- Email: [email protected]
Photon number statistics invariant manipulation of cavity fields based on a single atomic conditional measurement
1 November 1998
Optics Communications 156 Ž1998. 32–36
Photon number statistics invariant manipulation of cavity fields based on a single atomic conditional measurement A. Napoli, A. Messina
)
INFM and MURST, Instituto di Fisica dell’UniÕersita, ` Via Archirafi 36, 90123 Palermo, Italy Received 18 November 1997; revised 23 July 1998; accepted 27 July 1998
Abstract A single-atom conditional micromaser-based manipulation of the quantized bimodal field of a high-Q cavity is reported. The proposed experimental scheme makes it possible modifying only the phases of the initial probability amplitudes of finding a well-defined population in the cavity. Some examples are discussed bringing to light the application potentialities of the method as well as its inherent theoretical interest. The practical feasibility of the scheme is briefly analyzed. q 1998 Elsevier Science B.V. All rights reserved. PACS: 42.50.–p; 42.50.q x Keywords: Quantum state manipulation; Unitary transformation; Two-photon processes; Micromaser
1. Introduction Implementing unitary manipulations of a quantum state-signal is an issue of topical and central importance in the field of quantum communication w1x. In this context, for instance, the possibility of distinguishing at an appropriate receiver nonorthogonal quantum states represents an important practical objective w2x. More in general, methods aimed at controlling selected properties of a quantum state while it is unitarily transformed are of interest both from a fundamental and applicative point of view. In this paper, we report on a single-atom conditional experimental scheme realizing a particular photon number statistics invariant manipulation of the cavity field.
atoms and a bimodal ‘ideal’ microresonator w3x. Here ‘ideal’ means that the two cavity modes, having different frequencies v 1 and v 2 such that v 1 f v 2 f 10 10 Hz, are assumed with the same experimentally achievable high quality factor Q G 5 = 10 10 w4x. The upper, intermediate and lower atomic states are Rydberg states denoted by <2: ' < q :, < i : and <1: ' < y :, respectively. The corresponding energies Eq, Ei and Ey are such that < Eqy Ei y v 2 < f < Ei y Eyy v 1 < Ž " s 1. whereas the maximum energy separation Eqy Eys v 0 satisfies the two-photon resonance condition v 0 ( v 1 q v 2 . The states < q : Žor < y :. and < i : have opposite parity and l2 Ž l1 . denotes the coupling strength relative to the dipole-allowed transition between such two states. lm Ž m s 1,2., assumed real, may be expressed as lm s y² m d i : vm r2 ´ 0 V where ² m < d < i : is the matrix element of the electric dipole operator and the effective volume V has been taken coincident for both cavity modes w3,5x. One-photon processes are excluded supposing that the two Bohr transition frequencies involving the atomic intermediate state are highly detuned from v 1 and v 2 . A typical value of such detuning D in these conditions is D ( 2p = 39 MHz whereas simultaneously we have lm2 f 10y5D 2 w3,5x. Thus, the atom
(
2. The system and its Hamiltonian model The implementation of the experiment we are going to propose requires a low-flux beam of three-level Rydberg
) Corresponding author. E-mail address: [email protected]
0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 4 2 1 - 0
A. Napoli, A. Messinar Optics Communications 156 (1998) 32–36
behaves as an effective two-level system w3,5,6x. The Hamiltonian model describing such two-photon atom-field coupling contains intensity-dependent detuning terms and has the form w6x:
been shown w9x that the time evolution of the state < p,q,q : may be expressed as follows: eyi H t p,q,q :s C p, q Ž t . p,q,q :
2
Hs
q S p, q Ž t . p q 1,q q 1,y :
Ý v˜m am† am q Ž v 0 q b 2 a 2† a 2 y b1 a 1†a 1 . S z
Ž2.
where
ms1
q lw a 1 a 2 Sqq h.c. x
33
Ž1.
where v˜ m s vm q Ž1r2. bm . A typical value for the effective Stark shift coefficients b 1 and b 2 may be taken as f 2.4 = 10 3 Hz, being D bm s lm2 Ž m s 1,2.. We assume for simplicity b 1 f b 2 . The m th cavity mode quanta are created and annihilated by the operators am† and am , respectively. The real constant l measures the two-photon atom-field coupling and we assume l f 10 3 Hz, being lD s l1 l2 w6x. Finally, the atom is represented by pseudospin operators S z , S " such that 2 S z < " : s "< " :, S " < " : s 0 and S " < . : s < " :. Quite recently, it has been shown that the effective atom-field coupling mechanism described by Eq. Ž1., may be successfully used to generate single-mode w7x and bimodal w8x cavity Fock states in the context of a micromaser-based conditional measurement approach. The schemes proposed in Refs. w7,8x significantly differ from that we are going to present in this article, under two peculiar aspects. First of all, our way of manipulating the cavity field relies on a single atomic conditional measurement. As a consequence, our proposal is not plagued by the currently unavoidable low efficiency of the atomic detection apparatus which dramatically ruins all the schemes wherein the internal state measurement of the atoms of a long sequence is involved. Secondly, we propose a practical recipe which, differently from the projects discussed in Refs. w7,8x turns out to be applicable, with unmodified reliability and efficiency, to arbitrarily assigned initial conditions of the cavity field.
l
½
C p, q Ž t . s cos
t Ž p q q q 2.
