Collge de France abroad Lectures Quantum information with real or - - PowerPoint PPT Presentation

Collège de France abroad Lectures Quantum information with real or artificial atoms and photons in cavities

Lecture 3: Estimation and reconstruction of quantum states in Cavity QED experiments: Fock and Schrödinger cat states

Introduction

Ideally, the direct reconstruction of a quantum state is straightforward:

• ne accumulates statistics about the measurement of a complete set of
• bservables, performed on a large number of realizations of the system’s
• state. The results fully constrains the system’s density operator, which is

found by solving a set of equations equating the observed statistics with the theoretical ones. This ideal procedure may lead to difficulties. If the data are noisy, it may happen that the directly constrained density operator is found to be non- physical, e.g. with negative eigenvalues. The number of available copies of the system might be small, the data presenting then large fluctuations. Sometimes, the set of measured observables is incomplete, making it impossible to fully constrain the parameters defining the system’s state. In these realistic situations, state reconstruction is an estimation problem: how can we optimally guess the parameters defining the state from the information provided by incomplete measurements, performed on a finite set of copies and suffering limitations from noise? Inspired by classical estimation theory, recalled in &III-A, we will analyze the general method

• f maximum likelihood (&III-B) and maximum entropy (&III-C), before

describing their application in Cavity QED (&III-D), illustrated by the reconstruction of Schrödinger cats and Fock states of a field (&III-E).

III-A. Reminder about classical estimation theory

Measuring a random variable X (possibly a multi-component vector) yields a result x. The known probability law p(x|!) of X depends upon an unknown parameter !, which can also be a vector. For reasons made clear below,the function p(x|!) is called the likelihood of ! corresponding to result x. The measurement of X brings an information about ! that we want to quantify. We call estimator !(x) a function which associates to each x result an estimation of the true !. Many estimator choices are possible. For a given one, different results xi generally yield different estimations !(xi). The variance of !(x), averaged over measurements, defines the estimator precision. This variance has a lower limit independent of the estimator, called the Cramer-Rao bound. This limit is related to a function of the likelihood, called the Fisher Information. Let us briefly recall how these results are obtained.

Fisher Information & Cramer-Rao bound

We consider here unbiased estimators, whose average over a large number of measurements yields the true value !t of !. If X is a continuous variable, this condition writes (all results can be extended to discrete variables by replacing integrals by sums): !(x)"!t = p(x |!t )

#

!(x)"!t

( )dx = 0

(4 "1) Hence, by derivation (! is supposed here continuous and having a single component): !(x)"!t

( ) #p

#!t dx

$

" p(x |!t )dx = 0 %

$

!(x)"!t

( ) #p

#!t dx

$

=1 (4 " 2)

We then use the identity:

Cramer-Rao bound (cont’d)

!p !" = p !Log(p) !" (4 # 3) which leads to : "(x)#"t

( ) p !Log(p)

!"t

$

dx =1 % "(x)#"t

( )

p & ' ( )

$

p !Log(p) !"t & ' * ( ) +dx =1 (4 # 4) We then square the integral and use the Cauchy-Schwartz inequality:

f (x)g(x)dx

!

2 "

f (x)

!

2dx

g(x)

!

2dx

#1" $(x)%$t

( )

2 p(x |$t )dx

!

( )

&Log(p) &$t ' ( ) * + ,

2

p(x |$t )dx

!

' ( ) ) * + , , (4 % 5)

We then introduce the variance of !: I(!) is the expectation value of the square of the logarithmic derivative with respect to ! of the likelihood function (*). The larger I(!) is, the more the statistical law contains information allowing us to pin down ! and the smaller the variance bound is. An estimation is said to be optimal if its standard deviation (square root of variance) reaches the Cramer Rao bound, i.e. "!

= I-1/2(!t).

Cramer Rao bound (cont’d)

(*)The Fisher information definition is generalized to situations where ! is a multicomponent vector by introducing a Fisher matrix involving the expectation values of second order partial derivatives of Logp(x|!) with respect to the ! components. This generalization is beyond the scope of this lecture.

!"

2 =

"(x)#"t

( )

2 p(x |"t )dx

$

(4 # 6)

and we get the Cramer-Rao inequality:

!"

2 #

1 I("t ) (4 $ 7)

where we have defined the Fisher information (function of !):

I !

( ) =

"Logp x |!

( )

"! # $ % & ' (

2

p(x |!)dx

)

(4 * 8)

Additivity of Fisher information

Performing N independent measurements of X yields results xi with the global probability:

p(x1...xi...xN |!) = p(xi |!) " Log p(x1...xi...xN |!)

( )

i

#

= Log p(xi |!

( )

i

$

(4 % 9)

since !L og p(x) !"

#

p(x)dx = !p !" dx = 0

#

$ % & ' ( )

Hence the Fisher information generated by the N independent measurements:

IN (!) = "Logp x1..xi...xN

( )

"! # $ % & ' (

2

p x1..xi...xN |!

( )

)

dx1..dxi...dxN = "Logp xi

( )

"! # $ % & ' (

2

p(xi |!)dxi

)

i

*

+ "Logp xi

( )

"! # $ % & ' ( p(xi |!)dxi

)

i+ j

*

, "Logp x j

( )

"! # $ % % & ' ( ( p(x j |!)dx j

)

(4 -10)

which establishes the additivity of Fisher Information:

IN (!) = NI1(!) (4 "11)

and the N-dependence of the optimal estimation standard deviation:

!N = 1 NI1("t ) = !1 N ; !1 = 1 I1 "t

( )

# $ % % & ' ( ( (4 )12)

We retrieve the known result that the standard deviation of the estimation (error) decreases as the inverse of the square root of the number of measurements.

