A. Operator Relations

Home , Displacement operator, Squeeze operator

A.1 Theorem 1

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

α2 exp(αA)B exp(−αA)=B + α [A, B]+ [A, [A, B]] + .... (A.1) 2! Proof. Let f1(α)=exp(αA)B exp(−αA) , (A.2)

then, one can expand f1 in Taylor series about the origin. We ﬁrst evaluate the derivatives − − f1(α)=exp(αA)(AB BA)exp( αA) , so f1(0) = [A, B] . (A.3) Similarly − − f1 (α)=exp(αA)(A [A, B] [A, B] A)exp( αA) , so that f1 (0) = [A, [A, B]] . (A.4) Now, we write the Taylor’s expansion α2 f (α)=f (0) + αf (0) + f (0) + ... (A.5) 1 1 1 2! 1 or α2 exp(αA)B exp(−αA)=B + α [A, B]+ [A, [A, B]] + .. (A.6) 2! A particular case is when [A, B]=c,wherec 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–Haussdorf Relation

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

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

then α2 exp [α(A + B)] = exp αA exp αB exp − [A, B] (A.9) 2 α2 =expαB exp αA exp [A, B] . 2

Proof. Deﬁne f2(α) ≡ exp αA exp αB . (A.10) Then df (α) 2 =[A +exp(αA)B exp −(αA)]f (α) (A.11) dα 2 =(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

df (α) 2 =exp(αA)A exp αB +exp(αA)exp(αB)B (A.12) dα =expαA exp αB [exp(−αB)A exp αB + B]

= f2(α)(A + B + α [A, B]) .

By comparing (A.11) and (A.12), we can see that f2(α)commuteswith (A+B +α [A, B]), thus one can integrate as a c-number diﬀerential equation, getting α2 α2 f (α)=exp (A + B)α + [A, B] =expα(A+B)exp [A, B] , (A.13) 2 2 2

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 α2 exp(αn)a exp(−αn)=a − αa + a + ... =exp(−α)a. (A.16) 2 Similarly exp(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, the Theorem 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∗) . Also exp −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 da†l a, a†l = la†l−1 = , (A.22) da† dal a†,al = −lal−1 = − . 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† ∂f(a, a†) a, f(a, a†) = , (A.23) ∂a† ∂f(a, a†) a†,f(a, a†) = − . (A.24) ∂a 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

∂z ∂z p ≡ ,q ≡ , (B.2) ∂x ∂y

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 dx dy dz = = . (B.6) P Q R Proof. If (B.5) are solutions of (B.6), then the equations

∂u ∂u ∂u dx + dy + dz =0, (B.7) ∂x ∂y ∂z and dx dy dz = = , P Q R must be compatible; thus, we must have 368 B. The Method of Characteristics

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

and similarly for v Pvx + 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

∂z u + u =0, (B.10) x z ∂x ∂z u + u =0, y z ∂y

and substituting (B.10) into (B.8) we get ∂z ∂z ∂u −P − Q + R =0, ∂x ∂y ∂z

and (B.1) is satisfied. The second part of the proof is to show that the general solution of (B.1) is F (u, v)=0. (B.11) From (B.11), one writes ∂F ∂F ∂u ∂u ∂z ∂F ∂v ∂v ∂z = + + + =0, (B.12) ∂x ∂u ∂x ∂z ∂x ∂v ∂x ∂z ∂x ∂F ∂F ∂u ∂u ∂z ∂F ∂v ∂v ∂z = + + + =0. (B.13) ∂y ∂u ∂y ∂z ∂y ∂v ∂y ∂z ∂y We finally notice that (B.13) is satisfied considering (B.10). Example. Find the general solution of the equation ∂z ∂z x2 + y2 =(x + y)z. (B.14) ∂x ∂y In this case

P = x2 , (B.15) Q = y2 , R =(x + y)z,

and we have to ﬁnd the solution of dx dy dz = = . (B.16) x2 y2 (x + y)z Reference 369

Integrating, ﬁrst dx dy = , x2 y2 we get −1 −1 x + y = c1 . (B.17) On the other hand,

x2 − dx − dy ( 2 1)dy dy dz = y = = , x2 − y2 x2 − y2 y2 (x + y)z

from where we get x − y = c = v. (B.18) z 2 Combining (B.17) and (B.18), we get xy = c = u, (B.19) z 1 so the general solution can be put as xy x − y F , =0, (B.20) z z

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

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

then the solution is xy x − y = g . (B.22) z z

Reference

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

In this Appendix, we show the equation ⎡ ⎤ ⎣ − − − 2⎦ 2 − − δ(t tj)δ(t tk) S R ρaa = pRδ(t t ) . (C.1) j= k

For regular pumping, one can put tj = t0 + jτ, where τ is the constant time interval between the atoms and t0 some arbitrary time origin [1]. In this case, there are no pumping ﬂuctuations, and therefore, there are no correlations between the products of delta functions, that is δ(t − tj)δ(t − tk)S = δ(t − tj)S δ(t − tk)S (C.2) j,k j k = R2 . Now, we split the l.h.s. of the above equation in two parts 2 δ(t − tj)δ(t − tk)S + δ(t − tj)δ(t − tk)S = R , (C.3)