2
pqqq2 1 2
=exp yi
sin
2
t Ž p q q q 2.
5
v 1Ž 2 p q 1 . q 12 v 2 Ž 2 q q 1 . t 4
Ž3.
(
S p, q Ž t . s y2 i
Ž p q 1 .Ž q q 1 . Ž p q q q 2.
=exp yi
1 2
2
l sin
2
t Ž p q q q 2.
v 1Ž 2 p q 1 . q 12 v 2 Ž 2 q q 1 . t 4
Ž4. Denote by t the time spent by the atom inside the cavity prepared in a generic state < c : s Ý p,q a p,q < p,q : at t s 0. The time evolution of the initial state < c :< q : of the combined system may be written in the form eyi Ht c : q : s cq Ž t . : q : q cy Ž t . : y :
Ž5.
where the states < c "Žt .: may be explicitly given with the help of Eq. Ž2.. Suppose now that when the atom leaves the cavity, we detect it in its excited state. Then, the cavity field disentangles from the atom and in view of Eq. Ž5., the state of the radiation inside the microresonator is reduced to A < cqŽt .:, A being a suitable normalization constant such that Ay2 Ž t . ' Ý a p q
2
C p, q
2
pq
3. The experimental scheme An excited effective two-level atom is injected into the bimodal cavity prepared in a generic state < c :. The atom simultaneously exchanges one photon with each cavity mode. The atom-field interaction time may be controlled appropriately adjusting the atomic velocity with the help of a good velocity selector. A conditional highly efficient detection of the internal state of the two-level system immediately after it leaves the resonator, disentangles the atom from the cavity field with in turn collapses onto a new state < c˜ :. To put the discussion on a quantitative basis we study the time evolution of the coupled atom-field system. Denote by < p,q,s : ' < p,q :< s : ' < p :< q :< s : a generic common eigenstate of a 1† a 1, a 2† a 2 and S z pertaining to the eigenvalues p, q and s , respectively. It has
l
pyq yi
'
Ý p,q
q
apq
2
½
cos 2
Ž p y q .2 Ž p q q q 2.
2
l 2
t Ž p q q q 2.
sin2
l 2
t Ž p q q q 2.
5
Ž6.
coincides with the t-dependent probability of finding the atom in its upper state. An attractive aspect of this atom-field coupling adopted in this paper is that the equation Ay2 Žt . s 1 is exactly satisfied by t s 2prl whatever the initial state of the cavity is. Eqs. Ž3. and Ž4. show, that provided the atom-field interaction time may be fixed sharply coincident with t s 2prl, then a successful measurement of the internal
A. Napoli, A. Messinar Optics Communications 156 (1998) 32–36
34
atomic state projects the bimodal cavity field onto the following state: < c˜ : s e iq Ý a p q e iŽ g 1 pq g 2 q. < p,q :
Ž7.
p,q
wherein each probability amplitude differs only in phase from the corresponding one in the initial state < c :. In Eq. Ž7. q is an appropriate global phase constant whereas g k g w0,2p w and is related with the Hamiltonian model parameters in such a way that lg k differs from 2pv k for an integer multiple of 2pl. The meaning of Eq. Ž7. is that exploiting our conditional single-atom-based experimental procedure it is possible to manipulate the initial phases of the field state without modifying, however, the photon number statistics present at t s 0. This result may be explained in terms of a simple physical picture 1. The atom enters the cavity in the excited state and deposits two photons into the two different modes. Then it reabsorbs two photons, one from each mode, and leaves the cavity in the excited state. As the Hamiltonian allows for no direct communication between the two cavity modes, the photon number statistics cannot be affected and only the phases of the probability amplitudes can change. The key point that we wish to remark here, is that there indeed exists a transit time, independent from specific initial conditions of the cavity field, after which the atom leaves the cavity in its excited state with probability one. This notable circumstance stems directly from the two-photon two-mode coupling mechanism proposed in this paper. In fact, the existence of a well-defined atom-field interaction time t at which S p,q Žt . ŽEq. Ž4.. vanishes whereas simultaneously C p,qŽt . ŽEq. Ž3.. reduces to a pure phase factor, results from the linear Ž p q q .dependence of the infinitely-many Rabi frequencies of the atom-field system described by Eq. Ž1.. Of course such a peculiar dependence is an exact inherent aspect pertaining to the dynamics of the specific atom-field system discussed in this paper. Moreover, one should remark that the method of quantum state engineering proposed in this paper is characterized by a flexibility related to the ability of controlling the atom-cavity mode resonant conditions. Stated another way, in the context of our project, the coupling constant l plays the role of adjustable parameter. The structure of c˜ : expressed by Eq. Ž7. reveals that our experimental scheme realizes the following unitary transformation
c :™ exp i Ž g 1 a 1† a 1 . 4 exp i Ž g 2 a 2† a 2 . 4 c : ' U1U2 c :' U c :' c˜ :
Ž8.
