- Email: [email protected]
High-bit-rate quantum communication
1 June 2002
Optics Communications 206 (2002) 287–294 www.elsevier.com/locate/optcom
High-bit-rate quantum communication Victor V. Kozlov *, Matthias Freyberger Abteilung f€ur Quantenphysik, Universit€at Ulm, Ulm D-89081, Germany Received 25 January 2002; accepted 10 April 2002
Abstract We show how an entangled state can be generated within the internal quantum structure of a short picosecond pulse in the form of a soliton. One can then transmit such an intra-entangled soliton through a conventional communication fiber. We suggest basic principles of the transmission of a quantum state through non-linear and dispersive dielectrics. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 03.67.Hk; 42.81.Dp; 42.50.Md; 42.65.Tg
Quantum communication is a key element in the modern field of quantum information processing [1]. We approach this problem using methods already developed in classical communication. In particular, ultrafast all-optical technologies and communication lines made of silica fibers make it nowadays possible to speed up the signal transmission by utilizing the wide frequency bandwidth of optical radiation [2]. In analogy, we shall show that the ultrashort optical pulses are also able to provide high-bit rate fiber quantum communication where the object of transmission is a quantum state rather than a classical signal. The higher the transmission rate, the shorter are the pulses and the more they are distorted by the fiber dispersion and non-linearity. These effects, mainly associated with group-velocity dispersion
* Corresponding author. Tel.: +49-731-502-3088; fax: +49731-502-3086. E-mail address: [email protected] (V.V. Kozlov).
(GVD) and self-phase modulation (SPM) [2], severely limit the transmission rate. Distortion-free propagation is however possible with optical solitons, i.e., in the regime when SPM is balanced by GVD [2,3]. This feature of shape preservation makes the soliton a perfect candidate as an information carrier. In the following, we shall demonstrate how to deliver a desirable quantum state from a sender station Alice to a receiver Bob. When one transmits a quantum object through optical fibers its state will change under the action of dispersion and non-linearity. Therefore, Bob receives a state which is different from what has been sent by Alice though at the same time the classical shape remains intact (apart from a trivial position and phase shift). Such shape preservation would be good enough for classical fiber communication but it is not sufficient for quantum communication. One can however avoid the distortion of the quantum state by constructing a transmission line with zero path-averaged GVD and zero path-averaged SPM, as shown in [4,5]. This is accomplished by use of a combination of a two-level
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 4 5 6 - 6
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system and a conventional fiber. The two-level system supports solitons of self-induced trans parency (SIT) [6] and the ordinary fiber gives rise to the well-known solitons of a non-linear Schr€ odinger equation (NLSE) [2,3,7], see Fig. 1 for details. Using these two non-linear media the compensation mechanism works as follows. The faroff-resonance two-level medium can display negative non-linearity and positive dispersion. Therefore, it appears as a ‘‘mirror image’’ of the fiber with positive SPM and negative GVD [4,5]. Since SIT and NLSE solitons both have the same sech shape and provided their amplitudes are made equal, the soliton emerging from one of the media and entering the other (its ‘‘mirror image’’) does not experience any transient reshaping and associated interconnection losses. The important change is that at the boundary the SPM phase shift reverses the direction of rotation. Hence, we can get full dispersion and non-linearity cancellation when the lengths of the media are properly adjusted, and then the net non-linear phase shift accumulated through the total line becomes zero. The same effect of reversal with following full cancellation is also characteristic for the evolution of any classical or quantum variable. In the limit of vanishing losses, such a zero path-averaged GVD/SPM transmission line, if considered as the input–output block, acts like vacuum: any quan-
Fig. 1. Sketch of a two-component transmission line between Alice and Bob. The line has total length L consisting of a fiber of length lf and a SIT medium of length lr . The two-level (SIT) medium compensates for the GVD and SPM distortion of the quantum signal which was picked up during propagation through the fiber. The zero path-averaged GVD/SPM line requires lr ¼ jDf =Dr jlf . For example, 1 micron-long sample of InGaAs with d ¼ 2 1028 C m, d ¼ 1:0 THz, and N ¼ 1015 cm3 , see [20], is needed to compensate the SPM and GVD of a 4 km-long communication fiber with n2 ¼ 1:2 1022 m2 =V2 and k 00 ¼ 1:0 ps2 =km. Also, Ar ¼ 10Af . For notations see the main text.
tum state entering the system will non-trivially evolve inside but will eventually reproduce itself when reaching the output. Quantum information processing involves transmission and/or exchange of different types of quantum states. Since a conventional mode-locked laser generates pulses only in a coherent state, there is a need for an additional tool capable of engineering other basic types of states such as squeezed and entangled states. In the following we shall show how the on-resonant SIT-type interaction can be exploited as an entangler or a squeezer of initially coherent solitons. Given this tool and the link to Bob via zero path-averaged GVD/SPM transmission line, Alice can prepare on demand and then deliver to Bob coherent, squeezed, or entangled states. Here, the optical soliton plays the role of a classical carrier of quantum information. The single classical shape, which is preserved throughout the preparation–transmission process, allows us to encode in it different types of quantum states. The soliton will simply carry these states through the line. Note that the preparation and the transmission stage both include the SIT two-level medium but in two completely different regimes: first we utilize its on-resonant features and then we need the faroff-resonant interaction. Instead of using two different SIT media one can also combine the two stages. Then a single two-level transition now slightly detuned from resonance will serve both stages simultaneously. In this hybrid setup, the quantum state at Alice’s side is intentionally underdeveloped to such extent that further evolution in the fiber turns it into the desirable form exactly at the distance where the soliton emerges at Bob’s side. The paper is organized as follows. We start from a short overview of the quantum-mechanical description of SIT and NLSE solitons and put them on the same footing. We then describe the evolution of the soliton through the NLSE and the SIT medium, respectively. By referring to wellknown entanglement criteria and adapting them to the soliton context we show (i) how the SIT interaction produces an entangled state of the soliton from an initially coherent field; (ii) how to build the zero path-averaged GVD/SPM trans-