j=k j=k 2 δ(t − tj)δ(t − tk)S + Rδ(t − t )=R , j= k thus proving the relation ⎡ ⎤ ⎣ − − − 2⎦ 2 − − δ(t tj)δ(t tk) S R ρaa = pRδ(t t ) j= k for p =1. In the Poissonian case, tj is totally uncorrelated from tk(j = k),so 2 δ(t − tj)δ(t − tk)S = δ(t − tj)S δ(t − tk)S = R , (C.4) j= k j k 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 being very 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 quantum optical 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 of a 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 described by the time evolution of probability distributions. To show these ideas with an example, we take a look at the random walk in one dimension, by now, a classical problem [3]. A person moves in a line, taking random steps forward or backward, with equal probability, at fixed time intervals τ. Calling the position xn = na, then the probability that it occupies the site xn at time t is P (xn | t) and obeys the equation 1 1 P (x | t + τ)= P (x − | t)+ P (x | t) . (D.1) n 2 n 1 2 n+1 Now, we go to the continuum limit, letting τ and a become small, but a2 with finite τ . Then ∂ P (x | t + τ)=P (x | t)+τ P (x | t)+... (D.2) ∂t ∂ P (x ± | t)=P (x ± a | t)=P (x | t) ± a P (x | t) n 1 ∂x a2 ∂2 + P (x | t)+... , 2 ∂x2 and inserting the above expansions in (D.1), we get ∂ a2 ∂2 τ P (x | t)+O(τ 2)= P (x | t)+O(a4)+... (D.3) ∂t 2 ∂x2 Now, letting τ,a → 0with a2 D ≡ , (D.4) τ D being the diffusion coefficient, we get a diffusion or Fokker–Planck Equation: ∂ D ∂2 P (x | t)= P (x | t) . (D.5) ∂t 2 ∂x2 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) then P (∪iAi)= P (Ai) . (D.8) i 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)= , (D.10) P (B) 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 and B 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 P (A ∩ Bi)=P (∪i(A ∩ Bi)) = P (A) , (D.13) i thus P (A, Bi)= P (A | Bi)P (Bi)=P (A) , (D.14) i i or, put it in words, if we sum the joint probability over the mutually exclusive events Bi, it eliminates that variable. These ideas will be useful later 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 values x1,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 stochastic process. 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 conditional probability is entirely determined by the knowledge of the most recent condition, 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 joint probability can be written as

n−1 | P (x1,t1; x2,t2; ...xn,tn)= i=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 variables, eliminates that variable, in other words P (A ∩ B ∩ C...)=P (A ∩ C...) . (D.19) B Now, we apply this idea to a stochastic process 376 D. Stochastic Processes in a Nutshell

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–Kolmogorov Equation

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 ,ands is an intermediate time t0 ≤ s ≤ t,andx(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)

The transition probability can be split into two parts, one that does not change 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) = dyW1(x | y)P (y,t | x0,t0) (D.29) ∂t

− 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(x ,t) i = [W P (x ,t) − W P (x ,t)] . (D.30) ∂t ij j ji i j

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 378 D. Stochastic Processes in a Nutshell . 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 = Pn(t)expins , (D.36) n

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

. exp(ins) Pn= q [Pn−1 exp(ins) − Pn exp(ins)] n n or ∂G(s, t) = q(exp(is) − 1)G(s, t) . (D.37) ∂t It is simple to verify that the solution of (D.37) is

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

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

(tq)n P (t)=exp(−tq) , (D.39) n n! which is a Poisson distribution with n = tq.

D.4 The Fokker–Planck Equation

Sometimes, instead of discrete jumps, one chooses to describe the random process as a continuous one. If we take, for example, in the Chapman–Kolmogorov equation [3]:

Φ(w | x) ≡ W (x + w | x) , (D.40)

∞ n n ∂P(x, t | x0,t (−1) ∂ 0) = Q (x)P (x, t | x t , (D.42) ∂t n! ∂xn n 0, 0) n=1 with n Qn(x)= w Φ(w | x)dw. (D.43)

Many times, the above equation is truncated, keeping only the ﬁrst two terms, getting the Fokker–Planck Equation. In one dimension, with Q1 = A, Q2 = B,weget

∂P(x, t | x t ) ∂ 0, 0 = − [A(x, t)P (x, t | x t )] (D.44) ∂t ∂x 0, 0 1 ∂2 + [B(x, t)P (x, t | x t )] . 2 ∂x2 0, 0 A simple generalization to more variables leads to the Fokker Planck equation ∂P(x,t| x t ) ∂ 0, 0 = − [A (x,t)P (x,t | x t )] (D.45) ∂t ∂x i 0, 0 i i 1 ∂2 + [B (x,t)P (x,t | x t )] , 2 ∂x ∂x ij 0, 0 i,j i j where A is the drift vector and B the diﬀusion matrix. This equation can also be written as ∂P(x,t| x t ) ∂ 0, 0 + J (x,t)=0, (D.46) ∂t ∂x i i i

Ji(x,t)=[Ai(x,t)P (x,t | x0,t0)] (D.47) 1 ∂ − [B (x,t)P (x,t | x t )] . 2 ∂x ij 0, 0 j j

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 | w t ) 1 ∂2 0, 0 = [P (w, t | w t )] . (D.48) ∂t 2 ∂w2 0, 0 380 D. Stochastic Processes in a Nutshell

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 = −s2φ. (D.50) ∂t | | − 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 1 φ(s, t)=exp isw − s2(t − t ) , (D.51) 0 2 0

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

− 2 | 1 −(w w0) P (w, t w0,t0)= exp − . (D.52) 2π(t − t0) 2(t t0)

The ﬁrst two moments are

W = w0 , (D.53) 2 (ΔW ) = t − t0 .

This distribution spreads in time and corresponds precisely to Einstein’s model for Brownian motion. An important characteristic of Wiener’s process is the independence of the increments, which is interesting for stochastic integration purposes. We saw that, in general, for Markov Processes, one has

n−1 | P (wn,tn; wn−1,tn−1; ...w0,t0)= i=0 P (wi+1,ti+1 wi,ti)P (w0,t0) 2 − − 1 (w − w ) n 1 − 2 − i+1 i = i=0 [2π(ti+1 ti)] exp P (w0,t0) . (D.54) 2(ti+1 − ti) 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) 2 − 1 (Δw ) n 2 − i = i=1 [2πΔti] exp P (w0,t0) , (D.56) 2(Δti) 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)

T | T W(t)W(t0) W0,t0 = dwdw0P (w,t; w0,t0)ww0 | T = dw0 W(t) W0,t0 w0 P (w0,t0) .

