information analysis of quantum nondemolition measurement of a photon in a resonator

ISSN 0027�1349, Moscow University Physics Bulletin, 2009, Vol. 64, No. 6, pp. 611–616. © Allerton Press, Inc., 2009.Original Russian Text © D.N. Yanyshev, B.A. Grishanin, V.N. Zadkov, 2009, published in Vestnik Moskovskogo Universiteta. Fizika, 2009, No. 6, pp. 50–54.

611

INTRODUCTION

In connection with the rapid progress of the meth�ods of quantum information processing, storage, andtransmission [2] a number of the foundations of thetraditional quantum mechanics and quantum physicshave lately found new understanding. In particular, thenotion of quantum measurement has begun to beunderstood in a different way than its traditional defi�nition [3, 4], this being that the establishment of anunivocal correspondence between the proper states ofthe measured variable of a quantum object and thereadings of a quasi�classical device does not corre�spond to the modern understanding of the role ofquantum effects in a physical experiment. Along withthe problem of single�stage direct mining of the datafrom a quantum system in the classical form in prob�lems where the peculiarities of quantum informationare used, the problem of its mapping becomes the keyone, whereas its de�quantification is not obligatoryand, on the contrary, in many cases the practical valueof this mapping is attributed to the preservation of thecoherent properties of quantum states.

The concept of so�called quantum nondemolitionmeasurements is one of the most attractive theories fromthe physical point of view [5, 7]. As is shown in [8, 9] thenotion of nondemolition quantum measurement itself,when introduced in a more general form of entanglingmeasurement, can be considered irrespective of thedegree of quantum coherence loss, i.e. the degree ofdequantification of device readings (and, simulta�neously, objects) in the representation used during mea�

surements in the measured quantum

states of the object in the state of the object–device

system . Even in case of entirely coherent, i.e.,purely quantum representation of the resultant infor�mation, the most fundamental quality of classical infor�mation related to k indices in this measurement is, nev�ertheless, adequately represented. This is due to thesimultaneous membership of the k index in two differ�ent physical systems A and B, i.e., to the copying andmultiplication of the classical part of the initial quan�tum information related to a set of states in the ini�tial quantum ensemble of the source.

This concept of quantum measurement can bedemonstrated by the fundamental example of the non�demolition quantum measurement of a photon in aresonator using a probe atom, which was used in theexperiment described in [1] (Fig. 1). Here a probeatom (A) performs the entirely coherent entanglingmeasurement of the initial state of a number of resona�tor mode quanta (C). This state is further dequantifiedas the result of the atom�induced emission of a pho�ton, which is then recorded by the photodetector (D).

k| ⟩A k| ⟩A k| ⟩B

k| ⟩A

k| ⟩A k| ⟩B

k| ⟩A

Information Analysis of Quantum Nondemolition Measurementof a Photon in a Resonator

D. N. Yanysheva, B. A. Grishanin†, and V. N. Zadkovb

Faculty of Physics and International Laser Center, Moscow State University, Moscow, 119991 Russiae�mail: a [email protected]; b [email protected]

Received June 12, 2009, approved August 28, 2009

Abstract—The concept of entangled quantum measurement was demonstrated by the example of the funda�mental value of the experiment on quantum nondemolition measurement of a photon in a resonator with theuse of a probe atom [1]. The quantum�information analysis of this experiment was performed. A mechanismof information transmission in the nondemolition measurement scheme in the case of the classical and quan�tum formalism was demonstrated. The results were shown to coincide in both cases. This is the result of copy�ing the classical part of the initial quantum information attributed to a set of quantum object states.

Key words: theory of measurement, entangled states, nondemolition measurement

DOI: 10.3103/S0027134909060101

ACD

R1 R2 e

g

i

δ

R1, 2

Fig. 1. Scheme of the experiment on quantum nondemo�lition measurement of a photon (C) in a resonator using aprobe atom (A) passing through the resonator.

C

†Deceased.

