Chapter 7: Quantum States of Light - Cornell University 7: Quantum States of Light ... With this in mind, and the fact that (for a single mode cavity), ... The vacuum state 0 may also

Quantum Optics for Photonics and Optoelectronics (Farhan Rana, Cornell University)

1

Chapter 7: Quantum States of Light

7.1 Cavity Fields The operators for the fields inside a cavity are,

mmmm

momo

mmmm

mo

n

mmmm

mom

rUtatatrH

rUtataitrE

rUtatatrA

)()(ˆ)(ˆ2

1),(ˆ

)()(ˆ)(ˆ2

),(ˆ

)()(ˆ)(ˆ2

),(ˆ

The Hamiltonian is,

2

1ˆˆˆmm

mm aaH

The time dependence of the operators in the Heisenberg picture is,

ti

mm

timm

m

m

eata

eata

ˆ)(ˆ

ˆ)(ˆ

In this chapter we will mostly consider only a single mode of the field in a cavity and ignore the remaining modes. In order to keep the notation from getting too cumbersome, we will drop the mode number in the subscripts (e.g. "m" above) unless necessary, and write the Hamiltonian and the field operators as follows,

2

1ˆˆˆ aaH o

)()(ˆ)(ˆ2

1),(ˆ

)()(ˆ)(ˆ2

),(ˆ

)()(ˆ)(ˆ2

),(ˆ

rUtatatrH

rUtataitrE

rUtatatrA

ooo

o

o

oo

7.2 Fock States or Photon Number States Number states of a mode contain a definite number of photons. As discussed in an earlier Chapter, these are eigenstates of the photon number operator and are defined as,

Cavity


2

nnnaann

n

an

n

ˆˆˆ

0!

)(

and,

nnnH o

2

1ˆ

Also,

1ˆ

11ˆ

nnna

nnna

It follows that,

0|ˆ||ˆ| nannan

and,

0),(ˆ),(ˆ ntrHnntrEn

.

So surely n cannot be the quantum state of radiation coming out of, say, antennas, where the field is

expected to have a non-zero average value. 7.3 Coherent States Coherent states of light are the closest approximation to the classical radiation emitted from oscillating

currents. We define an operator )(ˆ D (called the displacement operator) as,

aaeD ˆ*ˆ)(ˆ

where is a complex number,

ie||

A coherent state of a radiation mode, , is defined as,

0)(ˆ D

Since,

0ˆ,ˆ,ˆ0ˆ,ˆ,ˆprovidedˆˆ2

]ˆ,ˆ[ˆˆ

BABBAAeeee BA

BABA

we have,

aa

aa

eeeD

eeeD

ˆˆ*2

||

ˆ*ˆ2

||

2

2

)(ˆ

)(ˆ

7.3.1 Properties of )(ˆ D

)(ˆ D has the following properties:

(i) aaaa eeeeD ˆ*ˆ2ˆ*ˆ

2

)(ˆ


3

)(ˆ)(ˆ

1)(ˆ)(ˆ

1

DD

DD

(ii) aDaD ˆ)(ˆ)(ˆ Proof:

aaaa eae ˆ*ˆˆ*ˆ ˆ

Recall that,

1ˆ,îfˆˆ ˆˆ BAAeAe BB

aDaD ˆ)(ˆˆ)(ˆ

(iii) *ˆ)(ˆˆ)(ˆ aDaD The above relation is obtained by taking the adjoint of the relation in (ii). 7.3.2 Properties of Coherent States Important properties of coherent states are as follows: (i) Coherent states are properly normalized,

10|00)(ˆ)(ˆ0| DD

(ii) A coherent state is a linear superposition of photon number states,

nn

n

a

n

n

a

a

aaD

n

n

n

nn

n

n

0

2

0

2

0

2

2

2

!2

||exp

0!

)ˆ(

!2

||exp

0!

)ˆ(

2

||exp

0)êxp(2

||exp

0)ˆ*(exp)ˆ(exp2

||exp0)(ˆ

(iii) If a photon number measurement is performed on a coherent state, the probability nP of finding n

photons in a coherent state is,

2

22

!2exp|

nnnP

n

)exp(!

2

2

n

n

The photon number distribution in a coherent state looks like a Poisson distribution. (iv) Coherent states are eigenstates of the destruction operator a ,


4

0)(ˆˆˆ Daa

Since,

aDaD ˆ)(ˆˆ)(ˆ

we get upon multipying both sides with D ,

)(ˆˆ)(ˆ

]ˆ)[(ˆ)(ˆˆ

DaD

aDDa

Therefore,

0)(ˆ

0)(ˆ0ˆ)(ˆ0)(ˆˆˆ

D

DaDDaa

a

The above equation also implies,

*ˆ a

(v) Mean photon number and variance in the photon number for a coherent state are as follows,

2

42

2

22

ˆˆ

ˆˆˆˆˆ

ˆˆˆˆˆ

ˆˆˆˆ,ˆˆˆˆˆ

ˆˆˆ

nn

aaaaaa

aaaaaa

aaaaaaaaaan

aann

Therefore,

nnnn ˆˆˆˆ 222

The variance in photon number is equal to the mean. This is not surprising since we saw earlier that the photon number distribution is Poissonian. (vi) Different coherent states are not orthogonal. Suppose and are two different complex numbers

and and are the corresponding coherent states. We now find the value of the inner product

| ,

0)(ˆ)(ˆ0| DD

Since,

*ˆ*ˆ*ˆˆ22

ˆ*ˆ2ˆ*ˆ2

ˆ*ˆˆ*ˆ

22

22

)(ˆ)(ˆ

eeeeee

eeeeee

eeDD

aaaa

aaaa

aaaa

Therefore,


5

*22

*22

22

22

|

0)(ˆ)(ˆ0

e

eDD

22 e

(vii) Coherent states form a complete set. The completeness relation can be written as,

11

ir dd

where ir i . (viii) Mean values of field operators are non-zero for coherent states. Note that,

*ˆ

ˆ

a

a

Therefore, if the field operator for a single field mode is written as,

)(ˆˆ2

),(ˆ rUeaeatrA titi

o

oo

and we have a coherent state, i.e.,

0ˆ*ˆ aa

e

then,

)(*2

),(ˆ rUeetrA titi

o

oo

and,

)(*2

),(ˆ rUeeitrE titio oo

)(*2

1),(ˆ rUeetrH titi

oo

oo

Notice that if ie then the phase is also the phase of the average values of the field oscillations.