For the Wiener process

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

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

| 2 − − W (t)W (s) W0,t0 = w0 +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| x t ) ∂ 0, 0 = − [A (x,t)P (x,t| x t )] ∂t ∂x i 0, 0 i i 1 ∂2 + [B (x,t)P (x,t| x t )] . (D.60) 2 ∂x ∂x ij 0, 0 i,j i j

As we mentioned before, the first term in the r.h.s is the drift term, which will rule the deterministic motion, and the second one is the diffusion term, which will cause the probability to broaden. This different role of the two terms can be easily seen if we calculate xi and xixj. One can easily show that dx i = A , (D.61) dt i dx x 1 i j = x A + x A + B + B . dt i j j i 2 ij ji

D.4.3 Steady-State Solution

Very often in optics and other areas of physics, one is not really interested in the time-dependent solution of the Fokker–Planck equation , but rather in the steady state. Thus, we set the time derivative to zero and get 382 D. Stochastic Processes in a Nutshell ⎡ ⎤ ∂ 1 ∂ ⎣−A (x,t)P (x,t| x t )+ [B (x,t)P (x,t| x t )]⎦ =0, ∂x i 0, 0 2 ∂x ij 0, 0 i i j j (D.62) and if the constant current is set to zero (detailed balance) , one gets

and deﬁning a Potential function V (x) by P (x,t| x0,t)=N exp(−V (x)), we get for V −∂V (x) −1 − −1 ∂Bjk =2 Bij Aj Bij . (D.64) ∂xi ∂xk j j,k Integrating (D.64), we get for the probability distribution ⎡ ⎤ ⎣ −1 − −1 ∂Bjk ⎦ PSS(x)=N exp 2Bij Aj(x)dxi Bij dxi . (D.65) ∂xk i,j i,j,k

In particular, for Bij = Dδij ,weget 2 P (x)=N exp A(x)dx (D.66) SS D

D.5 Stochastic Diﬀerential Equations

D.5.1 Introduction

One way of treating the motion of a Brownian particle, or any other problem with 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 ﬂuid over 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 that accounts for the collisions of the water molecules with the Brownian particle, 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 delta function is a constant, L(t) has a ﬂat spectrum or it correspond to white noise. Let us assume that the initial velocity of the Brownian particle is deterministic and given by V (0) = V0,thenfort>0, for each sample path t V (t)=V0 exp(−γt)+exp(−γt) exp(γt )L(t )dt . (D.70) 0 Using the properties of L, we can calculate V and V 2

V (t) | V0,t0 = V0 exp(−γt) , (D.71) V 2(t) | V ,t = V 2 exp(−2γt)+ 0 0 0 t t´ exp(−2γt) dt dt exp γ(t + t)L(t)L(t) 0 0 D V 2(t) | V ,t = V 2 exp(−2γt)+ [1 − exp(−2γt)] . (D.72) 0 0 0 2γ When t →∞ D V 2(t) | V ,t = , (D.73) 0 0 2γ On the other hand, for short times

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

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

ΔV (V − V ) | A = = 0 t=t0+Δt = −γV , (D.75) Δt Δt 384 D. Stochastic Processes in a Nutshell

(ΔV )2 B = = D, (D.76) Δt so that the corresponding Fokker–Planck equation is

∂P(V,t) ∂ D ∂2P = γ (VP)+ . (D.77) ∂t ∂V 2 ∂V 2 The above equation describes the so-called Ornstein–Uhlenbeck process, corresponding to a linear drift and a constant diﬀusion term. We now calculate the power spectrum of V . So we ﬁrst need the two time correlation function

2 − V (t)V (t ) = V0 exp [ γ(t + t )] t t +exp[−γ(t + t)] dt dt exp [γ(t + t)] L(t)L(t) 0 0

t 2 − − = V0 exp [ γ(t + t )] + exp [ γ(t + t )] D dt exp 2γt 0 Γ V (t)V (t) = V 2 exp [−γ(t + t)] + exp [−γ(t + t)] [exp(2γt) − 1] . 0 2γ (D.78) In steady state, for t, t →∞but with t − t = τ,weget D V (t + τ)V (t) = exp(−γ | τ |) . (D.79) 2γ Finally, taking the Fourier transform, we get the power spectrum of V 1 φ (ω)= exp(iωτ)V (t + τ)V (t)dτ (D.80) V 2π 1 D = . 2π ω2 + γ2

D.5.2 Ito Versus Stratonovich

A more general type of Langevin equation can be written as dx = a(x, t)+b(x, t)L(t) , (D.81) dt where the previous D factor can be absorbed in b,sothat

L(t)L(t) = δ(t − t) , (D.82) L =0.

Now we deﬁne D.5 Stochastic Diﬀerential Equations 385 t W (t)= L(t)dt , (D.83) 0 assumed to be continuous, so that t+Δt W (t +Δt) − W0(t) | W0,t = dsL(s) =0, (D.84) t

[W (t +Δt) − W (t)]2 | W ,t = (D.85) 0 0 t+Δt t+Δt t+Δt t+Δt ds1 ds2L(s1)L(s2) = ds1 ds2δ(s1 − s2)=Δt, t t t t 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 Wiener increment. The stochastic differential equation (D.81) is not fully defined unless one specifies how to integrate it. Normally, this would not be a problem, and the rules of ordinary calculus apply. However, here we must be careful because we are dealing with a rapidly varying function of time L(t). Thus, we define the integral the mean square limit of a Riemann–Stieltjes sum t n f(t )dW (t )=ms lim f(τi)[W (ti) − W (ti−1)] , (D.86) n→∞ t0 i=1 ≤ → where ti−1≤τi ti, and we have divided the time interval from t0 t in n 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. f(ti)+f(ti−1) b) Stratonovich: f(τi)= 2 . From the above assumptions, one learns how to calculate things with Ito and Stratonovich. For Stratonovich, we have for example t S W (t)dW (t) t0 n W (ti)+W (ti−1) = ms lim [W (ti) − W (ti−1)] n→∞ 2 i=1 n 1 2 2 1 2 2 = ms lim W (ti) − W (ti−1) = W (t) − W (t0) , 2 n→∞ 2 i=1 386 D. Stochastic Processes in a Nutshell

