Entangled Quantum Nonlinear Schro¨dinger Solitons
Yinchieh Lai1,2and Ray-Kuang Lee31Department of Photonics, National Chiao-Tung University, Hsinchu 300, Taiwan, Republic of China 2Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan, Republic of China 3
Institute of Photonics Technologies, National Tsing-Hua University, Hsinchu 300, Taiwan, Republic of China (Received 6 February 2009; published 2 July 2009)
Considered as a multipartite quantum system, time-multiplexed nonlinear Schro¨dinger solitons after collision are rigorously proved to become quantum entangled in the sense that their quadrature components of suitably selected internal modes satisfy the inseparability criterion. Clear physical insights for the origin of entanglement are given, and the required homodyne local oscillator pulse shape for optimum entanglement detection is determined.
DOI:10.1103/PhysRevLett.103.013902 PACS numbers: 42.65.Tg, 03.67.Bg, 05.45.Yv, 42.50.Dv
The quantum nonlinear Schro¨dinger equation (QNLSE) has been widely used as a model equation for studying the quantum effects of bosonic solitons. In particular, quantum optical solitons in photonic waveguides with Kerr nonline-arity can be accurately described by the QNLSE [1–3]. For weakly interacting ultracold atoms in a Bose-Einstein con-densate, the bosonic matter wave field evolves also in the form of QNLSE, possibly with additional terms of linear or periodic potentials [4,5]. Based on the studies of the QNLSE, the possibility of squeezing generation through quantum solitons was predicted by theoretical works in 1987 [1] and subsequently confirmed by fiber soliton ex-periments in 1991 [6]. Some of the important theoretical approaches for solving QNLSE and other more compli-cated quantum nonlinear pulse propagation problems in-clude the quantum stochastic simulation method [2], the Bethe’s ansatz method [3], the quantum perturbation the-ory [7], the back-propagation method [8], and the cumulant expansion technique [9]. Some of the important experi-ments include the quantum nondemolition measurement using solitons [10], the generation of amplitude squeezed states through optical filtering [11,12] or imbalanced non-linear interference [13], the intrasoliton photon number quantum correlation in both the spectra and time domain [14,15], and the generation of continuous variable Einstein-Podolsky-Rosen entangled states by adiabatically expanding an optical vector soliton [16]. Among them, the generation of entangled states using nonlinear Schro¨dinger solitons is of particular interest for possible quantum in-formation applications.
The known optical soliton scheme of generating con-tinuous variable Einstein-Podolsky-Rosen states is based on the generation and mixing of two independent squeezed vacuum states from a fiber squeezer [17]. In the soliton quantum nondemolition measurement schemes, soliton collision has been widely utilized to induce a quantum correlation between two solitons of different wavelengths or polarizations. The photon number noises of one soliton can be encoded to the phase noises of the other soliton
through the cross-phase modulation and thus create quan-tum correlation between the two solitons. Recently we have also shown that the photon number correlation of two time-multiplex solitons can be directly established through nonlinear interaction [15]. The whole system is a pure state of infinite modes if no optical loss is assumed. However, if one selects one mode for each soliton and traces out all the other modes to form a two-partite system, the reduced two-partite state will be a mixed state in general. Such a reduced two-partite state is thus not guar-anteed to be entangled even when the quantum correlation has been proven. The situation is also true for studying finite internal modes of a single soliton, where the quantum correlation is known to exist but the quantum entanglement is not for sure.
Even for squeezed state generation using solitons, there are still important issues left unanswered. Typically the generation of pulse squeezed vacuum states from solitons is through the use of a balanced nonlinear interferometer. The homodyne detection scheme is then used to detect the quadrature component of the output squeezed vacuum state with the mean field pulse from the other port of the interferometer as the local oscillator. Larger squeezing can be detected if one only detects the soliton parts and rejects all the continuum parts [7]. In several studies of squeezing generation using nonfundamental solitons, it has also been suggested that the continuum may help with achieving larger squeezing within some parameter space [18]. The determination of optimal local oscillators for squeezing detection has been investigated in the literature and has been related to eigenfunction problems based on the correlation properties or the transformation matrix of the multimode field operators [19–21]. Description of in-trasoliton photon number correlations in terms of three simply chosen internal modes has also been shown to be effective [21]. However, since the considered soliton sys-tems here are intrinsic, complicated multimode problems, the complete correlation properties or transformation ma-trices of the multimode field operators are not easy to PRL 103, 013902 (2009) P H Y S I C A L R E V I E W L E T T E R S 3 JULY 2009week ending
obtain as the starting point. It is thus very desirable to develop numerically efficient algorithms that can directly determine the optimal local oscillators for squeezing or entanglement detection as well as the optimal basis func-tions for the mode description of quantum solitons. Recently, experimental observation of squeezed lights with 10 dB quantum noise reduction from optical para-metric oscillation processes has been reported [22]. It will be interesting to see whether the soliton schemes can also generate and detect large quantum squeezing through optimization.