7.3.3 Quadrature Operators and Quadrature Fluctuations of Coherent States Recall from Chapter 6 that all narrow band real signals )(ty can be represented by phasors,

tititi etxetxetxty )(*)(2

1)(Re)(

If, )()()( 21 tixtxtx

then )(1 tx and )(2 tx are the quadratures of )(ty . With this in mind, and the fact that (for a single mode cavity),

titi eaeatrA ˆˆ2

1),(


6

we define the (Schrodinger picture) quadrature operators, 1x and 2x , for a mode as (mode subscript suppressed), 21 ˆˆˆ xixa

Note that 1x and 2x are Hermitian operators and observables. It follows that,

2

ˆ,ˆ

2

ˆˆˆ

2

ˆˆˆ

ˆˆˆ

21

21

21

ixx

i

aax

aax

xixa

If 111 ˆˆˆ xxx and 222 ˆˆˆ xxx , then the commutator result above implies the uncertainty

relation,

16

1ˆˆ 22

21 xx

For a coherent state , with ie , we have,

1*

4

11*2*

4

1ˆ

1*4

11*2*

4

1ˆ

sinIm2

*ˆ

cosRe2

*ˆ

22222

22221

2

1

x

x

ix

x

This implies,

4

1ˆ

ˆˆˆ

21

21

21

21

x

xxx

Also,

4

1ˆ22 x

Thus, for coherent states 161ˆˆ 2

221 xx . Coherent states satisfy the quadrature uncertainty relation

with equality and are therefore called minimum uncertainty states. 7.3.4 Vacuum Quadrature Fluctuations The vacuum state 0 may also be considered a coherent state with 0 . Therefore, the quadrature

fluctuations of the vacuum are the same as for the state ,

4

10ˆ0

4

10ˆ0 2

221 xx


7

7.3.5 Generalized Quadratures and Generalized Quadrature Fluctuations for Coherent States Recall that a narrowband real signal ty can be written as,

tietxty )(Re)( where, )()()( 21 txitxtx

The quadratures are the real and imaginary components of )(tx . In the complex plane, this means )(1 tx

and )(2 tx are components of )(tx along x -axis (real axis) and y -axis (imaginary axis), respectively.

We may also write )(tx as,

iii etxitxetxetxtx 22

2 )()()(

where now we are looking at the components of )(tx along the two perpendicular axis of a coordinate

system that is rotated at an angle with respect to the yx coordinate system. With this as motivation, we define the generalized quadrature operators as,

i

eaeax

eaeax

exixexexa

exixexexa

ii

ii

iii

iii

2

ˆˆˆ

2

ˆˆˆ

ˆˆˆˆˆ

ˆˆˆˆˆ

2

22

2

22

2

The generalized quadratures satisfy the commutation relation,

2

ˆ,ˆ 2i

xx

16

1ˆˆ 22

2 xx

For a coherent state,

4

1ˆ

4

1ˆ

2

2

x

x

For a coherent state, mean square quadrature fluctuations are the same no matter which "direction" you look. This can be graphically illustrated by drawing error diagrams. 7.3.6 Error Diagrams of Quantum States of Radiation The fluctuations in quantum optical states are sometimes depicted graphically in a 21 xx plane. An

arrow is drawn and the tip of the arrow is at the point where 1x equals 2 aa and 2x equals

iaa 2 . The dimensions of the shaded figure (or the error figure) drawn around the tip of the arrow

along different directions indicate the magnitude (root mean square value) of the fluctuations around the average value along those directions. As an example, consider a coherent state . We have,


8

Im2

*ˆ

Re2

*ˆ

2

1

ix

x

So for a coherent state we draw an arrow in the complex 21 xx plane whose tip is at the coordinates

Im,Re . For a coherent state we know that,

4

1ˆ4

1ˆ 22

21 xx

In fact, for a coherent state, as discussed earlier, the fluctuations are the same along any direction, i.e.,

41ˆ 2 x for any value of . Therefore, the fluctuations in a coherent state are represented by

drawing a circle of radius 21 , and area 4 , around the tip of the arrow, as shown below.