which obeys the rules of ordinary calculus. On the other hand, for Ito t I W (t)dW (t) t0

n = ms lim [W (ti−1)] [W (ti) − W (ti−1)] n→∞ i=1 n = ms lim [W (ti−1)ΔW (ti)] n→∞ i=1 n ' ( 1 2 2 2 = ms lim [W (ti−1)+ΔW (ti)] − W (ti−1) − ΔW (ti) n→∞ 2 i=1 n 1 2 2 1 2 = W (t) − W (t0) − ms lim ΔW (ti) , 2 n→∞ 2 i=1 and because n 1 2 ms lim ΔW (ti) = t − t0 , n→∞ 2 i=1 we ﬁnally get t 1 2 2 I W (t )dW (t )= W (t) − W (t0) − (t − t0) . (D.87) t0 2 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’s book [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)] 1 = f [x(t)] dx + f [x(t)] dx2 + ... 2 D.5 Stochastic Diﬀerential Equations 387

= f [x(t)] [a(x, t)dt + b(x, t)dW ] 1 + f [x(t)] b(x, t)dW 2 + ... 2 and using (D.88), we get 1 df [x(t)] = a(x, t)f [x(t)] + b(x, t)f [x(t)] dt 2

+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 df(x) = dx∂tP (x, t)f(x) dt b = dx a∂ f + ∂2f P (x, t) , x 2 x

and integrating by parts and discarding the surface terms, we get 1 dxf(x)∂ P (x, t)= dxf(x) −∂ aP + ∂2bP , t x 2 x

thus getting the Ito–Fokker–Planck equation

∂tP (x, t | x0,t0)=−∂x [a(x, t)P (x, t | x0,t0)] (D.90) 1 + ∂2 [b(x, t)P (x, t | x ,t )] (D.91) 2 x 0 0 Similarly, for many variables, if one has an Ito stochastic diﬀerential equation (I)dx = a(x,t)dt + b(x,t)dW , (D.92) where dW is an n-component Wiener process, then the corresponding Ito’s Fokker–Planck equation is: ∂tP (x,t| x0,t0)=− ∂i [ai(x,t)P (x,t | x0,t0)] (D.93) i 1 T + ∂i∂j bb (x,t) P (x,t| x0,t0) . (D.94) 2 ij i,j Thus, from our previous notation

B = bbT (D.95)

Similarly, for Stratonovich 388 D. Stochastic Processes in a Nutshell

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

we get a Stratonovich–Fokker–Planck equation | − S | ∂tP (x,t x0,t0)= ∂i ai (x,t)P (x,t x0,t0) i 1 + ∂ bS ∂ bST (x,t) P (x,t| x ,t ) . (D.97) 2 i ik j jk 0 0 i,j,k

By simple comparison between the two Fokker–Planck equations, we get 1 aS = a − b ∂ bT , (D.98) i i 2 kj k ij j,k S bik = bik .

This last relation tells us that if we have a given Fokker–Planck equation, it corresponds to a Langevin equation to be integrated a la Ito, with a and b drift and diffusion coefficients, and to a Langevin equation to be integrated 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 Kamp en [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 deﬁne X x = . (D.99) Ω The key point in Van Kampen’s expansion is that we can separate [1]: √ x(t)=x0(t)+ y(t) , (D.100)

where x0(t) is the deterministic part, y(t) represents the ﬂuctuations and 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 is the noise strength, so we write √ dx = a(x)dt + dW (t) , (D.101) D.6 Approximate Methods 389

and √ x(t)=x0(t)+ x1(t)+ x2(t)+... (D.102)

Diﬀerentiating x and expanding a(x) around x0,weget √ dx0(t)+ dx1(t)+ dx2(t)+... 1 √ = a(x )dt + a(x )(x − x )dt + a(x )(x − x )2dt + ... + dW (t) 0 0 0 2 0 0 √ = a(x0)dt + a (x0) x1(t)+ x2(t)+... dt + 1 √ a(x ) x2 + ... dt ++ dW (t) , 2 0 1 and by comparing diﬀerent orders of we get

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

dx1(t)=a (x0)x1(t)dt +dW (t) , (D.104) 1 dx (t)=a(x )x (t)dt + a(x )x2dt, (D.105) 2 0 2 2 0 1 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 dV a(x)=− , (D.107) dx with γ g V = − x2 + x4 . (D.108) 2 4 The stochastic diﬀerential equation, in this case is √ dx(t)=(γx − gx3)dt + dW (t) . (D.109)

The shape of the potential is described in the Fig. D.1 We will consider the case γ>0. Near the equilibrium positions, the drift is practically zero, and the noise term in the stochastic equation is quite important. On the other hand, very far from the equilibrium positions, the motion is dominated by a large drift and is practically a deterministic one. ± γ For γ>0,x= 0 is an unstable equilibrium position and x = g are stable ones. 390 D. Stochastic Processes in a Nutshell

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

Applying our method in this example, we get

− 3 dx0 =(γx0 gx0)dt, x0(0) = h, (D.110)

2 dx1 =(γ − 3gx )x1dt +dW, x1(0) = 0 , (D.111) 0 − 2 − 2 dx2 = (γ 3gx0)x2 3gx0x1 dt, x2(0) = 0 , (D.112) The solution for the deterministic motion is h exp(γt) x0(t)= , (D.113) g 2 − 1+ γ h [exp(2γt) 1]

so that if we choose the unstable equilibrium point, that is the initial condition h =0,thewegetx0(t) = 0, and for the stable equilibrium points, ± γ ± γ h = g ,wegetx0(t)= g , 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)= exp [γ(t − t )] dW (t ) , (D.116) o

x2(t)=0. ± γ In the stable case x0 = h = g ,weget References 391

dx (t)=−2γx dt +dW (t), (D.117) 1 1 γ dx (t)= −2γx ∓ 3g x2 dt, 2 2 g 1

and the solutions are t x1(t)= dW (t )exp[−2γ(t − t )] (D.118) 0 t ∓ γ 2 − − x2(t)= 3g dt x1(t )exp[ 2γ(t t )] . g 0