Bayes law and Maximum Likelihood estimator

A natural choice for the estimator !(x) is justified by Bayes law, combined with an assumption of minimal knowledge about ! prior to measurements. The joint probability p(x,!) for finding values x and ! can be expressed in terms of the a priori probabilities p(x) and p(!) and the conditional probabilities p(x|!) and p(!|x):

p(x,!) = p x |!

( ) p ! ( ) = p ! | x ( ) p x ( )

" p ! | x

( ) = p x |! ( ) p ! ( )

p(x) = p x |!

( )

p x |!

( ) p ! ( )d!

#

p !

( )

(4 $13)

If nothing is a priori known about !, we assume a flat p(!) distribution, leading to: p ! | x

( ) =

p x |!

( )

p x |!

( )d!

"

(4 #14) The probability distribution of ! after result x has been found is thus given by the likelihood function p(x|!) normalised on !. A natural estimator picks the value of ! which maximizes this probability distribution and hence the likelihood function p(x|!). We thus define the Maximum Likelihood estimator !ML(x) (abbreviated as «!Max Like!») by the implicit function: !ML(x) " solution of #p x |!

( )

#! = 0 $ #Logp x |!

( )

#! % & ' ( ) *

!=!ML

= 0 (4 +15)

It can be shown that the Max Like estimator is optimal at the limit N #∞.

Simple illustration: a coin game

Consider a heads or tails draw, X taking the bit values x= 0/ 1 with probabilities p and q=1-p, which we parametrize with an angle ! by defining p=cos2(!/2), q=sin2(!/2) with 0 $ !< %. We get:

p(0 |!) = cos2 ! 2 " #log p 0 |!

( )

#! $ % & ' ( )

2

= tg2 ! 2 ; p(1|!) = sin2 ! 2 " #log p 1|!

( )

#! $ % & ' ( )

2

= cotg2 ! 2 (4 * 23)

and the Fisher information generated by a draw, found to be !-independent:

I1(!) = p(0 |!) "log p 0 |!

( )

"! # $ % & ' (

2

+ p(1|!) "log p 1|!

( )

"! # $ % & ' (

2

= cos2 ! 2 tg2 ! 2 + sin2 ! 2 cotg2 ! 2 =1 (4 ) 24)

The precision of an optimal estimation of ! is thus for N draws; IN = NI1 = N ! "N #

( ) =

1 N (4 $ 25) We then deduce the standard deviation of p and q, and the standard deviation of X (whose expectation value is <X>=q): !N p

( ) = !N q ( ) = !N (X) = cos"

2 sin" 2 !N "

( ) =

pq N (4 # 26) This is a well known result. N draws yield ~ Np times 0 and ~Nq times 1 with a fluctuation &(Npq). The measurement of X, whose average is q, is performed with a precision &(Npq)/N = &(pq/N). These results apply to the two-element POVMs realizing the QND measurement of the photon number (see lecture 1).

III-B. Estimation of a quantum state by the Maximum Likelihood principle

Max Like reconstruction principle

Inspired by classical estimation theory, we want to infer the most likely density

• perator ' reproducing the statistics of N observations of a quantum system given

in N identical copies. We list all the POVM’s Ei elements (or generalized measurements operators Mi) associated to the measurement results, each POVM element being labelled by its index i. We express the result of a complete measurement by the list of all the i’s, each being associated to its frequency of

• ccurrence fi = ni/N. If i is a continuous parameter, we discretize it in bins. The

theoretical probability pi for finding the result i when the system state is ' writes: The Ei’s must be properly normalized, so that the sum over i of all the pi’s is unity. For instance, if we measure on N copies of a qubit, all prepared in the same state, the 3 Pauli observables (x, (y et (z, with one third of measurements performed on each, we have 6 POVM’s elements corresponding to the projectors )j+ , )j- on the |±1> eigenstates of (j (j=x,y,z) and Ej± = (1/3))j±. The combinatory factor in this expression, easily derived, is irrelevant in the

• following. According to Max Like, we must find ' which maximizes this functional.

pi = Tr !Ei

[ ]

(4 " 21) We define the likelihood function L(') as the functional of ' equal to the probability for finding the experimental frequencies fi on the system in state ': L(!) = p( fi, f j...fk...| !) = N!/ nj!

j

" # $ % & ' ( Tr !Ei

[ ]

( )

Nfi i

"

Max Like reconstruction principle (cont’d)

which correspond to the ‘’best possible coïncidence’’ between the measured frequencies and the theoretical probabilities. Setting xi=Tr{'Ei}/fi in last equ. and taking into account the first equ. and Log(*iTr['Ei] = Log1 =0 leads to the inequality for Log[L(')] and sets its upper-

• bound. Determining the maximum lilelihood solution amounts to finding the '
• perator such that Log[L(')] gets ‘’as close as possible’’ to the limit, satisfying ‘’as

well as possible’’ the relations: It satisfies the general inequality:

Log L(!)

[ ] " C + N

fiLog fi

( )

i

#

(4 $ 30)

which follows from the relation, true for any set of positive xi (concavity of log):

Log xi fi

i

!