Comparing < c : with c˜ : as given by Eq. Ž8., we note that U looks like the evolution operator for a free field
1
The authors thank one of the referees for remarking this point.
with appropriately modified interaction-dependent frequencies g 1ty1 and g 2ty1. This description is only a fictitious convenient point of view since the form c˜ : exhibited by the cavity field at t s t after the conditional atomic measurement, reflects on the contrary the role played by the coupling model adopted in this paper. It should be noted, in fact, that since the operator describing the matter–cavity interaction in this paper does not commute with the total Hamiltonian given by Eq. Ž1., the change of the phases of a given initial state of the field is not an effect trivially predictable in the time evolution of the system. Its occurrence is indeed a consequence of the fact that the infinitely many Rabi frequencies determining the dynamical behaviour of the system turn out to be multiple of a fundamental one equal to lr2p . Just this peculiar circumstance, regardless of the initial cavity field, guarantees the possibility of controlling the atom–cavity interaction time realizing the transformation from < c : to c˜ : in accordance with Eq. Ž7.. It should be noted that, in contrast with the factorizability of U into U1 and U2 , it is not possible to implement the unitary transformation Ž8. with the help of two two-level atoms separately interacting with the two nondegenerate quantized cavity modes in accordance with the single-photon Jaynes–Cummings model. The reason is that this fundamental interaction introduces in the dynamics of each single-atom single-mode system a peculiar characteristic infinite set of Rabi frequencies such that no matter-field disentangling measurement, made on the system at any t ) 0, might reproduce the same initial photon statistics. Turning back to Eqs. Ž8. and Ž9., we point out that starting with a disentangled state < c : s < w 1 :< w 2 : then c˜ :s w˜ 1 : w˜ 2 : where w˜ k :s Uk w : with k s 1,2.
4. Some possible applications of our method The project presented in Section 3 may be useful both from a fundamental and applicative point of view. A reliable generation of c˜ : from < c : may for instance be exploited: Ž1. to generate a manipulated final state orthogonal or quasiorthogonal Ž ² c c˜ : < 1 . to the initial one; Ž2. to control the squeezing parameters of an initial squeezed state. The examples presented here have been chosen to illustrate in a concrete way the application potentialities of our method. We wish, however, to emphasize that applications of the same kind discussed in this section may be realized with the help of different schemes w10x. 4.1. Generation of orthogonal states There exists a class of initial states of the bimodal cavity field which after manipulation may be transformed
A. Napoli, A. Messinar Optics Communications 156 (1998) 32–36
in such a way that ² c c˜ :s 0. Suppose that at t s 0
Letting this initial cavity field interact with the two-level atom in accordance with our proposal yields
`
< w 1: s
Ý
a p < p : ' state y signal,
c˜ :s w˜ 1 : 0 2 :
ps 0
Ý ' qs 0 N q 1
< q : ' auxiliary mode
Ž9.
With the help of our unitary manipulation the cavity field is reduced to the state N
c˜ :s w˜ 1 : Ý qs 0
e iq Ž q . e ig 2 q
'N q 1
q:
Ž 10.
The condition² c c˜ :s 0 is satisfied choosing atomfield coupling conditions such that g 2 s Ž2prN q 1. l where l may assume any value in the set 1,2, . . . , N. Eq. Ž10. says that exploiting the auxiliary mode 2 we may easily orthogonalize the state of the bimodal field whatever the quantum state-signal associated with the mode 1. Even when it is not possible to manipulate the cavity state in such a way to get an orthogonal output at the end of the experiment, it might, however, be possible to control the projection of the transformed state onto the input quantum state in such a way to make <² c < c˜ :< < 1. Example:
c :s b 1 : 0 2 :,
b 1 :s single-mode coherent state
Ž 11. x
c˜ :s b 1e ig 1 : 0 2 :
Ž 12 .
x
² c c˜: 2 s exp y2
b 1 2 Ž 1 y cosg 1 . 4
Ž 13.
It is easy to persuade oneself that the quasiorthogonality condition between the initial state < c : and the final state c˜ :, that is <² c < c :< 2 < 1, may be realized when values of g 1 appropriately near to p are selected. 4.2. Controlling the squeezing parameters As a final example, we present a manipulation of a quantum state signal initially injected into the cavity as a squeezed state. We put
c :s w 1 : 0 2 :
Ž 14.
and
w˜ :s S Ž z1e 2 ig 1 . D Ž b 1e ig 1 . 0 1 :.