612

MOSCOW UNIVERSITY PHYSICS BULLETIN Vol. 64 No. 6 2009

YANYSHEV et al.

The state of the latter is the result of measurement in atraditional entirely dequantified form. Entanglingmeasurements of a more general form correspond tothe time�intermediate state of the atom–photon–detector system when the emitted quantum is not yetfully absorbed and a corresponding quantum deriva�tive of this complex system, which transmits informa�tion on the measured number of resonator quanta,performs an entangled measurement with an interme�diate degree of coherence.

Taking into account the importance of this experi�ment and its close relationship to the problem of quan�tum information processing, the authors of this studyperformed information analysis of quantum commu�nication channels corresponding to this experiment,viz., they calculated the classical Shannon informa�tion applicable to classical channel resonator quantumnumber�photodetector readings and the quantumcoherent information applicable to the quantumchannel quantum resonator states–quantum atomstates.

1. A MODEL OF QUANTUM NONDEMOLITION MEASURMENT

To study the process of the quantum nondemoli�tion measurement of a photon in a resonator we ana�lyzed the fundamental experimental scheme [1]. Herethe probe atom (A), which is considered in the ele�mentary three�level model, serves as the device. Theprobe atom in the experiment drifts through a resona�tor C, whose mode frequency is quasi�resonant to the e

g transition of the atom. In this case, the atom issubject at the points R1 and R2 to the effects of auxiliarypulses of radiation with a frequency that is quasi�reso�nant to the atom transition frequency ωig. The mea�surement procedure is four�staged: preparation of theatom in a specified state (interaction of the atom withthe field of the auxiliary radiation pulse), interactionwith the resonator field, and measurement of the stateafter interaction with the second auxiliary pulse anddetection. Let us give a mathematical description ofthese processes.

First of all, let us determine the structure of thestate space HA ⊗ HC of the atom (A)–resonator (C)system [10]. For that let us define the following basis

(1)

using three possible states of the atom (indices a) andtwo resonator states (is there a photon in the resonatoror not; indices c). In case the atom is in the lower state

1| ⟩a 0| ⟩c

1| ⟩a 1| ⟩c

2| ⟩a 0| ⟩c

2| ⟩a 1| ⟩c

3| ⟩a 0| ⟩c

3| ⟩a 1| ⟩c⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

,

and there is one photon in the resonator, i.e., the

resonator state, when the atom is irradiated withthe first auxiliary φ1 radiation pulse we get the follow�ing state of the system:

(2)

For the first stage, namely for preparation of theatom in the state required for performing the measure�ment, the auxiliary radiation pulse (R1), whose fre�quency is tuned to the frequency of atom transitioni g, is used. If the time of interaction correspondsto the π/2 pulse the atom will pass to the ( +

)/ state. In the matrix form it can be written asfollows:

(3)

where t1 = φ1/ωig is the time of atom’s interaction withthe field of the first auxiliary radiation pulse (R1) andthe vector (c1, c2, c3, c4, c5, c6) in basis (1) is the initialstate of the system. If there is no photon in the resona�tor that corresponds to the resonator state, we getthe following state of the system:

(4)

The second stage is interaction of the probe atomwith the resonator field. The photon frequency in theresonator corresponds to the frequency of the atomictransition g e and here δ will be assumed to bezero. In this case, the state of the system will have theform:

1| ⟩a

1| ⟩c

ψ1

0

φ1/2( )cos

0

i φ1/2( )sin–

0

0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

.=

1| ⟩

2| ⟩ 2

c1 φ1/2( )cos ic3 φ1/2( )sin–

c2 φ1/2( )cos ic4 φ1/2( )sin–

c3 φ1/2( )cos ic1 φ1/2( )sin–

c4 φ1/2( )cos ic2 φ1/2( )sin–

c5

c6⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

,

0| ⟩c

ψ1

φ1/2( )cos

0

i φ1/2( )sin–

0

0

0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

.=


INFORMATION ANALYSIS OF QUANTUM NONDEMOLITION MEASUREMENT 613

(5)

where t2 = φ2/ωeg is the time of the atom’s interactionwith the resonator field and δ = 0. For the case whereφ2 = 2π we get