" # $ % & ' ( fiLogxi

i

!

fi ( 0 ; f1 =1

i

!

" # $ % & ' (4 ) 31)

Tr !Ei

[ ] = fi

(4 " 32)

It is useful to introduce the log of L(') (C being the log of the combinatory factor): Log L !

( )

( ) = C + N

fiLog Tr !Ei

{ }

( )

i

"

(4 # 29)

Example: qubit tomography by Max Like

Let us estimate by Max Like the density operator of a qubit defined by its Bloch vector: Note that this analytical result does not always yield a physical Bloch vector. If N is small, we may find, due to statistical fluctuations, |P

(ML)| >1. One must then

modify Max Like and search the maximum of the likelihood function for P restricted inside or on the Bloch sphere (see later). ! = 1 2 1+ Pj" j

j

#

$ % & & ' ( ) ) ; j = x,y,z (4 * 33) when we have found the 6 POVM elements Ej±=)j± /3 with the frequencies fj±. We show easily the identities: Tr{!" j±} = j ± ! ± j = 1 2 1± Pj

( )

(4 # 34) which leads to the log of the likelihood function: Log L(!))

( )"

f j+Log 1+ Pj 2 # $ % & ' (+ f j)Log 1) Pj 2 # $ % & ' ( * + ,

• .

/

j

(4 ) 35)

We get the components Pj of the Bloch vector by cancelling the partial derivatives:

! !Pj Log L("))

( ) = 0 # $

# f j+ 1+ Pj % f j% 1% Pj = 0 $ P

J (ML) = f j+ % f j%

f j+ + f j% (4 % 36)

When N#∞, the frequencies converge towards the statistical probabilities and we retrieve the result given in lecture I:

Pj = pj+ ! pj! (4 ! 37)

Iterative search of the Max Like solution

In general an analytic search of the Max Like solution 'ML is impossible. We describe here a general iterative method well suited for a computer [A.Lvovsky, Journ.of Optics B-Quantum and semiclassical optics 6, S556 (2004)]. We define first, based on the measurement results, an operator in the Hilbert space of S, which is a non linear function of ': To converge towards 'ML, we start from a zero order solution, '0 (for instance the normalized unity operator) and compute successive approximations '1, '2, '3…..'k by the following iterative sequence of operations, where N1, N2…N k+1 are normalization factors insuring that the 'khave a trace equal to 1:

(6 ! 50)

The method generally converges after a finite number of cycles, yielding a matrix 'k remaining practically stable thereafter. This fixed point satisfies at best the conditions fi=Tr'Ei and is the Max Like solution 'ML. This solution is by construction a positive and normalized operator and is thus a «bona fide» density operator, even if the number N of measurements is small.

!0 " R !0

( )!OR !0 ( ) = N 1 !1 " R !1 ( )!1R !1 ( ) = N 2!2!"!!k " R !k ( )!kR !k ( ) = N k +1 !k+1!" !ML

R !

( ) =

fiEi Tr !Ei

[ ]

i

"

(4 # 38) It is for '#'ML and Tr{'Ei} # fi a good approximation of *iEi= I. We infer that R('ML) must satisfy at best the identities: R !ML

( )!ML = !MLR !ML ( ) = R !ML ( )!MLR !Ml ( ) = !ML

(4 " 39)

Example of convergence of iterative method

Assume that a measurement of (z and (x on N=2 realizations of a qubit whose Bloch vector is in the xOz plane has given the results {+1,+1}. The exact Max Like solution restricted to physical density operators (|P|"1) corresponds then to the pure state:

!ML = 1 2 I + " x +" z 2 # $ % & ' ( = 1 2 2 2 +1 1 1 2 )1 * + , ,

• .

/ / = 0,853 0,353 0,353 0,147 * + ,

• .

/ (4 ) 40)

Compare this with the result given by the iterative method starting from '0=I/2 :

R(!0) = 1 2 + z z +

z + I / 2

( ) + z

+ 1 2 + x x +

x + I / 2

( ) + x

= + z z + + + x x + = 1 " # $ % & ' + 1 2 1 1 1 1 " # $ % & ' = 1 2 3 1 1 1 " # $ % & ' (4 ( 41)

which gives at first iteration:

!1 " + z z + + + x x +

( )!0 + z z + + + x x + ( ) = 1

2 + z z + + + x x +

( )

2

# !1 = 1 N 1 3 1 1 1 $ % & ' ( )

2

= 1 N 1 10 4 4 2 $ % & ' ( ) = 1 6 5 2 2 1 $ % & ' ( ) = 0,833 0,333 0,333 0,167 $ % & ' ( ) (4 * 42)

…and the fidelity with respect to exact Max like:

F

1 = Tr !ML!1

( ) = 1

2 + 2 3 = 1 2 1+ 8 9 " # $ % & ' = 0,971 (4 ( 43)

and at the second iteration:

R(!1) = 1 2 + z z +

z + !1 + z

+ 1 2 + x x +

x + !1 + x

" + z z + + + x x + = R !0

( )" 3

1 1 1 # $ % & ' ( ) !2 " 3 1 1 1 # $ % & ' (!1 3 1 1 1 # $ % & ' ( = 3 1 1 1 # $ % & ' (

4

) !2 = 1 N 2 116 48 48 20 # $ % & ' ( = 1 34 29 12 12 5 # $ % & ' ( (4 * 44)

…the fidelity has become:

F

2 = Tr !ML!2

( ) = 1

2 + 6 2 17 = 1 2 1+ 288 289 " # $ % & ' = 0,99913... (4 ( 45)

Accounting for measurement errors

The reconstructed ' summarizes all the information obtained by measuring multiple realizations of the system, taking into account the imperfections (classical noise). Max Like can easily account for the classical imperfections of the measuring

• process. The measurement of an observable on a system prepared in an eigenstate j

yields the result i with the probability pi= +ij which reduces to ,ij only in the ideal error-free case. In general {+ij} is a matrix with non-diagonal terms describing error rates. This matrix characterizes the measuring apparatus and can be determined by calibrations. The probability to get a result being normalized, we have:

!ij =1 "j

( )

(4 # 46)

i

$

The probability for finding the result i conditionned to the system being in ' is now:

p(i | !) = "ijTr E j! # $ % &

j

'

= Tr Ei

(")!

# $ % & (4 ( 47)

with: Ei

(!) =

!ij

j

"

E j (4 # 48) Taking into account the first eq, the Ei

(+) sum-up to I and thus also form a POVM:

Ei

(!) =

!ijE j = E j = I

j

"

i, j

"

i

"

(4 # 49)

The Max Like iteration must now be carried out with the modified operator R(+)('): R(!)(") = fiEi

(!)

Tr Ei

(!)"

# $ % &

i

'

(4 ( 50)

III-C State reconstruction by maximum entropy principle

[Vladimír Bu!ek and Gabriel Drobn# :Quantum tomography via MaxEnt principle

• J. Mod. Opt. 47 , 823-2840 (2000)]

Maximum entropy reconstruction (Max Ent)

We want to exploit the information obtained on a system by the knowledge of the expectation values gi of NG observables Gi which must satisfy: The search of ' thus amounts to a variational calculation under NG+1 contraints (the NG ones described by the eq. above, plus Tr' = 1). The usual method to solve this problem is the Lagrange multiplier procedure. We introduce NG+1 real coefficients

• 0, -1,.... -NG and look for the extremum of the linear combination of S and the

constraints multiplied by these coefficients and considered as functionals of ': Whereas Max Like searches ' which optimises the probability to reproduce all the frequencies of observed results, Max Ent obeys to a different logic. It searches, among all operators satisfying the above constraints, the one whose entropy S(") =

• Tr{"log"} is maximum. This condition is natural when the set of measured Gi‘s does

not yield complete information about the system’s state, since it amounts to exploit

• nly available information, without making any guess about unmeasured quantities.

It assumes maximum disorder compatible with the constraints imposed by the data.

Tr !Gi

{ } = gi

(4 " 51)

!S " #0! Tr$

[ ] "

#i! Tr$Gi

[ ]

i

%

= 0 (4 " 52) hence: ! Tr " Log" + #0I + #iGi

i

$

% & ' ( ) * + ,

• .
• /
• 1
• 2

3 4 4 5 6 7 7 = Tr !" Log" + #0I + #iGi

i

$

% & ' ( ) * + ,

• .
• /
• 1
• +Tr "! Log"

( ) { } = 0

(4 8 53)

Max Ent reconstruction (cont’d)

(**) The same argument is used in statistical physics to write the state of an oscillator under the exponential form '~exp(- H/kBT) when we fix only its average energy i.e. its temperature (<H> = kBT).

From the identity(*) Tr[',Log']=Tr[I,'] we get: Tr !" Log" + (#0 +1)I + #iGi

i

$

% & ' ( ) * + ,

• .

/ 0 = 0 (4 1 54) This equation must be true for ,'=,.|m><m’|, |m>,|m’> which imposes that the term in { } must be 0 and puts ' under the canonical form of an exponential operator(**):

!ME = e

" #0 +1

( )I "

#iGi

i

$ = 1 Z e

" #iGi

i

$ ; Z = Tr e

" #iGi

i

$ % & ' ( ' ) * ' + ' (4 " 55)

The -i are determined by satisfying at best the constraints Tr'Gi=gi, using a least square method. We look by iteration for the set of -i‘s which minimises the sum:

! 2 = Tr (Gi " gi) 1 Z e

" #iGi

i

$ % & ' ( ' ) * ' + '

2 i

$

(4 " 56)

(*) On écrit ! dans sa base propre :! = m

m

"

#m m (7 $ 4a) d'où Log! = m Log(

m

"

#m) m (7 $ 4b) et %Log! = m %#m #m

m

"

m + Log(#m)( %m m + m %m )

m

"

(7 $ 4c) Write ' in its eigenbasis:

(4 ! 54a) (4 ! 54b)

(4 ! 54c)

• ù !m vérifie la condition de normalisation :

!m m + m !m = ! m m = 0 (7 " 4d) On déduit alors de (7 " 4a,b,c,d) ) : Tr#!Log# = !$m + $mL og$m

m

%

! m m = Tr I!#

[ ]

m

%

satisfying the normalization condition: Then from

(4 ! 54d)

(4 ! 54 a,b,c,d) :

Max Ent with binary measurements

Let us apply Max Ent to the treatment of binary data, the Gi ‘s being a set of

• bervables having each only two eigenvalues ±1. This is the case of a qubit when we

measure the Pauli operators (i (i=x,y,z) or else of a field mode when we measure the parity of the photon number (see below). In these situations, giving the expectation values <Gi> =[fi(+1)-fi(-1)]/ [fi(+1)+fi(-1)] provides the same information as giving the frequencies fi(±1) of the measurement results +1 et -1. We thus expect that on large number of measurements, Max Like (which relies on frequencies) and Max Ent (which exploits expectation values) give identical