V.V. Kozlov, M. Freyberger / Optics Communications 206 (2002) 287–294
289
Z
o/
dt i 0 v^ðt; zÞ þ H:c:; on0
ð3Þ
Z
o/ 0 v^ðt; zÞ þ H:c:; dt i n0 ox0
ð4Þ
Z
o/ 0 dt i v^ðt; zÞ þ H:c:; n0 op0
ð5Þ
mission line and (iii) how to combine the preparation and the transmission stage by sharing one SIT medium, such that an initially coherent state turns into an entangled (or squeezed) state when reaching Bob. An elegant way of analyzing the evolution of quantum NLSE and SIT solitons is the Inverse Scattering Method [7–10]. Since optical solitons contain a large number of photons (108 ), the total quantum field /^ðt; zÞ can be decomposed in /^ ¼ / þ v^ with a large classical mean value / and a small quantum perturbation v^ such that all higher terms than linear in v^ can then be neglected, [11]. The classical fundamental soliton /ðt; zÞ has the shape of sech and is fully characterized by four parameters: photon number n0 , initial phase h0 , frequency detuning (momentum per photon) p0 , and initial position x0 . This observation suggests an expansion of the quantum perturbation v^ðt; zÞ into a complete set of temporal–spatial modes as o/0 o/ o/ o/ v^ ¼ D^ n þ 0 Dh^ þ 0 D^ p þ 0 D^ x þ v^c eiH0 ; on0 oh0 op0 ox0 ð1Þ
Dh^ðzÞ ¼ eiH0
where the first four terms describe the perturbations localized within the soliton while v^c ðt; zÞ accounts for the continuum, i.e., the perturbations delocalized in space and time. Here we shall use the Heisenberg picture in which the operators D^ n etc. evolve with distance z. The mode functions o/0 =on0 , etc. are obtained as derivatives of the initial classical shape /0 ¼ /ðt T0 ; 0Þ. The position shift T0 is chosen such that the peak of the /0 coincides with the peak of the soliton at any z. Additionally, in order to always have a reference frame which is moving with the soliton, we explicitly extract from the expansion Eq. (1) the classical SPM-induced phase shift H0 . The soliton operators have the simple interpretation as quantum perturbations at a distance z of the corresponding classical variables, e.g., D^ n stands for the perturbation of the mean photon number n0 . Each operator from the decomposition (1) can be extracted by appropriately projecting the total field v^ðt; zÞ. Thus we can write [8–11], Z o/
D^ nðzÞ ¼ eiH0 dt i 0 v^ðt; zÞ þ H:c:; ð2Þ oh0
when the total field obeys the relations ½/^ðt0 ; zÞ; /^y ðt; zÞ ¼ dðt t0 Þ and ½/^ðt; z0 Þ; /^ðt; zÞ ¼ 0 [12,13]. Eq. (6) suggest to regard the four operators as two collective modes of the quantized soliton: ðD^ n; Dh^Þ is the ‘‘wave mode’’ and ðn0 Dp^; D^ xÞ is the ‘‘particle mode’’ [11,14]. Though the two-mode Hilbert space is only a sub-space of the infinitely dimensional Hilbert space of the soliton, it has an important property. The evolution in the two-mode sub-space is decoupled from the dynamics of other degrees of freedom. This assures that during propagation of the soliton any quantum information imprinted on the four soliton variables will not leak into other modes (continuum). The four variables are also operationally sound. After reaching Bob, a desirable soliton variable or any linear combination of them can be projected out of the total field /^ðz; tÞ by a homodyne measurement with properly shaped local oscillators. This way of measurement [10] is deduced from the special projection given by the integrals in Eqs. (2)–(5). Note that such a measurement as well as the very possibility of the effective two-mode description is due to the unique
D^ pðzÞ ¼ e
iH0
D^ xðzÞ ¼ eiH0
and similarly for a mode from the continuum. The advantages of the set of modes used in (1) show up when we substitute this expansion into NLSE or SIT equations as shown in [9–11]. On this way we shall derive evolution equations for the operators, and what is important for our analysis, the evolution of the localized perturbations appears to be decoupled from the evolution of the continuum. We can therefore exclude the continuum from our further considerations and follow only the dynamics of the four soliton variables (2)–(5). Moreover, one finds from Eqs. (2)–(5) that they obey simple commutators ½D^ n; Dh^ ¼ i
and ½D^ x; n0 Dp^ ¼ i;
ð6Þ
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V.V. Kozlov, M. Freyberger / Optics Communications 206 (2002) 287–294
feature of solitons – the orthogonality between the localized and continuum modes at all distances. Using this two-mode description we can introduce two pairs of rotated quadrature amplitudes Xw sin ww cos ww qw Dh^ ¼ ; ð7Þ Pw cos ww sin ww q1 n w D^
Xp Pp
¼
1=2 n0
cos wp sin wp
sin wp cos wp
"
# x qp D^ ; ^ q1 p Dp
ð8Þ
where qp and qw provide a convenient scaling and ww (wp ) is the wave (particle) rotation angle. Quantum states in the Hilbert space which is spanned by the eigenstates of these quadratures can be used for the transmission of quantum signals. In particular, an initially coherent state j/i can be transformed into an entangled state or squeezed state, and then transported to Bob through a zero path-averaged GVD/SPM line. Given this effective two-mode description of solitons, we now turn to the effective preparation of the non-classical quantum states in the NLSE and then in the SIT medium. The crucial step is to recognize that the decomposition (1) puts the description of the two solitons on the same footing. The output shape after propagating in one medium is simultaneously the input shape for the other medium. Thus, by using the classical NLSE soliton shape /nls in Eq. (1) and substituting it into the NLSE we shall derive evolution equations for the four soliton operators. A similar procedure can be applied in the SIT case but now with the SIT soliton shape /sit in Eq. (1). Under appropriate matching conditions explained below we get /nls ðt; 0Þ / /sit ðt; 0Þ and therefore the mode functions in expansion (1) are the same for the two problems. The corresponding operators (2)–(5), however, evolve differently. We start with the NLSE medium for which the evolution equation reads [15]. o/^ k 00 o2 /^ ¼ i þ Nf /^y /^2 ; oz 2 os2
ð9Þ
where z is the propagation distance, s ¼ t ðnf =v0 Þz is the retarded time, v0 is the speed of light in vacuum, nf is the group index at frequency x0 , k 00 is the (negative) GVD at x0 , and Nf ¼ hx20 n2 =Af v0
