II. CONSIDERACIONES Y FUNDAMENTOS 1 Competencia
3. Consideraciones preliminares
Known as the Schr¨odinger inequality [97], the second term represents the contribution of the covariance of the two observables. For particular choices of Hermitian observables (including the EM quadrature operators ˆX1 and ˆX2 of the previous section) this second
covariance term is vanishing yielding the more commonly quoted Heisenberg–Robertson
uncertainty product
( A)2( B)2 1
4h[ ˆA,Bˆ]i
2, (3.11)
a relation that holds true for all of the states and observables of interest to this thesis, but not for any arbitrary chosen ˆAand ˆB2. We refer to this second expression simply as the Heisenberg Uncertainty Principle (HUP).
The HUP for the electromagnetic field
The commutator for the quadrature operators of the electromagnetic field may be found from definition equations 3.7 and the creation and annihilation operators equations 3.3 to obtain
[ ˆX1,Xˆ2] = 2i (3.12)
Which leads to the Heisenberg product ˆ
X1 Xˆ2 1. (3.13)
Thus there is some lower bound to the combination of uncertainties in determining both the phase and amplitude of the electromagnetic field. This is an important result. It means that neither quadrature field may be determined to ultimate precision without losing all information of the other. For repeated measurements of a light field the state of the field takes on a statistical interpretation where for an ensemble of measurements certainty is limited by the standard deviations dictated by equation 3.13. A similar more general result follows for any arbitrary pair of orthogonal quadrature operators
( Xˆ✓)2( Xˆ✓+⇡/2)2 1. (3.14)
When the product of uncertainties is minimised such that Xˆ1 Xˆ2= 1, a state is known
as a minimum uncertainty state.
3.1.3 Minimum uncertainty states of light
Minimum uncertainty states are of particular interest in the quantum optics laboratory and for high precision metrology, such as in interferometric gravitational wave detectors
2Of course from equation 3.10 it is true that ( A)2( B)2 >1
4|h[ ˆA,Bˆ]i|
2, but equality is only reached
like Advanced LIGO [75]. They are idealised states of light for which classical noise is entirely absent and quantum noise is the only limit to precision, leaving variance in the fields at a minimum, i.e.,
( Xˆ1)2( Xˆ2)2= 1. (3.15)
The uncertainty relation of equation 3.13 places a general lower bound applicable to any state, but the strict equality is achieved only with a certain special subset of states. Minimum uncertainty states represent the best possible precision a field of light may be determined.
The vacuum state
The vacuum state |0i is the lowest energy ground state of the electromagnetic field. It contains no coherent field, i.e. no photons h0|nˆ|0i = 0 but still has fluctuations in its quadratures enforced by the HUP (equation 3.13). Uncertainty in both quadrature can be found from the variance (equation 3.9) and the use of the commutator
V( ˆX1|0i) = h0|Xˆ12|0i h0|Xˆ1|0i2
= h0|(ˆa†ˆa†+ ˆaˆa+ ˆa†ˆa+ ˆaˆa†)|0i h0|(ˆa+ ˆa†)|0i2 = h0|([ˆa,ˆa†] + 2ˆa†aˆ+ ˆa†ˆa†+ ˆaˆa|0i h0|(ˆa+ ˆa†)|0i2 = 1 + 0 + 0 + 0 0
= 1 (3.16)
Likewise V( ˆX2|0i) = 1. The vacuum state is therefore a minimum uncertainty state with
variance distributed equally between the amplitude and phase quadratures
2Xˆ
1= 2Xˆ2 = 1. (3.17)
The presence of fluctuations in the absence of any coherent amplitude is a subtle but important fact, leading to a non-zero variance in the field even at its zero point energy. Direct detection of the vacuum field will yield no photons, however, the field may interfere and mix with coherent ‘bright’ fields that do. Where vacuum fields enter in to experiments of fragile prepared states, they partially replace noise statistics of that state with uncor- related fluctuations of the vacuum, destroying or degrading the coherence of that state. The mixing of these vacuum fields fluctuations with coherent fields introduces additional uncorrelated quantum noise in interferometric experiments and is the source of quantum noise in interferometric gravitational wave detectors such as Advanced LIGO and beyond.
The coherent state
In order form a realistic quantum model of a laser field it is necessary to form a state of light that closely mimics the harmonic evolution of the electric field as a function of time. Clearly the HUP places some fundamental limits on the statistical distribution of measurement events of this field. However, the expectation value of the state should resemble that of a classical equivalent field albeit the average outcome of many samples. We define a ‘coherent’ state of the electric field to be an eigenstate of the boson annihilation operator
ˆ
The expectation of the field operator (equation 3.5) yields a classically equivalent expres- sion for the field
hE(t)i = ✓ ~! 2✏0 ◆1/2 h↵|(ˆae i!t+ ˆa†ei!t)|↵i = ✓ ~! 2✏0 ◆1/2 (↵e i!t+↵⇤ei!t) (3.19) where the eigenvalue ↵ represents the complex amplitude of the field.