Now, we notice that in all cases x1 =0,so 2 2 2 x (t) = x0 + x1 +2x0 x2 + ... (D.119)

and in the two cases, we can write

x2(t) = [exp(2γt) − 1] , (D.120) uns 2γ γ 3 x2(t) = + [1 − exp(−4γt)] − [1 − exp(−2γt)]2 + ... stable g 4γ 4γ

→∞ 2 ∞ 2 → γ − In the limit t , x ( ) uns diverges, while x (t) stable g 2γ . As we can see, the perturbative expansion gives the correct answer when starting from stable equilibrium points but it diverges when starting from an unstable equilibrium. In this last case, the perturbative expansion is no longer 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. World Scientific, 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 Schrödinger Differential Equation

Here we present the detailed derivation of the Homodyne Schr¨odinger differential equation. We start from the expansion given by (16.36), which in the two-jump situation, and neglecting the commutators between the jump operators and the no-jump evolution, can be expressed as

∞ m1+m2 (Δt) m2 m1 ρ(Δt)= S(Δt)J2 J1 ρ(0) . (E.1) m1!m2! m1,m2=0

The probability of m1and m2 quantum jumps of the respective types is given by

m1+m2 (Δt) m2 m1 Pm1,m2 (Δt)= Tr[S(Δt)J2 J1 ρ(0)] . (E.2) m1!m2! The master equation of the ﬁeld, corresponding to a lossy cavity at temper- ature T , may be written as

dρ γ =(J + J )ρ − ρ[a†a(1 + 2n )+2ε(1 + n )a† dt 1 2 2 th th 2 +2εntha + nth + ε (1 + 2nth)] γ − [a†a(1 + 2n )+2ε(1 + n )a +2εn a† 2 th th th 2 +nth + ε (1 + 2nth)]ρ (E.3)

Therefore, according to the discussion given in Chap. 16, one possible way of writing S(Δt)is S(Δt)ρ = N(Δt)ρN †(Δt) , (E.4) with γ(Δt) N(Δt)=exp − [a†a(1 + 2n )+2ε(1 + n )a† 2 th th 2 +2εntha + nth + ε (1 + 2nth)] . (E.5) 394 E. Derivation of the Homodyne Stochastic Schr¨odinger Diﬀerential Equation

Using (E.2) and (E.5), we can write m1 m2 exp(μ1)(μ1) exp(μ2)(μ2) Pm1,m2 (Δt)= m1! m2! † m2 † m1 a a m1 a Tr[exp(β´) 1+ 1+ ρ 1+ ε ε ε a m2 1+ exp(β†´) ] ( E . 6 ) ε ´with 2 μ1 = γ(Δt)ε (1 + nth) , (E.7) 2 μ2 = γ(Δt)ε nth , γ(Δt) β´= − {a†a(1 + 2n ) 2 th † +2[ε(1 + nth)a + εntha ]+nth} . 3 2 2− 2 1 2 From (E.6), we can calculate mi and σi = mi mi up to order ε . The result is 2 m = μ (1 + X) , i i ε 2 σi = μi . (E.8) Now, we turn to the ﬁnal step of this calculation, which yields the time evolution of the state vector. After repeated jump and no-jump events, the unnormalized wavefunction for the ﬁeld can be written as * | ψf (Δt)=N(Δt − tm)C2N(tm − tm−1)C1N... | ψf (0) or, except for an overall phase factor,

| * m2 m1 ψ f (Δt)=N(Δt)C2 C1 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 γ(Δt) | ψ* (Δt)=exp − a†a(1 + n ) f 2 th † +2[ε(1 + nth)a +2εntha] † m2 a a m1 1+ 1+ | ψ (0) (E.10) ε ε f − 3 or expanding up to ε 2 E Derivation of the Homodyne Stochastic Schr¨odinger Diﬀerential Equation 395 γ(Δt) | ψ* (Δt)= 1 − [a†a(1 + n )+aa†n ] f 2 th th † −γ(Δt)ε[a(1 + nth)+a nth)] 1 × 1+ (m a + m a†) | ψ (0) (E.11) ε 1 2 f We are interested in the limit ε →∞. In deriving (E.11), we considered ε − 3 1 large, γ(Δt) ∼ ε 2 and m1,m2,μ1,μ2 ∼ ε 2 . Now we consider two random numbers with non-zero average m1,m2

σ1 m1 = m1 + √ ΔW1 , Δt

σ2 m2 = m2 + √ ΔW2 , (E.12) Δt which satisfy 2 2 (ΔW1) = (ΔW2) =Δt. (E.13)

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

m1,m2 * * * Δ | ψf (Δt)=| ψf (Δt)−|ψf (0) ' γ γ = − (a†a(1 + n ) − aa†n 2 th 2 th

+2γXa(1 + nth) † † +a nth Δt + a γnthΔW2 ( * +a γ(1 + nth)ΔW1 | ψf (0) , which is the desired result. F. Fluctuations

† 2 † 2 We want to calculate d (Δa a) and Md (Δa a) . √ 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 +2a†aa†aa†a − 2a†aa†aa†a + a†aa†aa†a} −a†aa†aδN + a†aa†aδN a†a†aa†aaa†a−a†a†aaa†a†aa + δN , (F.1) a†aa†a or

d (Δa†a)2 = −γδt(Δa†a)(Δa†a)(Δa†a) −(Δa†a)2δN a†a†aa†aaa†a−a†a†aaa†a†aa + δN (F.2) a†aa†a

Now, we apply the above results to the more interesting case T>0, † † † O = a a; C1 = (nth +1)γa,C2 = γntha ; H = ωa a:

† 2 † † † d (Δa a) = −γ(nth +1)(Δa a))(Δa a)(Δa a)dt † 2 −(Δa a) δN1 (a†aa†aa†aa†a−a†aa†aa†aa†a)δN + 1 a † aa†a † † † † † + γnthdt[−aa aa aa† +2aa aa −aa +2aa†aa†a†a−2aa†a†a−aa†a†aa†a + a†aa†aaa†−a†aa†aaa†] † 2 −(Δa a) δN2 (aa†aa†aa†aa†−aa†aa†aa†aa†)δN + 2 . (F.3) aa†aa†

In the above expression, neither the deterministic nor the stochastic term is deﬁnitely non-increasing. But in the mean, it does decrease 398 F. Fluctuations

d(Δa†a)2 (Δa†a)a†aa†a(Δa†a) M = −γ(n +1) dt th a † a (Δaa†)aa†aa†(Δaa†) − γn ≤ 0 . (F.4) th aa† G. The No-Cloning Theorem [1]

We assume that we have a device able to duplicate an arbitrary quantum state. 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 | ψα | β = φ | φ´α | βα | β .