ψ2

0

φ1/2( )cos

0

i φ2/2( ) φ1/2( )sincos–

φ2/2( ) φ1/2( )sinsin–

0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

,= (6)

If the photon frequency in the resonator is tunedout by δ from the frequency of the probe atom transi�tion g e, we get a more complex state of the sys�tem, which is described as follows:

ψ2

0

φ1/2( )cos

0

i φ1/2( )sin

0

0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

.=

(7)ψ2

0

φ1/2( )cos

0

i φ1/2( ) e

12�� 2iπδ κ–( )t2

2iπδ– κ–( )2κ

��– e

12�� 2iπδ κ+( )t2

2iπδ– κ+( )2κ

��+⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

sin–

i φ1/2( )ie

12�� 2iπδ κ–( )t2

φ2

2κt2

��ie

12�� 2iπδ κ+( )t2

φ2

2κt2

��–⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

sin–

0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

,=

where κ = . If there is no photon inthe resonator, it does not actually change the state ofthe system and it will remain the same as that in (4).

The third experimental stage is analogous to thefirst one, i.e., the probe atom is subject to the effects of

4π2δ2– φ2

2/t2

2– the field of the second auxiliary radiation pulse (R2)

with a frequency that is resonant with that of theatomic transition i g. If the time of atom’s interac�tion with the field is t1 = φ1/ωig and there is a photon inthe resonator, the atom will pass to the state

(8)ψ3

0

φ1/2( )2cos φ2/2( ) φ1/2( )2

sincos–

0

i φ1/2( )cos φ1/2( )sin– i φ1/2( ) φ2/2( ) φ1/2( )sincoscos–

φ2/2( ) φ1/2( )sinsin–

0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

.=

If there was no photon in the resonator, the state ofthe system will be as follows:

(9)ψ4

φ1/2( )2cos φ1/2( )2

sin–

0

i φ1/2( ) φ1/2( )sincos–

0

0

0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

.=

Thus, if φ1 = π/2 and φ2 = 2π and δ = 0, the systemstate (in the presence or in the absence of a single pho�ton) will be

(10)

0

1

0

0

0

0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

or

0

0

i–

0

0

0⎝ ⎠⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎛ ⎞

,

614


YANYSHEV et al.

respectively. It follows from Eq. (10) that at δ = 0 andif the conditions φ1 = π/2 and φ2 = 2π are met, theatom passes either to the state i or g depending onwhether there is a photon in the resonator or not. Inthis case, the photon after such a measurementremains in the resonator if it was there before the inter�action or is absent in the opposite case, i.e., the state ofthe probe atom after the measurement depends on theresult of quantum nondemolition measurement of thephoton’s presence in the resonator.

The last and fourth stage corresponds to classicaldequantification of quantum information during itsmeasurement with a classical detector (D), which isdescribed in detail in [1].

2. CLASSICAL INFORMATION ANALYSIS

2.1 Calculation of the Conditional Probability of the Measurement Event

In our experiment the amount of information spec�ified by the Shannon formula, which is defined asinformation entropy serving as a measure of uncer�tainty of the messages of a given source (the messagesare described by the totality of magnitudes x1, x2, …, xn

and corresponding probabilities p1, p2, …, pn of x1,x2, …, xn the appearance in the message) will be used asthe criteria for evaluating the amount of the mineddata for a single measurement of the photon’s pres�ence/absence in the resonator with the use of a probeatom. In the case of a definite (discrete) statistical dis�tribution of the probabilities pk the information entropyor Shannon information (classical information) is thefollowing magnitude

(11)

if = 1. In our case we have a conditional

probability of the measurement event which can bepresented as follows:

Ish pk pklnk 1=

n

∑–=

pk1k 1=n

∑

(12)

where pc is the probability of the photon’s presence inthe resonator. Then the Shannon information willhave the form:

(13)

Figure 2 shows classical information as a functionof dimensionless tuning out of the photon frequencyin the resonator at the frequency of the atomic transi�tion g e. It is apparent from the plot that the max�imum is reached at zero tuning�out and is equal to 1bit. The sign of the frequency tuning out does notdepend on the result and thus only the positive part ofthe dependence is plotted in the figure.

