A. Operator Relations

A. Operator Relations

A.1 Theorem 1

Let A and B be two non-commuting operators, then [1]

exp(αA)B exp(−αA) = B + α [A,B] +α2

2![A, [A,B]] + .... (A.1)

Proof.Let

f1(α) = exp(αA)B exp(−αA) , (A.2)

then, one can expand f1 in Taylor series about the origin. We first evaluatethe derivatives

f ′1(α) = exp(αA)(AB −BA) exp(−αA) ,

sof ′1(0) = [A,B] . (A.3)

Similarly

f ′′1 (α) = exp(αA)(A [A,B] − [A,B]A) exp(−αA) ,

so thatf ′′1 (0) = [A, [A,B]] . (A.4)

Now, we write the Taylor’s expansion

f1(α) = f1(0) + αf ′1(0) +

α2

2!f ′′1 (0) + ... (A.5)

or

exp(αA)B exp(−αA) = B + α [A,B] +α2

2![A, [A,B]] + .. (A.6)

A particular case is when [A,B] = c, where c is a c-number, then

exp(αA)B exp(−αA) = B + αc , (A.7)

in which case exp(αA) acts as a displacement operator.

364 A. Operator Relations

A.2 Theorem 2: The Baker–Campbell–HaussdorfRelation

Let A and B be two non-commuting operators such that

[A, [A,B]] = [B, [A,B]] = 0 , (A.8)

then

exp [α(A+B)] = expαA expαB exp[

−α2

2[A,B]

]

(A.9)

= expαB expαA exp[

α2

2[A,B]

]

.

Proof.Define

f2(α) ≡ expαA expαB . (A.10)

Then

df2(α)dα

= [A+ exp(αA)B exp−(αA)]f2(α) (A.11)

= (A+B + α [A,B])f2(α) ,

where in the last step, we used (A.6).Also, from the definition of f2(α), we can write

df2(α)dα

= exp(αA)A expαB + exp(αA) exp(αB)B (A.12)

= expαA expαB [exp(−αB)A expαB + B]= f2(α)(A +B + α [A,B]) .

By comparing (A.11) and (A.12), we can see that f2(α) commutes with(A+B+α [A,B]), thus one can integrate as a c-number differential equation,getting

f2(α) = exp[

(A+B)α +α2

2[A,B]

]

= expα(A+B) expα2

2[A,B] , (A.13)

thus obtaining the desired result.Another application of the Theorem 1 is taking

A = aa†, (A.14)B = a or a† .

As[n, a] = −a (A.15)

A.3 Theorem 3: Similarity Transformation 365

and the higher order commutators also give a with alternating signs, thus

exp(αn)a exp(−αn) = a− αa+α2

2a+ ... = exp(−α)a . (A.16)

Similarlyexp(αn)a† exp(−αn) = exp(α)a† . (A.17)

A.3 Theorem 3: Similarity Transformation

exp(αA)f(B) exp−(αA) = f(exp(αA)(B) exp−(αA)) . (A.18)

Proof.We start with the following identity

[exp(αA)(B) exp−(αA)]n = exp(αA)B exp(−αA) exp(αA)B exp(−αA)...= exp(αA)Bn exp(−αA) .

Then, for any function f(B) that can be expanded in power series, theTheorem 3 follows.

As an interesting application, let us calculate

exp(−αa† + α∗a

)

f(a, a†) exp(

αa† − α∗a)

= f [exp(−αa† + α∗a

)

a exp(

αa† − α∗a)

,

exp(−αa† + α∗a

)

a† exp(

αa† − α∗a)

]

= f(a+ α, a† + α∗) .

Alsoexp(−αa†) f(a, a†) exp

(

αa†)

= f(a+ α, a†) , (A.19)

exp (α∗a) f(a, a†) exp (−α∗a) = f(a, a† + α∗) , (A.20)

exp (αn) f(a, a†) exp (−αn) = f [a exp(−α), a† exp(α)] (A.21)

Other properties:One can easily show that

[

a, a†l]

= la†l−1 =da†l

da†, (A.22)

[

a†, al]

= −lal−1 = −dal

da.

A more general version of the above relations is for a function f(a, a)which may be expanded in power series of a and a†

[

a, f(a, a†)]

=∂f(a, a†)∂a†

, (A.23)

[

a†, f(a, a†)]

= −∂f(a, a†)∂a

. (A.24)

366 A. Operator Relations

Reference

1. Louisell, W.H.: Quantum Statistical Properties of Radiation. John Wiley,New York (1973)

B. The Method of Characteristics

We have a first-order partial differential equation: [1]

Pp+Qq = R , (B.1)

where P = P (x, y, z), Q = Q(x, y, z), R = R(x, y, z), and

p ≡ ∂z

∂x, q ≡ ∂z

∂y, (B.2)

and we wish to find a solution of (B.1), of the form

z = f(x, y) . (B.3)

The general solution of (B.1) is

F (u, v) = 0 , (B.4)

where F is an arbitrary function, and

u(x, y, z) = c1 , (B.5)v(x, y, z) = c2 ,

is a solution of the equations

dxP

=dyQ

=dzR

. (B.6)

Proof.If (B.5) are solutions of (B.6), then the equations

∂u

∂xdx+

∂u

∂ydy +

∂u

∂zdz = 0 , (B.7)

anddxP

=dyQ

=dzR

,

must be compatible; thus, we must have

368 B. The Method of Characteristics

Pux +Quy +Ruz = 0 , (B.8)

and similarly for vPvx +Qvy +Rvz = 0 . (B.9)

On the other hand, if x and y are independent variables and z = z(x,y),then from (B.5), we get

ux + uz∂z

∂x= 0 , (B.10)

uy + uz∂z

∂y= 0 ,

and substituting (B.10) into (B.8) we get[

−P ∂z∂x

−Q∂z

∂y+R

]

∂u

∂z= 0 ,

and (B.1) is satisfied.The second part of the proof is to show that the general solution of

(B.1) isF (u, v) = 0 . (B.11)

From (B.11), one writes

∂F

∂x=∂F

∂u

(

∂u

∂x+∂u

∂z

∂z

∂x

)

+∂F

∂v

(

∂v

∂x+∂v

∂z

∂z

∂x

)

= 0 , (B.12)

∂F

∂y=∂F

∂u

(

∂u

∂y+∂u

∂z

∂z

∂y

)

+∂F

∂v

(

∂v

∂y+∂v

∂z

∂z

∂y

)

= 0 . (B.13)

We finally notice that (B.13) is satisfied considering (B.10).Example.Find the general solution of the equation

x2 ∂z

∂x+ y2 ∂z

∂y= (x+ y)z . (B.14)

In this case

P = x2 , (B.15)Q = y2 ,

R = (x+ y)z ,

and we have to find the solution of

dxx2

=dyy2

=dz

(x+ y)z. (B.16)

Reference 369

Integrating, firstdxx2

=dyy2

,

we getx−1 + y−1 = c′1 . (B.17)

On the other hand,

dx− dyx2 − y2

=(x2

y2 − 1)dy

x2 − y2=

dyy2

=dz

(x+ y)z,

from where we getx− y

z= c2 = v . (B.18)

Combining (B.17) and (B.18), we get

xy

z= c1 = u , (B.19)

so the general solution can be put as

F

(

xy

z,x− y

z

)

= 0 , (B.20)

or if we write (B.20) in the equivalent form

u = g(v) , (B.21)

then the solution isxy

z= g

(

x− y

z

)

. (B.22)

Reference

1. Sneddon, I.: Elements of Partial Differential Equations. (Mc-Graw Hill,New York (1957)

C. Proof

In this Appendix, we show the equation⎡

⎣〈∑

j =k

δ(t− tj)δ(t′ − tk)〉S −R2

⎤

⎦ ρ2aa = −pRδ(t− t′) . (C.1)

For regular pumping, one can put tj = t0 + jτ, where τ is the constanttime interval between the atoms and t0 some arbitrary time origin [1].

In this case, there are no pumping fluctuations, and therefore, there areno correlations between the products of delta functions, that is

∑

j,k

〈δ(t− tj)δ(t′ − tk)〉S =∑

j

〈δ(t− tj)〉S∑

k

〈δ(t′ − tk)〉S (C.2)

= R2 .

Now, we split the l.h.s. of the above equation in two parts∑

j =k

〈δ(t− tj)δ(t′ − tk)〉S +∑

j=k

〈δ(t− tj)δ(t′ − tk)〉S = R2 , (C.3)

∑

j =k

〈δ(t− tj)δ(t′ − tk)〉S +Rδ(t− t′) = R2 ,

thus proving the relation⎡

⎣〈∑

j =k

δ(t− tj)δ(t′ − tk)〉S −R2

⎤

⎦ ρ2aa = −pRδ(t− t′)

for p = 1.In the Poissonian case, tj is totally uncorrelated from tk(j �= k) , so∑

j =k

〈δ(t− tj)δ(t′ − tk)〉S =∑

j

〈δ(t− tj)〉S∑

k

〈δ(t′ − tk)〉S = R2 , (C.4)

which proves (C.1) for p = 0.Notice that in the above result, we are missing an atom in the second

summation, so the above result is approximate, the approximation beingvery good when R >> 1. (The error is of the order of R compared to R2.)