In the this Letter we start by developing a theory that can efficiently determine the true optimum homodyne local oscillator pulse shape for soliton squeezing detection. Our theory leads to the discovery of the ‘‘natural’’ internal squeezing modes that are minimum-uncertainty states. By thinking in terms of these internal squeezing modes, one can easily understand why there are intra- and intersoliton quantum correlations and how to optimally detect the correlation. Most importantly, based on the theory we can rigorously prove that the time-multiplexed optical solitons after nonlinear interaction are indeed quantum mechanically entangled in the sense that the ‘‘quadrature components’’ of the specially selected multipartite state can satisfy the following inseparability criterion: the un-certainty product of the inferred quadrature components is below the Heisenberg uncertainty product limit.
We start from the well known QNLSE given below:
@ ^U @z ¼ i12 @
2U^ @t2 þ i ^U
y ^
U ^U . By assuming the quantum noises are much less than the mean fields, the linearization ap-proximation can be justified. The quantum noise part of the soliton can be described by the following linearized opera-tor equation: @ ^u @z ¼ i 1 2 @2^u @t2 þ i2U0U0^u þ iU20^uy: (1)
Here U0ðz; tÞ is the classical solution and ^uðz; tÞ, ^uyðz; tÞ
are the perturbed quantum field operators. In order to calculate the quantum noises by the back-propagation method [8], the adjoint system of Eq. (1) is intro-duced by requiring the inner product of the solutions of the two systems to be a conserved quantity along z. Here the definition of the inner product is given by huAðz; tÞj ^uðz; tÞi ¼R1
2½uAðz; tÞ ^uðz; tÞ þ H:c:dt. This
leads to the following classical linear adjoint evolution equation: @uA @z ¼ i 1 2 @2uA @t2 þ i2U0 U0uA iU20uA: (2)
Since both Eqs. (1) and (2) are linear, their solutions can be formally written as ^uðz; tÞ ¼ Lz 0 ^uð0; tÞ and
uAðz; tÞ ¼ A
z 0 uAð0; tÞ. Here Lz 0 and Az 0 are the
formal evolution operators of the two systems (linear and adjoint) from 0 to z. The symbol is introduced to remind us of the fact that these formal linear differential operators operate on both^u and ^uy. With such compact notations and
by assuming the initial input state is a coherent state, the detected squeezing ratio after the propagation distance z can be nicely expressed as
RðzÞ ¼hA0 z fðtÞjA0 z fðtÞi
hfðtÞjfðtÞi : (3)
Here fðtÞ is the local oscillator pulse used in the homo-dyne detection, and A0 z fðtÞ is the back-propagated
local oscillator pulse through the adjoint system. Mathematically Eq. (3) can be viewed as a functional of fðtÞ, and the condition for its stationary solutions can be determined by performing a variation with respect to fðtÞ. Using the fact that the inner product of two solutions is conserved along z, one has hA0 z fðtÞjA0 z fðtÞi ¼
hfðtÞjLz 0A0 z fðtÞi. It is then easy to show that the
variational equation RðzÞ ¼ 0 leads to the following ei-genvalue problem with the eiei-genvalue equal to the optimum squeezing ratio:
Lz 0A0 z fðtÞ ¼ fðtÞ: (4)
Equation (4) is one of the main results in this Letter. It elegantly describes the necessary condition that the opti-mal local oscillator pulse shape must satisfy. Since we are mainly interested in the solution with the globally mini-mum eigenvalue , the numerical inverse power method can be applied to Eq. (4) for iteratively approaching the eigensolutions we want to find.