It should be noted here that unlike classical signals, where fluctuations represented the random motion in time of the tip of the phasor or the variations in an ensemble of signals, the fluctuations represented in the error diagram above are of quantum origin. The error diagram means that coherent states do not have a well defined value of, say, the 1x quadrature. As will be shown later, a coherent state is in fact a superposition of quadrature eigenstates. The vacuum state 0 is also a coherent state for which the average values of both the field quadratures

are zero and the mean square fluctuations in each quadrature equal 41 . Therefore, a vacuum state is

represented in the 21 xx plane as shown below.

x1

A coherent state x2

x1

A vacuum state

x2


9

7.3.7 Coherent States as Displaced Vacuum States In this Section we will show that coherent states are “displaced” vacuum states, and clarify the meaning

of the term “displaced.” We saw earlier in Chapter 5 that the average values of fields )(ˆ rA

, )(ˆ rE

and )(ˆ rH

in photon number states are zero. For example, if the field operator )(ˆ rA

for a single mode is,

)(ˆˆ2

))(ˆ

)(ˆ rUaarUq

rAo

then for a state with n photons,

0)(ˆ)(ˆ nrAnrA

We also showed that coherent states have non-zero average values of the field operators,

)(*2

)(ˆ rUrAo

We now explore the reason behind the non-zero average values of the field operators for coherent states from a different angle. Let q be the eigenstates of the field amplitude operator q , such that,

qqqq ˆ

and,

1

qqdq

It follows that,

ntrAqqndqnrAn ),(ˆ)(ˆ

)()(

)()()(

2

*

rUqqdq

qrUq

qdq

n

nn

where )(qn is the n -th Hermite-Gaussian. The above equation implies that the field operator will have

a non-zero average value if q has a non-zero value when averaged with respect to the wavefunction

)(qn . Recall that for the vacuum state the ground state wavefunction in q-space is Gaussian,

o

q

o

eqq

2

1)(0

20

2

Since 2

0 )(q is centered at 0q , the average value for the q operator is zero. In fact, the Hermite-

Gaussian wavefunctions 2

)(qn for all "n " are centered at zero, and so all photon number states have

zero average field values.

What if we consider a "displaced" version of 2

0 )(q , say 2

0 )(' q , such that,


10

o

oqq

o

eq

2)(2

01

)('

Then, of course, with respect to 2

)(' qo the average value of the field operator is not zero,

)()(')(ˆ 20

rUqrUqqdqrA

o

The question then is, which quantum state corresponds to 2

0 )(' q ? In other words, for what state

will 2

|q equal 2

0 )(' q ? Recall that,

3

33

2

22

!3!21

q

q

q

q

qqe ooo

qqo

Therefore,

)(

)('

0

/24

1

0

2

qe

eq

qq

qq

o

o

o

o

Since the momentum operator p acts like a derivative on a q-space wavefunction,

)(0ˆ 0 qqipq

we can write,

0

)()('|

ˆ

00

piq

qq

o

eq

qeqq

This implies,

0p

iqo

e

But,

i

aap o

2ˆ

Therefore,

0ˆ0

0

0

*

22

2

De

e

e

aa

aqaq

i

aaiqo

where,


11

2

* oq

Therefore, the sought after state is a coherent state! The Figure below shows the q-space probability

distribution of a vacuum state and a coherent state and the action of the displacement operator )(ˆ D .

Since )(ˆ D displaces the vacuum state, it is called a displacement operator. Coherent states are thus "displaced" vacuum states.

7.3.8 Quadrature Eigenstates, Quadrature Fluctuations, and Error Figures We had said earlier that a coherent state is a superposition of quadrature eigenstates. Here we quantify this notion. First note that both 1x and 2x quadratures cannot be measured simultaneously since they do

not commute, 2)(ˆ),(ˆ 21 itxtx . Recall that for a particle ipx ]ˆ,ˆ[ , so we have a wavefunction in

position )(|),( txtx , and we have a wavefunction in momentum )(|),( tptp , and the

probabilities of finding a particular value of position or momentum (but not both) upon measurement are

given by 2

),( tx and 2

),( tp , respectively. Similarly, quantum states of radiation (e.g. coherent

states) can be expanded in the eigenstates of the 1x quadrature or in the eigenstates of the 2x quadrature (but not both). We start from (mode number subscript is suppressed),

piqa

piqa

oo

oo

ˆˆ2

1ˆ

ˆˆ2

1ˆ

But we also have,

21

21

ˆˆ

ˆˆˆ

xixa

xixa

Therefore,

p

qx

o

o

ˆ2

1x

ˆ2

ˆ

2

1

q qo 0

Vacuum state

Coherent state

Displacement operator


12

Since the two quadrature operators are proportional to the operators q and p , therefore, the

eigenstates of q (i.e q ) are also eigenstates of 1x with eigenvalue qo2

, and eigenstates of p (i.e.

p ) are also eigenstates of 2x with eigenvalue po2

1. To avoid confusion below, I will write q as

qq ˆ and p as

pp ˆ where the subscripts indicate the operators of which the states are eigenstates.

From the above discussion,

2

1

ˆˆ

ˆˆ

2

1

2

xop

x

oq

pp

qq

Let,

2

1

ˆˆ

ˆˆ

2

1

2

xop

x

oq

cp

qbq

where c and b will be determined to properly normalize the eigenstates of 1x and 2x . We have,

)(22

)( 2

ˆˆˆˆ

21

qqbqqqqqqxx

qq

Let,

)2(2

)1(2

x

1

1

qx

q

o

o

Then,

)(2

)(2

11112

ˆ11ˆ 11xxxxbxx

xx

We need, )( 11ˆ11x 11

xxxxx

so b must equal,

4

1

2

ob

Finally, we have,

1ˆ

4

1

ˆ 22x

ooq

qq


13

or, using (1) above,

qoo

xxx

ˆ1

4

1

ˆ122

1

Similarly, one can show that,

4

1

2

1

oc

and,

poox

xxˆ24

1

ˆ2 222

)( 22ˆ22x 22xxxx

x

Completeness: We have.