• reconstructions. Let us show this in the simple case of a qubit measurement. The

Max Ent solution writes: ! = 1 Z e

" #i$ i

i

% with #i$ i = #

i

%

&i$ i

i

%

= # $ & ; &i = #i # # = #i

2 i

%

' ( ) ) * + , , (4 " 57) We then develop the exponential, using the identity (/

2=I:

! = 1 Z e

" # $% = 1

Z cosh # I " sinh # $ % & ' ( ) = 1 2 I " tanh # $ % & ' ( ) (4 " 58)

and we determine the -i by identifying Tr"#i with the measured expectation values Pi of the (i’s. We get, as for Max Like: ! = 1 2 I + P

i" i i

#

$ % & ' ( ) (4 * 59)

Max Ent and observables with more than 2 values

If the Gi ‘s have more than two eigenvalues, the information exploited by Max Ent is a priori less rich than Max Like which uses complete experimental histograms and not only expectation values. For instance, if we measure field quadratures X0 as in unsual homodyne methods the frequencies f(x0) included in the Max Like method contain more information than the expectations <X0>= ∫x0f(x0)dx0. We can however improve Max Ent by adding to the Gi ‘s the higher order moments Gi

2, Gi 3 etc…with

their Lagrange multipliers and the searched density operator becomes: The moments are given by the measurement histograms and if these histograms are registered, there is no reason not to exploit their full information. When these moments are introduced, Max Ent and Max Like should yield equivalent results. Note that, as the iterative version of Max Like, the exponential solution of Max Ent is automatically a bona fide density operator (positive, with unit trace). The best reconstruction method to use in each situation depends upon practical considerations (length of calculation notably). We will not discuss here these technical details. We will compare later Max Like and Max Ent on an example (reconstruction of Fock states).

! = 1 Z exp " #i

(1)Gi "

#i

(2)Gi 2 "... i

$

i

$

% & ' ( ) * (4 " 60)

III-D Field state reconstructions in Cavity QED

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

? ?

Repeated QND photon counting

• n copies of field determines

the diagonal 'nn elements of the field density operator in Fock state basis, but leaves the off- diagonal coherences 'nn’ unknown Recipe to determine the off-diagonal elements and completely reconstruct ': translate the field in phase space by homodyning it with coherent fields of different complex amplitudes and measures (on many copies) a function of the photon number in translated fields: Tomography of trapped light

QND photon counting and field state reconstruction

" $ " (%)

) =

= D(% ) " D(&% &% )

"?

Field translation operator (Glauber): D (% )= exp (% a† - %'a) "(%)

) nn = (n’n’’ Dnn’ (% ) "n’n’’ Dn’’n (&%

&%) )

The homodyning translation in phase space admixes field coherences "n’n’’ into the diagonal matrix elements "(%)

) nn of

the translated field:

We determine "(%)

) nn by QND photon counting on translated fields, for

many 1’s, and get a set of linear equations constraining all the "n’n’’ s. By inverting these equations, we get the full density operator of the field. This direct reconstruction method has its problems.

Reconstructing field state by homodyning and QND photon counting

Requires many copies: quantum state is a statistical concept

measured

(4 ! 61)

State reconstruction is analoguous to CAT(*) SCAN medical tomography

Mixing with coherent fields of different complex amplitudes is equivalent to rotating the direction of observation in X ray cat scans. By a mathematical transform, a computer fully reconstructs the quantum state. (*): CAT is

here acronym

• f

Computer Assisted Tomography (not Schrödinger CAT!)

A direct way to reconstruct the quantum state via its Wigner function

W(1) can also be defined as the expectation value of the parity operator P in the state displaced by 21 in phase space: W (!) = 2 " Tr D(#!)$D(!)P

[ ]

(6 # 27) P is defined either as the operator inverting the sign of the quadratures, or as the photon number parity operator:

P x = !x

• r

P n = !1

( )

n n

The two definitions are equivalent since the Hermite polynomials describing the eigenstates of |n> in the |x> basis have the parity of n. The density operator ' in the quadrature basis and W can be obtained from each

• ther by reciprocal formulae and thus contain equivalent information about the

field state: W (x + ip) = 1 ! due"2ipu

#

x + u / 2 $ x " u / 2 x + u / 2 $ x " u / 2 = dp e2ipuW (x + ip)

#

exp2i pX ! xP

[ ] = e2ipXe!2ixPe

!2xp X,P

[ ] = e2ipXe!2ixPe!ixp = e2ip(X!x/2)e!2ixP

Relation between W and P

D(! = x + ip) = exp (x + ip)a† " (x " ip)a # $ % & = exp2i pX " xP

[ ]

(X = a + a† 2 ; P = a† " a 2i )

In order to establish the relation between W and P, we perform simple manipulations of the displacement operator replacing a and a† by X and P and using Glauber formula and the commutation relation [X,P]=iI/2: D(!) "u / 2 = D(x + ip) "u / 2 = e2ip(X"x/2)e"2ixP "u / 2 = e2ip(X"x/2) x " u / 2 = eip(x"u) x " u / 2 We the recognize that exp(-2ixP) translates the wave function by +x, which yields, by applying D(1) to the state |-u/2>: u / 2 D(!") = x + u / 2 e!ip(x+u) Then, by conjugating and replacing u by -u:

W (!) = 1 " du u / 2

#

D($!)%D(!) $u / 2 = 2 " dv v

#

D($!)%D(!)P v ; (v = u / 2)

and by using these expressions in the definition of W given in previous slide: which demonstrates the connexion between W and P. The field state is thus fully determined by its mean photon number parity in all its translations in phase space.