A more general proof is found in the reference [2].

372 C. Proof

References

1. An excellent discussion on this point, as well and on noise supression in quantumoptical systems is found in: Davidovich, L.: Rev. Mod. Phys., 68, 127 (1996)

2. Benkert, C., Scully, M.O., Bergou, J., Davidovich, L., Hillery, M., Orszag, M.:Phys. Rev. A, 41, 2756 (1990)

D. Stochastic Processes in a Nutshell

D.1 Introduction

Classical Mechanics gives a deterministic view of the dynamical variables ofa system. This of course is true, when one is not in a chaotic regime.

On the other hand, in many cases, the system under study is only de-scribed by the time evolution of probability distributions.

To show these ideas with an example, we take a look at the random walkin one dimension, by now, a classical problem [3].

A person moves in a line, taking random steps forward or backward, withequal probability, at fixed time intervals τ.

Calling the position xn = na, then the probability that it occupies thesite xn at time t is P (xn | t) and obeys the equation

P (xn | t+ τ) =12P (xn−1 | t) +

12P (xn+1 | t) . (D.1)

Now, we go to the continuum limit, letting τ and a become small, butwith finite a2

τ . Then

P (x | t+ τ) = P (x | t) + τ∂

∂tP (x | t) + ... (D.2)

P (xn±1 | t) = P (x± a | t) = P (x | t) ± a∂

∂xP (x | t)

+a2

2∂2

∂x2P (x | t) + ... ,

and inserting the above expansions in (D.1), we get

τ∂

∂tP (x | t) + O(τ2) =

a2

2∂2

∂x2P (x | t) + O(a4) + ... (D.3)

Now, letting τ, a→ 0 with

D ≡ a2

τ, (D.4)

D being the diffusion coefficient, we get a diffusion or Fokker–PlanckEquation:

∂

∂tP (x | t) =

D

2∂2

∂x2P (x | t) . (D.5)

374 D. Stochastic Processes in a Nutshell

D.2 Probability Concepts

Let us call ω an event and let A describe a set of events , thus

ω ∈ A , (D.6)

meaning that the event ω belongs to the set of events A [2] .Also, we call Ω the set of all the events and Φ the set of no events.We now introduce the probability of A, P (A), satisfying the following

axioms

i) P (A) ≥ 0 for all A.ii) P (Ω) = 1.iii) If Ai(i = 1, 2, 3...) is a countable collection of non-overlapping sets, such

that

Ai ∩Aj = Φ, i �= j , (D.7)

thenP (∪iAi) =

∑

i

P (Ai) . (D.8)

Now, we are ready to define the joint and conditional probabilities.Joint probability

P (A ∩B) = P {ω ∈ A and ω ∈ B} . (D.9)

Conditional probability

P (A | B) =P (A ∩B)P (B)

, (D.10)

which satisfies the intuitive idea of a conditional probability that ω ∈ A(given that we know that ω ∈ B) is given by the joint probability of A andB divided by the probability of B.

Now, suppose we have a collection of sets Bi, such that

Bi ∩Bj = Φ , (D.11)

∪i(A ∩Bi) = A ∩ (∪iBi) = A . (D.12)

Now, by the axiom iii∑

i

P (A ∩Bi) = P (∪i(A ∩Bi)) = P (A) , (D.13)

thus∑

i

P (A,Bi) =∑

i

P (A | Bi)P (Bi) = P (A) , (D.14)

or, put it in words, if we sum the joint probability over the mutuallyexclusive events Bi, it eliminates that variable. These ideas will be usefullater to derive the Chapman–Kolmogorov Equation.

D.3 Stochastic Processes 375

D.3 Stochastic Processes

We have a time-dependent random variable X(t) and measure the valuesx1, x2, x3... at times t1, t2, t3..., then the joint probability densities

P (x1, t1;x2, t2; ...)

describe completely the system, which is referred to as a stochasticprocess.

One can also define the conditional probability densities as

P (x1, t1;x2, t2; ... | y1, τ1; y2, τ2; ...) = (D.15)

P (x1, t1;x2, t2; ...y1, τ1; y2, τ2; ..)/P (y1, τ1; y2, τ2; ..) ,

where the time sequence increases as

t1 ≥ t2 ≥ ... ≥ τ1 ≥ τ2...

Some simple examples:

a) Complete independence.In this case X(t) is completely independent of past and future, or

P (x1, t1;x2, t2; ...) = �iP (xi, ti) . (D.16)

b) The next simplest case is the Markov Process, where the conditionalprobability is entirely determined by the knowledge of the most recentcondition, that is

P (x1, t1;x2, t2; ... | y1, τ1; y2, τ2; ...) = P (x1, t1;x2, t2; ... | y1, τ1) . (D.17)

It is simple to show, that for the Markovian case, an arbitrary jointprobability can be written as

P (x1, t1;x2, t2; ...xn, tn) = �n−1i=1 P (xi, ti | xi−1, ti−1)P (xn, tn) . (D.18)

D.3.1 The Chapman–Kolmogorov Equation

As we saw in the previous section, summing over all mutually exclusive vari-ables, eliminates that variable, in other words

∑

B

P (A ∩B ∩ C...) = P (A ∩ C...) . (D.19)

Now, we apply this idea to a stochastic process


P (x, t | x0, t0) =∫

dyP (x, t; y, s | x0, t0) (D.20)

=∫

dyP (x, t | y, s;x0, t0)P (y, s | x0, t0) .

Next, we apply the Markov condition, getting the Chapman–KolmogorovEquation

P (x, t | x0, t0) =∫

dyP (x, t | y, s)P (y, s | x0, t0) . (D.21)

In the above analysis, t0 is any initial time for which x(t0) = x0 , and sis an intermediate time t0 ≤ s ≤ t, and x(s) = y.

At this point, we observe that P (x, t | x0, t0) is a probability density,satisfying the initial condition

P (x, t | x0, t0) |t=t0= δ(t− t0) , (D.22)

and the normalization condition∫

dxP (x, t | x0, t0) = 1 . (D.23)

Now, going back to (D.21), we write t = s+ Δt, and expand in Δt

P (x, s+Δt | x0, t0)=∫

dy[

P (x, s | y, s)+Δt∂P (xt | y, s)

∂t|t=s

]

P (y, s | x0, t0) ,

or

P (x, s+ Δt | x0, t0) = P (x, s | x0, t0) + Δt∫

dyW (x | y)P (y, s | x0, t0) ,

(D.24)where W (x | y) is the transition rate, defined as

W (x | y) =∂P (x, s | y, s)

∂t|t=s . (D.25)

Letting Δt→ 0, (D.24) becomes

∂P (x, t | x0, t0)∂t

=∫

dyW (x | y)P (y, t | x0, t0) . (D.26)

This is the forward Chapman–Kolmogorov equation.By integrating (D.26), one can easily verify that

∫

dxW (x | y) = 0 . (D.27)

The transition probability can be split into two parts, one that does notchange plus the change, that is

D.3 Stochastic Processes 377

W (x | y) = W0(x)δ(x − y) +W1(x | y) , (D.28)

and integrating the above equation in x and using (D.27), we get

W0(y) = −∫

W1(x | y)dx ,

so the forward Chapman–Kolmogorov equation now reads as

∂P (x, t | x0, t0)∂t

=∫

dyW1(x | y)P (y, t | x0, t0) (D.29)

−∫

dyW1(y | x)P (x, t |x0 , t0) ,

which has the form of a rate equation.If the random variable X can take discrete values, the forward Chapman–

Kolmogorov equation can be written as

∂P (xi, t)∂t

=∑

j

[WijP (xj , t) −WjiP (xi, t)] . (D.30)

This equation is known as the Master Equation.Many stochastic processes are of a special type called ‘birth and death

process’ or one-step process [4].They correspond to

Wij = rjδi,j−1 + gjδi,j+1, (i �= j) (D.31)

which permits jumps to adjacent sites.Also, for the diagonal part

Wn = −(rn + gn) , (D.32)

so the master equation reads.

Pn= rn+1Pn+1 + gn−1Pn−1 − (rn + gn)Pn , (D.33)

where rn represents the probability per unit time to jump from n→ n−1,and gn the probability per unit time to go from n→ n+ 1.

Typically, one-step processes occur in atomic transition via one photon(emission and absorption), nuclear excitation and de-excitation, fission, etc.

An interesting example is the Poisson process, defined as

rn = 0 , (D.34)gn = q .

Pn(0) = δn,0,

and the Master Equation is


.