1ˆˆ

qqdq

qq

1222

11 ˆˆ

qqdq o

xx

oo

Define a change of variables,

qo2

x1

and obtain,

11ˆˆ1111

xxdx

xx

Similarly, one can obtain,

12ˆˆ2222

xxdx

xx

from,

1ˆˆ-

ppdp

pp

Now that we have the eigenstates of the quadrature operators, we will try to express coherent states in terms of these eigenstates. We can write,

22

11

ˆ22ˆ2

ˆ11ˆ1

1

1

xx

xx

xxdx

xxdx

We need the values of 1ˆ1x

x and 2ˆ2

xx

. We start from,

00)(ˆ ˆ*ˆ aaeD

Let,


14

ie

We also know that,

piqa

piqa

oo

oo

ˆˆ2

1ˆ

ˆˆ2

1ˆ

Therefore,

0ˆ2cosexpˆ

2sinexp)2(sin

2exp

0ˆcos2ˆsin

2exp

0ˆˆ2

ˆˆ2

exp

2

piqii

piqi

piqe

piqe

o

o

o

o

oo

i

oo

i

The last line follows from using,

constant if ]ˆˆ[

ˆˆ2

ˆ,ˆˆˆ

B,Aeeee BABA

BA

Take the inner product with the bra qq

on both sides to get,

)(

)(

0

2sin)2sin(

2

0

2cos2

sin)2sin(2

ˆ

2

2

o

qii

qqii

q

qqee

qeeeq

where ooq 2cos . Similarly, one can show that,

)(

)(

0

2cos)2sin(

2

0

2sin2cos)2sin(

2ˆ

2

2

o

pii

ppii

p

ppee

peeep

where op 2sin . Now recall that,

o

q

o eq

24

1

0

2

)(

o

p

o

pqieq

edqp

24

1

00

2

1)(

2)(

Finally,


15

211

2

1

cossin2)2sin(24

1

1ˆ

4

1

1ˆ

2

22

xxii

oqox

eee

xx

And similarly,

222

2

2

sincos2)2sin(24

1

2ˆ4

1

2ˆ

2

2)2(

xxii

op

ox

eee

xx

So we have,

412

cos

2

12

1ˆ

21

1

2

x

x ex

and,

412

sin

2

12

2ˆ

22

2

2

x

x ex

The above expressions show that a coherent state can be considered as a superposition of the eigenstates of the 1x operator. This superposition has a Gaussian probability distribution with a mean value centered

at cos and the variance of this distribution is 41 . Similarly, one may also consider a coherent state

as a superposition of the eigenstates of the 2x operator. This superposition also has a Gaussian

probability distribution with a mean value centered at sin and the variance of this distribution is also

41 . These results justify the error diagram for the coherent states discussed earlier and show below.

Averages Using the Quadrature Distributions: Knowing the expansions of a coherent state in terms of the quadrature eigenstates, one can calculating averages of quantities of interest using these expansions. For example,

x1

A coherent state x2

||cos

||sin


16

cos

ˆˆ1ˆˆ

12

1ˆ1

11ˆˆ11111

1

11

xxdx

αxxxdxααxααxαx

x

xx

4

1

cos

ˆˆ

ˆˆx

2221

21ˆ1

21

21

111

1

xxdx

xx

xx

x

And similarly,

sinˆ1ˆˆ 22

2ˆ2222 2

xxdxxxx x

4

1sinˆˆx 222

22

22222

22

xxxdxx

Note that the above results were obtained earlier in a different way without the knowledge of the probability distribution functions associated with the expansion of a coherent state in quadrature eigenstates. If one were to write as a linear super position of the eigenstates of 1x , the resulting

(root-mean-square) uncertainty would equal the extent of the figure in the direction of 1x . Similarly, if

one were to write as a linear super position of the eigenstates of 2x , the resulting (root-mean-square)

uncertainty would equal the extent of the figure in the direction of 2x .

One can define the eigenstates of the generalized quadrature operators, x and 2ˆ x (for any value of

), and expand in terms of these eigenstates,

xx

xxdx ˆˆ

or,

22 ˆ22ˆ2

xxxxdx

x1

A coherent state x2

||cos

||sin


17

Since the error figure for a coherent state is a circle, it means that the fluctuations in all directions are equal and one may safely guess that,

412

sin

2

12

2ˆ

412

cos

2

12

ˆ

22

2

2

2

2

x

x

x

x

ex

ex

7.3.9 Time Dependence of Coherent States Suppose the quantum state of a single mode field at time 0t is a coherent state,

00ˆ0 ˆˆ aaeDt

We need to find t . Suppose the Hamiltonian is,

aaH o ˆˆ

We have ignored the vacuum energy since it plays no interesting part in the discussion that follows. We have,

0

0ˆ

0ˆ

0)(

)(ˆ)(ˆ

ˆˆˆ

ˆ

ˆˆ

tata

tHi

tHi

tHi

tHi

tHi

tHi

e

eeDe

De

e(tet

but,

ti

ti

o

o

eata

eata

ˆ)(ˆ

ˆ)(ˆ

Therefore,

0)( ˆ)(ˆ)( atatet

where,

ti

ti

o

o

et

et

One can write, )(tt

Note that all the time dependence goes into the definitions of the complex numbers and .


