Journal of Low Temperature Physics, Vol. 106, Nos. 3/4, 1997

Quantum Nondemol i t ion Measurement of the Photon Number in a Josephson-Junction Cavity

N o r i y u k i H a t a k e n a k a and Tet suo Ogawa *

NTT Basic Research Laboratories, Atsugi-shi, Kanagawa 243-01, Japan �9 Department of Physics, Tohoku University, Aoba-ku, Sendal 980-77, Japan

This paper proposes a quantum nondemotition measurement of the photon number in a Josephson-junction cavity. Under a current-biased Josephson junction with small capacitance, the Josephson phase fluctuates quantum- mechanically around its classical value due to the charging effect, and it couples to the photons in the junction cavity nonlinearly, which is necessary for the quantum nondemolition measurement. We show that the photon num- ber in the junction cavity can be nondestructively measured by detecting the fluctuation of Yosephson supercurrent through the junction.

PACS numbers: 42.50.Dv, 42.50.Lc, 74.50. +r.

1. I N T R O D U C T I O N

There have been much attention to quantum dynamics of mesoscopic tunnel junctions because of the interest in the elementary processes of tun- neling as well as the potential for applications. In a mesoscopic junction, a charging energy can be much larger than other relevant energies like thermal energy, and it dominates the single-electron-tunneling (SET) process. In the presence of photon fields, the SET process is greatly modified. On the other hand, the photon fields might be controlled by the tunneling electrons. In particular, nonlinear interaction under the strong coupling between them causes various types of quantum processes, builds the quantum correlations, and then creates nonclassical states of lights. In view of this, we investi- gated such nonclassical states of light in Josephson systems. We suggested a generation scheme of squeezed photon states in Josephson junctions, and studied numerically how an initial photon state (the vacuum or the coher-

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516 N. Hatakenaka and T. Ogawa

ent state) evolves to quadrature-phase amplitude squeezed states and/or the sub-Poissonian photon states.i, 2 This paper discusses a measurement scheme of quantum states of photon fields built in the Josephson junction. We show that the photon number in the junction cavity can be nondestructively mea- sured by measuring the fluctuation of the Josephson supercurrent through the junction.

2. Q U A N T U M N O N D E M O L I T I O N M E A S U R E M E N T

In a general quantum measurement, the observable As of the signal sys- tem fls is measured by detecting the change in the observable 2p of the probe system flp using the proper interaction between the signal and probe systems expressed by an interaction Hamiltonian flim. The measurement destroys the quantum states of the signal system. In a quantum nondemoli- tion (QND) measurement, precise detection of an observable is accomplished at the expense of an increase in uncertainty of its canonical conjugate ob- servable. A QND measurement requires the following conditions: 3

1. flint is a function of ./is, i.e., cgflint/cgfts # 0

2. Ap] 0,

3. [flint, As] = 0, and

4. fl.~ is not a function of tile conjugate observable of ,4.,s, i.e., afl,,,<lOA = o

The first two items are necessary for a measurement of ~i.s by using .zip. The third and last items are required for evading back-action of the mea- surement. The QND measurements of the photon number have predom- inantly been performed in nonlinear optical mediums. There have been recent studies investigating other systems, such as an atom-photon system in a microcavity 4'5 and an electron-photon system with interferometers. 7 In the atom-photon system, the probe is no longer a field but a beam of atoms interacting nonlinearly and nonresonantly with the signal field. In the fol- lowing sections, we show that the photon number in the junction cavity is the QND observable, and the supercurrent through the Josephson junction is the readout observable in our system.

3. H A M I L T O N I A N

Suppose that a Josephson junction operates at low temperatures under current bias (IB). The junction itself provides a good resonating cavity for

Quantum Nondemol i t ion Measurement o f . . . 517

confined photon fields due to a large impedance mismatch with the outside world. 6 For the sake of simplicity, we ignore the loss of radiation through leakage to the outside. We observe here that tunneling Cooper pairs interact with photon fields in a Josephson-junction cavity.

The tunnel Hamiltonian for Cooper pairs interacting with the photon fields in the Josephson-junction cavity is given ass,9

with

/:/T = /:/(To) cos r + 5 4(0) sin r 2ee ~' T

(i)

/:/(T o) = E j cos 0, (2)

~(o) 2e ^ = -~EjsinO, (3)

= V (4)

and ~ are respectively the Josephson coupling energy and the where E j phase-difference operator between superconductors. r is the phase due to the coupling between tunneling Cooper pairs and confined photon fields in the junction cavity. ~ and ~t are annihilation and creation operators of the photon field with frequency w. The electrostatic contribution to the junction energy is given by (2e~t)2/2C =- Ecc~h 2 with the junction capacitance C" and the number-difference opeator/t that conjugates to the phase-difference operator. Adding the bias current contribution - E j x ( ~ - r with x = IB/Ic, and shifting the origin of the potential, the total Hamiltonian of our system is then described as

The second term shows the potential of a current-biased Josephson junction (the Josephson potential). The last term denotes the Hamiltonian for a single-mode photon field in the junction cavity.