is the fiber non-linearity with the Kerr coefficient n2 and the effective fiber core area Af . The field operators /^ and /^y are slowly varying amplitudes of positive and negative frequency parts ffi of the electric pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi field operator E^ðt; zÞ ¼ hx0 =Af /^ðt; zÞ exp½iðx0 t kzÞ centered near frequency x0 . Substituting /^ ¼ / þ v^ into Eq. (9) and then linearizing this equation near the classical solution we arrive at
o^ v k 00 o2 v^ 2 2 y ^ ^ i ¼ þ N 2 j/j v þ / v : ð10Þ f oz 2 os2 As mean field / in Eq. (10) we choose the fundamental soliton shape 1=2 n0 jcf j n0 jcf j z s x0 sech /nls ðs; zÞ ¼ 2 vf 2 exp ½iðp0 s kf z þ h0 Þ :
ð11Þ
Here p0 ¼ x x0 Rdenotes the mean frequency 2 detuning and n0 ¼ ds j/nls j is the mean photon number. We have also used cf ¼ Nf =2Df with Df 1 00 k . The dispersion relation for the fiber reads 2
2
kf ðp0 ; n0 Þ ¼ p02 14 n20 jcf j k 00 and the group ve-
locity in the retarded frame is given by vf ¼ 1 ðokf =op0 Þ . Expanding v^ðt; zÞ around /nls as suggested in Eq. (1), then substituting the resulting decomposition into the linearized NLSE (10), and finally projecting both sides of the equation by an appropriate projection function, as demonstrated in Eqs. (2)–(5), we get a self-contained set of evolution equations whose solutions are D^ nðzÞ ¼ D^ n0 ; Dh^ðzÞ ¼ Dh^0
ð12Þ okf zD^ n0 ; on0
ð14Þ
Dp^ðzÞ ¼ Dp^0 ; D^ xðzÞ ¼ D^ x0 þ
ð13Þ
ov1 f zDp^0 ; op0
ð15Þ
with the initial fluctuations D^ n0 D^ nð0Þ, etc. Here the derivatives are evaluated at p0 ¼ 0, h0 ¼ 0, and x0 ¼ 0. The evolution takes place separately within the wave and the particle mode and it looks like free propagation of two independent massive particles. Since the last terms in Eqs. (13) and (15) are re-
V.V. Kozlov, M. Freyberger / Optics Communications 206 (2002) 287–294
sponsible for coupling the variables within the same mode, we shall refer to them as self-terms. They can be used for producing squeezed states, as shown in [11,15–17]. However, the equations do not induce correlations between the two modes, thereby pointing to the inability of the NLSE model to generate entangled states. Such a desirable coupling of the wave and the particle mode is possible by the SIT-type interaction [10,12]. Given the classical shape n0 jcr j1=2 n0 jcr j z /sit ðt; zÞ ¼ t x0 sech 2 vr 2 exp ½i½ðp0 þ dÞt kr z þ h0
ð16Þ
of the SIT soliton and following similar lines as in the NLSE case, it is possible to formulate evolution equations for the four soliton operators in the SIT medium. These soliton operators are again defined via the mode decomposition, Eq. (1). As in the NLSE model, the derivatives in the mode functions in the expansion (1) are evaluated at p0 ¼ 0, h0 ¼ 0, and x0 ¼ 0. The trivial (classical) position and phase shifts are contained in the constants T0 and H0 entering the decomposition of Eq. (1). The corresponding solutions are [10,12] D^ nðzÞ ¼ D^ n0 ;
ð17Þ
oKr oKr zD^ n0 zDp^0 ; Dh^ðzÞ ¼ Dh^0 on0 op0
ð18Þ
Dp^ðzÞ ¼ Dp^0 ;
ð19Þ
D^ xðzÞ ¼ D^ x0 þ
ov1 ov1 r zDp^0 þ r zD^ n0 ; op0 on0
ð20Þ
with Kr kr p0 v1 r . The SIT-type interaction offers richer dynamics [cf. Eqs. (17)–(20) with Eqs. (12)–(15)] due to more complex frequency and intensity dependencies of the group velocity vr ðp0 ; n0 Þ ¼ v0 =ð1 þ SÞ and a richer dispersion relation kr ðp0 ; n0 Þ ¼ ðp0 þ dÞð1 SÞ=v0 . Both are controlled by a dimensionless coupling strength 2pxab Nd 2 1 ; S¼ 2 2 1 2 h n jc j þ ðp0 þ dÞ 4 0 r
ð21Þ
291
with the dipole moment d of the transition between the two levels separated by an optical frequency xab , the concentration N of the two-level systems and the frequency detuning ðp0 þ dÞ ¼ x xab . We also assume vanishing inhomogeneous R broadening. As in the NLSE model, n0 ¼ dt 2 j/sit j is the mean photon number. The coefficient cr ¼ Nr =2Dr is determined by the non-linearity Nr ¼
d 2 hx oKr n0 jcr j h2 Ar on0 8
2
ð22Þ
and the dispersion Dr ¼ 4
ovr ; op0
ð23Þ
with the cross-section Ar of the beam in the SIT medium. The two constants are of resonant origin which is in clear contrast to the purely dispersive nature of Nf and Df in the NLSE model. The last terms in Eqs. (18) and (20) are the desired cross-terms which correlate the two modes. In the Schr€ odinger picture this would correspond to the generation of an entangled state. The degree of the correlations is the important issue. Here we apply the criteria proposed by Reid and Drummond in [18] in the context of the Einstein–Podolsky–Rosen (EPR) paradox [19]. By construction, our quadratures (7) and (8) obey the commutation relations ½X^w ; P^w ¼ i ¼ ½X^p ; P^p . They cannot be both simultaneously specified more accurately than allowed by the Heisenberg uncertainty principle: D2 X^w D2 P^w > 14 and analogously for the ‘‘particle mode’’. On the other hand, if strong correlations between X^w and X^p and simultaneously between P^w and P^p are induced, then one can infer X^w and P^w from measurements of X^p and P^p . If the errors of these inferences are quantified by the two-mode variances
2 2 ^ ^ ^ Xw Xp Dinf Xw ;
2 P^w þ P^p ð24Þ D2inf P^w and result in 1 D2inf X^w D2inf P^w < ; 4
ð25Þ
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V.V. Kozlov, M. Freyberger / Optics Communications 206 (2002) 287–294
then this means a ‘‘contradiction’’ with local realistic theories [18]. It is this contradiction which constitutes the EPR argument. The quantum state of such EPR-correlated pairs of variables belongs to the class of entangled states. Note that the original formulation of the EPR argument was based on two spatially separated quantum particles with strongly correlated positions and momenta. In our case, the ‘‘wave mode’’ and the ‘‘particle mode’’ of the soliton can be similarly correlated, but in contrast to the EPR proposal they belong to the same object, and hence we characterize the corresponding states as intra-entangled. The EPR criteria Eq. (25) is now used to check whether the evolution described by Eqs. (17)–(20) leads to an EPR-type correlation. With Eqs. (17)– (20), we formulate the two pairs of quadrature amplitudes according to the definitions (7) and (8). Then we substitute them into Eq. (24), and define the total inference error D2tot D2inf X^w þ D2inf P^w :
ð26Þ
One can check that the condition D2tot < 1 necessarily implies the fulfillment of the EPR inequality Eq. (25). The formulation of the EPR paradox in terms of the sum (and not the product) of the inferences errors simplifies the calculations. In particular, it turns out that rotation angles ww and wp enter D2tot simply in the form W ww wp . The EPR correlations can be indeed obtained between the wave and the particle modes as demonstrated in Fig. 2 and they are maximal when the soliton is
Fig. 2. Inference error D2tot as the function of propagation depth z=Lres with Lres 2v0 =n0 jcjS, for the case of exact resonance, d ¼ 0. The plot is obtained by minimizing the D2tot with respect to three free parameters W, qw , and qp .