18

7.3.10 Time Dependence of the Quadrature Operators

For a real signal tietxty Re writing txitxtx 21 implied that the fast time

dependence of )(ty given by was not included in the definitions of )(1 tx and )(2 tx but was

explicitly factored out. For quantum fields we know that the Heisenberg operator trA ,

is,

)(ˆ)(ˆ2

1),(ˆ tatatrA

We write this as,

titititi oooo eetaeetatrA ˆˆ2

1,

Therefore, we define time dependent quadrature operators tx1ˆ and tx2ˆ as,

)(ˆ)(ˆ)(ˆ

)(ˆ)(ˆ)(ˆ

21

21

txitxeta

txitxetati

ti

o

o

For free fields (i.e. fields whose time dependence is governed by the Hamiltonian aaH o ˆˆ ),

ti

ti

o

o

eata

eata

ˆ)(ˆ

ˆ)(ˆ

and therefore )(ˆ1 tx and )(ˆ2 tx will be independent of time. We can also write,

i

etaetatx

etaetatx

titi

titi

oo

oo

2

)(ˆ)(ˆ)(ˆ

2

)(ˆ)(ˆ)(ˆ

2

1

And for the generalized quadratures we get,

i

eetaeetatx

eetaeetatx

tiitii

tiitii

oo

oo

2

)(ˆ)(ˆ)(ˆ

2

)(ˆ)(ˆ)(ˆ

2

Note of Caution: The time dependence of a quadrature operator is not governed by the Heisenberg equation,

Htx

dt

txdi ˆ,ˆ

ˆ

t

Hit

Hi

exetx

ˆˆ

ˆˆ

The time dependence of a quadrature operator is defined only through the equation,

2

)(ˆ)(ˆ)(ˆ

tiitii oo eetaeetatx

The explicit presence of time exponentials in the above relation implies that the time dependence of the quadrature operators is not given directly by the Heisenberg equation. For any quantum state, txtttxt ˆ0)(ˆ0

The question then is how does one compute averages related to the quadrature operators in the Schrodinger and Heisenberg pictures. Below we discuss how to calculate the average of )(ˆ tx in the two pictures.

tie


19

Heisenber Picture: Suppose one has the quantum state 0t at time 0t . One first computes the

Heisenebrg operators tata ˆ,ˆ at time t , find the desired quadrature operator )(ˆ tx , and then compute,

0)(ˆ0 ttxt

Schrodinger Picture: Suppose one has already computed the quantum state t at time t . The

quantity 0)(ˆ0 ttxt found above is then the same as,

teeaeeat

tiitii oo

2

ˆˆ

Note that in the expression above the creation and destruction operators are in the Schrodinger picture. However, the time dependent complex exponentials remain. Coherent States: For a coherent state, one obtains,

i

eetx

ee

eetaeetatx

ii

ii

tiitii

2)(ˆ

2

2

)(ˆ)(ˆ)(ˆ

2

And,

4

1)(ˆ)(ˆ 2

22 txtx

Note that the quadrature averages and variances are independent of time. This is true only for non-interacting free fields.

7.4 Squeezed States of Light 7.4.1 Introduction For coherent states,

4

1ˆ

4

1ˆ

2

2

x

x

Coherent states satisfy the uncertainty relation,

16

1ˆˆ 22

2 xx

with equality. Squeezed states also satisfy the uncertainty condition with equality but have different fluctuations in quadratures x and 2

ˆ x . Obviously, they do that by decreasing the fluctuations in one

quadrature at the expense of increasing the fluctuations in the other quadrature. 7.4.2 The Squeezing Operator Squeezed states , are obtained by first squeezing the vacuum state 0 by the squeezing operator

)(ˆ S , where,


20

22 )ˆ(2

ˆ2)(ˆ

aa

eS

and then displacing it with )(ˆ D ,

0)(ˆ)(ˆ, SD

The squeezing parameter is a complex number,

2ier

)(ˆ S has the followng properties:

(i) )(ˆ)(ˆ)(ˆ1)(ˆ)(ˆ 1 SSSSS

(ii) If 2ier , then,

rearaSaS

rearaSaSi

i

sinhˆcoshˆ)(ˆˆ)(ˆ

sinhˆcoshˆ)(ˆˆ)(ˆ

2

2

The proof follows from application of the formula,

BAABABABA ˆ,ˆ,ˆ!2

1ˆ,ˆˆ)ˆ(expˆ)ˆ(exp

and then summing up the resulting series. 7.4.3 Properties of Squeezed States

In what follows, we will assume that 2ier . Squeezed states have the following properties: (i) Averages of creation and destruction operators in squeezed states are as follows,

,ˆ,

0)(ˆˆ)(ˆ0

0)(ˆ)(ˆˆ)(ˆ)(ˆ0,ˆ,

a

SaS

SDaDSa

It follows that,

0,0ˆ,0

00)(ˆˆ)(ˆ0,0ˆ,0

a

SaSa

Also,

re

SaSSaS

SaaS

SaS

SDaDDaDS

SDaDSa

i sinhrcosh

0)(ˆˆ)(ˆ)(ˆˆ)(ˆ0

0)(ˆˆ2ˆ)(ˆ0

0)(ˆ)ˆ()(ˆ0

0)(ˆ)(ˆˆ)(ˆ)(ˆˆ)(ˆ)(ˆ0

0)(ˆ)(ˆˆ)(ˆ)(ˆ0,ˆ,

22

2

22

2

22

And,

rea i sinhrcosh,)ˆ(, 222

(ii) The number operator average is,


21

2

20ˆˆˆˆˆˆ0

0ˆˆˆˆ0

0ˆˆˆˆˆˆˆˆ0,ˆˆ

r

SaSSaS

SaaS

SDaDDaDSaa

2sinh

,

(iv) The quadrature operator average is,

2

,2

ˆˆ,,ˆ,

ii

ii

ee

eaeax

i

eex

ii

2,

2ˆ,

(v) The most distinguishing property of squeezed states compared to coherent states is the average fluctuations in the quadrature operators,

4

1sinh

4

1cosh

4sinhcosh

4sinhcosh

,ˆˆˆˆˆ,4

1,ˆ,

22

222

22

222

222 22

r

re

rre

erre

aaaaeaeax

ii

ii

ii

So if we let (i.e., look at the quadratures x and 2

ˆ x ) where is the angle associated with the

squeezing parameter , then,

r

r

errx

errx

2222

22

4

1sinhcosh

4

1,ˆ,

4

1sinhcosh

4

1,ˆ,

2

The squeezed state ire 2, has reduced fluctuations in the quadrature x compared to the

quadrature 2ˆ x . The error region in the 21 xx plane for a squeezed state with 0 is shown in the

Figure below. The reduced fluctuations in one quadrature and the increased fluctuations in the other quadrature, make the error region look like an ellipse.