Pn= q(Pn−1 − Pn) . (D.35)

This is a one-sided random walk.To solve it, we use the characteristic function:

G(s, t) = 〈exp ins〉 =∑

n

Pn(t) exp ins , (D.36)

with boundary condition G(s, 0) = 1.Multiplying the Master Equation by exp ins and summing over n, we get

∑

n

exp(ins).

Pn= q∑

n

[Pn−1 exp(ins) − Pn exp(ins)]

or∂G(s, t)∂t

= q(exp(is) − 1)G(s, t) . (D.37)

It is simple to verify that the solution of (D.37) is

G(s, t) = exp {tq [exp(is) − 1]} (D.38)

= exp(−tq)∑

n

(exp is)n(tq)n

n!,

thus comparing with (D.36), we finally get

Pn(t) = exp(−tq) (tq)n

n!, (D.39)

which is a Poisson distribution with 〈n〉 = tq.

D.4 The Fokker–Planck Equation

Sometimes, instead of discrete jumps, one chooses to describe the randomprocess as a continuous one.

If we take, for example, in the Chapman–Kolmogorov equation [3]:

Φ(w | x) ≡W (x+ w | x) , (D.40)

then

∂P (x, t | x0,t0)

∂t=∫

dwΦ(w | x− w)P (x − w, t | x0,t0) (D.41)

=∫

exp(

−w ∂

∂x

)

[

Φ(w | x)P (x, t | x0,t0)]

dw

=∫ [

1 − w∂

∂x+

12w2 ∂

2

∂x2+ ..

]

[

Φ(w | x)P (x, t | x0,t0)]

dw ,

D.4 The Fokker–Planck Equation 379

and because∫

dwΦ(w | x) = 0, we get

∂P (x, t | x0,t0)

∂t=

∞∑

n=1

(−1)n

n!∂n

∂xn

[

Qn(x)P (x, t | x0,t0)]

, (D.42)

withQn(x) =

∫

wnΦ(w | x)dw . (D.43)

Many times, the above equation is truncated, keeping only the first twoterms, getting the Fokker–Planck Equation.

In one dimension, with Q1 = A,Q2 = B, we get

∂P (x, t | x0,t0)∂t

= − ∂

∂x[A(x, t)P (x, t | x0,t0)] (D.44)

+12∂2

∂x2[B(x, t)P (x, t | x0,t0)] .

A simple generalization to more variables leads to the Fokker Planckequation

∂P (x, t | x0,t0)∂t

= −∑

i

∂

∂xi[Ai(x, t)P (x, t | x0,t0)] (D.45)

+12

∑

i,j

∂2

∂xi∂xj[Bij(x, t)P (x, t | x0,t0)] ,

where A is the drift vector and B the diffusion matrix. This equation canalso be written as

∂P (x, t | x0,t0)∂t

+∑

i

∂

∂xiJi(x, t) = 0 , (D.46)

Ji(x, t) = [Ai(x, t)P (x, t | x0,t0)] (D.47)

−12

∑

j

∂

∂xj[Bij(x, t)P (x, t | x0,t0)] .

Ji(x, t) is interpreted as a probability current.Let us take a one-dimensional example.

D.4.1 The Wiener Process

We take the articular case A = 0, B = 1, so the Fokker–Planck now reads [2]

∂P (w, t | w0,t0)∂t

=12∂2

∂w2[P (w, t | w0,t0)] . (D.48)


Once more, we use the characteristic function

φ(s, t) =∫

dw exp(isw)P (w, t | w0,t0) . (D.49)

The differential equation for φ is

∂φ

∂t= −s2φ . (D.50)

We also notice that as P (w, t | w0,t0) |t=t0= δ(w − w0), so φ(s, t0) =exp isw0, and the solution is

φ(s, t) = exp[

isw0 − 12s2(t− t0)

]

, (D.51)

which is a Gaussian, whose inverse transform is also a Gaussian

P (w, t | w0,t0) =1

√

2π(t− t0)exp− (w − w0)2

2(t− t0). (D.52)

The first two moments are

〈W 〉 = w0 , (D.53)〈(ΔW )2〉 = t− t0 .

This distribution spreads in time and corresponds precisely to Einstein’smodel for Brownian motion.

An important characteristic of Wiener’s process is the independence ofthe increments, which is interesting for stochastic integration purposes.

We saw that, in general, for Markov Processes, one has

P (wn, tn;wn−1, tn−1; ...w0, t0) = �n−1i=0 P (wi+1, ti+1 | wi, ti)P (w0, t0)

= �n−1i=0

{

[2π(ti+1 − ti)]− 1

2 exp[

− (wi+1 − wi)2

2(ti+1 − ti)

]}

P (w0, t0) . (D.54)

Now we define the Wiener increments as

ΔWi = W (ti) −W (ti−1) , (D.55)Δti − ti − ti−1 ,

so the joint probability density for the increments is

P (Δwn; Δwn−1; ...Δw1;w0)

= �ni=1

{

[2πΔti]− 1

2 exp[

− (Δwi)2

2(Δti)

]}

P (w0, t0) , (D.56)

D.4 The Fokker–Planck Equation 381

thus they are statistically independent.If we define the mean and autocorrelation functions as

〈W(t) | W0, t0〉 =∫

dwP (w, t | w0, t0)w , (D.57)

〈W(t)W(t0)T | W0, t0〉 =∫

dwdw0P (w, t;w0, t0)wwT0

=∫

dw0〈W(t) | W0, t0〉wT0 P (w0, t0) .

For the Wiener process

〈W (t)W (s) |W0, t0〉 = 〈[W (t) −W (s)]W (s) |W0, t0〉 + 〈W (s)2〉 , (D.58)

and due to the independence of the increments, the first term is zero and

〈W (t)W (s) |W0, t0〉 = w20 + min(t− t0, s− t0) . (D.59)

D.4.2 General Properties of the Fokker–Planck Equation

The general Fokker–Plank equation reads

∂P (x, t | x0,t0)∂t

= −∑

i

∂

∂xi[Ai(x, t)P (x, t | x0,t0)]

+12

∑

i,j

∂2

∂xi∂xj[Bij(x, t)P (x, t | x0,t0)] . (D.60)

As we mentioned before, the first term in the r.h.s is the drift term, whichwill rule the deterministic motion, and the second one is the diffusion term,which will cause the probability to broaden. This different role of the twoterms can be easily seen if we calculate 〈xi〉 and 〈xixj〉. One can easily showthat

d〈xi〉dt

= 〈Ai〉 , (D.61)

d〈xixj〉dt

= 〈xiAj〉 + 〈xjAi〉 +12〈Bij + Bji〉 .

D.4.3 Steady-State Solution

Very often in optics and other areas of physics, one is not really interestedin the time-dependent solution of the Fokker–Planck equation , but rather inthe steady state. Thus, we set the time derivative to zero and get


∑

i

∂

∂xi

⎡

⎣−Ai(x, t)P (x, t | x0,t0) +12

∑

j

∂

∂xj[Bij(x, t)P (x, t | x0,t0)]

⎤

⎦ = 0 ,

(D.62)and if the constant current is set to zero (detailed balance) , one gets

Ai(x, t)P (x, t | x0,t0) =12

∑

j

∂

∂xj[Bij(x, t)P (x, t | x0, t0)] (D.63)

or

2Ai −∑

j

∂Bij

dxj=∑

j

Bij1

P (x, t | x0,t0)∂P (x, t | x0,t0)

∂xj

=∑

j

Bij∂In [P (x, t | x0,t0)]

∂xj,

and defining a Potential function V (x) by P (x, t | x0,t) = N exp(−V (x)),we get for V

−∂V (x)∂xi

= 2∑

j

B−1ij Aj −

∑

j,k

B−1ij

∂Bjk

∂xk. (D.64)

Integrating (D.64), we get for the probability distribution

PSS(x) = N exp

⎡

⎣

∫∑

i,j

2B−1ij Aj(x)dxi −

∫∑

i,j,k

B−1ij

∂Bjk

∂xkdxi

⎤

⎦ . (D.65)

In particular, for Bij = Dδij , we get

PSS(x) = N exp∫

2D

A(x)dx (D.66)

D.5 Stochastic Differential Equations

D.5.1 Introduction

One way of treating the motion of a Brownian particle, or any other problemwith a random force, is via a Langevin or Stochastic differential equation

.

V= −γV + L(t) , (D.67)

where, in the case of a Brownian particle, the r.h.s. is the force of the fluidover the particle and is made up of two components:

a) The damping force −γV

D.5 Stochastic Differential Equations 383

b) A rapidly varying force L(t), independent of the particle’s velocity thataccounts for the collisions of the water molecules with the Brownian par-ticle, whose average is zero. Thus

〈L(t)〉 = 0 (D.68)〈L(t)L(t′)〉 = Dδ(t− t′) .