How to measure directly the Wigner function of the field in Cavity QED

W?

1. Translate field in phase space by -1 2. Send a non-resonant atom with a %-phase shift per photon across Ramsey

• interferometer. If atom exits in level e (g),

parity is even (+1) (resp.odd =-1). 3. Repeat with a large number of atoms and average between +1 (even) and -1 (odd) to find out the mean parity. This yields W(1) 4. Repeat procedure for different 1 values to reconstruct W and thus '. The method, proposed by Lutterbach and Davidovich (PRL 78, 2547 (1997)) is elegant in its principle. In practice, it requires to be able to realize a %- phase shift per photon for arbitrary fields, which is not possible due to non-linear atom-field coupling. The Ramsey interferometer measures in fact a «!generalized parity operator!» from which the state can be reconstructed using estimation procedures. See below.

Simple test: reconstructing a coherent state (using Max Ent, see below for details)

Fidelity F=0.98 Requires a submicrometer mirror stability

Two injections (preparing the state, then translating it)

'

W

III-E Preparing and reconstructing a Schrödinger cat state

Schrödinger cat Wigner function

3

!"#$%"&"'#"

()*+,++#*-./01)2&" (#%&3!0'4"&.#*) The |4><24| and |24><4| terms of the «!cat!» density operator correspond to the non-gaussian oscillations of its Wigner function. Decoherence destroys these

• scillations and reduces W to a sum of functions associated to the coherent states

|4> et |24> transforming the cat into a statistical mixture (see later). Superposition of two coherent states with

• pposite phases

= ! ! + "! "! + ! "! + "! ! 2

56 7'8"#+$'.$9.*.#$%"&"').:"-!.0'.;.< #$2=-0'4.)$.*.#-*,,0#*-.,$2&#" >6 ?*&0)@./"*,2&"/"').< 0')"&*#+$'.A0)%.*'.*)$/.A0)%.) )0

0=%

=% B)$/ B)$/ !")"#)"!. !")"#)"!.0'." 0'." B)$/ B)$/ !")"#)"!. !")"#)"!.0'.4 0'.4 5C%".(#%&3!0'4"&.#*).0,.=&$!2#"!.D@.)%" D*#EF*#+$'.$9.*.=*&0)@./"*,2&"/"').$'.)%" :"-!.=%*,".G,"".'"1).=*4"H6 I!!.=*&) JK"'.=*&)

Preparing a photonic Schrödinger cat by parity measurement

In fact, due to non-linearity, a ‘’distorted’’ cat is prepared (see below)

R1 R2

1.Coherent field is prepared in C

• 2. Single atom is prepared

in R1 in a superposition of e and g

• 3. Atom shifts the

field phase in two

• pposite directions as

it crosses C: superposition leads to entanglement in typical Schrödinger cat situation

• 4. Atomic states mixed again in R2 maintains cat’s ambiguity:

| ,e > + | ,g > #( | > + | >)|e>+(| >- | >)|g> Detecting atom in e or g projects field into + or - cat state superposition! How single atom prepares Schrödinger cat state of light: single atom index effect

LBIM.N5OPNOPQ

?&"=*&0'4.*'!.&"#$',)&2#+'4.*.R.#*).S

e or g T4U

% 2 % 2

g or e

V5 V> ; V> V5 ; W W

%/2 %/2 %/2 %/2 %/2 %/2 %/2 %/2

X

%/2 %/2

V",$'*').*)$/,< R.K*#22/.#-"*'"&,.S .=&"=*&*+$'.*)$/ ("Y2"'#".$9.*)$/,./"*,2&", 4"'"&*-0Z"!.=*&0)@.$9.!0,=-*#"!.:"-!

7'8"#+'4.*.#$%"&"').:"-!..[ W0,=-*#0'4.:"-!.D@.F\

=$,0+$' +/"

4 21 21

Taking into account light-shifts non-linearity

We must chose a small * value to get a large phase-shift per photon. The ratio +0/* (+0=Vacuum Rabi frequency at C center) is of the order of 1, and non-linear terms in n in the expansion of the atom-field states make phase- shift per photon n-dependent: about % for n = 0, it is ~ 0.5 % for n=5. This (exactly computable) non-linearity has two effects: it distorts the «!cat!» state prepared by the first atom and modifies the direct reconstruction procedure which no longer directly determines the Wigner function W, but a «!generalized W function!»

1

0.8 0.6 0.4 0.2 Phase-shift d6(n)/dn (in units of %) versus photon number

• 1. The cat prepared by 1st atom is distorted:

Ideal even cat Expected even cat taking non- linearity into account

Preparation distortion

Effects of non-linear phase shift

Distortion produced by direct reconstruction

• 2. The function reconstructed by measuring the probability differences

%e-%g is not the Wigner function W(1), but a « generalized W function », from which the density operator, and hence the true W function can be

• btained:

Wigner function

• f

prepared even cat Generalized W function directly measured («!even cat!»)

?