tuned to exact resonance where the self-terms [/ oKr =on0 and / ov1 r =op0 ] in Eqs. (18) and (20) vanish. For computing the plot in Fig. 2 we assume that the soliton is initially (i.e., at z ¼ 0) in a coherent state. The initial variances of the four involved operators are calculated using the projection definitions (2)–(5) taken at z ¼ 0, as shown in [10]. The result reads Z o/ 2 2 hðD^ n0 Þ i ¼ dt 0 ¼ n0 ; ð27Þ oh0 o/0 2 1 p2 1 1þ dt ¼ 3 on0 12 n0 0:6 ; n0
2 hðDh^0 Þ i ¼
Z
1 hðDp^0 Þ i ¼ 2 n0 2
1 hðD^ x0 Þ i ¼ 2 n0 2
ð28Þ
Z
o/0 2 n0 jcj2 ¼ dt ; ox0 12
ð29Þ
Z
o/0 2 p2 1 ¼ ; dt op0 3 n30 jcj2
ð30Þ
and all cross-correlations vanish. Here c stands for cf when the NLSE medium is considered and c ¼ cr in case of the SIT medium. Note that below we shall have to equalize cf and cr in order to satisfy the two models with one soliton solution. 2 One may check that neither the product of hðD^ n0 Þ i 2 2 and hðDh^0 Þ i nor the product of hðD^ p0 Þ i and hðD^ x0 Þ2 i minimizes the Heisenberg uncertainty relation. Therefore, the noise of the coherent soliton is always above the vacuum noise. This noise level is shown in Fig. 2 by the upper straight line. It is also seen that in order to build up the EPR-type correlations, the noise first has to be decreased to the level of vacuum fluctuations (denoted as the EPR boundary) and only then the correlations can be characterized as that of the EPR type. Until now the two media were treated separately. We finally turn our attention to the consideration of a transmission line as a combination of the two, see Fig. 1. The obvious requirement for such a combination is that the soliton in one medium appears simultaneously as the soliton in the other, that is, numbers of photons and frequencies of the fundamental solitons, Eqs. (11) and (16),
V.V. Kozlov, M. Freyberger / Optics Communications 206 (2002) 287–294
should be equal. That is we have the requirement [4,5] jcf j ¼ jcr j or Ar d 2 v0 jk 00 j ¼ ; Af 2p h2 x 0 n2
ð31Þ
when written as a relation between the beam crosssections. See also the caption to Fig. 1 for a numerical example. Then, for the two-component transmission line with a fiber of length lf and a two-level medium of length lr we get with Eqs. (13), (15), (18), and (20) oKr okf oKr Dh^ðLÞ ¼ Dh^0 lr þ lf D^ lr Dp^0 ; n0 on0 on0 op0 ð32Þ
ov1 ov1 ov1 r f lr þ lf D^ n0 : D^ xðLÞ ¼ D^ x0 þ p0 þ r lr D^ op0 op0 on0 ð33Þ The formation of the zero path-averaged GVD/ SPM transmission line is possible in the far-offresonant limit. We use the fact that the last terms in Eqs. (32) and (33) asymptotically decrease with detuning / d4 while the SIT terms inside the square brackets approach zero only as the inverse cube of the detuning. Therefore, for sufficiently large detunings the last terms in Eqs. (32) and (33) can be neglected while the slower decreasing SITterms in square brackets are used for canceling the NLSE contribution. The term in square brackets in Eq. (32) differs from the corresponding term in Eq. (33) only by a constant factor, so that they can be made zero simultaneously by an appropriate choice of lengths lr ¼ jDf =Dr jlf . Then, the propagation through such a line yields the evolution Dh^ðLÞ ¼ Dh^0 and D^ xðLÞ ¼ D^ x0 . Or, in other words, the initial quantum state is completely restored at the output. Hence such a compensated transmission line would be a perfect candidate for sending quantum information which is encoded in the twomode structure (‘‘wave mode’’ and ‘‘particle mode’’) of a quantum soliton. The same equations, (32) and (33), describe a hybrid line where the transmission and the preparation stage are combined so that an initially coherent state from Alice turns into an intraentangled state at Bob’s side. By choosing an ap-
293
propriate detuning (close to resonance), we can keep the bracketed terms zero, while the crossterms have a substantial value producing the desired entanglement as shown above. In conclusion, we have demonstrated how entangled states which are essential states for quantum information, can be generated within the internal quantum structure of an optical soliton and then transported through a transmission line consisting of a conventional fiber and a SIT medium. From the infinite number of terms in the expansion (1) we have used only those four which form two localized excitation modes. They remain decoupled from all other modes (continuum) during the evolution in the fiber and the SIT medium. Therefore they are perfect candidates for quantum information carriers. Since the solitons in contrast to other pulses are not distorted by fiber dispersion, one can produce intra-entangled quantum 10-ps solitons and carry 10 billion entangled quantum states per second through the transmission line.
Acknowledgements Financial support from the programme ‘‘QUBITS’’ of the European Commission is acknowledged. V.V.K. is grateful to Prof. Eberly and RTC members for warm hospitality in Rochester Theory Center at the University of Rochester and to the support from a grant to the RTC by Corning Inc. which made these visits possible.
References [1] For recent books on the subject, see: H.-K. Lo, T. Spiller, S. Popescu (Eds.), Introduction to Quantum Computation and Information, World Scientific, Singapore, 1998; J. Gruska, Quantum Computing, McGraw-Hill, London, 1999; D. Bouwmeester, A. Ekert, A. Zeilinger (Eds.), The Physics of Quantum Information, Springer, Berlin, 2000; M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000; G. Alber, T. Beth, M. Horodecki, P. Horodecki, R. Horodecki, M. R€ otteler, H. Weinfurter, R. Werner,
294
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
V.V. Kozlov, M. Freyberger / Optics Communications 206 (2002) 287–294 A. Zeilinger, Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments, Springer, Berlin, 2001. G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, New York, 2001. A. Hasegawa, Appl. Opt. 23 (1984) 3302. V.V. Kozlov, A.B. Matsko, J. Opt. Soc. Am. B 16 (1999) 519. A.B. Matsko, V.V. Kozlov, M.O. Scully, Phys. Rev. Lett. 82 (1999) 3244. S.L. McCall, E.L. Hahn, Phys. Rev. Lett. 18 (1967) 908; Phys. Rev. 183 (1969) 457. V.E. Zakharov, A.B. Shabat, Z. Eksp. Teor. Fiz. 61 (1972) 118 [Sov. Phys. – JETP 34 (1972) 62]. D.J. Kaup, J. Math. Phys. 16 (1975) 2036. H.A. Haus, Y. Lai, J. Opt. Soc. Am. B7 (1990) 386. Y. Lai, H.A. Haus, Phys. Rev. A 42 (1990) 2925. H.A. Haus, Electromagnetic Noise and Quantum Optical Measurements, Springer, Berlin, 2000. For the validity of the equal-space commutation rules in SIT media see: V.V. Kozlov, A.B. Matsko, J. Opt. Soc. Am. B 17 (2000) 1031.
[13] For the validity of the equal-space commutation rules in fibers see A.B. Matsko, V.V. Kozlov, Phys. Rev. A 62 (2000) 033811. [14] V.V. Kozlov, A.B. Matsko, Europhys. Lett. 54 (2001) 592. [15] For the first prediction of squeezing of solitons see P.D. Drummond, S.J. Carter, J. Opt. Soc. Am. B 4 (1987) 1565. [16] For reviews on quantum properties of optical solitons including squeezing see P.D. Drummond, R.M. Shelby, S.R. Friberg, Y. Yamamoto, Nature 365 (1993) 307; A. Sizmann, G. Leuchs, in: E. Wolf (Ed.), Progress in Optics, XXVIII, Elsevier, Amsterdam, 1999, p. 373; G. Leuchs, N. Korolkova, Opt. Photon. News 13 (2002) 64. [17] For recent observation of multi-mode quantum correlations in fiber solitons see S. Sp€alter, N. Korolkova, F. K€ onig, G. Leuchs, Phys. Rev. Lett. 81 (1998) 786. [18] M.D. Reid, P.D. Drummond, Phys. Rev. Lett. 60 (1988) 2731; M.D. Reid, Phys. Rev. A 40 (1989) 913. [19] A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47 (1935) 777. [20] H. Gotoh et al., Appl. Phys. Lett. 72 (1998) 1341.