In the above equation, ψ | ψ =1and0<|<α| β>|< 1, so we conclude that 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

| 0a | Qx →| 0a | 0b | Q0x +(| 0a | 1b+ | 1a | 0b) | Y0x , (H.1)

| 1a | Qx →| 1a | 1b | Q1x +(| 0a | 1b+ | 1a | 0b) | Y1x . (H.2) As there are several free parameters, we can impose some conditions, namely xQi | Qix +2xYi | Yix =1,i=0, 1 ,

xY0 | Y1x =x Y1 | Y0x =0,

xQi | Yix =0,i=0, 1 ,

xQ0 | Q1x =0, (H.3) where the QsandY s are states of the copying machine. (out) With the above assumptions, one can write ρab , describing the modes a and b after the copying of a pure state | ψ = α | 0 + β | 1 as √ (out) 2 | | | | | | ρab = α 00 00 x Q0 Q0 x + 2αβ 00 + x Y1 Q0 x √ + 2αβ | +00 |x Q0 | Y1x 2 | 2 | | | + 2αx Y0 Y0 x +2βx Y1 Y1 x + + √ √ + 2αβ | +11 |x Q1 | Y0x + 2αβ | 11+ |x Y0 | Q1x 2 +β | 1111 |x Q1 | Q1x . (H.4)

Now, if we trace over the b mode, we get the density matrix for the a-mode (out) | | 2 2 | − 2 | ρa = 0 a 0 α + βx Y1 Y1 x αx Y0 Y0 x

+ | 0a1 | αβ (xQ1 | Y0x +x Y1 | Q0x)

+αβ | 1a0 | (xQ0 | Y1x +x Y0 | Q1x) | | 2 2 | − 2 | + 1 a 1 β + αx Y0 Y0 x βx Y1 Y1 x . (H.5) 402 H. The Universal Quantum Cloning Machine

(out) (out) The density operator ρb = ρa , in other words the two output modes are equal, but diﬀerent to the input state. To quantify the diﬀerence, we use the ‘distance’ (22.7), giving as a result

2 4 2 2 2 2 Dα =2x (4α − 4α +1)+2α (1 − α )(e − 1) , (H.6)

with xY0 | Y0x =x Y1 | Y1x ≡ x, (H.7) e Y | Q = Q | Y = Q | Y = Y | Q ≡ , (H.8) x 0 1 x x 0 1 x x 1 0 x x 1 0 x 2 which are the two free parameters, with 0≤ x ≤ 1 , 0 ≤ e ≤ √1 . 2 2 The ﬁrst requirement is that the distance Dα be independent of the input, that is of α. So we impose ∂D α =0, (H.9) ∂(α2) which gives us a relation between the parameters

e =1− 2x,

2 so that Dα becomes input independent Dα =2x . We also require a condition on the two-mode density. The distance between the density operator and its ideal version should be input independent, (out) − (id) 2 that is Dαb = Tr(ρab ρab ) satisﬁes ∂D αb =0. (H.10) ∂(α2)

After some algebra, one gets

2 2 2 Dαb =(f11) +2(f12) +2(f13)

2 2 2 (f22) +2(f23) +(f33) , √ 4 − 2 − 2 − 1 − 2 with f11 = α α√(1 2x),f12 = 2αβ(α 2 (1 2x)),f13 =(αβ) ,f22 = 2 − 2 − 1 − 4 − 2 − 2((αβ) x),f23 = 2αβ(β 2 (1 2x)),f33 = β β (1 2x). 2 Now, the (H.9) can be solved, giving x = 9 . (out) If we write ρa in the basis | ψ = α | 0+ β | 1 and | ψ⊥ = α | 0−β | 1, we readily get

(out) 5 1 ρ = | ψ ψ | + | ψ⊥ ψ⊥ | . a1 6 a1 6 a1

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 A 3.3 See Appendix A 3.4 Follow the text from (3.39) to (3.45) 3.5 Use (3.44) and: ∞ 2 ∂ T ∂ i k3k1 · δ11(ρ)= δ(ρ)+ 3 2 exp(ik ρ)dk ∂z´ ∂z´ (2π) −∞ k ∞ 2 ∂ T i k3k1 · δ13(ρ)= 3 2 exp(ik ρ)dk. ∂x´ (2π) −∞ k

Chapter4

4.1 Deﬁne a† | β = β | β, and follow the same procedure as in (4.2) to (4.6). 4.2 Use αα∗ | α =exp(− )exp(αa†) | 0, 2 and write

| αα |=exp(−αα∗)exp(αa†) | 00 | exp(α∗a). 404 I. Hints to Solve the Problems

4.3 Use (4.6) 4.4 Convert the sums into integrals 4.5 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 + )lF (n)(a†,a), ∂a† ∂ F (n)(a†,a)a†l = N (a† + )lF (n)(a†,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