The imperfect Ramsey apparatus directly measures a generalized Wigner function

Fringe offset and contrast are characteristic of a real imperfect apparatus. Labelling with index i the phase space translations, the information extracted from field is given by the expectation values of the “error-affected” operators:

Fringe offset (ideally=0) Fringe contrast (ideally=1) Ideally : !1 " !0 = D(#)P D("#) = D(#)cos !a†a

( ) D("#) = !

2 W (#) non " linearity effect : !1 " !0 = D(#)cos $ a†a

( ) "%r

& ' ( )D("#) = D(#)PgenD("#) = ! 2 Wgen(#) Errors (calibrated defects of Ramsey apparatus) : !1

(errors) = *11!1 +*10!0

; !0

(errors) = *01!1 +*00!0

; (*00 +*10 =1 ; *01 +*11 =1) + !1

(errors) " !0 (errors) = *11 "*00

+ (*11 +*00 "1) D(#)cos $ a†a

( ) "%r

& ' ( )D("#) = ! 2 Wgen

errors(#)

(4 ! 63)

(4 ! 64)

The experimental probabilities for finding a qubit atom in state j=0 or 1 can also be expressed separately, as expectation values of «error affected!» POVM elements: (4 ! 67)

! j

(errors) = Tr "D(#)E j ($,%)D(&#)

{ }

with 2E j

($,%) =1+ &1

( )

j

%11 &%00

( )+ %11 +%00 &1 ( )cos ' a†a

( ) &$r

( )

( ) * +

Gi

(errors) = !11 "!00

( )I

+ (!11 +!00 "1) D(#i)cos $ a†a

( ) "%r

& ' ( )D("#i)

( )

Direct measurement of generalized W function

Measured %g-%e for fields translated along real axis of phase space, for even and odd cats Direct observation

• f fringes with
• pposite signs for

even and odd cat is a signature of quantum coherence

Direct measurement of generalized W function (ctn’d)

Measuring %e-%g along cercles around the origin of phase space for «!even!» and «!odd!» cats Points give experimental values of ,g-,e and lines are theoretical fits (see later)

Since the measured observables are «!close!» to parity, they are «!almost binary!» and the Max Ent method applies well. We have performed NG= 500 field displacements and measured the expectation values of the corresponding errors affected generalized parity operators (with one phase 0r). We have averaged over 400 realizations per displacement. Since the measurements do not change n, we have used in each realization the information provided by~10 atoms (which reduces the number of realizations necessary per displacement). The searched 'ME is the exponential of a linear combination of 500 Gi

(errors) operators. The

coefficients of this combination are 500 Lagrange multipliers. These multipliers are determined by a least square fit minimizing the 72 sum given in the discussion of the Max Ent method. The theoretical curves superimposed to the experimental points in the graphs of previous pages are fits obtained with the values of these multipliers.The agreement between the experiments and the fits is quite good. Once 'ME has been determined, we compute the true W. We thus go from W(gen)

(errors), given by

the direct data, to the true W by an indirect route. On next page, we show the 3D graphs of the true Wigner functions obtained by this method.

Reconstructing Schrödinger cat states by Max Ent

Reconstructed Wigner function of cat |4>+|24 24>

Gaussian components

Quantum interference (cat’s coherence)

D

2

D2= 8 Fidelity: 0.72

Non-classical states of freely propagating fields with similar W function (and smaller photon numbers) have been generated in a different way

(Ourjoumtsev et al., Nature 448, 784 (2007))

Similar W-functions reconstructions of synthesised superpositions of Fock states by J. Martinis et al (UCSB) in Circuit QED. See Lecture 6.

V"#$',)&2#)"!.(#%&3!0'4"&.#*),.G]*1.J')H

W

Deléglise et al. Nature 455, 510-514 (2008).

JK"'.#*) G5"&.*)$/.!")"#)"!.0'.4H I!!.;*) G5"&.*)$/.!")"#)!"!.0'."H !>"^._Mc.=%6 !>"^.5>.=%6

Snapshots of Schrödinger cat Wigner function at increasing times shows decoherence in action

Schrödinger cat decoherence

and energy conservation imposes:

!i

(") i

#

2 = " 2 (1$ e$%t ) = n(1$ e$%t )

Decoherence is due to the field’s coupling to its environment. We can model this environment as a reservoir of field modes, initially in vacuum, linearly coupled to the cavity field. Let us first describe how this model describes the damping of a coherent state. At time t=0, the field is in state |1> and all the reservoir modes are in vacuum. At time t, the field, still coherent, has decayed to|1exp(-8t/2)>, each reservoir mode being in a small coherent field with an amplitude .i

(1)(t):

! " 0 i

i

#

$ !e%&t /2 " 'i

(!) i i

#

; & = 1 Tc

If the field is initially in the cat state |1> +|4> with <1|4> ~ 0, we get, by linearity at time t: 1 2 ! + "

( ) #

0 i $ 1 2

i

%

!e&'t / 2 # E(!)(t) + "e&'t /2 # E(")(t)

( )

with E(!)(t) = (i

(!) i

;

i

%

E(")(t) = (i

(") i i

%

The cavity mode gets entangled with the reservoir. This entanglement is responsible for decoherence. As soon as the two environment states get

• rthogonal, there is a «which path» information in the environment which lifts the

quantum ambiguity of the state superposition and destroys the quantum coherence.