Optics Communications 206 (2002) 287–294 www.elsevier.com/locate/optcom
High-bit-rate quantum communication Victor V. Kozlov *, Matthias Freyberger Abteilung f€ur Quantenphysik, Universit€at Ulm, Ulm D-89081, Germany Received 25 January 2002; accepted 10 April 2002
Abstract We show how an entangled state can be generated within the internal quantum structure of a short picosecond pulse in the form of a soliton. One can then transmit such an intra-entangled soliton through a conventional communication fiber. We suggest basic principles of the transmission of a quantum state through non-linear and dispersive dielectrics. Ó 2002 Elsevier Science B.V. All rights reserved. PACS: 03.67.Hk; 42.81.Dp; 42.50.Md; 42.65.Tg
Quantum communication is a key element in the modern field of quantum information processing [1]. We approach this problem using methods already developed in classical communication. In particular, ultrafast all-optical technologies and communication lines made of silica fibers make it nowadays possible to speed up the signal transmission by utilizing the wide frequency bandwidth of optical radiation [2]. In analogy, we shall show that the ultrashort optical pulses are also able to provide high-bit rate fiber quantum communication where the object of transmission is a quantum state rather than a classical signal. The higher the transmission rate, the shorter are the pulses and the more they are distorted by the fiber dispersion and non-linearity. These effects, mainly associated with group-velocity dispersion
* Corresponding author. Tel.: +49-731-502-3088; fax: +49731-502-3086. E-mail address: [email protected] (V.V. Kozlov).
(GVD) and self-phase modulation (SPM) [2], severely limit the transmission rate. Distortion-free propagation is however possible with optical solitons, i.e., in the regime when SPM is balanced by GVD [2,3]. This feature of shape preservation makes the soliton a perfect candidate as an information carrier. In the following, we shall demonstrate how to deliver a desirable quantum state from a sender station Alice to a receiver Bob. When one transmits a quantum object through optical fibers its state will change under the action of dispersion and non-linearity. Therefore, Bob receives a state which is different from what has been sent by Alice though at the same time the classical shape remains intact (apart from a trivial position and phase shift). Such shape preservation would be good enough for classical fiber communication but it is not sufficient for quantum communication. One can however avoid the distortion of the quantum state by constructing a transmission line with zero path-averaged GVD and zero path-averaged SPM, as shown in [4,5]. This is accomplished by use of a combination of a two-level
0030-4018/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 ( 0 2 ) 0 1 4 5 6 - 6
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system and a conventional fiber. The two-level system supports solitons of self-induced trans parency (SIT) [6] and the ordinary fiber gives rise to the well-known solitons of a non-linear Schr€ odinger equation (NLSE) [2,3,7], see Fig. 1 for details. Using these two non-linear media the compensation mechanism works as follows. The faroff-resonance two-level medium can display negative non-linearity and positive dispersion. Therefore, it appears as a ‘‘mirror image’’ of the fiber with positive SPM and negative GVD [4,5]. Since SIT and NLSE solitons both have the same sech shape and provided their amplitudes are made equal, the soliton emerging from one of the media and entering the other (its ‘‘mirror image’’) does not experience any transient reshaping and associated interconnection losses. The important change is that at the boundary the SPM phase shift reverses the direction of rotation. Hence, we can get full dispersion and non-linearity cancellation when the lengths of the media are properly adjusted, and then the net non-linear phase shift accumulated through the total line becomes zero. The same effect of reversal with following full cancellation is also characteristic for the evolution of any classical or quantum variable. In the limit of vanishing losses, such a zero path-averaged GVD/SPM transmission line, if considered as the input–output block, acts like vacuum: any quan-
Fig. 1. Sketch of a two-component transmission line between Alice and Bob. The line has total length L consisting of a fiber of length lf and a SIT medium of length lr . The two-level (SIT) medium compensates for the GVD and SPM distortion of the quantum signal which was picked up during propagation through the fiber. The zero path-averaged GVD/SPM line requires lr ¼ jDf =Dr jlf . For example, 1 micron-long sample of InGaAs with d ¼ 2 1028 C m, d ¼ 1:0 THz, and N ¼ 1015 cm3 , see [20], is needed to compensate the SPM and GVD of a 4 km-long communication fiber with n2 ¼ 1:2 1022 m2 =V2 and k 00 ¼ 1:0 ps2 =km. Also, Ar ¼ 10Af . For notations see the main text.
tum state entering the system will non-trivially evolve inside but will eventually reproduce itself when reaching the output. Quantum information processing involves transmission and/or exchange of different types of quantum states. Since a conventional mode-locked laser generates pulses only in a coherent state, there is a need for an additional tool capable of engineering other basic types of states such as squeezed and entangled states. In the following we shall show how the on-resonant SIT-type interaction can be exploited as an entangler or a squeezer of initially coherent solitons. Given this tool and the link to Bob via zero path-averaged GVD/SPM transmission line, Alice can prepare on demand and then deliver to Bob coherent, squeezed, or entangled states. Here, the optical soliton plays the role of a classical carrier of quantum information. The single classical shape, which is preserved throughout the preparation–transmission process, allows us to encode in it different types of quantum states. The soliton will simply carry these states through the line. Note that the preparation and the transmission stage both include the SIT two-level medium but in two completely different regimes: first we utilize its on-resonant features and then we need the faroff-resonant interaction. Instead of using two different SIT media one can also combine the two stages. Then a single two-level transition now slightly detuned from resonance will serve both stages simultaneously. In this hybrid setup, the quantum state at Alice’s side is intentionally underdeveloped to such extent that further evolution in the fiber turns it into the desirable form exactly at the distance where the soliton emerges at Bob’s side. The paper is organized as follows. We start from a short overview of the quantum-mechanical description of SIT and NLSE solitons and put them on the same footing. We then describe the evolution of the soliton through the NLSE and the SIT medium, respectively. By referring to wellknown entanglement criteria and adapting them to the soliton context we show (i) how the SIT interaction produces an entangled state of the soliton from an initially coherent field; (ii) how to build the zero path-averaged GVD/SPM trans-