Ž 18 .
Thus, we see that in general, our experimental scheme enables both the rotation of the axis of the uncertainty ellipse relative to the signal mode and the shift of its centre on a circle of radius < b 1 <. In particular, we note that the choice 2g 1 s p realizes the inversion of the role of the squeezed quadrature. We wish to stress that the transformations of the bimodal field from the states given by Eqs. Ž11. and Ž14. to those given by Eqs. Ž12. and Ž17., respectively, are such that the initial and the final states of the second mode coincide. Thus, these specific transformations might also be realized with the help of an appropriate single atom– single mode cavity coupling.
5. Discussion and conclusions The practical feasibility of the method presented in this paper relies on some more or less tacit assumptions which we wish to discuss explicitly. Let us begin by examining the effects stemming from the unavoidable nonideal control of the transit time of the atom through the cavity. Current velocity selectors guarantee an accuracy which may reach values up to 99%. This means that the time t spent by the atom inside the cavity, in the context of our single-atom experimental scheme, can be controlled within an uncertainty of 1%, that is tstqdt,
ts
2p
Ž 19.
l
where
Ž 20 .
p,q
with C p q Ž t q d t . s exp yi
1 2
v 1 Ž 2 p q 1 . q 12 v 2 Ž 2 q q 1 .
= Žt q d t . 4
w 1 :s S Ž z1 . D Ž b 1 . 0 1 :
Ž15 . = Ž y1 .
with S Ž z 1 . s exp
Ž 17 .
where
e iq Ž q .
N
< w2: s
35
½
1 2
5
z 1)a 12 q h.c. ,
D Ž b 1 . s exp b 1 a 1† q h.c. 4 .
"i
pqq
pyq pqqq2
Ž 16 .
cos
sin
ž ž
l 2
l 2
/ ./
d t Ž p q q q 2. dtŽ pqqq2
.
Ž 21.
A. Napoli, A. Messinar Optics Communications 156 (1998) 32–36
36
The condition 2
C p q Ž t q d t . f 1 ; p,q
Ž 22 .
plays the central role in the context of our approach. Unfortunately, it restricts the class of the cavity states which can be manipulated in accordance with Eq. Ž7., since it requires that pqqq2<
2
l
Ž 23.
Considering that in the worst situation, that is
1
gp
y2
Ž 24.
we may read Eq. Ž23. saying that the initial cavity states for which the transformation described by Eq. Ž7. may be successfully realized, must contain at the most a g-dependent finite number of Fock states of the bimodal field. Stated another way, the higher the degree of precision g with which we are able to control the atom-field interaction time, the wider the set of initial cavity states on which our conditional scheme may operate with an experimentally interesting level of success. It should be noted in fact that the probability of success Ay2 Žt q d t . of the experiment, in presence of nonideal performance of the velocity selectors, may be easily seen to be not significantly different from that relative to the ideal case provided that the initial state consistently satisfies Eq. Ž23.. In our opinion, reducing the uncertainty of the atom-field interaction time represents a reasonable goal which, for example, might be achieved with the help of an interesting recently proposed new scheme w11x. Before concluding, it seems appropriate to point out other aspects of our method. First of all, let us observe that the atomic velocity may be chosen in such a way that it is legitimate neglecting the spontaneous lifetime Žf 30 ms. of the two atomic levels with respect to the flight time Ž t int - 1 ms.. The cavity damping time Ž Qrv f 10 ms. is moreover high enough to guarantee acceptable probabilities for the success of the experiment. We finally remark that in spite of the fact that
the efficiency of the field ionization technique commonly used for detecting the internal atomic state is not higher than 50%–60%, the practical manipulation of the cavity field in accordance with our scheme is not dramatically ruined. The probability of success of the method here proposed reaches, in fact, experimentally interesting values both because it requires the internal state detection of only one atom and because t may be selected in such a way that the probability of finding the exiting atom in its upper state is very near to 1. In conclusion, we believe that the implementation of our unitary manipulation technique is reliable and therefore, might be of some application and fundamental interest.
Acknowledgements The authors wish to thank Dr. M.G. Palma and Dr. D. Bruss for stimulating conversation on the subject of this paper. The financial support from CRRNSM-Regione Sicilia is greatly acknowledged.