22

Also note that,

16

1,ˆ,,ˆ, 22

22222

iiii rexrerexre

Squeezed states, like coherent states, are minimum uncertainty states since they satisfy the uncertainty

relation with equality. The error region is an ellipse of semi-minor axis equal to rr ee 2

1

4

1 2 , and

semi-major axis equal to rr ee2

1

4

1 2 . The Figure below shows the error region for a squeezed state

ire 2, for 6 . The squeezed state has reduced fluctuations in the quadrature x compared to

the quadrature 2ˆ x .

Squeezed Vacuum: A squeezed vacuum state ,0 is obtained by applying the squeezing operator to a

vacuum state,

0)(ˆ,0 S

For squeezed vacuum, the average values of the quadratures are zero and the quadrature fluctuations are

dictated by the squeezing parameter 2ier . The figure below shows the squeezed vacuum state for

2 .

x1

A squeezed state

x2

x1

A squeezed state x2


23

7.4.4 Squeezed States and Two-Photon Coherent States Suppose we define two new operators b and b as a linear combination of the operators a and a , as follows,

aab

aab

ˆ*ˆ*ˆ

ˆˆˆ

Here, and are complex numbers. If we also want to enforce the commutation relations for b and

b (i.e. 1ˆ,ˆ bb ), then we must have,

122 .

Now suppose we define b and b as,

rearab

SaSSaSbi sinhˆcoshˆˆ

)(ˆˆ)(ˆ)(ˆˆ)(ˆˆ

2

and,

rareab

SaSbi coshˆsinhˆˆ

)(ˆˆ)(ˆˆ

2

You can verify that 1ˆ,ˆ bb . We now want to find eigenstates of the operator b . We know that

coherent states are eigenstates of the operator a . ai.e. . We try the following state,

0)(ˆ)(ˆ DS

In the above expression, the vacuum state is displaced first and then squeezed later. The resulting state is not exactly a squeezed state but it is closely related. Note that,

0)(ˆ)(ˆ

)(ˆ

ˆ)(ˆ

0)(ˆˆ)(ˆ

)(ˆ)(ˆ)(ˆˆ)(ˆ0)(ˆ)(ˆˆ

DS

S

aS

DaS

oDSSaSDSb

Therefore, the state 0)(ˆ)(ˆ DS is an eigenstate of the operator b with eigenvalue . These states are

called two-photon coherent states. The 0 state is the vacuum (or ground state) of a in the sense that

x1

A squeezed vacuum state

x2


24

00ˆ a . The state ,00)(ˆ0)0(ˆ)(ˆ SDS , or the squeezed vacuum state, is the ground

state of the operator b , i.e,

00)(ˆˆ Sb

So we now have the following two sets of states:

Two-photon coherent states: p

DS ,0)(ˆ)(ˆ

Squeezed states: s

SD ,0)(ˆ)(ˆ or just ,

The question that arises now is whether the above two sets represent the same or different states? We will

answer this question next. One can also generate an eigenstate of b with eigenvalue by operating with

bb ˆêxp * on the ground state 0)(ˆ S of b , i.e.,

0)(ˆˆˆ * Se bb

But,

rareab

rearabi

i

coshˆsinhˆˆ

sinhˆcoshˆˆ

2

2

Therefore,

0)(ˆˆcoshsinhˆsinhcoshexp

0)(ˆ

*22*

ˆˆ *

Sarrearer

Se

ii

bb

So,

0)(ˆ Sinh)cosh(ˆ0)(ˆ)(ˆ 2 SrerDDS i

The above expression establishes the relationship between two-photon coherent states and squeezed states,

s

ip

rer ,sinhcosh, 2

This implies that for two-photon coherent states with squeezing parameter 2ier ,

rpp

r

pp

ex

ex

22

22

4

1,ˆ,

4

1,ˆ,

7.4.5 Time Dependence of Squeezed States Suppose the quantum state of a single mode field at time 0t is a squeezed state,

0ˆˆ,0 SDt

We need to find t . Suppose the Hamiltonian is,

aaH o ˆˆ

We have,


25

tt

tStD

eeSeeDe

SDe

e(tet

tHi

tHi

tHi

tHi

tHi

tHi

tHi

tHi

,

0ˆˆ

0ˆˆ

0ˆˆ

,0)(

ˆˆˆˆˆ

ˆ

ˆˆ

where,

ti

ti

o

o

et

et

2

One can write, ttt ),(

Note that all the time dependence goes into the definitions of the complex numbers and .

7.5 Phase of Quantum States of Radiation 7.5.1 Introduction We begin by asking the following question: What is the phase of a photon? And does this question even make sense? Let us start from what we already know about phase. We know that classical electric and magnetic fields inside a cavity oscillate and if one plots the strength of electric field at any location in the cavity one obtains a curve in time that looks like as shown in the Figure below.