Variations in the time of the atom’s flight throughthe resonator yield the dependence of Ish on thedimensionless time of the atom’s drift through the res�onator shown in Fig. 2b. It is apparent from the plotthat the maximum is reached at the drift time corre�sponding to 2π Rabi pulse and is equal to 1 bit.

2.2. Taking Resonator Quantum Efficiency into Account

A detector can determine the state of the atom gwith some probability pd. To calculate the amount ofinformation transmitted by the detector let us con�struct a matrix of the conditional probability of deter�mining the atomic state

pijpc ψ3 4[ ] 2

1 pc–( ) ψ4 3[ ] 2

pc 1 ψ3 4[ ] 2–( ) 1 pc–( ) 1 ψ4 3[ ] 2

–( )⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

,=

Ish

pij

plj pik

k 1=

2

∑l 1=

2

∑

��

⎩ ⎭⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎧ ⎫

pij.2logj 1=

2

∑i 1=

2

∑=

543210δ/ωeg, rel. un.

1.0

0.8

0.6

0.4

0.2

0

Ish, bit

(a)

543210φ2/ωeg, rel. un.

1.0

0.8

0.6

0.4

0.2

0

Ish, bit

(b)

6 7 1.00.80.60.40.2pd, rel. un.

1.0

0.8

0.6

0.4

0.2

0

Ish bit

(c)

0

Fig. 2. The amount of classical information as a function of dimensionless photon frequency tuning out in the resonator from thefrequency of atomic transition e g (a), dimensionless time of the atom’s passage through the resonator (b) and quantumdetector efficiency pd (c).


INFORMATION ANALYSIS OF QUANTUM NONDEMOLITION MEASUREMENT 615

(14)

Using the Shannon formula (13) from the matrix ofconditional probability we get the dependence of theamount of information for all the states on the quan�tum efficiency of the detector pd shown in Fig. 2c.

3. QUANTUM INFORMATION ANALYSIS

3.1. Calculation of Coherent Information in the Resonator–Atom System Taking the Presence

of an Idle Atomic Level Into Account

Transformation in the entire HA ⊗ HC system withspace dimensions dimHA = 3, dimHC = 2 is unitary inthe considered model and thus the transformed state ispure and its wave function has the form

(15)

where, according to the standard assumption on thestructure of the measuring device (a probe atom in ourcase) its initial state is considered to be preset.

For the informational characteristics of coherentinformation contained at the system output to bedetermined it is necessary to take into account thatonly two atomic levels iA = 1,2 are used to further pro�cess the measured quantum information. At theatomic output this information is presented in theform of quantum entanglement with the measuredobject, which is an atom. This means that the informa�tion related to the noninformative level iA = 3 can berepresented only noncoherently using the procedureof projection [11] at the atom output to correspondingtwo�dimensional and one�dimensional subspaces � =

� + � .

For the coherent information functional Ic[ ]

[11], where = �Ψt , taking into account the

pij = 1 pd–( ) ψ3 2[ ] 2

pd ψ3 4[ ] 21 pd–( ) ψ3 6[ ] 2

1 pd–( ) ψ4 1[ ] 2pd ψ4 3[ ] 2

1 pd–( ) ψ4 5[ ] 2

⎝ ⎠⎜ ⎟⎜ ⎟⎛ ⎞

.