References w1x C.M. Caves, P.D. Drummond, Rev. Mod. Phys. 66 Ž1994. 481. w2x A. Ekert, R. Jozsa, Rev. Mod. Phys. 68 Ž1996. 733. w3x P.A. Maia-Neto, L. Davidovich, J.M. Raimond, Phys. Rev. A 43 Ž1991. 5073. w4x A. Joshi, A. Kremid, N. Nayak, B.V. Thompson, R.K. Bullough, J. Mod. Optics 43 Ž1996. 971. w5x M. Brune, J.M. Raimond, S. Haroche, Phys. Rev. A 35 Ž1987. 154. w6x S.-C. Gou, Phys. Lett. A 147 Ž1990. 218. w7x A. Napoli, A. Messina, J. Mod. Optics 44 Ž1997. 2093. w8x A. Napoli, A. Messina, J. Mod. Optics 44 Ž1997. 2075. w9x A. Napoli, A. Messina, J. Mod. Optics 43 Ž1996. 64910. w10x M. Brune, S. Haroche, V. Lefevre, J.M. Raimond, N. Zagury, Phys. Rev. Lett. 65 Ž1990. 976. w11x A. Kozhekin, G. Kurizki, B. Sherman, Phys. Rev. A 54 Ž1996. 3535.
Optics Communications 156 Ž1998. 32–36
Photon number statistics invariant manipulation of cavity fields based on a single atomic conditional measurement A. Napoli, A. Messina
)
INFM and MURST, Instituto di Fisica dell’UniÕersita, ` Via Archirafi 36, 90123 Palermo, Italy Received 18 November 1997; revised 23 July 1998; accepted 27 July 1998
Abstract A single-atom conditional micromaser-based manipulation of the quantized bimodal field of a high-Q cavity is reported. The proposed experimental scheme makes it possible modifying only the phases of the initial probability amplitudes of finding a well-defined population in the cavity. Some examples are discussed bringing to light the application potentialities of the method as well as its inherent theoretical interest. The practical feasibility of the scheme is briefly analyzed. q 1998 Elsevier Science B.V. All rights reserved. PACS: 42.50.–p; 42.50.q x Keywords: Quantum state manipulation; Unitary transformation; Two-photon processes; Micromaser
1. Introduction Implementing unitary manipulations of a quantum state-signal is an issue of topical and central importance in the field of quantum communication w1x. In this context, for instance, the possibility of distinguishing at an appropriate receiver nonorthogonal quantum states represents an important practical objective w2x. More in general, methods aimed at controlling selected properties of a quantum state while it is unitarily transformed are of interest both from a fundamental and applicative point of view. In this paper, we report on a single-atom conditional experimental scheme realizing a particular photon number statistics invariant manipulation of the cavity field.
atoms and a bimodal ‘ideal’ microresonator w3x. Here ‘ideal’ means that the two cavity modes, having different frequencies v 1 and v 2 such that v 1 f v 2 f 10 10 Hz, are assumed with the same experimentally achievable high quality factor Q G 5 = 10 10 w4x. The upper, intermediate and lower atomic states are Rydberg states denoted by <2: ' < q :, < i : and <1: ' < y :, respectively. The corresponding energies Eq, Ei and Ey are such that < Eqy Ei y v 2 < f < Ei y Eyy v 1 < Ž " s 1. whereas the maximum energy separation Eqy Eys v 0 satisfies the two-photon resonance condition v 0 ( v 1 q v 2 . The states < q : Žor < y :. and < i : have opposite parity and l2 Ž l1 . denotes the coupling strength relative to the dipole-allowed transition between such two states. lm Ž m s 1,2., assumed real, may be expressed as lm s y² m d i : vm r2 ´ 0 V where ² m < d < i : is the matrix element of the electric dipole operator and the effective volume V has been taken coincident for both cavity modes w3,5x. One-photon processes are excluded supposing that the two Bohr transition frequencies involving the atomic intermediate state are highly detuned from v 1 and v 2 . A typical value of such detuning D in these conditions is D ( 2p = 39 MHz whereas simultaneously we have lm2 f 10y5D 2 w3,5x. Thus, the atom
(
2. The system and its Hamiltonian model The implementation of the experiment we are going to propose requires a low-flux beam of three-level Rydberg
) Corresponding author. E-mail address: [email protected]
0030-4018r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 8 . 0 0 4 2 1 - 0
A. Napoli, A. Messinar Optics Communications 156 (1998) 32–36
behaves as an effective two-level system w3,5,6x. The Hamiltonian model describing such two-photon atom-field coupling contains intensity-dependent detuning terms and has the form w6x:
been shown w9x that the time evolution of the state < p,q,q : may be expressed as follows: eyi H t p,q,q :s C p, q Ž t . p,q,q :
2
Hs
q S p, q Ž t . p q 1,q q 1,y :
Ý v˜m am† am q Ž v 0 q b 2 a 2† a 2 y b1 a 1†a 1 . S z
Ž2.
where
ms1
q lw a 1 a 2 Sqq h.c. x
33
Ž1.