∂ 1 ∂ 1 exp(−iθ1) ∂ = exp(−iθ1) + , ∂α1 2 ∂ρ1 2i ρ1 ∂θ1

∂ 1 ∂ 1 exp(−iθ2) ∂ = exp(−iθ2) + , ∂α2 2 ∂ρ2 2i ρ2 ∂θ2 ∂ 1 ∂ = +, ∂θ1 2 ∂μ ∂ 1 ∂ ∂ = − , ∂θ2 2 ∂μ ∂θ where θ + θ μ = 1 2 , 2 θ − θ θ = 1 2 . 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. 1 14.4 Part (b). Use the quadratic part of the formula for 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 A 18.4 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 duplicator 22.3 Apply the two gates to the input data and programs. Index

absorption, 1, 2 Boltzmann distribution, 3 annihilation operator, 24 Bose-Einstein distribution, 4 atom -field interaction bound entangled state, 341 semiclassical theory, 11 boundary condition atom optics, 247 input-output theory, 194 diffraction, 248 broad-band spectrum, 14 experiments in diffraction, 249 optical elements, 247 C-NOT Gate sources, 247 2 qubit gate, 331 theory of diffraction, 250 C-NOT gate, 352, 359, 360 atomic decay, 18 CEL, 176 atomic diffraction holographic laser, 181 large detuning, 252 two photon laser, 181 no detuning, 252 Chapman-Kolmogorov equation, 375 atomic focusing, 255 forward, 376 aberrations, chromatic, 261 characteristic function, 76 aberrations,isotopic, 261 normally ordered, 78 aberrations,spherical, 261 circuit, 351 classical focus, 259 coherence experiment, 255 first order, 62 quantum focal curve, 259 n-th order, 62 quantum focus, 258 second order effects, 62 theory, 256 second order, classical, 63 initial conditions and solution, 257 second order, quantum mechanical, thin versus thick lenses, 258 65 coherence function Baker-Campbell-Hausdorff relation, first order, 54 364 n-th order, 54 Bell inequalities, 341 coherent squeezed state, 43 Bell state, 344 coherent state, 31 Bell states, 340 coordinate representation, 35 entangled states, 334 displacement operator, 33 BHSH inequality, 343, 344 minimum uncertainty states, 32 birth and death process, 377 non-orthogonality, 32 blackbody energy, 4 normalization, 31 Bloch equations, 16, 17 overcompleteness, 33 Bloch Sphere photon statistics, 34 qubit, 331 coincidence rate 410 Index

n-th fold, 58 dephasing, 320, 321 collapse, 91 detailed balance, 382 time, 91 detector collective dephasing, 320 ideal, 53 commutation relation natom,57 between the electric field and the one atom, 54 vector potential, 27 DFS, 323, 324, 326, 327 Dirac´s commutator, 210 dipole approximation, 85 Loisell´s trigonometric functions, 204 distance between two operators, 350 number-phase, 203 distance btween operators, 350 Susskind-Glogower phase, 206 dressed states, 88, 89 commutation relations, 23, 24, 26, 27 drift and diffusion coefficients, 383 Condition for DFS, 323 duplicator, 354 conditional probability, 374 continuous measurements, 278 electric field phase narrowing, 280, 283 per photon, 86 control, 352 positive frequency component, 60 correlation function emission first order, 60, 62 spontaneous, 1, 2 n-th order, 59 stimulated,1,2 second order, 62, 65 energy second order, examples, 65 multimode radiation field, 23 correlations energy density, 2 dynamics of, 314 entangled, 339 creation operator, 24 entangled state, 333, 357 entangled states, 334, 335 data Hilbert space, 357 entanglement, 337, 340, 344 decimation, 277 entanglement between copies, 356 in vibrational states, 305 entanglement measurement, 337 decoherence ergodic hypothesis, 69 how long it takes?, 316 events, 374 in phase sensitive reservoirs, 319 Fermi golden rule, 188 Decoherence Free Subspace, 320, 322, generalized, 189 325 Fidelity, 355 Decoherence Free Subspaces, 320 fidelity, 350, 354, 356 decoherence time, 319 Fock state, 25 degenerate parametric oscillator Fock states, 24 input-output theory, 195 Fokker-Planck equation, 373, 379 quadrature fluctuations-input output general properties, 381 theory, 197 of the damped harmonic oscillator, density matrix 105 multimode thermal state, 37 several dimensions, 379 time evolution of elements, 102 steady state solution, 381 density of modes, 26 time dependence-damped harmonic density operator oscillator, 106 in P-representation, 105 free radiation field, 83 modified with continuous measurements, 280 gage thermal state, 36 Coulomb, 84 Index 411