If we expand the coherent state in the basis of number states|↵i=P1n=0cn|niand apply
the definition, equation 3.18, then
ˆ a 1 X n=0 cn|ni=↵ 1 X n=1 cn 1pn|ni, (3.20)
where we may identify the recursive relation cn= p↵ncn 1 with normalisation factorc0= e |↵|2/2
to obtain a general expression for the coherent state in the number state basis
|↵i=e |↵|2/2 1 X n=0 ↵n p nk!| ni. (3.21)
More generally we may think of the coherent states as being generated by a displacement operation from the vacuum state. Using the expression|ni= (ˆa†)n/pn!|0i, we may recast equation 3.21 in terms of the vacuum ground state |0i and a sum of creation operators,
|↵i=e |↵|2/2 1 X n=0 ↵n(ˆa†)n n! |0i. (3.22) We may identify this summation as an exponential form to arrive at
|↵i=e|↵|2/2e↵ˆa†|0i. (3.23) Because e ↵⇤ˆa|0i =|0i, without loss of generality we may introduce an additional term that gives the expression
|↵i=e|↵|2/2e ↵⇤ˆae+↵aˆ†|0i=e(↵aˆ† ↵⇤aˆ)|0i (3.24) where we have used the identity eAˆ+ ˆB = eAˆeBˆe [ ˆA,Bˆ]/2 to arrive at the displacement operator, such that
|↵i= ˆD(↵)|0i, where Dˆ(↵) =e(↵ˆa† ↵⇤ˆa). (3.25) Thus any coherent state may be formed by application of this displacement operator to the vacuum state.
An important property of this displacement operation is that it preserves central moments of a state’s quadrature statistics. For the coherent state |↵i formed from the vacuum by
displacement ˆD(↵) we may make use of the fact that ˆ
D†(↵) ˆX1Dˆ(↵) = ˆX1+↵+↵⇤ (3.26)
to find the variance
V( ˆX1|↵i) = h↵|( ˆX1 h↵|Xˆ1|↵i)2|↵i = h0|(D†(↵)( ˆX1 h0|D†(↵) ˆX1D(↵)|0i)2D(↵)|0i = h0|( ˆX1+ 2Re[↵] h0|Xˆ1+ 2Re[↵]|0i)2|0i = h0|( ˆX1+ 2Re[↵] h0|Xˆ1|0i 2Re[↵])2|0i = h0|( ˆX1 hXˆ1i)2|0i = 1 (3.27)
Where we have used the result of equation 3.16 for the vacuum state. A similar result follows for ˆX2 and we find the coherent states, like the vacuum state, are minimum un-
certainty states
X1(↵)= X2(↵)= 1. (3.28) Thus the coherent state is the ideal approximation of the classical electromagnetic field having an expectation (average) that evolves identically to the classical field and the minimum allowable noise distributed equally amongst its quadratures. The coherent and vacuum states share identical statistics in the fluctuating quadratures, the coherent fields however have some non-zero mean number of photons in that field.
The squeezed states
The squeezed states are minimum uncertainty states of light for which the quadrature vari- ances are no longer equal. Like the coherent and vacuum states they obey the constriction of equation 3.15, but may allow greater precision in one quadrature at the expense of the other, i.e.
( ˆX1)<1,by allowing ( ˆX2)>1. (3.29)
Whilst coherent states are formed by a displacement operations from the vacuum, pre- serving the statistics of the light, squeezed state generation typically makes use of a multi- photon process to skew the statistics in a favourable way. For example, by simultaneously creating or destroying a pair of photons into a light mode, their correlated statistics may lead to a quadrature variance that is smaller than that available for a vacuum or coher- ent state. Consider a Hamiltonian for the two photon process where pairs of degenerate photons are generated from some auxiliary mode ˆc,
H=Hsys+ [ˆa2cˆ† ˆa†2cˆ], (3.30)
where Hsys is the system Hamiltonian (containing all other dynamics) and is the inter-
action strength. In the interaction picture the corresponding unitary evolution operator for the state due to the two photon process is
ˆ
Uint = exp h
where " is the interaction strength scaled by the relevant interaction time. If the mode ˆ
c is provided by some strong coherent state that isn’t depleted by the interaction, we may make a parametric approximation replacing it with the complex amplitude⇢=rei , giving a “squeezing operator”
ˆ
S(⇢) = exph1/2(⇢⇤ˆa2 ⇢aˆ†2)i (3.32) where r is the squeeze parameter (r 1) and is the phase angle of the interaction.
As in the case of generating coherent states from vacuum, general bright and vacuum squeezed states may be generated by the application of the squeeze operator to the vacuum followed by a displacement,
|↵,⇢i= ˆD(↵) ˆS(⇢)|0i. (3.33)
As was discussed in relation coherent states, central statistical moments are invariant under displacement operations. Modifications to the vacuum state statistics are therefore due wholly to the above squeezing operator. It can be shown that
ˆ
S†(⇢)aSˆ(⇢) =acoshr a†e 2i sinhr (3.34) ˆ
S†(⇢)a†Sˆ(⇢) =a†coshr ae 2i sinhr (3.35) The corollary of this is that quadratures operators are transformed as
ˆ
S†(⇢)X1,2Sˆ(⇢) = (coshr⌥e2i sinhr) ˆX1,2. (3.36)
Thus for a squeezed state formed by the application of the parametric squeezing operator to the vacuum, the variances may be described by
h 2Xˆ i = e 2r (3.37)
h 2Xˆ +⇡/2i = e2r. (3.38) Clearly these satisfy 2Xˆ 2Xˆ +⇡/2 = 1 and are minimum uncertainty states.
The squeezed states may be bright squeezed states with coherent amplitude or vacuum states where↵= 0. Thus it is possible that by invoking a two-photon generating device it is possible to alter the distribution of statistics between quadratures from a regular vac- uum state to produce less quantum noise in a chosen quadrature at the expense of another.
As an aside, it should be noted that squeezed vacuum state contains a mean photon number
¯
N =|↵|2+ sinh2r. (3.39) This means that in the absence of a coherent magnitude the squeezed vacuum still has a non-zero number of photons.