where v˜ m s vm q Ž1r2. bm . A typical value for the effective Stark shift coefficients b 1 and b 2 may be taken as f 2.4 = 10 3 Hz, being D bm s lm2 Ž m s 1,2.. We assume for simplicity b 1 f b 2 . The m th cavity mode quanta are created and annihilated by the operators am† and am , respectively. The real constant l measures the two-photon atom-field coupling and we assume l f 10 3 Hz, being lD s l1 l2 w6x. Finally, the atom is represented by pseudospin operators S z , S " such that 2 S z < " : s "< " :, S " < " : s 0 and S " < . : s < " :. Quite recently, it has been shown that the effective atom-field coupling mechanism described by Eq. Ž1., may be successfully used to generate single-mode w7x and bimodal w8x cavity Fock states in the context of a micromaser-based conditional measurement approach. The schemes proposed in Refs. w7,8x significantly differ from that we are going to present in this article, under two peculiar aspects. First of all, our way of manipulating the cavity field relies on a single atomic conditional measurement. As a consequence, our proposal is not plagued by the currently unavoidable low efficiency of the atomic detection apparatus which dramatically ruins all the schemes wherein the internal state measurement of the atoms of a long sequence is involved. Secondly, we propose a practical recipe which, differently from the projects discussed in Refs. w7,8x turns out to be applicable, with unmodified reliability and efficiency, to arbitrarily assigned initial conditions of the cavity field.
l
½
C p, q Ž t . s cos
t Ž p q q q 2.
2
pqqq2 1 2
=exp yi
sin
2
t Ž p q q q 2.
5
v 1Ž 2 p q 1 . q 12 v 2 Ž 2 q q 1 . t 4
Ž3.
(
S p, q Ž t . s y2 i
Ž p q 1 .Ž q q 1 . Ž p q q q 2.
=exp yi
1 2
2
l sin
2
t Ž p q q q 2.
v 1Ž 2 p q 1 . q 12 v 2 Ž 2 q q 1 . t 4
Ž4. Denote by t the time spent by the atom inside the cavity prepared in a generic state < c : s Ý p,q a p,q < p,q : at t s 0. The time evolution of the initial state < c :< q : of the combined system may be written in the form eyi Ht c : q : s cq Ž t . : q : q cy Ž t . : y :
Ž5.
where the states < c "Žt .: may be explicitly given with the help of Eq. Ž2.. Suppose now that when the atom leaves the cavity, we detect it in its excited state. Then, the cavity field disentangles from the atom and in view of Eq. Ž5., the state of the radiation inside the microresonator is reduced to A < cqŽt .:, A being a suitable normalization constant such that Ay2 Ž t . ' Ý a p q
2
C p, q
2
pq
3. The experimental scheme An excited effective two-level atom is injected into the bimodal cavity prepared in a generic state < c :. The atom simultaneously exchanges one photon with each cavity mode. The atom-field interaction time may be controlled appropriately adjusting the atomic velocity with the help of a good velocity selector. A conditional highly efficient detection of the internal state of the two-level system immediately after it leaves the resonator, disentangles the atom from the cavity field with in turn collapses onto a new state < c˜ :. To put the discussion on a quantitative basis we study the time evolution of the coupled atom-field system. Denote by < p,q,s : ' < p,q :< s : ' < p :< q :< s : a generic common eigenstate of a 1† a 1, a 2† a 2 and S z pertaining to the eigenvalues p, q and s , respectively. It has
l
pyq yi
'
Ý p,q
q
apq
2
½
cos 2
Ž p y q .2 Ž p q q q 2.
2
l 2
t Ž p q q q 2.
sin2
l 2
t Ž p q q q 2.
5
Ž6.
coincides with the t-dependent probability of finding the atom in its upper state. An attractive aspect of this atom-field coupling adopted in this paper is that the equation Ay2 Žt . s 1 is exactly satisfied by t s 2prl whatever the initial state of the cavity is. Eqs. Ž3. and Ž4. show, that provided the atom-field interaction time may be fixed sharply coincident with t s 2prl, then a successful measurement of the internal
A. Napoli, A. Messinar Optics Communications 156 (1998) 32–36
34
atomic state projects the bimodal cavity field onto the following state: < c˜ : s e iq Ý a p q e iŽ g 1 pq g 2 q. < p,q :
Ž7.