This curve has a “phase” associated with it. So we do have a concept of a phase for a field. The phase of a “photon” is an ill-defined concept. The phase of the “field” is a more meaningful concept. We also know that since the electric field operator is,

)()(ˆ)(ˆ2

),(ˆ rUtataitrE

the electric field can have an average value that is non-zero only for states that are linear superpositions of photon number states. Photon number states , i.e. states with a definite number of photons, give an

average value of zero for the electric field. For example, if then 0,ˆ trE

, but if

n

n

t

E


26

12

1 nn then 0,ˆ trE

. Therefore, states for which the concept of “phase” should

make sense must not be states of a definite photon number. These basic ingredients must be reflected in whatever operator we finally construct to describe the phase of the field. Let us first look at a coherent state,

0

2

!0)(ˆ

2

n

nn

neD

)(2

),(ˆ rUeeitrE titi

If,

ie

then,

)(sin2

2),(ˆ rUttrEo

So the phase of the complex parameter defines the phase of the average field. The average values of the quadrature operators,

i

etaetatx

etaetatx

titi

titi

2

)(ˆ)(ˆ)(ˆ

2

)(ˆ)(ˆ)(ˆ

2

1

are,

sin)(ˆ

cos)(ˆ

2

1

tx

tx

In the 21 xx the state phasor is drawn as shown below.

One can see the problem in identifying the phase of the average field if 1 . In that case, the error

circle would be close to the origin and the quadrature fluctuations would be larger than the mean quadrature values.

x1

A coherent state

x2

|| cos

|| sin


27

7.5.2 Phase Fluctuation Operator A Hermitian phase operator for a field is not particularly easy to construct. This difficulty should not be a surprise since the absolute value of phase cannot be a measurable or an observable property. However, phase differences are measurable. In what follows, we will concentrate on constructing a phase fluctuation operator for quantum states. Consider any arbitrary quantum state of radiation for which the average values of the quadrature

operators have the following non-zero values, oiti eAeta )(ˆ

o

o

Atx

AAtx

sin)(ˆ

real is cos)(ˆ

2

1

The average value o of the phase for such a quantum state is well defined as long as 1A . If a

measurement is made of any one quadrature, the relative uncertainty 22 ˆˆ txtx will be

inversely proportional to 2A and, therefore, small if, as assumed, 1A .

We can write,

txitxeta

txitxtxitxtxitxeta

ti

ti

21

212121

ˆˆˆ

ˆˆˆˆˆˆˆ

where,

txtxtx

txtxtx

222

111

ˆˆˆ

ˆˆˆ

We can also write using generalized quadratures,

iititi etxietxetaeta 2ˆˆˆˆ

where,

txtxtx

txtxtx

222 ˆˆˆ

ˆˆˆ

If the average phase of the field is o , i.e.,

x1

An arbitrary state of radiation

x2

A cos

A sin


28

oiti eAeta )(ˆ

then, as discussed earlier in the case of classical signals, the quadrature fluctuation operator txo

ˆ in

the direction o will describe amplitude fluctuations and the quadrature fluctuation operator

txo 2ˆ in the direction perpendicular to the direction o will be proportional to the phase

fluctuations. This is shown in the Figure below.

One can write,

ooo

oo

oo

i

iititi

etxitxA

etxietxetaeta

2

2

ˆˆ

ˆˆˆˆ

If the average value of the phase is o , and if 1A , then a phase fluctuation operator, t , can be defined as follows,

A

txt o

)(ˆ)(ˆ

2

The amplitude fluctuation operator is simply txo

ˆ . As an example, we calculate the phase fluctuations

of a coherent state . Suppose that,

oiti eeta )(ˆ

Then,

4

1)(ˆ)(ˆ

0)(ˆ

)(ˆ

22

2

2

txt

txt

o

o

The larger the magnitude of a coherent state, the smaller the mean square phase fluctuations.

7.5.3 Photon Number Fluctuation Operator Suppose for a quantum state the average photon number is on and the average phase is o ,

oiti eAeta )(ˆ

ontatatn )(ˆˆ)(ˆ

We assume that 1A and so the average phase o is well defined. Then, as before, one can write,


29

ooo

oo

oo

i

iititi

etxitxA

etxietxetaeta

2

2

ˆˆ

ˆˆˆˆ

The photon number operator becomes,

txAA

txitxAtxitxAtata

tntntn

o

oooo

ˆ2

ˆˆˆˆˆˆ

ˆˆˆ

2

22

We have ignored terms that contain squares of the quadrature fluctuation operators. The above relation

shows that the average photon number on must approximately equal 2A and the photon number

fluctuation operator tn must approximately equal txAo

ˆ2 . These approximations are excellent

when 1 onA . So we define an approximate photon number fluctuation operator as,

txntxAtnoo o ˆ2ˆ2ˆ

One can then write,

oio

ti eneta )(ˆ

o

ooo

io

oo

io

ti

etnin

tnn

etxitxneta

ˆ2

ˆ

ˆˆˆ 2

7.5.4 Photon Number and Phase Uncertainty Relation There is an interesting uncertainty relation between the photon number fluctuation operator and the phase fluctuation operator. The photon number fluctuation and the phase fluctuation operators for a quantum state with an average number of photons equal to on are,

txntnoo ˆ2ˆ

on

txt o

)(ˆ)(ˆ

2

The commutation relation between the photon number fluctuation operator and the phase fluctuation operator follows from the commutation relation between the quadrature operators,

itxtxttnoo

)(ˆ,)(ˆ2)(ˆ,ˆ 2

4

1)(ˆ)(ˆ

)(ˆ),(ˆ

22

ttn

ittn

Therefore, the photon number and the phase (with respect to a reference) of a radiation field cannot be measured simultaneously with high accuracy. In other words, measurement of one will necessarily disturb the other. The other way to interpret the same result is that if a quantum state of radiation has a well defined value for the phase of the field then this quantum state cannot have a well defined number of photons and it must be a superposition of different photon number states. On the other hand, if a quantum state has a well defined number of photons then it cannot have a well defined value for the phase of the field.