Schrödinger cat decoherence (ctn’d)

The field density operator at time t is obtained by tracing over the environment. As long as the two field components remain nearly orthogonal, we get: !(t) = 1 2 "(t) "(t) + #(t) #(t) + E(#)(t) E(")(t) "(t) #(t) + E(")(t) E(#)(t) #(t) "(t) $ % & ' Consider now the special case 4=21 which entails .i

(4)=- .i (1). The overlap of the

final environment states becomes:

E(!)(t) E(")(t) = #$i

(") $i (") i

%

= exp(#2 $i

(") 2) = exp #2

$i

(") 2 i

&

' ( ) * + , = exp(#2n 1# e#-t . / 1)

i

%

!(t) = 1 2 "(t) "(t) + #"(t) #"(t) + exp #2n 1# e#$t % & ' (

( ) "(t) #"(t) + #"(t) "(t)

( )

% & ' (

and: 1 2 ! + "

( ) #

0 i $ 1 2

i

%

!e&'t / 2 # E(!)(t) + "e&'t /2 # E(")(t)

( )

At short times (t <<1/8), the amplitude of the coherence term simplifies as:

exp(!2n 1! e!"t # $ % &) ' exp !2n"t

( ) = exp !2n t / Tc ( )

Decoherence rate proportional to cat’s components «!distance!» The cat’s coherence decays exponentially with time constant:

TD = Tc 2n = 2Tc D2 D = ! " #

( )

Fifty milliseconds in the life of a Schrödinger cat (a movie of decoherence)

Lifetime of Schrödinger cat state

Pd.C%"$&@ Je"#).$9.)%"&/*-.=%$)$',.GP6QdH

]"*'.'2/D"&.$9 )%"&/*-.=%$)$',

W"#$%"&"'#".+/" C!"#f.5_.g.N./,

M . S . K i m a n d V . B u ž e k , Sc h r ö d i n g e r

• c

a t s t a t e a t f i n i t e t e m p e r a t u r e , P h y s . R e v . A 4 6 , 4 2 3 9 (1 9 9 2 )

;$%b&"'#".G26*6H C0/".G/,H

FI!!.#*) FJK"'.#*)

Fock state reconstruction: Max Ent vs Max Like

We prepare an |n> state by running a sequence of POVMs realized with probe qubits atoms (see lecture 2). We then displace it by 21i (400 different values distributed

• n 20 lines passing by phase space origin). We then measure (x or (y on ~10 atoms

with a phase shift 60~%/2 per photon. We average over ~100 to 200 realization for each 1i. The same data are used to obtain ' by Max Ent and by Max Like. For Max Like, we use the first 3 atoms out of the about 10 crossing C after the field displacement and we measure for each 1i the frequencies fi(p) for detecting 3-p atoms in j=1 and p in j=0. We thus exploit not only atom count averages, but also atom correlations on a single realization. The operator R(') of the iteration procedure is defined as: where the Ej

(0,+) are given by an equation above [we give ' here in terms of the

separate Ej

(0,+) POVMs, which is equivalent to the expression in terms of the Gi]

For Max Ent we average the difference of count rates in j=1 and j=0 for each 1i value and we look for the optimal ' reproducting these averages under the form:

! = 1 Z exp " #i D($i) E0

%; &

( ) " E1

%; &

( )

' ( ) *D("$i) ' ( ) *

i

+

,

• .

/ 1 (4 " 68) R(!) = fi(p)D("i) E0

#i ; $

( )

% & ' (

p

E1

#i ; $

( )

% & ' (

3) p

D()"i) Tr !D("i) E0

#i ; $

( )

% & ' (

p

E1

#i ; $

( )

% & ' (

3) p

D()"i)

{ }

i p=0,1,2,3

*

(4 ) 69)

Reconstructing Fock states by Max Ent and Max Like: density operators

Max Ent Max Like n=0 n=1 n=2 n=3 n=4

F='00=0,890 F='11=0,976 F='22=0,936 F='33=0,821 F='44=0,51 F='00=0,945 F='11=0,966 F='22=0,920 F='33=0,823 F='44=0,556 Contamination by state n-1=3 due to cavity losses and by n=0 (projection modulo 4)

Reconstructing Fock states by Max Ent and Max Like: Wigner functions

Max Ent Max Like n=0 n=1 n=2 n=3 n=4

Max Like exploits signal fluctuations neglected in Max Ent. There is less phase noise, even if all the data are not exploited (3 atoms out of ~10 per measuring sequence) Reconstructions by Xing Xing & I.Dotsenko (unpublished)

Conclusion of third lecture

In this lecture, I have described the reconstruction in Cavity QED of non classical Shrödinger cats and Fock states of a field trapped in a Cavity, using the Max Like and Max Ent estimation procedures.The method has also been extended to the study of decoherence, by reconstructing snapshots of the field state at increasing times following the state

• preparation. Movies of the decoherence of Schrödinger cats have been
• btained in this way. In these studies, the preparation of the non-classical

states was non-deterministic (it was based on the random projection of the field state induced by a measurement). In the next lecture, I will describe a deterministic method to generate Fock state based on quantum feedback, which also corrects the effects of decoherence. In the 6th lecture, I will analyse a general method to prepare arbitrary states of a field oscillator which has been recently demonstrated in Circuit QED (this method requiring negligible decoherence during the preparation time). Reference about state reconstruction and decoherence study in Cavity QED: S.Deléglise, I.Dotsenko, C.Sayrin, J.Bernu, M.Brune, J-M.Raimond and S.Haroche; «!Reconstruction of non-classical field states with snapshots of their decoherence!», Nature, 455, 510 (2008).