〈L(t)L(t′)〉 is referred to as the two time correlation function.If one defines the spectrum as the Fourier transform of the two time

correlation function

S(ω) =∫ +∞

−∞dτ exp(iωτ)〈L(t+ τ)L(t)〉 , (D.69)

we immediately notice that, because the Fourier transform of a deltafunction is a constant, L(t) has a flat spectrum or it correspond to whitenoise.

Let us assume that the initial velocity of the Brownian particle is deter-ministic and given by V (0) = V0, then for t > 0, for each sample path

V (t) = V0 exp(−γt) + exp(−γt)∫ t

0

exp(γt′)L(t′)dt′ . (D.70)

Using the properties of L, we can calculate 〈V 〉 and 〈V 2〉

〈V (t) | V0, t0〉 = V0 exp(−γt) , (D.71)〈V 2(t) | V0, t0〉 = V 2

0 exp(−2γt) +

exp(−2γt)∫ t

0

dt′∫ t

0

dt′′ expγ(t′ + t′′)〈L(t′)L(t′′)〉

〈V 2(t) | V0, t0〉 = V 20 exp(−2γt) +

D

2γ[1 − exp(−2γt)] . (D.72)

When t→ ∞〈V 2(t) | V0, t0〉 =

D

2γ, (D.73)

On the other hand, for short times

〈(ΔV )2(t0 + Δt) | V0, t0〉 = DΔt+ O(Δt)2.. (D.74)

We also notice, that in this case, the drift and diffusion coefficients are

A =〈ΔV 〉Δt

=(〈V − V0〉) |t=t0+Δt

Δt= −γV , (D.75)


B =〈(ΔV )2〉

Δt= D , (D.76)

so that the corresponding Fokker–Planck equation is

∂P (V, t)∂t

= γ∂

∂V(V P ) +

D

2∂2P

∂V 2. (D.77)

The above equation describes the so-called Ornstein–Uhlenbeckprocess, corresponding to a linear drift and a constant diffusion term.

We now calculate the power spectrum of V . So we first need the two timecorrelation function

〈V (t)V (t′)〉 = V 20 exp [−γ(t+ t′)]

+ exp [−γ(t+ t′)]∫ t

0

dt′′∫ t′

0

dt′′′ exp [γ(t′′ + t′′′)] 〈L(t′′)L(t′′′)〉

= V 20 exp [−γ(t+ t′)] + exp [−γ(t+ t′)]D

∫ t′

0

dt′′′ exp 2γt′′′

〈V (t)V (t′)〉 = V 20 exp [−γ(t+ t′)] + exp [−γ(t+ t′)]

Γ

2γ[exp(2γt′) − 1] .

(D.78)In steady state, for t, t′ → ∞ but with t− t′ = τ , we get

〈V (t+ τ)V (t)〉 =D

2γexp(−γ | τ |) . (D.79)

Finally, taking the Fourier transform, we get the power spectrum of V

φV (ω) =12π

∫

exp(iωτ)〈V (t+ τ)V (t)〉dτ (D.80)

=12π

D

ω2 + γ2.

D.5.2 Ito Versus Stratonovich

A more general type of Langevin equation can be written as

dxdt

= a(x, t) + b(x, t)L(t) , (D.81)

where the previous D factor can be absorbed in b, so that

〈L(t)L(t′)〉 = δ(t− t′) , (D.82)〈L〉 = 0 .

Now we define


W (t) =∫ t

0

L(t′)dt′ , (D.83)

assumed to be continuous, so that

〈W (t+ Δt) −W0(t) | W0, t〉 = 〈∫ t+Δt

t

dsL(s)〉 = 0 , (D.84)

〈[W (t+ Δt) −W0(t)]2 | W0, t〉 = (D.85)

〈∫ t+Δt

t

ds1∫ t+Δt

t

ds2L(s1)L(s2)〉 =∫ t+Δt

t

ds1∫ t+Δt

t

ds2δ(s1 − s2) = Δt ,

therefore, one could write a Fokker–Planck equation for W with

A = 0, B = 1 ,

which correspond to a Wiener process , and Ldt = dW becomes a Wienerincrement.

The stochastic differential equation (D.81) is not fully defined unless onespecifies how to integrate it. Normally, this would not be a problem, and therules of ordinary calculus apply. However, here we must be careful becausewe are dealing with a rapidly varying function of time L(t).

Thus, we define the integral the mean square limit of a Riemann–Stieltjessum

∫ t

t0

f(t′)dW (t′) = ms limn→∞

n∑

i=1

f(τi) [W (ti) −W (ti−1)] , (D.86)

where ti−1≤τi ≤ ti, and we have divided the time interval from t0 → t inn intermediate times t1t2...tn.

One can verify that it does matter which f(τi) we choose.Two popular choices are:

a) Ito with τi = ti−1.

b) Stratonovich: f(τi) = f(ti)+f(ti−1)2 .

From the above assumptions, one learns how to calculate things with Itoand Stratonovich.

For Stratonovich, we have for example

S

∫ t

t0

W (t′)dW (t′)

= ms limn→∞

n∑

i=1

[

W (ti) +W (ti−1)2

]

[W (ti) −W (ti−1)]

=12ms lim

n→∞

n∑

i=1

[

W 2(ti) −W 2(ti−1)]

=12[

W 2(t) −W 2(t0)]

,


which obeys the rules of ordinary calculus.On the other hand, for Ito

I

∫ t

t0

W (t′)dW (t′)

= ms limn→∞

n∑

i=1

[W (ti−1)] [W (ti) −W (ti−1)]

= ms limn→∞

n∑

i=1

[W (ti−1)ΔW (ti)]

= ms limn→∞

12

n∑

i=1

{

[W (ti−1) + ΔW (ti)]2 −W (ti−1)2 − ΔW (ti)2

}

=12[

W (t)2 −W (t0)2]−ms lim

n→∞12

n∑

i=1

ΔW (ti)2 ,

and because

ms limn→∞

12

n∑

i=1

ΔW (ti)2 = t− t0 ,

we finally get

I

∫ t

t0

W (t′)dW (t′) =12[

W (t)2 −W (t0)2 − (t− t0)]

. (D.87)

Finally, for the Ito integration, one can prove that

dW (t)2 = dt, (D.88)dW (t)2+N = 0, N = 1, 2, 3..

The details and proof of the above properties are found in Gardiner’sbook [2].

From these properties, we can see that dW ∼ √dt, and we have to keep

terms up to (dW )2, which differs from the ordinary calculus.

D.5.3 Ito’s Formula

Consider a function f [x(t)]. We will derive the basic formula for Ito’s calculus:

df [x(t)] = f [x(t) + dx] − f [x(t)]

= f ′ [x(t)] dx+12f ′′ [x(t)] dx2 + ...


= f ′ [x(t)] [a(x, t)dt+ b(x, t)dW ]

+12f ′′ [x(t)]

[

b(x, t)dW 2 + ...]

and using (D.88), we get

df [x(t)] ={

a(x, t)f ′ [x(t)] +12b(x, t)f ′′ [x(t)]

}

dt

+b(x, t)f ′ [x(t)] dW . (D.89)

The above formula can be easily generalized for many dimensions.Now, we take the average of Ito’s formula

d〈f(x)〉dt

=∫

dx∂tP (x, t)f(x)

=∫

dx[

a∂xf +b

2∂2

xf

]

P (x, t) ,

and integrating by parts and discarding the surface terms, we get∫

dxf(x)∂tP (x, t) =∫

dxf(x)[

−∂xaP +12∂2

xbP

]

,

thus getting the Ito–Fokker–Planck equation

∂tP (x, t | x0, t0) = −∂x [a(x, t)P (x, t | x0, t0)] (D.90)

+12∂2

x [b(x, t)P (x, t | x0, t0)] (D.91)

Similarly, for many variables, if one has an Ito stochastic differential equa-tion

(I)dx = a(x, t)dt+ b(x, t)dW , (D.92)

where dW is an n-component Wiener process, then the correspondingIto’s Fokker–Planck equation is:

∂tP (x, t | x0, t0) = −∑

i

∂i [ai(x, t)P (x, t | x0, t0)] (D.93)

+12

∑

i,j

∂i∂j

[

bbT (x, t)]

ijP (x, t | x0, t0) . (D.94)

Thus, from our previous notation

B = bbT (D.95)

Similarly, for Stratonovich


(S)dx = aS(x, t)dt+ bS(x, t)dW, (D.96)

we get a Stratonovich–Fokker–Planck equation

∂tP (x, t | x0, t0) = −∑

i

∂i

[

aSi (x, t)P (x, t | x0, t0)

]

+12

∑

i,j,k

∂i

[

bSik∂jbST

jk (x, t)]

P (x, t | x0, t0) . (D.97)

By simple comparison between the two Fokker–Planck equations, we get

aSi = ai −

12

∑

j,k

bkj∂kbTij , (D.98)

bSik = bik .