V.V. Kozlov, M. Freyberger / Optics Communications 206 (2002) 287–294
289
Z
o/
dt i 0 v^ðt; zÞ þ H:c:; on0
ð3Þ
Z
o/ 0 v^ðt; zÞ þ H:c:; dt i n0 ox0
ð4Þ
Z
o/ 0 dt i v^ðt; zÞ þ H:c:; n0 op0
ð5Þ
mission line and (iii) how to combine the preparation and the transmission stage by sharing one SIT medium, such that an initially coherent state turns into an entangled (or squeezed) state when reaching Bob. An elegant way of analyzing the evolution of quantum NLSE and SIT solitons is the Inverse Scattering Method [7–10]. Since optical solitons contain a large number of photons (108 ), the total quantum field /^ðt; zÞ can be decomposed in /^ ¼ / þ v^ with a large classical mean value / and a small quantum perturbation v^ such that all higher terms than linear in v^ can then be neglected, [11]. The classical fundamental soliton /ðt; zÞ has the shape of sech and is fully characterized by four parameters: photon number n0 , initial phase h0 , frequency detuning (momentum per photon) p0 , and initial position x0 . This observation suggests an expansion of the quantum perturbation v^ðt; zÞ into a complete set of temporal–spatial modes as o/0 o/ o/ o/ v^ ¼ D^ n þ 0 Dh^ þ 0 D^ p þ 0 D^ x þ v^c eiH0 ; on0 oh0 op0 ox0 ð1Þ
Dh^ðzÞ ¼ eiH0
where the first four terms describe the perturbations localized within the soliton while v^c ðt; zÞ accounts for the continuum, i.e., the perturbations delocalized in space and time. Here we shall use the Heisenberg picture in which the operators D^ n etc. evolve with distance z. The mode functions o/0 =on0 , etc. are obtained as derivatives of the initial classical shape /0 ¼ /ðt T0 ; 0Þ. The position shift T0 is chosen such that the peak of the /0 coincides with the peak of the soliton at any z. Additionally, in order to always have a reference frame which is moving with the soliton, we explicitly extract from the expansion Eq. (1) the classical SPM-induced phase shift H0 . The soliton operators have the simple interpretation as quantum perturbations at a distance z of the corresponding classical variables, e.g., D^ n stands for the perturbation of the mean photon number n0 . Each operator from the decomposition (1) can be extracted by appropriately projecting the total field v^ðt; zÞ. Thus we can write [8–11], Z o/
D^ nðzÞ ¼ eiH0 dt i 0 v^ðt; zÞ þ H:c:; ð2Þ oh0
when the total field obeys the relations ½/^ðt0 ; zÞ; /^y ðt; zÞ ¼ dðt t0 Þ and ½/^ðt; z0 Þ; /^ðt; zÞ ¼ 0 [12,13]. Eq. (6) suggest to regard the four operators as two collective modes of the quantized soliton: ðD^ n; Dh^Þ is the ‘‘wave mode’’ and ðn0 Dp^; D^ xÞ is the ‘‘particle mode’’ [11,14]. Though the two-mode Hilbert space is only a sub-space of the infinitely dimensional Hilbert space of the soliton, it has an important property. The evolution in the two-mode sub-space is decoupled from the dynamics of other degrees of freedom. This assures that during propagation of the soliton any quantum information imprinted on the four soliton variables will not leak into other modes (continuum). The four variables are also operationally sound. After reaching Bob, a desirable soliton variable or any linear combination of them can be projected out of the total field /^ðz; tÞ by a homodyne measurement with properly shaped local oscillators. This way of measurement [10] is deduced from the special projection given by the integrals in Eqs. (2)–(5). Note that such a measurement as well as the very possibility of the effective two-mode description is due to the unique
D^ pðzÞ ¼ e
iH0
D^ xðzÞ ¼ eiH0
and similarly for a mode from the continuum. The advantages of the set of modes used in (1) show up when we substitute this expansion into NLSE or SIT equations as shown in [9–11]. On this way we shall derive evolution equations for the operators, and what is important for our analysis, the evolution of the localized perturbations appears to be decoupled from the evolution of the continuum. We can therefore exclude the continuum from our further considerations and follow only the dynamics of the four soliton variables (2)–(5). Moreover, one finds from Eqs. (2)–(5) that they obey simple commutators ½D^ n; Dh^ ¼ i
and ½D^ x; n0 Dp^ ¼ i;
ð6Þ
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feature of solitons – the orthogonality between the localized and continuum modes at all distances. Using this two-mode description we can introduce two pairs of rotated quadrature amplitudes Xw sin ww cos ww qw Dh^ ¼ ; ð7Þ Pw cos ww sin ww q1 n w D^
Xp Pp
¼
1=2 n0
cos wp sin wp
sin wp cos wp
"
# x qp D^ ; ^ q1 p Dp
ð8Þ
where qp and qw provide a convenient scaling and ww (wp ) is the wave (particle) rotation angle. Quantum states in the Hilbert space which is spanned by the eigenstates of these quadratures can be used for the transmission of quantum signals. In particular, an initially coherent state j/i can be transformed into an entangled state or squeezed state, and then transported to Bob through a zero path-averaged GVD/SPM line. Given this effective two-mode description of solitons, we now turn to the effective preparation of the non-classical quantum states in the NLSE and then in the SIT medium. The crucial step is to recognize that the decomposition (1) puts the description of the two solitons on the same footing. The output shape after propagating in one medium is simultaneously the input shape for the other medium. Thus, by using the classical NLSE soliton shape /nls in Eq. (1) and substituting it into the NLSE we shall derive evolution equations for the four soliton operators. A similar procedure can be applied in the SIT case but now with the SIT soliton shape /sit in Eq. (1). Under appropriate matching conditions explained below we get /nls ðt; 0Þ / /sit ðt; 0Þ and therefore the mode functions in expansion (1) are the same for the two problems. The corresponding operators (2)–(5), however, evolve differently. We start with the NLSE medium for which the evolution equation reads [15]. o/^ k 00 o2 /^ ¼ i þ Nf /^y /^2 ; oz 2 os2
ð9Þ
where z is the propagation distance, s ¼ t ðnf =v0 Þz is the retarded time, v0 is the speed of light in vacuum, nf is the group index at frequency x0 , k 00 is the (negative) GVD at x0 , and Nf ¼ hx20 n2 =Af v0