p,q
wherein each probability amplitude differs only in phase from the corresponding one in the initial state < c :. In Eq. Ž7. q is an appropriate global phase constant whereas g k g w0,2p w and is related with the Hamiltonian model parameters in such a way that lg k differs from 2pv k for an integer multiple of 2pl. The meaning of Eq. Ž7. is that exploiting our conditional single-atom-based experimental procedure it is possible to manipulate the initial phases of the field state without modifying, however, the photon number statistics present at t s 0. This result may be explained in terms of a simple physical picture 1. The atom enters the cavity in the excited state and deposits two photons into the two different modes. Then it reabsorbs two photons, one from each mode, and leaves the cavity in the excited state. As the Hamiltonian allows for no direct communication between the two cavity modes, the photon number statistics cannot be affected and only the phases of the probability amplitudes can change. The key point that we wish to remark here, is that there indeed exists a transit time, independent from specific initial conditions of the cavity field, after which the atom leaves the cavity in its excited state with probability one. This notable circumstance stems directly from the two-photon two-mode coupling mechanism proposed in this paper. In fact, the existence of a well-defined atom-field interaction time t at which S p,q Žt . ŽEq. Ž4.. vanishes whereas simultaneously C p,qŽt . ŽEq. Ž3.. reduces to a pure phase factor, results from the linear Ž p q q .dependence of the infinitely-many Rabi frequencies of the atom-field system described by Eq. Ž1.. Of course such a peculiar dependence is an exact inherent aspect pertaining to the dynamics of the specific atom-field system discussed in this paper. Moreover, one should remark that the method of quantum state engineering proposed in this paper is characterized by a flexibility related to the ability of controlling the atom-cavity mode resonant conditions. Stated another way, in the context of our project, the coupling constant l plays the role of adjustable parameter. The structure of c˜ : expressed by Eq. Ž7. reveals that our experimental scheme realizes the following unitary transformation
c :™ exp i Ž g 1 a 1† a 1 . 4 exp i Ž g 2 a 2† a 2 . 4 c : ' U1U2 c :' U c :' c˜ :
Ž8.
Comparing < c : with c˜ : as given by Eq. Ž8., we note that U looks like the evolution operator for a free field
1
The authors thank one of the referees for remarking this point.
with appropriately modified interaction-dependent frequencies g 1ty1 and g 2ty1. This description is only a fictitious convenient point of view since the form c˜ : exhibited by the cavity field at t s t after the conditional atomic measurement, reflects on the contrary the role played by the coupling model adopted in this paper. It should be noted, in fact, that since the operator describing the matter–cavity interaction in this paper does not commute with the total Hamiltonian given by Eq. Ž1., the change of the phases of a given initial state of the field is not an effect trivially predictable in the time evolution of the system. Its occurrence is indeed a consequence of the fact that the infinitely many Rabi frequencies determining the dynamical behaviour of the system turn out to be multiple of a fundamental one equal to lr2p . Just this peculiar circumstance, regardless of the initial cavity field, guarantees the possibility of controlling the atom–cavity interaction time realizing the transformation from < c : to c˜ : in accordance with Eq. Ž7.. It should be noted that, in contrast with the factorizability of U into U1 and U2 , it is not possible to implement the unitary transformation Ž8. with the help of two two-level atoms separately interacting with the two nondegenerate quantized cavity modes in accordance with the single-photon Jaynes–Cummings model. The reason is that this fundamental interaction introduces in the dynamics of each single-atom single-mode system a peculiar characteristic infinite set of Rabi frequencies such that no matter-field disentangling measurement, made on the system at any t ) 0, might reproduce the same initial photon statistics. Turning back to Eqs. Ž8. and Ž9., we point out that starting with a disentangled state < c : s < w 1 :< w 2 : then c˜ :s w˜ 1 : w˜ 2 : where w˜ k :s Uk w : with k s 1,2.
4. Some possible applications of our method The project presented in Section 3 may be useful both from a fundamental and applicative point of view. A reliable generation of c˜ : from < c : may for instance be exploited: Ž1. to generate a manipulated final state orthogonal or quasiorthogonal Ž ² c c˜ : < 1 . to the initial one; Ž2. to control the squeezing parameters of an initial squeezed state. The examples presented here have been chosen to illustrate in a concrete way the application potentialities of our method. We wish, however, to emphasize that applications of the same kind discussed in this section may be realized with the help of different schemes w10x. 4.1. Generation of orthogonal states There exists a class of initial states of the bimodal cavity field which after manipulation may be transformed
A. Napoli, A. Messinar Optics Communications 156 (1998) 32–36
in such a way that ² c c˜ :s 0. Suppose that at t s 0
Letting this initial cavity field interact with the two-level atom in accordance with our proposal yields
`
< w 1: s
Ý
a p < p : ' state y signal,
c˜ :s w˜ 1 : 0 2 :
ps 0
Ý ' qs 0 N q 1
< q : ' auxiliary mode
Ž9.
With the help of our unitary manipulation the cavity field is reduced to the state N
c˜ :s w˜ 1 : Ý qs 0
e iq Ž q . e ig 2 q
'N q 1
q:
Ž 10.
The condition² c c˜ :s 0 is satisfied choosing atomfield coupling conditions such that g 2 s Ž2prN q 1. l where l may assume any value in the set 1,2, . . . , N. Eq. Ž10. says that exploiting the auxiliary mode 2 we may easily orthogonalize the state of the bimodal field whatever the quantum state-signal associated with the mode 1. Even when it is not possible to manipulate the cavity state in such a way to get an orthogonal output at the end of the experiment, it might, however, be possible to control the projection of the transformed state onto the input quantum state in such a way to make <² c < c˜ :< < 1. Example:
c :s b 1 : 0 2 :,
b 1 :s single-mode coherent state
Ž 11. x
c˜ :s b 1e ig 1 : 0 2 :
Ž 12 .
x
² c c˜: 2 s exp y2
b 1 2 Ž 1 y cosg 1 . 4
Ž 13.