This last relation tells us that if we have a given Fokker–Planck equa-tion, it corresponds to a Langevin equation to be integrated a la Ito, witha and b drift and diffusion coefficients, and to a Langevin equation to beintegrated a la Stratonovich with aS and bS drift and diffusion coefficients,and the relation between the Ito and the Stratonovich coefficients is given by(D.98).

D.6 Approximate Methods

Non-linear Langevin equations are difficult to solve exactly.We present here the Ω-expansion of Van Kampen [5], where Ω is the

size or number of particles of our system.We consider a variable X that is proportional to the particle number, and

definex =

X

Ω. (D.99)

The key point in Van Kampen’s expansion is that we can separate [1]:

x(t) = x0(t) +√εy(t) , (D.100)

where x0(t) is the deterministic part, y(t) represents the fluctuations andε = 1

Ω .This decomposition is based on the Central Limit Theorem that says that

for large Ω, the fluctuations of X around its mean value go as Ω.Of course, this expansion fails, as we shall see, near an instability point.We also assume that in the stochastic equation, the small parameter

√ε

is the noise strength, so we write

dx = a(x)dt+√εdW (t) , (D.101)

D.6 Approximate Methods 389

andx(t) = x0(t) +

√εx1(t) + εx2(t) + ... (D.102)

Differentiating x and expanding a(x) around x0, we get

dx0(t) +√εdx1(t) + εdx2(t) + ...

= a(x0)dt+ a′(x0)(x− x0)dt+12a′′(x0)(x − x0)2dt+ ...+

√εdW (t)

= a(x0)dt+ a′(x0)[√εx1(t) + εx2(t) + ...

]

dt+

12a′′(x0)

[

εx21 + ...

]

dt+ +√εdW (t) ,

and by comparing different orders of ε we get

dx0(t) = a(x0)dt, (D.103)

dx1(t) = a′(x0)x1(t)dt+ dW (t) , (D.104)

dx2(t) = a′(x0)x2(t)dt+12a′′(x0)x2

1dt , (D.105)

and so on.The initial conditions for x1, x2, x3... are

x1(0) = 0, x2(0) = 0, etc : (D.106)

Now, we take a non-trivial example.A particle, in one dimension, under the action of a double-well potential

a(x) = −dVdx

, (D.107)

withV = −γ

2x2 +

g

4x4 . (D.108)

The stochastic differential equation, in this case is

dx(t) = (γx− gx3)dt+√εdW (t) . (D.109)

The shape of the potential is described in the Fig. D.1We will consider the case γ > 0. Near the equilibrium positions, the drift

is practically zero, and the noise term in the stochastic equation is quiteimportant.

On the other hand, very far from the equilibrium positions, the motion isdominated by a large drift and is practically a deterministic one.

For γ > 0, x = 0 is an unstable equilibrium position and x = ±√

γg are

stable ones.


Fig. D.1. Double-well potential for the cases γ < 0 (upper curve) and γ > 0 (lowercurve)

Applying our method in this example, we get

dx0 = (γx0 − gx30)dt, x0(0) = h , (D.110)

dx1 = (γ − 3gx20)x1dt+ dW,x1(0) = 0 , (D.111)

dx2 =[

(γ − 3gx20)x2 − 3gx0x

21

]

dt, x2(0) = 0 , (D.112)

The solution for the deterministic motion is

x0(t) =h exp(γt)

√

1 + gγh

2 [exp(2γt) − 1], (D.113)

so that if we choose the unstable equilibrium point, that is the initialcondition h = 0, the we get x0(t) = 0, and for the stable equilibrium points,h = ±

√

γg , we get x0(t) = ±

√

γg , as it should.

For the first case: h = x0(t) = 0, we get

dx1 = γx1dt+ dW (t) , (D.114)

dx2 = γx2dt , (D.115)

and the solution to the above equations are

x1(t) =∫ ∞

o

exp [γ(t− t′)] dW (t′) , (D.116)

x2(t) = 0.

In the stable case x0 = h = ±√

γg , we get

References 391

dx1(t) = −2γx1dt+ dW (t), (D.117)

dx2(t) =(

−2γx2 ∓ 3g√

γ

gx2

1

)

dt ,

and the solutions are

x1(t) =∫ t

0

dW (t′) exp [−2γ(t− t′)] (D.118)

x2(t) = ∓3g√

γ

g

∫ t

0

dt′x21(t

′) exp [−2γ(t− t′)] .

Now, we notice that in all cases 〈x1〉 = 0, so

〈x2(t)〉 = x20 + ε

(〈x21〉 + 2x0〈x2〉

)

+ ... (D.119)

and in the two cases, we can write

〈x2(t)〉uns =ε

2γ[exp(2γt) − 1] , (D.120)

〈x2(t)〉stable =γ

g+

ε

4γ[1 − exp(−4γt)] − 3ε

4γ[1 − exp(−2γt)]2 + ...

In the limit t→ ∞, 〈x2(∞)〉uns diverges, while 〈x2(t)〉stable → γg − ε

2γ .As we can see, the perturbative expansion gives the correct answer when

starting from stable equilibrium points but it diverges when starting froman unstable equilibrium. In this last case, the perturbative expansion is nolonger valid.

References

1. Tombesi P.: In: Gomez, B., Moore, S.M., Rodriguez-Vargas, A.M., Rueda, A.(eds) Stochastic Processes Applied to Physics and Other Related Fields. WorldScientific, Singapore (1983)

2. Gardiner, C.W.: Handbook of Stochastic Methods. Springer Verlag, Berlin(1983)

3. Stenholm, S.: In: Meystre, P., Scully, M.O. (eds) Quantum Optics, Experimen-tal Gravitation and Measurement Theory. Plenum Press, NY (1983)

4. Van Kampen, N.G.: Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam (1981).

5. Van Kampen, N.G.: Can. J. Phys., 39, 551 (1961)

E. Derivation of the Homodyne Stochastic

Schrodinger Differential Equation

Here we present the detailed derivation of the Homodyne Schrodinger dif-ferential equation. We start from the expansion given by (16.36), which inthe two-jump situation, and neglecting the commutators between the jumpoperators and the no-jump evolution, can be expressed as

ρ(Δt) =∞∑

m1,m2=0

(Δt)m1+m2

m1!m2!S(Δt)Jm2

2 Jm11 ρ(0) . (E.1)

The probability of m1and m2 quantum jumps of the respective types isgiven by

Pm1,m2(Δt) =(Δt)m1+m2

m1!m2!Tr [S(Δt)Jm2

2 Jm11 ρ(0)] . (E.2)

The master equation of the field, corresponding to a lossy cavity at temper-ature T , may be written as

dρdt

= (J1 + J2)ρ− γ

2ρ[a†a(1 + 2〈n〉th) + 2ε(1 + 〈n〉th)a†

+2ε〈n〉tha+ 〈n〉th + ε2(1 + 2〈n〉th)]

−γ2[a†a(1 + 2〈n〉th) + 2ε(1 + 〈n〉th)a+ 2ε〈n〉tha†

+〈n〉th + ε2(1 + 2〈n〉th)]ρ (E.3)

Therefore, according to the discussion given in Chap. 16, one possible way ofwriting S(Δt) is

S(Δt)ρ = N(Δt)ρN †(Δt) , (E.4)

with

N(Δt) = exp{

−γ(Δt)2

[a†a(1 + 2〈n〉th) + 2ε(1 + 〈n〉th)a†

+2ε〈n〉tha+ 〈n〉th + ε2(1 + 2〈n〉th)]}

. (E.5)

394 E. Derivation of the Homodyne Stochastic Schrodinger Differential Equation

Using (E.2) and (E.5), we can write

Pm1,m2(Δt) =[

exp(μ1)(μ1)m1

m1!

] [

exp(μ2)(μ2)m2

m2!

]

Tr[exp(β)(

1 +a†

ε

)m2 (

1 +a

ε

)m1

ρ

(

1 +a†

ε

)m1

(

1 +a

ε

)m2

exp(β†)] (E.6)

´withμ1 = γ(Δt)ε2(1 + 〈n〉th) , (E.7)

μ2 = γ(Δt)ε2〈n〉th ,

β = −γ(Δt)2

{a†a(1 + 2〈n〉th)

+2[ε(1 + 〈n〉th)a+ ε〈n〉tha†] + 〈n〉th} .From (E.6), we can calculate 〈mi〉 and σ2

i = 〈m2i 〉 − 〈mi〉2 up to order

(1ε

) 32 .