Now let us consider the small oscillation of the phase particle at the bottom of the Josephson potential. The Hamiltonian for the Josephson potential, [~lrj, is approximated as

S18 N . H a t a k e n a k a a n d T . O g a w a

Thus the total Hamiltonian is described as

/:/ = + Hph +

where

[-Iel = { Ecc h2 +

(7)

Ej cos(0)2 (50)22! } 4-Ej (1-cos(0)- x(O)) + Ej (sin(0)- x)50, (8)

Hph ^ ( 1) { - r ( s i n ( 0 ) _ x ) ~ } cos(0) ~_.T , (9) = hw ,,~ta+ + Ej

hint = - E j cos(0) - sin(0)~0 ~._22 2~

The Hamiltonian for an electronic system, f/el, consists of three parts; the first part shows the small oscillations of the phase particle at one of the local minima in the Josephson potential described in the second term of Eq. (8). This part is bosonized in the next section. The last term is the bias current contributions for the system, and disappears at x = sin(0). Hereafter we neglect the last two parts of the electronic Hamiltonian. For the photon part of the Hamiltonian, fIvh , the first term describes the confined electromagnetic field in the junction cavity. The second term is due to the coupling with the electronic system and is renormalized into the first term, resulting in the frequency shift, flint is the interaction part of the Hamiltonian between photon fields and tunneling Cooper pairs up to the second order of v'~cc/liw.

4. Q N D M E A S U R E M E N T OF T H E P H O T O N N U M B E R

In this section, we demonstrate that the photon number (n~) in the junction cavity is the QND observable, and the supercurrent through the Josephson junction is the readout observable in our system. The electronic system can be bosonized as

EJ c ~ (11) f'Iel = Ecc h2 + 2 T -

with the Josephson plasma frequency

= ~/2EccEj cos(0)/5, (12)

Q u a n t u m N o n d e m o l i t i o n M e a s u r e m e n t o f . . . 519

by introducing the following operators:

1

v ~ \ 2Ecc ] ( "l\b+b*], (13)

~O - ,/~i \E jcos (O>}

Note that the Josephson plasma frequency is controlled by the bias current. Physically, the dispersive interaction is required for the QND measure-

ment. Under the nonresonance condition t2 # w, the interaction Hamiltonian becomes

= h~btb + hCoata - ha ^ &taDtb (15) Ej cos(0>

by using Eqs. (4) and (14), ignoring the nonconservative terms on energy. This Hamiltonian, the central result of this paper, is similar to the Hamilto- nian for the QND measurement of the photon number via the optical Kerr effect derived by Imoto et al. 3 Since

OHint [/Iint,fia] = 0 a n d ~ # 0 , (16)

it turns out that the number operator ga = at& of the confined photons in the junction cavity is the QND observable. According to Imoto et al., the readout observable is one of the quadratures, (b - bt)/2i, and is equivalent to the phase operator described in Eq. (14). The phase operator satisfies the condition for the readout observable of the QND measurement, and is related to the supercurrent operator as follows. The supercurrent operator due to small oscillations of the Josephson phase is defined as

d~ 2e i = 2 ~ = ~[~,/:/e~ +//z~t] (17)

I t , 0 /1 - Etc. ) 2-E~oj,. (is) k

Therefore, in principle, the photon number in the junction cavity can be non- destructively measured by measuring the Josephson supercurrent through the junction. However, the expectation values of the Josephson supercur- rent due to small oscillations of the Josephson phase is zero since (50} = 0. The Josephson supercurrent is, of course, X = sin(0}. Practically, the current fluctuations, A] = _T - <]>, is the best candidate for the readout observable. It can be calculated by using the above current operator. Supercurrent fluc- tuations due to small oscillation of the Josephson phase are calculated as

<(~i)~> = ~<~0~> {i - E~ (~o>; ~ 2h~ J > o, (19)

520 N. Hatakenaka and T. Ogawa

if the photon is in the number state. The supercurrent fluctuations are also related to the photon number in the junction cavity. Therefore, the photon number in the junction cavity can be nondestructively measured by detecting the fluctuations of the Josephson supercurrent flowing through the junction.

5. S U M M A R Y

We have theoretically studied the quantum nondemolition measurement of the photon numbers in the Josephson-junction cavity. In our proposed scheme, the photon number in the junction cavity is nondestructively mea- sured by observing the supercurrent fluctuation flowing through the junction. Moreover, in contrast to the other systems, a special interferometer is not needed.

A C K N O W L E D G M E N T S

We would like to thank Professor Susumu Kurihara of Waseda Univer- sity for valuable discussions. One of the authors (T.O:) is grateful to financial support through a Grant-in Aid for Scientific Research on Priority Areas, "Mutual Quantum Manipulation of Radiation Field and Matter," from the Ministry of Education, Science and Culture of Japan.

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John Wiley & Sons, New York, 1982, Chapter 6 and the references thererin. 7. A. Shimizu, Phys. Rev. A 43, 3819 (1991). 8. G. -L. Ingold and Yu. V. Nazarov, in Single Charge Tunneling, Edited by

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