Ψt UCA t( )Ψ0, Ψ0 ψC 0| ⟩A,⊗= =

P̂1 2+ P̂1 2+ P̂3 P̂3

ρ̂CA

ρ̂CA Ψt+

orthogonality of the projection components in �, thefollowing obvious relation occurs

Here the second term vanishes since for it (with theequation Ic = – taken into account)both terms of this equation coincide due to the one�dimensionality of the projector. As a result only theprojection of the wave function onto the two�dimen�

sional subspace of informative atomic states =

is left in the calculations and the equation forthe coherent information takes an elementary form:

(16)

This equation modifies that for the entanglementwith the resonator of a three�level atom and coincideswith it for the case when the output state Ψt does notcontain the contribution of the level . In the gen�eral case the last term does not vanish due to the break�

ing of the normalization of the wave function .

3.2. Results of the Calculation of Coherent Information

Taking Eq. (16) into account we can plot thedependence of the coherent information on thedimensionless time of the atom’s passage through theresonator in a manner similar to that in Section 2.1.This dependence is shown in Fig. 3a. It is apparentfrom the plot that the maximum is reached at the pas�sage time corresponding to the 2[phi]�pulse and isequal to 1 bit.

Using the results of the performed calculations wecan trace the dependence of the coherent informationIc on the dimensionless time of the atom’s passagethrough the field of the first auxiliary radiation pulse(R1) (Fig. 3b), from which it is apparent that the max�

Ic Ic P̂1 2+ ΨtΨt+

P̂1 2+[ ] Ic P̂3ΨtΨt+

P̂3[ ].+=

S ρ̂A[ ] S ρ̂CA[ ]

ΨtP

P̂1 2, Ψt

Ic S TrCΨtPΨt

P+

[ ] S ΨtPΨt

P+

[ ].–=

3| ⟩A

ΨtP

74321φ2/ωig, rel. un.

1.0

0.8

0.6

0.4

0.2

0

Ic, bit(a)

5 60 74321φ1/ωig, rel. un.

1.0

0.8

0.6

0.4

0.2

0

Ic, bit (b)

5 60 62024δ/ωig, rel. un.

1.0

0.8

0.6

0.4

0.2

0

Ic, bit(c)

46

Fig. 3. The amount of coherent information as a function of dimensionless time of the atom’s passage though the resonator (a),dimensionless time of the atom’s passage through the field of the first auxiliary pulse (R1) (b) and dimensionless tuning out of thephoton frequency in the resonator from the frequency of atomic transition e g (c).

616


YANYSHEV et al.

imum is reached at the points π/2 and 3π/2, whichcorresponds to the φ/2�pulse.

When the frequency of the resonator mode is tunedout from that of the probe atom transition e g theamount of the transmitted coherent informationdecreases with the increase in the tuning out (Fig. 3c).

CONCLUSIONS

The concept of the previously�proposed entanglingquantum measurement is demonstrated by the exam�ple of the fundamental experiment on quantum non�demolition measurement of a photon in a resonatorwith the use of a probe atom [1]. A quantum�informa�tional analysis of this experiment was performed. Amechanism of information transmission in a schemeof nondemolition measurement was demonstrated forthe case of classic and quantum formalism and theresults are shown to be similar in both cases. This isattributed to the copying of the classical part of the ini�tial quantum information related to a set of states in aquantum object.

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3. J. von Neumann, Mathematical Foundations of Quan�tum Mechanics (Princeton Univ., Princeton, 1955).

4. A. Sadbery, Quantum Mechanics and the Particles ofNature (Cambridge Univ., Cambridge, 1986; Mir,Moscow, 1989).

5. Quantum Theory and Measurement, Ed. by J. A. Weelerand W. H. Zurek (Princeton Univ., Princeton, 1983).

6. V. B. Braginsky and F. Ya. Khalili, Rev. Mod. Phys. 68, 1(1996).

7. P. Grangier, J. A. Levenson, and J.�P. Poizat, Nature396, 537 (1998).

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information analysis of quantum nondemolition measurement of a photon in a resonator

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