The result is〈mi〉 = μi(1 +

2ε〈X〉) ,

σ2i = 〈μi〉 . (E.8)

Now, we turn to the final step of this calculation, which yields the timeevolution of the state vector. After repeated jump and no-jump events, theunnormalized wavefunction for the field can be written as

| ˜ψ〉f (Δt) = N(Δt− tm)C2N(tm − tm−1)C1N... | ψ〉f (0)

or, except for an overall phase factor,

| ˜ψ〉f (Δt) = N(Δt)Cm22 Cm1

1 ψ〉f (0) , (E.9)

where the symbol ∼ indicates that the vector is not normalized.Using (E.5) and (16.51), one can write, up to a normalization constant

| ˜ψ〉f (Δt) = exp(

−γ(Δt)2{

a†a(1 + 〈n〉th)

+2[ε(1 + 〈n〉th)a† + 2ε〈n〉tha]}

)

(

1 +a†

ε

)m2 (

1 +a

ε

)m1 | ψ〉f (0) (E.10)

or expanding up to ε−32

E Derivation of the Homodyne Stochastic Schrodinger Differential Equation 395

| ˜ψ〉f (Δt) ={

1 − γ(Δt)2

[a†a(1 + 〈n〉th) + aa†〈n〉th ]

−γ(Δt)ε[a(1 + 〈n〉th) + a†〈n〉th)]}

×[

1 +1ε(m1a+m2a

†)]

| ψ〉f (0) (E.11)

We are interested in the limit ε → ∞. In deriving (E.11), we considered ε

large, γ(Δt) ∼ ε−32 and m1,m2, μ1, μ2 ∼ ε

12 . Now we consider two random

numbers with non-zero average m1,m2

m1 = 〈m1〉 +σ1√Δt

ΔW1 ,

m2 = 〈m2〉 +σ2√Δt

ΔW2 , (E.12)

which satisfy〈(ΔW1)2〉 = 〈(ΔW2)2〉 = Δt . (E.13)

We notice that ΔWi are two independent Wiener Processes.Finally, (E.11) can be written as

Δm1,m2 | ˜ψ〉f (Δt) =| ˜ψ〉f (Δt)− | ˜ψ〉f (0)

={[

−γ2(a†a(1 + 〈n〉th) − γ

2aa†〈n〉th

+2γ〈X〉a(1 + 〈n〉th)

+a†〈n〉th]

Δt+ a†√

γ〈n〉thΔW2

+a√

γ(1 + 〈n〉th)ΔW1

}

| ˜ψ〉f (0) ,

which is the desired result.

F. Fluctuations

We want to calculate d〈(Δa†a)2〉 and Md〈(Δa†a)2〉.We do it first in a simple case T=0, O = a†a;C =

√γa;H = �ωa†a.

d 〈(Δa†a)2〉 = γδt{−〈a†aa†aa†a〉 + 2〈a†aa†a〉〈a†a〉− 2〈a†a〉〈a†a〉〈a†a〉 + 〈a†aa†a〉〈a†a〉}− 〈a†aa†a〉δN + 〈a†a〉〈a†a〉δN

+〈a†a†aa†aa〉〈a†a〉 − 〈a†a†aa〉〈a†a†aa〉

〈a†a〉〈a†a〉 δN , (F.1)

or

d 〈(Δa†a)2〉 = −γδt〈(Δa†a)(Δa†a)(Δa†a)〉− 〈(Δa†a)2〉δN+

〈a†a†aa†aa〉〈a†a〉 − 〈a†a†aa〉〈a†a†aa〉〈a†a〉〈a†a〉 δN (F.2)

Now, we apply the above results to the more interesting case T > 0,O = a†a;C1 =

√

(〈n〉th + 1)γa, C2 =√

γ〈n〉tha†;H = �ωa†a:

d 〈(Δa†a)2〉 = −γ(〈n〉th + 1)〈(Δa†a))(Δa†a)(Δa†a)〉dt− 〈(Δa†a)2〉δN1

+(〈a†aa†aa†a〉〈a†a〉 − 〈a†aa†a〉〈a†aa†a〉)δN1

〈a † a〉〈a†a〉+ γ〈n〉thdt[−〈aa†aa†aa†〉 + 2〈aa†aa†〉 − 〈aa†〉+ 2〈aa†aa†〉〈a†a〉 − 2〈aa†〉〈a†a〉 − 〈aa†〉〈a†a〉〈a†a〉+ 〈a†aa†a〉〈aa†〉 − 〈a†a〉〈a†a〉〈aa†〉]− 〈(Δa†a)2〉δN2

+(〈aa†aa†aa†〉〈aa†〉 − 〈aa†aa†〉〈aa†aa†〉)δN2

〈aa†〉〈aa†〉 . (F.3)

In the above expression, neither the deterministic nor the stochastic termis definitely non-increasing. But in the mean, it does decrease

398 F. Fluctuations

Md〈(Δa†a)2〉

dt= −γ(〈n〉th + 1)

〈(Δa†a)a†a〉〈a†a(Δa†a)〉〈a † a〉

− γ〈n〉th 〈(Δaa†)aa†〉〈aa†(Δaa†)〉

〈aa†〉 ≤ 0 . (F.4)

G. The No-Cloning Theorem [1]

We assume that we have a device able to duplicate an arbitrary quantumstate. That is, if the system is initially in the state | ψ〉

| ψ〉⊗ | α〉 → U(| ψ〉⊗ | α〉) =| φ〉⊗ | α〉⊗ | α〉 ,

where | φ〉 is the state of the system after performing its copying. Similarly,for a different input state, we would have

| ψ〉⊗ | β〉 → U(| ψ〉⊗ | β〉) =| φ´〉⊗ | β〉⊗ | β〉 .

Taking the inner product of these two states, we get

〈ψ | ψ〉〈α | β〉 = 〈φ | φ´〉〈α | β〉〈α | β〉 .

In the above equation, 〈ψ | ψ〉 = 1 and 0<|< α | β >|< 1, so we concludethat 〈φ | φ´〉〈α | β〉 = 1, which is impossible, because | 〈φ | φ´〉 |≤ 1.

Thus, the system represented by | ψ〉 cannot exist.

Reference

1. Peres, A. Quantum Theory; Concepts and Methods. Kluwer, The Netherlands(1995)

H. The Universal Quantum Cloning

Machine [1]

We will develop the Universal copying machine, as proposed by Buzek et al.The following transformation is proposed

| 0〉a | Q〉x →| 0〉a | 0〉b | Q0〉x + (| 0〉a | 1〉b+ | 1〉a | 0〉b) | Y0〉x , (H.1)

| 1〉a | Q〉x →| 1〉a | 1〉b | Q1〉x + (| 0〉a | 1〉b+ | 1〉a | 0〉b) | Y1〉x . (H.2)

As there are several free parameters, we can impose some conditions,namely

x〈Qi | Qi〉x + 2x〈Yi | Yi〉x = 1, i = 0, 1 ,

x〈Y0 | Y1〉x =x 〈Y1 | Y0〉x = 0 ,

x〈Qi | Yi〉x = 0, i = 0, 1 ,

x〈Q0 | Q1〉x = 0 , (H.3)

where the Qs and Y s are states of the copying machine.With the above assumptions, one can write ρ(out)

ab , describing the modesa and b after the copying of a pure state | ψ〉 = α | 0〉 + β | 1〉 as

ρ(out)ab = α2 | 00〉〈00 |x 〈Q0 | Q0〉x +

√2αβ | 00〉〈+ |x 〈Y1 | Q0〉x

+√

2αβ | +〉〈00 |x 〈Q0 | Y1〉x+[

2α2x〈Y0 | Y0〉x + 2β2

x〈Y1 | Y1〉x] | +〉〈+ |

+√

2αβ | +〉〈11 |x 〈Q1 | Y0〉x +√

2αβ | 11〉〈+ |x 〈Y0 | Q1〉x+β2 | 11〉〈11 |x 〈Q1 | Q1〉x . (H.4)

Now, if we trace over the b mode, we get the density matrix for the a-mode

ρ(out)a = | 0〉a〈0 | (α2 + β2

x〈Y1 | Y1〉x − α2x〈Y0 | Y0〉x

)

+ | 0〉a〈1 | αβ (x〈Q1 | Y0〉x +x 〈Y1 | Q0〉x)

+αβ | 1〉a〈0 | (x〈Q0 | Y1〉x +x 〈Y0 | Q1〉x)

+ | 1〉a〈1 | (β2 + α2x〈Y0 | Y0〉x − β2

x〈Y1 | Y1〉x)

. (H.5)

402 H. The Universal Quantum Cloning Machine

The density operator ρ(out)b = ρ

(out)a , in other words the two output modes

are equal, but different to the input state. To quantify the difference, we usethe ‘distance’ (22.7), giving as a result

Dα = 2x2(4α4 − 4α2 + 1) + 2α2(1 − α2)(e− 1)2 , (H.6)

withx〈Y0 | Y0〉x =x 〈Y1 | Y1〉x ≡ x , (H.7)

x〈Y0 | Q1〉x =x 〈Q0 | Y1〉x =x 〈Q1 | Y0〉x =x 〈Y1 | Q0〉x ≡ e

2, (H.8)

which are the two free parameters, with 0≤ x ≤ 12 , 0 ≤ e ≤ 1√

2.