It is easy to persuade oneself that the quasiorthogonality condition between the initial state < c : and the final state c˜ :, that is <² c < c :< 2 < 1, may be realized when values of g 1 appropriately near to p are selected. 4.2. Controlling the squeezing parameters As a final example, we present a manipulation of a quantum state signal initially injected into the cavity as a squeezed state. We put
c :s w 1 : 0 2 :
Ž 14.
and
w˜ :s S Ž z1e 2 ig 1 . D Ž b 1e ig 1 . 0 1 :.
Ž 18 .
Thus, we see that in general, our experimental scheme enables both the rotation of the axis of the uncertainty ellipse relative to the signal mode and the shift of its centre on a circle of radius < b 1 <. In particular, we note that the choice 2g 1 s p realizes the inversion of the role of the squeezed quadrature. We wish to stress that the transformations of the bimodal field from the states given by Eqs. Ž11. and Ž14. to those given by Eqs. Ž12. and Ž17., respectively, are such that the initial and the final states of the second mode coincide. Thus, these specific transformations might also be realized with the help of an appropriate single atom– single mode cavity coupling.
5. Discussion and conclusions The practical feasibility of the method presented in this paper relies on some more or less tacit assumptions which we wish to discuss explicitly. Let us begin by examining the effects stemming from the unavoidable nonideal control of the transit time of the atom through the cavity. Current velocity selectors guarantee an accuracy which may reach values up to 99%. This means that the time t spent by the atom inside the cavity, in the context of our single-atom experimental scheme, can be controlled within an uncertainty of 1%, that is tstqdt,
ts
2p
Ž 19.
l
where
Ž 20 .
p,q
with C p q Ž t q d t . s exp yi
1 2
v 1 Ž 2 p q 1 . q 12 v 2 Ž 2 q q 1 .
= Žt q d t . 4
w 1 :s S Ž z1 . D Ž b 1 . 0 1 :
Ž15 . = Ž y1 .
with S Ž z 1 . s exp
Ž 17 .
where
e iq Ž q .
N
< w2: s
35
½
1 2
5
z 1)a 12 q h.c. ,
D Ž b 1 . s exp b 1 a 1† q h.c. 4 .
"i
pqq
pyq pqqq2
Ž 16 .
cos
sin
ž ž
l 2
l 2
/ ./
d t Ž p q q q 2. dtŽ pqqq2
.
Ž 21.
A. Napoli, A. Messinar Optics Communications 156 (1998) 32–36
36
The condition 2
C p q Ž t q d t . f 1 ; p,q
Ž 22 .
plays the central role in the context of our approach. Unfortunately, it restricts the class of the cavity states which can be manipulated in accordance with Eq. Ž7., since it requires that pqqq2<
2
l
Ž 23.
Considering that in the worst situation, that is
1
gp
y2
Ž 24.
we may read Eq. Ž23. saying that the initial cavity states for which the transformation described by Eq. Ž7. may be successfully realized, must contain at the most a g-dependent finite number of Fock states of the bimodal field. Stated another way, the higher the degree of precision g with which we are able to control the atom-field interaction time, the wider the set of initial cavity states on which our conditional scheme may operate with an experimentally interesting level of success. It should be noted in fact that the probability of success Ay2 Žt q d t . of the experiment, in presence of nonideal performance of the velocity selectors, may be easily seen to be not significantly different from that relative to the ideal case provided that the initial state consistently satisfies Eq. Ž23.. In our opinion, reducing the uncertainty of the atom-field interaction time represents a reasonable goal which, for example, might be achieved with the help of an interesting recently proposed new scheme w11x. Before concluding, it seems appropriate to point out other aspects of our method. First of all, let us observe that the atomic velocity may be chosen in such a way that it is legitimate neglecting the spontaneous lifetime Žf 30 ms. of the two atomic levels with respect to the flight time Ž t int - 1 ms.. The cavity damping time Ž Qrv f 10 ms. is moreover high enough to guarantee acceptable probabilities for the success of the experiment. We finally remark that in spite of the fact that
the efficiency of the field ionization technique commonly used for detecting the internal atomic state is not higher than 50%–60%, the practical manipulation of the cavity field in accordance with our scheme is not dramatically ruined. The probability of success of the method here proposed reaches, in fact, experimentally interesting values both because it requires the internal state detection of only one atom and because t may be selected in such a way that the probability of finding the exiting atom in its upper state is very near to 1. In conclusion, we believe that the implementation of our unitary manipulation technique is reliable and therefore, might be of some application and fundamental interest.
Acknowledgements The authors wish to thank Dr. M.G. Palma and Dr. D. Bruss for stimulating conversation on the subject of this paper. The financial support from CRRNSM-Regione Sicilia is greatly acknowledged.
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