is the fiber non-linearity with the Kerr coefficient n2 and the effective fiber core area Af . The field operators /^ and /^y are slowly varying amplitudes of positive and negative frequency parts ffi of the electric pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi field operator E^ðt; zÞ ¼ hx0 =Af /^ðt; zÞ exp½iðx0 t kzÞ centered near frequency x0 . Substituting /^ ¼ / þ v^ into Eq. (9) and then linearizing this equation near the classical solution we arrive at
o^ v k 00 o2 v^ 2 2 y ^ ^ i ¼ þ N 2 j/j v þ / v : ð10Þ f oz 2 os2 As mean field / in Eq. (10) we choose the fundamental soliton shape 1=2 n0 jcf j n0 jcf j z s x0 sech /nls ðs; zÞ ¼ 2 vf 2 exp ½iðp0 s kf z þ h0 Þ :
ð11Þ
Here p0 ¼ x x0 Rdenotes the mean frequency 2 detuning and n0 ¼ ds j/nls j is the mean photon number. We have also used cf ¼ Nf =2Df with Df 1 00 k . The dispersion relation for the fiber reads 2
2
kf ðp0 ; n0 Þ ¼ p02 14 n20 jcf j k 00 and the group ve-
locity in the retarded frame is given by vf ¼ 1 ðokf =op0 Þ . Expanding v^ðt; zÞ around /nls as suggested in Eq. (1), then substituting the resulting decomposition into the linearized NLSE (10), and finally projecting both sides of the equation by an appropriate projection function, as demonstrated in Eqs. (2)–(5), we get a self-contained set of evolution equations whose solutions are D^ nðzÞ ¼ D^ n0 ; Dh^ðzÞ ¼ Dh^0
ð12Þ okf zD^ n0 ; on0
ð14Þ
Dp^ðzÞ ¼ Dp^0 ; D^ xðzÞ ¼ D^ x0 þ
ð13Þ
ov1 f zDp^0 ; op0
ð15Þ
with the initial fluctuations D^ n0 D^ nð0Þ, etc. Here the derivatives are evaluated at p0 ¼ 0, h0 ¼ 0, and x0 ¼ 0. The evolution takes place separately within the wave and the particle mode and it looks like free propagation of two independent massive particles. Since the last terms in Eqs. (13) and (15) are re-
V.V. Kozlov, M. Freyberger / Optics Communications 206 (2002) 287–294
sponsible for coupling the variables within the same mode, we shall refer to them as self-terms. They can be used for producing squeezed states, as shown in [11,15–17]. However, the equations do not induce correlations between the two modes, thereby pointing to the inability of the NLSE model to generate entangled states. Such a desirable coupling of the wave and the particle mode is possible by the SIT-type interaction [10,12]. Given the classical shape n0 jcr j1=2 n0 jcr j z /sit ðt; zÞ ¼ t x0 sech 2 vr 2 exp ½i½ðp0 þ dÞt kr z þ h0
ð16Þ
of the SIT soliton and following similar lines as in the NLSE case, it is possible to formulate evolution equations for the four soliton operators in the SIT medium. These soliton operators are again defined via the mode decomposition, Eq. (1). As in the NLSE model, the derivatives in the mode functions in the expansion (1) are evaluated at p0 ¼ 0, h0 ¼ 0, and x0 ¼ 0. The trivial (classical) position and phase shifts are contained in the constants T0 and H0 entering the decomposition of Eq. (1). The corresponding solutions are [10,12] D^ nðzÞ ¼ D^ n0 ;
ð17Þ
oKr oKr zD^ n0 zDp^0 ; Dh^ðzÞ ¼ Dh^0 on0 op0
ð18Þ
Dp^ðzÞ ¼ Dp^0 ;
ð19Þ
D^ xðzÞ ¼ D^ x0 þ
ov1 ov1 r zDp^0 þ r zD^ n0 ; op0 on0
ð20Þ
with Kr kr p0 v1 r . The SIT-type interaction offers richer dynamics [cf. Eqs. (17)–(20) with Eqs. (12)–(15)] due to more complex frequency and intensity dependencies of the group velocity vr ðp0 ; n0 Þ ¼ v0 =ð1 þ SÞ and a richer dispersion relation kr ðp0 ; n0 Þ ¼ ðp0 þ dÞð1 SÞ=v0 . Both are controlled by a dimensionless coupling strength 2pxab Nd 2 1 ; S¼ 2 2 1 2 h n jc j þ ðp0 þ dÞ 4 0 r
ð21Þ
291
with the dipole moment d of the transition between the two levels separated by an optical frequency xab , the concentration N of the two-level systems and the frequency detuning ðp0 þ dÞ ¼ x xab . We also assume vanishing inhomogeneous R broadening. As in the NLSE model, n0 ¼ dt 2 j/sit j is the mean photon number. The coefficient cr ¼ Nr =2Dr is determined by the non-linearity Nr ¼
d 2 hx oKr n0 jcr j h2 Ar on0 8
2
ð22Þ
and the dispersion Dr ¼ 4
ovr ; op0
ð23Þ
with the cross-section Ar of the beam in the SIT medium. The two constants are of resonant origin which is in clear contrast to the purely dispersive nature of Nf and Df in the NLSE model. The last terms in Eqs. (18) and (20) are the desired cross-terms which correlate the two modes. In the Schr€ odinger picture this would correspond to the generation of an entangled state. The degree of the correlations is the important issue. Here we apply the criteria proposed by Reid and Drummond in [18] in the context of the Einstein–Podolsky–Rosen (EPR) paradox [19]. By construction, our quadratures (7) and (8) obey the commutation relations ½X^w ; P^w ¼ i ¼ ½X^p ; P^p . They cannot be both simultaneously specified more accurately than allowed by the Heisenberg uncertainty principle: D2 X^w D2 P^w > 14 and analogously for the ‘‘particle mode’’. On the other hand, if strong correlations between X^w and X^p and simultaneously between P^w and P^p are induced, then one can infer X^w and P^w from measurements of X^p and P^p . If the errors of these inferences are quantified by the two-mode variances
2 2 ^ ^ ^ Xw Xp Dinf Xw ;
2 P^w þ P^p ð24Þ D2inf P^w and result in 1 D2inf X^w D2inf P^w < ; 4
ð25Þ
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V.V. Kozlov, M. Freyberger / Optics Communications 206 (2002) 287–294
then this means a ‘‘contradiction’’ with local realistic theories [18]. It is this contradiction which constitutes the EPR argument. The quantum state of such EPR-correlated pairs of variables belongs to the class of entangled states. Note that the original formulation of the EPR argument was based on two spatially separated quantum particles with strongly correlated positions and momenta. In our case, the ‘‘wave mode’’ and the ‘‘particle mode’’ of the soliton can be similarly correlated, but in contrast to the EPR proposal they belong to the same object, and hence we characterize the corresponding states as intra-entangled. The EPR criteria Eq. (25) is now used to check whether the evolution described by Eqs. (17)–(20) leads to an EPR-type correlation. With Eqs. (17)– (20), we formulate the two pairs of quadrature amplitudes according to the definitions (7) and (8). Then we substitute them into Eq. (24), and define the total inference error D2tot D2inf X^w þ D2inf P^w :
ð26Þ
One can check that the condition D2tot < 1 necessarily implies the fulfillment of the EPR inequality Eq. (25). The formulation of the EPR paradox in terms of the sum (and not the product) of the inferences errors simplifies the calculations. In particular, it turns out that rotation angles ww and wp enter D2tot simply in the form W ww wp . The EPR correlations can be indeed obtained between the wave and the particle modes as demonstrated in Fig. 2 and they are maximal when the soliton is
Fig. 2. Inference error D2tot as the function of propagation depth z=Lres with Lres 2v0 =n0 jcjS, for the case of exact resonance, d ¼ 0. The plot is obtained by minimizing the D2tot with respect to three free parameters W, qw , and qp .