General single qubit transformation, photon statistics, 140 331 Lindblad form, 324 Generalized Einstein´s relations, 160 Lindblad operators, 324, 325 generalized Toffoli gate, 361 Liouville´s equation, 98 generating function, 68 local realism, 342 locality, 341 Hadamard gate, 329 locality and reality, 341 Hadamard Transform, 330 hidden variables, 341 Markoffian assumption, 100 Markov approximation, 109 input field Master equation, 99, 219, 377 definition, 193 damped harmonic oscillator, 100 input-output theory, 191 damped oscillator in squeezed bath, Ito versus Stratonovich, 384 111 Ito´s Fokker Planck equation, 387 generalized with pump statistics, 138 Ito´s formula, 387 Lindblad form, 219 two-level atom in thermal bath, 110 Jaynes- Cummings model, 149 Maxwell´s equations, 21 Jaynes-Cummings Hamiltonian, 87, 91, Maxwellian velocity distribution, 274 135, 278 measurements, 263 joint probability, 374 in a dynamical sense, 314 Von Neumann, 311 Kraus operators, 323 measuring the atomic phase, 272 method of characteristics, 367 Lamb-Dicke expansion, 299 micromaser, 136 Lamb-Dicke parameter, 298 cooperative effects, 152 Langevin equatiins, 108 Master equation, 148 Langevin equation noise reduction, 156 of the damped harmonic oscillator, operation, 147 109 photon statistics, 149 laser theory, 4, 5 squeezing in trapping states, 152 rate equations, 6 tangent and cotangent trapping stability analysis, 8 states, 151 steady-state, 7 trapping condition, 150 threshold condition, 6 trapping states, 149 laser theory-quantum mechanical mixed state, 35, 340 adiabatic approximation, 166 density operator for thermal state, 36 atomic noise correlation, 163 thermal distribution, 36 c-number Langevin equations, 165 mixed states, 339 Fokker Planck equation, 143 momentum distribution general Master equation, 138 atomic diffraction, 252 injection statistics-heuristic discus- multimode squeezed state, 47 sion, 136 multiple photon transitions, 185 Langevin equations, 163 noise reduction, 156 no cloning theorem, 349, 350 noise supression via pump statistics, noise operator 170 damped harmonic oscillator, 110 phase and intensity fluctuations, 168 non commuting operators, 363 phase diffusion, 143 non-classical vibrational states, 305 412 Index non-Hermitian Hamiltonian, 220 photon count distribution, 71 non-linear Langevin equation, 388 chaotic state, 72 coherent state, 72 one atom laser photon statistics Raman, 222 thermal state, 36 Hamiltonian, 222 photons, 1 one qubit stochastic processor, 359 plane waves, 22 optical nutation, 18 Poisson distribution Optical Parametric Amplifier, 338 of incoming atoms, 64 Ornstein-Uhlenbeck process, 384 polarized photon, 329 output field population inversion, 4 definition, 193 potential condition, 382 program Hilbert space, 357 P-representation, 73, 76 Pual trap averages of normally ordered stability analysis, 290 products, 77 coherent state, 79 Q-representation, 73 examples, 78 anti-normally ordered products, 74 Fourier transform, 77 density operator in terms of, 75 normalization, 76 examples, 74 number state, 79 normalization, 73 parametric amplification, 189 QND measurement of photon number, parametric amplifier 275 degenerate, 190 QND measurements non-degenerate, 190 of vibrational states, 303 quadrature fluctuations-ideal case, quadrature operators, 41 191 quadrature squeezing in degenerate Partial transposition criterion, 340 parametric oscillator Paul trap, 287 experimental evidence, 198 boundary between stable and quantum and classical foci, 258 unstable solutions, 293 quantum beat laser, 175 equations of motion, 288 Master equation, 179 Mathieu´s equation, 291 photon statistics, 179 oscillation frequencies, 289 rotating wave approximation, 176 Pauli matrices, 86, 334 quantum bit Peres Horodecki criteria, 340 qubit, 329 phase fluctuations quantum circuit, 351 Pegg and Barnett quantum correlations, 340 coherent state, 213 quantum efficiency, 50, 68 Fock state, 212 quantum gate arrays, 357 laser, 215 quantum gates, 357 Phase Gate quantum jump, 220 Single qubit gate, 330 quantum jumps, 219 photoelectric emission quantum logic gate, 329 fluctuations, 69 Quantum Logic Gates, 330 photoemission, 68 Quantum Network photon Quantum Gates, 330 antibunching, 66 quantum non-demolition measurement bunching, 66 condition, 268 Index 413

in cavity QED, 270 Schmidt decomposition, 335, 338 quantum non-demolition measure- Schmidt form, 336 ments, 266 Schmidt number, 337 experiments, 267 second harmonic generation in cavity QED, 267 early experiment, 185 quantum non-demolition observable, separability, 339 268 separable, 339, 340 quantum non.demolition measurement spectrum, 14 effect of measuring apparatus, 269 spontaneous decay quantum phase in squeezed bath, 114 Dirac, 203 squeeze operator, 43 Louisell, 204 squeezed state Pegg and Barnett balanced detection, 48 coherent states, 212 balanced homodyne detection, 50 Fock states, 211 calculation of moments, 44 hermitian phase operator, 208 direct detection, 47 orthonormal states, 208 heterodyne detection, 48, 50 phase state, 208 ordinary homodyne detection, 47, 48 properties, 211 photon statistics, 46 Susskind-Glogower, 203, 204 pictorial representation, 44 expectation values, 207 quadrature fluctuations, 45 quantum processor, 357 second order correlation function, 66 quantum stochastic processor, 357 squeeze operator, 43 Quantum teleportation, 344 squeezed vacuum state, 338 quantum teleportation, 344 squeezing entanglement, 329 degenerate parametric oscillator, 197 Quantum Teleportation Protocol, 347 double ended cavity, 197 quantum trajectories, 220 single ended cavity, 197 qubit, 329 squeezing spectrum degenerate parametric oscillator, 197 Rabi oscillations, 16 standard quantum limit, 264 quantum mechanical, 90 harmonic oscillator, 264 Raman cooling, 299, 300 thermal effects, 265 Raman effect, 189 stochastic differential equations, 382 RamanNathapproximation stochastic equation atomic diffraction, 253 instability point, 388 Ramsey fields, 272 stochastic processes, 375 random walk independence, 375 one sided, 378 Stochastic Schrodinger equation, 220, rate equation, 3 224 rate equations, 6 Stratonovich ´s Fokker-Planck reality, 342 equation, 388 revival, 93 subPoissonian, 66 rotating-wave approximation, 87 superPoissonian, 66 rotation of a qubit, 359 system-apparatus coupling, 315 Rydberg atom, 146 target, 352 Schawlow-Townes linewidth, 169 the no cloning theorem, 349 Schimdt form, 339 thermal energy, 3 414 Index

Toffoli Gate Two-mode squeezed states 3 qubit gate, 333 entangled states, 338 Toffoli gate, 360 transition probability, 57 Universal Quantum Copying Machine, transverse momentum change 350, 351 atom optics, 250 Van Kampen´s expansion, 388 trapped ion, 329 vector potential, 23 trapped ions, 294 visibility, 62 effective Hamiltonian, 296 observation of non-classical vibra- white noise, 383 tional states, 294 Wiener increments, 380 theory, 294 Wiener Process, 379, 385 triplicator, 355 Wiener process, 223 two mode squeeze operator, 47 Wigner representation, 73, 79 Two mode squeezed states, 338 moments, 80 two partite entanglement, 334 Wigner-Weisskopf theory for sponta- two-level atom, 14, 85 neous emission, 111