30

For a coherent state with average number of photons equal to on ,

o

o

nt

ntn

4

1

4

1)(ˆ

)(ˆ

22

22

The photon number and phase uncertainty relation is satisfied with equality. Note that larger the average photon number, the smaller the uncertainty in the phase. The definitions for the phase fluctuation operator and the photon number fluctuation operator presented here work well only when the quantum state has a large average photon number on and a well defined

average phase o . As 0)(ˆ ontn , the phase fluctuations described by the operator

ontxto

)(ˆ)(ˆ 2 can become very large. The problem is that phase of a field is not a well

defined concept when the photon number is very small. For example, consider two different coherent

states, with the same phase, but for one 1)(ˆ2 tn and for the other 1)(ˆ

2 tn , as shown

in the Figure below.

When 1 , the quadrature fluctuations are comparatively large, and its difficult to categorize

quadrature fluctuations as amplitude or phase fluctuations. For example, )(ˆ 2 txo now also

contributes to amplitude fluctuations. Although one can still talk about quadrature fluctuations, ),(ˆ),(ˆ 2 txtx but no quadrature fluctuation, for any value of , can be labled as an amplitude

fluctuation and no quadrature fluctuation can be labled as a phase fluctuation. Phase Noise in Squeezed States: Consider now a squeezed state , with a large average photon

number,

ro enrtn 22222

sinh)(ˆ Assuming

If,

oo ii ere 2

then,

x1

Two different coherent states

x2


31

)(sin

2,),(ˆ,

,)(ˆ,

rUttrE

eta

o

ti

The average phase of the field is o . It is not necessary that the angles o and o be the same. First,

we assume that oo . We also assume that 2 is large and therefore on is large. The fluctuation

operators are,

txntnoo ˆ2ˆ

on

txt o

)(ˆ)(ˆ

2

We get,

r

ro

e

etxntno

22

2222

4

14,)(ˆ,4,)(ˆ,

The above result agrees reasonably well with the exact result for )(ˆ 2 tn for a squeezed state,

rretn r 22222 sinhcosh2,)(ˆ,

provided 2 is large. For phase fluctuations we get,

2

222

4

1

4

1,)(ˆ,

r

o

r e

n

et

The number-phase uncertainty relation is satisfied with equality (approximately),

4

1)(ˆ)(ˆ 22 ttn

Compared to a coherent state, the photon number fluctuations in this squeezed state are reduced (or squeezed) at the expense of increased phase fluctuations (see the Figure above). Now suppose, that the squeezing parameter’s phase o is 2 o . As before,

txntnoo ˆ2ˆ

x1

A squeezed state with increased phase fluctuations and squeezed photon number fluctuations

x2

|| cos

|| sin


32

on

txt o

)(ˆ)(ˆ

2

It follows that,

4

1)(ˆ)(ˆ

4

,ˆ,,)(ˆ,

,)(ˆ,4,)(ˆ,

22

2

2222

22

22

ttn

e

n

txt

e

txntn

r

o

r

o

o

o

Now phase fluctuations are squeezed at the expense of increased photon number (or amplitude) fluctuations, as shown in the Figure below.

7.6 Thermal Radiation: A Statistical Mixture So far we have looked at quantum states of radiation that could be described using a pure state. One can also have states of radiation that are statistical mixtures and are describable only by a density operator. One such state is the thermal state. Consider a single radiation mode inside a cavity. We know that at any temperature the radiation must be in thermal equilibrium with the walls of the cavity and the walls of the cavity must be continuously absorbing from and loosing photons to the radiation mode. Therefore, the radiation mode cannot have a fixed well defined number of photons. We assume that the density operator for the radiation mode can be written as,

nnnPn

0

In the photon number state basis, the density operator has no off-diagonal elements but only diagonal elements, nP , that give the probability of there being n photons in the radiation mode. From statistical physics we know that for a canonical ensemble the probability for a system at temperature T to have

energy E is proportional to TKE Be . Therefore,

TK

n

BenP

21

x1

A squeezed state with increased photon number fluctuations and squeezed phase fluctuations

x2

|| cos

|| sin


33

The constant of proportionality is obtained by requiring that,

10

nnP

which gives,

TKTKn BB eenP 1 The average number of photons in the mode equals,

1

1ˆˆˆ0

TKn BenPnnn

Trace

The expression for the average photon number is called the Bose-Einstein factor, and the thermal probability distribution,

TKTKn BB eenP 1 is called the Bose-Einstein distribution. Bose-Einstein distribution is sometimes also written in terms of the average photon number as,

n

TKTKn

n

n

neenP BB

ˆ1

ˆ

ˆ1

11

The fluctuations in the photon number are,

nnnnn ˆ1ˆˆˆˆ 222

Note that for the same average number of photons, a thermal state has larger photon number fluctuations than a coherent state. Since, the thermal state is a statistical mixture of different photon number states (and not a linear superposition of different photon number states), it has no well defined phase. The thermal or Bose-Einstein distribution played an important role in the development of the quantum theory. Max Plank studied the frequency distribution of radiation coming out of a blackbody cavity radiator and, in order to explain the experimental observations, postulated that radiation must be emitted and absorbed in discrete packets or “quanta” of energy. Max Plank was the first one to show that for a thermal state the energy in a radiation mode is distributed according to the Bose-Einstein distribution.

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Chapter 7: Quantum States of Light - Cornell University 7: Quantum States of Light ... With this in mind, and the fact that (for a single mode cavity), ... The vacuum state 0 may also