The variance of the phase is more difficult to determine, and the curve which is shown here is a result of a numeric simulation. The dashed curve shows the large n approximatiom.
Squeezed vacuum A field excitation said to be quadrature squeezed, when its field uncerainty keeps: For some values of the measurement phase angle χ. Of course this “squeeze” has to be compensated by a variance larger than ¼ at the phase angle perpendicular to χ. We will start with the squeezed vacuum state which is defined: With the squeeze unitary operator: That has amplitude and phase defined as following:
When those definition are similar to those were made for the coherent states ( with operator D) From the exponent in the squeezed state definition it is clear the squeezed state consists only from even n states, but in order to show it is necessary to use general operator ordering methods. The number of state expansion will there for be brought directly: We will now turn for calculating some of the expectation values. First, for the mean number of photons we can write: Where we use unitarity. By operator relations, we can get to:
And now we can calculate the mean photon number: It is shown that if we don’t have any squeezing (s=0), the mean number of photons vanishes, and we return to ordinary vacuum state. Using the same methods of algebra we can derive the second moment: And the variance is accordingly: The degree of second order coherence will be The photon number flactuations of the squeezed vacuum are thus superpoissonian .
The interest of the squeezed vacuum lies in the quadrature operators properties. For destruction and creation operators : Which again shows that squeezed vacuum states are superpositions of n states with even n only. Now it is easy to show that: The variance of quadrature operator can be calculated with the help of: And thus be:
A presentation of the quadrature expectation values is shown in the picture. S is given by exp(s)=2. The lengths of the ellipse axis and the inclination angle are written. The black disc is the uncertainty area. if s=0 this is a coherent state but otherwise the values of the variances are either greater or smaller than ¼.
The expectation values of the field can be also calculated in a similar way. The mean field can be then shown to vanish: And the noise is: Which means completely phase dependent, unlike coherent states. That can be shown from the graph: When s equals 0 we get constant noise, like in coherent states.
It can be shown from the noise equation that: And this is a quadrature squeezed field. The max. value of the field variance is: And the product satisfy the uncertainty relations: The state can’t be squeezed for all values of s. the condition for squeezing is: The range of angles for which squeezing is possible thus decreases when s increases.
Squeezed coherent states Those states are defined by: D is the displacement operator: We will see that squeezed coherent states keep our possibility of reducing the noise in the system, but can also acquire a non-zero signal, unlike the squeezed vacuum.
In order to find the mean photon number we have to calculate some important relations, similar to those we calculate for the vacuum. With the help of the properties of D operator, we get: These transformations provide eigen value relations: And they reduce to squeezed vacuum relations where α=0. the mean photons number is then calculated: This is actually a sum between the coherent and squeezed vacuum mean numbers.
The photon number variance is then obtained: We can see that taking s to zero will give us a coherent state, and taking α to zero will give us a squeezed vacuum. The main interest lies in the expectation values of the quadrature operators. We start with: Which are identify to the coherent states result. The quadrature variances will thus be: Identical to the results for the squeezed vacuum state.
In the figure below we see a comparison between ordinary vacuum squeezed vacuum state and the coherent squeezed state. The figure well shows how the mean quadrature values are solely depended on the coherent parameter α and while the variance values solely determined by ζ the vacuum parameter.
From the former results, we easily obtain the expectation values of the electric field operator: Which is identical to the coherent state signal. The field variance, or noise is: Which is identical to the phase dependent squeezed vacuum noise. The signal to noise ratio reads: The SNR is dependent by distinct phase angles whose relative values depend of the method of generation of the squeezed coherent state. The source of light can be adjusted to optimize the SNR : The coherent squeezed state there for enjoys both world. His coherent signal noise is such of a coherent state, and his SNR can be improved, matching the noise of the squeezed vacuum state.
In the figure we can see a representation for a single mode amplitude squeezed coherent state with <n>=4 θ=v/2 and exp(s)=2, showing the mean and uncertainty of the electric field. The maximum SNR is achieved. This is an example for an amplitude coherent squeezed state.
The uncertainties occur in the limit when the coherent contribution is much larger: Then the uncertainty in photons number is: The uncertainty in phase is: And the product of photon number and phase uncertainties is again: The amplitude squeezed coherent state has reduced photon number uncertainty and an enhanced phase uncertainty.
In the figure bellow we can see the mean field representation of the amplitude squeezed state in the former figure. The dashed lines represent the noise band. The increased phase uncertainty and the decreased amplitude uncertainty make this noise band deferent from the one in the coherent state graph where the noise band was constant.
In this figure we present the case of . The light in this case is said to be phase squeezed coherent state. The effects of the squeezing on the amplitude and phase uncertainties are now interchanged: The major axis of the noise ellipse is enhanced to show an increased amplitude uncertainty, and the minor axis is reduced to show a decreased amplitude uncertainty.
This figure shows experimental results for the noise. The measurements are made by homodyne detection, with a local oscillator phase angle that varies linearly with the time. (a) is a coherent state (b) is a squeezed vacuum state (c) Is amplitude squeezed state (d) phase squeezed state (e) A squeezed state with 48 degrees between the coherent vector, and the axis of the noise ellipse.