tuned to exact resonance where the self-terms [/ oKr =on0 and / ov1 r =op0 ] in Eqs. (18) and (20) vanish. For computing the plot in Fig. 2 we assume that the soliton is initially (i.e., at z ¼ 0) in a coherent state. The initial variances of the four involved operators are calculated using the projection definitions (2)–(5) taken at z ¼ 0, as shown in [10]. The result reads Z o/ 2 2 hðD^ n0 Þ i ¼ dt 0 ¼ n0 ; ð27Þ oh0 o/0 2 1 p2 1 1þ dt ¼ 3 on0 12 n0 0:6 ; n0
2 hðDh^0 Þ i ¼
Z
1 hðDp^0 Þ i ¼ 2 n0 2
1 hðD^ x0 Þ i ¼ 2 n0 2
ð28Þ
Z
o/0 2 n0 jcj2 ¼ dt ; ox0 12
ð29Þ
Z
o/0 2 p2 1 ¼ ; dt op0 3 n30 jcj2
ð30Þ
and all cross-correlations vanish. Here c stands for cf when the NLSE medium is considered and c ¼ cr in case of the SIT medium. Note that below we shall have to equalize cf and cr in order to satisfy the two models with one soliton solution. 2 One may check that neither the product of hðD^ n0 Þ i 2 2 and hðDh^0 Þ i nor the product of hðD^ p0 Þ i and hðD^ x0 Þ2 i minimizes the Heisenberg uncertainty relation. Therefore, the noise of the coherent soliton is always above the vacuum noise. This noise level is shown in Fig. 2 by the upper straight line. It is also seen that in order to build up the EPR-type correlations, the noise first has to be decreased to the level of vacuum fluctuations (denoted as the EPR boundary) and only then the correlations can be characterized as that of the EPR type. Until now the two media were treated separately. We finally turn our attention to the consideration of a transmission line as a combination of the two, see Fig. 1. The obvious requirement for such a combination is that the soliton in one medium appears simultaneously as the soliton in the other, that is, numbers of photons and frequencies of the fundamental solitons, Eqs. (11) and (16),
V.V. Kozlov, M. Freyberger / Optics Communications 206 (2002) 287–294
should be equal. That is we have the requirement [4,5] jcf j ¼ jcr j or Ar d 2 v0 jk 00 j ¼ ; Af 2p h2 x 0 n2
ð31Þ
when written as a relation between the beam crosssections. See also the caption to Fig. 1 for a numerical example. Then, for the two-component transmission line with a fiber of length lf and a two-level medium of length lr we get with Eqs. (13), (15), (18), and (20) oKr okf oKr Dh^ðLÞ ¼ Dh^0 lr þ lf D^ lr Dp^0 ; n0 on0 on0 op0 ð32Þ
ov1 ov1 ov1 r f lr þ lf D^ n0 : D^ xðLÞ ¼ D^ x0 þ p0 þ r lr D^ op0 op0 on0 ð33Þ The formation of the zero path-averaged GVD/ SPM transmission line is possible in the far-offresonant limit. We use the fact that the last terms in Eqs. (32) and (33) asymptotically decrease with detuning / d4 while the SIT terms inside the square brackets approach zero only as the inverse cube of the detuning. Therefore, for sufficiently large detunings the last terms in Eqs. (32) and (33) can be neglected while the slower decreasing SITterms in square brackets are used for canceling the NLSE contribution. The term in square brackets in Eq. (32) differs from the corresponding term in Eq. (33) only by a constant factor, so that they can be made zero simultaneously by an appropriate choice of lengths lr ¼ jDf =Dr jlf . Then, the propagation through such a line yields the evolution Dh^ðLÞ ¼ Dh^0 and D^ xðLÞ ¼ D^ x0 . Or, in other words, the initial quantum state is completely restored at the output. Hence such a compensated transmission line would be a perfect candidate for sending quantum information which is encoded in the twomode structure (‘‘wave mode’’ and ‘‘particle mode’’) of a quantum soliton. The same equations, (32) and (33), describe a hybrid line where the transmission and the preparation stage are combined so that an initially coherent state from Alice turns into an intraentangled state at Bob’s side. By choosing an ap-
293
propriate detuning (close to resonance), we can keep the bracketed terms zero, while the crossterms have a substantial value producing the desired entanglement as shown above. In conclusion, we have demonstrated how entangled states which are essential states for quantum information, can be generated within the internal quantum structure of an optical soliton and then transported through a transmission line consisting of a conventional fiber and a SIT medium. From the infinite number of terms in the expansion (1) we have used only those four which form two localized excitation modes. They remain decoupled from all other modes (continuum) during the evolution in the fiber and the SIT medium. Therefore they are perfect candidates for quantum information carriers. Since the solitons in contrast to other pulses are not distorted by fiber dispersion, one can produce intra-entangled quantum 10-ps solitons and carry 10 billion entangled quantum states per second through the transmission line.
Acknowledgements Financial support from the programme ‘‘QUBITS’’ of the European Commission is acknowledged. V.V.K. is grateful to Prof. Eberly and RTC members for warm hospitality in Rochester Theory Center at the University of Rochester and to the support from a grant to the RTC by Corning Inc. which made these visits possible.
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294
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
V.V. Kozlov, M. Freyberger / Optics Communications 206 (2002) 287–294 A. Zeilinger, Quantum Information: An Introduction to Basic Theoretical Concepts and Experiments, Springer, Berlin, 2001. G.P. Agrawal, Nonlinear Fiber Optics, Academic Press, New York, 2001. A. Hasegawa, Appl. Opt. 23 (1984) 3302. V.V. Kozlov, A.B. Matsko, J. Opt. Soc. Am. B 16 (1999) 519. A.B. Matsko, V.V. Kozlov, M.O. Scully, Phys. Rev. Lett. 82 (1999) 3244. S.L. McCall, E.L. Hahn, Phys. Rev. Lett. 18 (1967) 908; Phys. Rev. 183 (1969) 457. V.E. Zakharov, A.B. Shabat, Z. Eksp. Teor. Fiz. 61 (1972) 118 [Sov. Phys. – JETP 34 (1972) 62]. D.J. Kaup, J. Math. Phys. 16 (1975) 2036. H.A. Haus, Y. Lai, J. Opt. Soc. Am. B7 (1990) 386. Y. Lai, H.A. Haus, Phys. Rev. A 42 (1990) 2925. H.A. Haus, Electromagnetic Noise and Quantum Optical Measurements, Springer, Berlin, 2000. For the validity of the equal-space commutation rules in SIT media see: V.V. Kozlov, A.B. Matsko, J. Opt. Soc. Am. B 17 (2000) 1031.
[13] For the validity of the equal-space commutation rules in fibers see A.B. Matsko, V.V. Kozlov, Phys. Rev. A 62 (2000) 033811. [14] V.V. Kozlov, A.B. Matsko, Europhys. Lett. 54 (2001) 592. [15] For the first prediction of squeezing of solitons see P.D. Drummond, S.J. Carter, J. Opt. Soc. Am. B 4 (1987) 1565. [16] For reviews on quantum properties of optical solitons including squeezing see P.D. Drummond, R.M. Shelby, S.R. Friberg, Y. Yamamoto, Nature 365 (1993) 307; A. Sizmann, G. Leuchs, in: E. Wolf (Ed.), Progress in Optics, XXVIII, Elsevier, Amsterdam, 1999, p. 373; G. Leuchs, N. Korolkova, Opt. Photon. News 13 (2002) 64. [17] For recent observation of multi-mode quantum correlations in fiber solitons see S. Sp€alter, N. Korolkova, F. K€ onig, G. Leuchs, Phys. Rev. Lett. 81 (1998) 786. [18] M.D. Reid, P.D. Drummond, Phys. Rev. Lett. 60 (1988) 2731; M.D. Reid, Phys. Rev. A 40 (1989) 913. [19] A. Einstein, B. Podolsky, N. Rosen, Phys. Rev. 47 (1935) 777. [20] H. Gotoh et al., Appl. Phys. Lett. 72 (1998) 1341.