The first requirement is that the distance Dα be independent of the input,that is of α. So we impose

∂Dα

∂(α2)= 0 , (H.9)

which gives us a relation between the parameters

e = 1 − 2x ,

so that Dα becomes input independent Dα = 2x2.We also require a condition on the two-mode density. The distance be-

tween the density operator and its ideal version should be input independent,that is Dαb = Tr(ρ(out)

ab − ρ(id)ab )2 satisfies

∂Dαb

∂(α2)= 0 . (H.10)

After some algebra, one gets

Dαb = (f11)2 + 2(f12)2 + 2(f13)2

(f22)2 + 2(f23)2 + (f33)2 ,

with f11 = α4−α2(1−2x), f12 =√

2αβ(α2− 12 (1−2x)), f13 = (αβ)2, f22 =

2((αβ)2 − x), f23 =√

2αβ(β2 − 12 (1 − 2x)), f33 = β4 − β2(1 − 2x).

Now, the (H.9) can be solved, giving x = 29 .

If we write ρ(out)a in the basis | ψ〉 = α | 0〉+β | 1〉 and | ψ⊥〉 = α | 0〉−β |

1〉, we readily get

ρ(out)a1

=56| ψ〉a1〈ψ | +

16| ψ⊥〉a1〈ψ⊥ | .

Reference

1. Buzek, V., Hillery, M.: Phys. Rev. A, 54, 1844 (1996)

I. Hints to Solve the Problems

Chapter1

1.1 Use (1.11) and (1.12).1.2 Calculate 〈n2〉1.3 Verify the solution using (1.3)

Chapter2

2.1 Use (2.36)

Chapter3

3.1 Iterate (3.27) many times.3.2 See Appendix A3.3 See Appendix A3.4 Follow the text from (3.39) to (3.45)3.5 Use (3.44) and:

∂

∂zδT11(ρ) =

∂

∂zδ(ρ) +

i

(2π)3

∫ ∞

−∞

k3k21

k2exp(ik · ρ)dk

∂

∂xδT13(ρ) =

i

(2π)3

∫ ∞

−∞

k3k21

k2exp(ik · ρ)dk.

Chapter4

4.1 Definea† | β〉 = β | β〉,

and follow the same procedure as in (4.2) to (4.6).4.2 Use

| α〉 = exp(−αα∗

2) exp(αa†) | 0〉,

and write

| α〉〈α |= exp(−αα∗) exp(αa†) | 0〉〈0 | exp(α∗a).

404 I. Hints to Solve the Problems

4.3 Use (4.6)4.4 Convert the sums into integrals4.5 Use (4.31) and (4.32)4.6 Start from (4.2)4.7 To prove the last property, use the second one for continuous spectrum.4.8 Use the results of Problem (4.7)

Chapter5

5.1 Use a procedure similar to the one leading to (5.27)5.2 Use the results of Problem (5.1)5.3 See Reference [1]5.4 See Reference [1]5.5 Use the results of Problem (5.1)

Chapter6

6.1 Calculate 〈n2〉 as we did for 〈n〉 in (6.95)

Chapter7

7.1 Use (4.16)7.2 Use (3.19)7.3 Use the commutation relation

[a, a†n] = na†(n−1).

7.4 Use (A.23) and (A.24)7.5 First show that

alF (n)(a†, a) = N (a+∂

∂a†)lF (n)(a†, a),

F (n)(a†, a)a†l = N (a† +∂

∂a)lF (n)(a†, a).

Chapter8

8.1 Find the eigenvalues and eigenvectors of Hn.8.2 Use (8.46) and (8.38)8.3 Approximate (8.55)

Chapter9

9.2 Verify the definitions, using the results of Problem (9.1)9.3 Use the rules given by (9.49)9.4 Use the rules given by (9.49)9.5 Use the rules given by (9.49)9.6 Find 〈a2〉 and 〈a†2〉 from an equation similar to (9.21)

I Hints to Solve the Problems 405

Chapter10

10.1 Use (10.59), (10.60), (10.61)10.2 Calculate the Fourier Transform of the result of Problem (10.1)10.3 Use (10.84), (10.85), (10.86).

Chapter11

11.1 Use (11.5) and (11.6)11.2 Use (11.8)11.3 Start from (11.22) and approximate the trigonometric functions.11.4 Start from (11.23)

Chapter12

12.1 Use the Generalized Einstein relations.12.2 Use a procedure similar to (12.35–12.38)12.3 See Reference [1]12.4 Take ε2 and differentiate with respect to time and use (12.65)

Chapter13

13.1 Star from (13.12)13.2 Use (13.24) and follow the rules given by (9.49). Then one gets a Fokker-

Planck equation in terms of α1 and α2. To go to polar coordinates, define

α1 = ρ1 exp(iθ1);α2 = ρ2 exp(iθ2),

then, one has

∂

∂α1=

12

exp(−iθ1) ∂

∂ρ1+

12i

exp(−iθ1)ρ1

∂

∂θ1,

∂

∂α2=

12

exp(−iθ2) ∂

∂ρ2+

12i

exp(−iθ2)ρ2

∂

∂θ2,

∂

∂θ1=

12∂

∂μ+,

∂

∂θ2=

12∂

∂μ− ∂

∂θ,

whereμ =

θ1 + θ22

,

θ =θ1 − θ2

2.

13.3 Use the results of the Problem (13.2)

406 I. Hints to Solve the Problems

Chapter14

14.1 Start from (14.62)14.2 Start from (14.66)14.3 Integrate (14.38) over ω.14.4 Part (b). Use the quadratic part of the formula for 1

τ .14.5 Use the results of the problem (14.4) for the case ω � ωj

Chapter15

15.1 Use (15.5), (15.6) and (15.7)15.2 Use (15.7)15.3 Use (15.5)15.4 See Reference [5]15.5 Use (15.49)15.6 See Reference [17]

Chapter16

16.1 See Reference [new29]16.2 See Reference [11]

Chapter17

17.1 Use (17.7)17.2 Verify that [H, c] = 0.17.4 See Reference [18]

Chapter18

18.1 Use (18.49) and (18.50)18.2 Use (A.16) and (A.17)18.3 See Appendix A18.4 See Reference [1]18.5 See Reference [1]18.6 See Reference [1]

Chapter19

19.1 See Reference [19]19.2 See Reference [19]

I Hints to Solve the Problems 407

Chapter20

20.1 Use (20.34), (20.41) to verify (20.42)20.2 See References [9], [10], [11], [12].

Chapter21

21.1 Apply the definition of K.21.2 Check the signs of the eigenvalues of the partially transposed density

matrix.21.3 Apply the NPT criterion.

Chapter22

22.2 Follow a procedure similar to the duplicator22.3 Apply the two gates to the input data and programs.

Index

absorption, 1, 2annihilation operator, 24atom -field interaction

semiclassical theory, 11atom optics, 247

diffraction, 248experiments in diffraction, 249optical elements, 247sources, 247theory of diffraction, 250

atomic decay, 18atomic diffraction

large detuning, 252no detuning, 252

atomic focusing, 255aberrations, chromatic, 261aberrations,isotopic, 261aberrations,spherical, 261classical focus, 259experiment, 255quantum focal curve, 259quantum focus, 258theory, 256

initial conditions and solution, 257thin versus thick lenses, 258

Baker-Campbell-Hausdorff relation,364

Bell inequalities, 341Bell state, 344Bell states, 340

entangled states, 334BHSH inequality, 343, 344birth and death process, 377blackbody energy, 4Bloch equations, 16, 17Bloch Sphere

qubit, 331

Boltzmann distribution, 3Bose-Einstein distribution, 4bound entangled state, 341boundary condition

input-output theory, 194broad-band spectrum, 14

C-NOT Gate2 qubit gate, 331

C-NOT gate, 352, 359, 360CEL, 176

holographic laser, 181two photon laser, 181

Chapman-Kolmogorov equation, 375forward, 376

characteristic function, 76normally ordered, 78

circuit, 351coherence

first order, 62n-th order, 62second order effects, 62second order, classical, 63second order, quantum mechanical,

65coherence function

first order, 54n-th order, 54

coherent squeezed state, 43coherent state, 31

coordinate representation, 35displacement operator, 33minimum uncertainty states, 32non-orthogonality, 32normalization, 31overcompleteness, 33photon statistics, 34

coincidence rate

410 Index

n-th fold, 58collapse, 91

time, 91collective dephasing, 320commutation relation

between the electric field and thevector potential, 27

Dirac´s commutator, 210Loisell´s trigonometric functions, 204number-phase, 203Susskind-Glogower phase, 206

commutation relations, 23, 24, 26, 27Condition for DFS, 323conditional probability, 374continuous measurements, 278

phase narrowing, 280, 283control, 352correlation function

first order, 60, 62n-th order, 59second order, 62, 65second order, examples, 65

correlationsdynamics of, 314

creation operator, 24

data Hilbert space, 357decimation, 277

in vibrational states, 305decoherence

how long it takes?, 316in phase sensitive reservoirs, 319

Decoherence Free Subspace, 320, 322,325

Decoherence Free Subspaces, 320decoherence time, 319degenerate parametric oscillator

input-output theory, 195quadrature fluctuations-input output

theory, 197density matrix

multimode thermal state, 37time evolution of elements, 102

density of modes, 26density operator

in P-representation, 105modified with continuous measure-

ments, 280thermal state, 36

dephasing, 320, 321detailed balance, 382detector

ideal, 53n atom, 57one atom, 54

DFS, 323, 324, 326, 327dipole approximation, 85distance between two operators, 350distance btween operators, 350dressed states, 88, 89drift and diffusion coefficients, 383duplicator, 354

electric fieldper photon, 86positive frequency component, 60

emissionspontaneous, 1, 2stimulated, 1, 2

energymultimode radiation field, 23

energy density, 2entangled, 339entangled state, 333, 357entangled states, 334, 335entanglement, 337, 340, 344entanglement between copies, 356entanglement measurement, 337ergodic hypothesis, 69events, 374

Fermi golden rule, 188generalized, 189

Fidelity, 355fidelity, 350, 354, 356Fock state, 25Fock states, 24Fokker-Planck equation, 373, 379

general properties, 381of the damped harmonic oscillator,

105several dimensions, 379steady state solution, 381time dependence-damped harmonic

oscillator, 106free radiation field, 83

gageCoulomb, 84

Index 411

General single qubit transformation,331

Generalized Einstein´s relations, 160generalized Toffoli gate, 361generating function, 68

Hadamard gate, 329Hadamard Transform, 330hidden variables, 341

input fielddefinition, 193

input-output theory, 191Ito versus Stratonovich, 384Ito´s Fokker Planck equation, 387Ito´s formula, 387

Jaynes- Cummings model, 149Jaynes-Cummings Hamiltonian, 87, 91,

135, 278joint probability, 374

Kraus operators, 323

Lamb-Dicke expansion, 299Lamb-Dicke parameter, 298Langevin equatiins, 108Langevin equation

of the damped harmonic oscillator,109

laser theory, 4, 5rate equations, 6stability analysis, 8steady-state, 7threshold condition, 6

laser theory-quantum mechanicaladiabatic approximation, 166atomic noise correlation, 163c-number Langevin equations, 165Fokker Planck equation, 143general Master equation, 138injection statistics-heuristic discus-

sion, 136Langevin equations, 163noise reduction, 156noise supression via pump statistics,

170phase and intensity fluctuations, 168phase diffusion, 143

photon statistics, 140Lindblad form, 324Lindblad operators, 324, 325Liouville´s equation, 98local realism, 342locality, 341locality and reality, 341

Markoffian assumption, 100Markov approximation, 109Master equation, 99, 219, 377

damped harmonic oscillator, 100damped oscillator in squeezed bath,

111generalized with pump statistics, 138Lindblad form, 219two-level atom in thermal bath, 110

Maxwell´s equations, 21Maxwellian velocity distribution, 274measurements, 263

in a dynamical sense, 314Von Neumann, 311

measuring the atomic phase, 272method of characteristics, 367micromaser, 136

cooperative effects, 152Master equation, 148noise reduction, 156operation, 147photon statistics, 149squeezing in trapping states, 152tangent and cotangent trapping

states, 151trapping condition, 150trapping states, 149

mixed state, 35, 340density operator for thermal state, 36thermal distribution, 36

mixed states, 339momentum distribution

atomic diffraction, 252multimode squeezed state, 47multiple photon transitions, 185

no cloning theorem, 349, 350noise operator

damped harmonic oscillator, 110non commuting operators, 363non-classical vibrational states, 305

412 Index

non-Hermitian Hamiltonian, 220non-linear Langevin equation, 388

one atom laserRaman, 222

Hamiltonian, 222one qubit stochastic processor, 359optical nutation, 18Optical Parametric Amplifier, 338Ornstein-Uhlenbeck process, 384output field

definition, 193

P-representation, 73, 76averages of normally ordered

products, 77coherent state, 79examples, 78Fourier transform, 77normalization, 76number state, 79

parametric amplification, 189parametric amplifier

degenerate, 190non-degenerate, 190quadrature fluctuations-ideal case,

191Partial transposition criterion, 340Paul trap, 287

boundary between stable andunstable solutions, 293

equations of motion, 288Mathieu´s equation, 291oscillation frequencies, 289

Pauli matrices, 86, 334Peres Horodecki criteria, 340phase fluctuations

Pegg and Barnettcoherent state, 213Fock state, 212laser, 215

Phase GateSingle qubit gate, 330

photoelectric emissionfluctuations, 69

photoemission, 68photon

antibunching, 66bunching, 66

photon count distribution, 71chaotic state, 72coherent state, 72

photon statisticsthermal state, 36

photons, 1plane waves, 22Poisson distribution

of incoming atoms, 64polarized photon, 329population inversion, 4potential condition, 382program Hilbert space, 357Pual trap

stability analysis, 290

Q-representation, 73anti-normally ordered products, 74density operator in terms of, 75examples, 74normalization, 73

QND measurement of photon number,275

QND measurementsof vibrational states, 303

quadrature operators, 41quadrature squeezing in degenerate

parametric oscillatorexperimental evidence, 198

quantum and classical foci, 258quantum beat laser, 175

Master equation, 179photon statistics, 179rotating wave approximation, 176

quantum bitqubit, 329

quantum circuit, 351quantum correlations, 340quantum efficiency, 50, 68quantum gate arrays, 357quantum gates, 357quantum jump, 220quantum jumps, 219quantum logic gate, 329Quantum Logic Gates, 330Quantum Network

Quantum Gates, 330quantum non-demolition measurement

condition, 268

Index 413

in cavity QED, 270quantum non-demolition measure-

ments, 266experiments, 267in cavity QED, 267

quantum non-demolition observable,268

quantum non.demolition measurementeffect of measuring apparatus, 269

quantum phaseDirac, 203Louisell, 204Pegg and Barnett

coherent states, 212Fock states, 211hermitian phase operator, 208orthonormal states, 208phase state, 208properties, 211

Susskind-Glogower, 203, 204expectation values, 207

quantum processor, 357quantum stochastic processor, 357Quantum teleportation, 344quantum teleportation, 344

entanglement, 329Quantum Teleportation Protocol, 347quantum trajectories, 220qubit, 329

Rabi oscillations, 16quantum mechanical, 90

Raman cooling, 299, 300Raman effect, 189Raman Nath approximation

atomic diffraction, 253Ramsey fields, 272random walk

one sided, 378rate equation, 3rate equations, 6reality, 342revival, 93rotating-wave approximation, 87rotation of a qubit, 359Rydberg atom, 146

Schawlow-Townes linewidth, 169Schimdt form, 339

Schmidt decomposition, 335, 338Schmidt form, 336Schmidt number, 337second harmonic generation

early experiment, 185separability, 339separable, 339, 340spectrum, 14spontaneous decay

in squeezed bath, 114squeeze operator, 43squeezed state

balanced detection, 48balanced homodyne detection, 50calculation of moments, 44direct detection, 47heterodyne detection, 48, 50ordinary homodyne detection, 47, 48photon statistics, 46pictorial representation, 44quadrature fluctuations, 45second order correlation function, 66squeeze operator, 43

squeezed vacuum state, 338squeezing

degenerate parametric oscillator, 197double ended cavity, 197single ended cavity, 197

squeezing spectrumdegenerate parametric oscillator, 197

standard quantum limit, 264harmonic oscillator, 264thermal effects, 265

stochastic differential equations, 382stochastic equation

instability point, 388stochastic processes, 375

independence, 375Stochastic Schrodinger equation, 220,

224Stratonovich ´s Fokker-Planck

equation, 388subPoissonian, 66superPoissonian, 66system-apparatus coupling, 315

target, 352the no cloning theorem, 349thermal energy, 3

414 Index

Toffoli Gate

3 qubit gate, 333

Toffoli gate, 360

transition probability, 57

transverse momentum change

atom optics, 250

trapped ion, 329

trapped ions, 294

effective Hamiltonian, 296

observation of non-classical vibra-tional states, 294

theory, 294

triplicator, 355

two mode squeeze operator, 47

Two mode squeezed states, 338

two partite entanglement, 334

two-level atom, 14, 85

Two-mode squeezed statesentangled states, 338

Universal Quantum Copying Machine,350, 351

Van Kampen´s expansion, 388vector potential, 23visibility, 62

white noise, 383Wiener increments, 380Wiener Process, 379, 385Wiener process, 223Wigner representation, 73, 79

moments, 80Wigner-Weisskopf theory for sponta-

neous emission, 111

A. Operator Relations

Documents

Transcript of A. Operator Relations