It is not difficult to prove that if fðtÞ is the eigenstate of Lz 0A0 z with the eigenvalue , then i fðtÞ is also the
eigenstate of Lz 0A0 z with the eigenvalue 1=. This
implies that if one uses fðtÞ as the basis to project out the corresponding internal mode field operator ^aðzÞ ¼ R
fðtÞ ^uðz; tÞdt=RjfðtÞj2dt, the projected mode will be a
minimum-uncertainty state with one quadrature squeezed and the other quadrature antisqueezed. So the results in Eq. (4) physically imply that the minimum-uncertainty state requirement is the necessary condition to achieve optimum squeezing detection. This is a very meaningful result that can provide us with deeper physical insights about the internal squeezing modes of the solitons. This set of internal modes is the natural basis set for describing the quantum noise properties of the soliton, in the sense that there is no quantum correlation among these internal modes. This can be easily seen from the formula for calculating the quantum correlation of two measured op-erators by the homodyne detection: C12ðzÞ / hA0 z
f1ðtÞjA0 z f2ðtÞi, where f1ðtÞ and f2ðtÞ are the two local
oscillator functions.
With the above physical insights, we now demonstrate how to determine the optimum local oscillator pulse shapes for detecting entanglement. To illustrate, let us consider the intersoliton case and assume the classical time-multiplexed two-soliton solution is symmetric in time t. Assume foptðtÞ
is the found (symmetric) eigenfunction with the smallest eigenvalue opt and we choose the two normalized local
oscillator functions for detecting the two solitons to be PRL 103, 013902 (2009) P H Y S I C A L R E V I E W L E T T E R S 3 JULY 2009week ending
f1ðtÞ / foptðtÞ for t > 0 and f2ðtÞ / foptðtÞ for t < 0. Since
f1ðtÞ is zero for t < 0 and f2ðtÞ is zero for t > 0, the two
functions do not overlap in time and correspond to the measurements on each soliton. The four related quadrature components are ^q1¼ hf1j ^ui, ^p1 ¼ hif1j ^ui, ^q2 ¼ hf2j ^ui,
and p^2¼ hif2j ^ui. From the minimum-uncertainty state
property stated above, it is not difficult to prove that the squeezing ratio for Var½ ^q1þ ^q2 is just opt 1, and the
squeezing ratio of Var½ ^p1þ ^p2 is just 1=opt 1. So ^q1
and ^q2 are anticorrelated, whilep^1 andp^2 are correlated.
To more accurately estimate how much p^1 and ^p2 are
correlated, we need to estimate the squeezing ratio of Var½ ^p1 ^p2. Note that the projection function for ^p1
^
p2 is antisymmetric, and thus it will be orthogonal to i
foptðtÞ. Therefore ^p1 ^p2 does not contain any
contribu-tion from the optimum internal mode. The squeezing ratio of Var½ ^p1 ^p2 is thus upper bounded by 1=snd, with snd
being the second smallest eigenvalue of the system. Based on these observations, we now have the following impor-tant result:
Squeezing Ratio of Var½ ^q1þ ^q2
Var½ ^p1 ^p2
opt
snd
< 1: Here the definition of the squeezing ratio of the uncertainty product is to compare the uncertainty product with the case of two independent coherent states.
The above result is a sufficient condition for proving that the two solitons after collision are indeed entangled in the sense that the inseparability criterion for bipartite continu-ous variables is satisfied [23]. The proof can be easily gen-eralized to the more general soliton collision/interaction cases. It can also be applied to the single soliton case to determine the optimum local oscillator for detecting intra-soliton entanglement. For the case of multipartite N iden-tical solitons, the proof still can be applied by simply noting that the squeezing ratio for Var½c1^q1þ . . . þ
ck^qkþ . . . þ cN^qN is opt, if ck¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rtkþ=2 tk=2jfoptðtÞj 2dt q . Here the projection function for the kth soliton is chosen to be foptðtÞ=ck within its time and to be
zero elsewhere. The squeezing ratio for Varf ^pk
½ðcffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1^p1 þ . . . þ ck1p^k1 þ ckþ1^pkþ1 þ . . . þ cN^pNÞ=
jc1j2 þ . . . þ jck1j2 þ jckþ1j2 þ . . . þ jcNj2
p
g is
then upper bounded by 1=snd since its projection
func-tion is orthogonal to foptðtÞ. Therefore one can use
ðc1^q1 þ . . . þ ck1^qk1þ ckþ1^qkþ1þ . . . þ cN^qNÞ=ck and ðc1^p1þ . . . þ ck1p^k1þ ckþ1p^kþ1þ . . . þ cN^pNÞ= ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jc1j2þ . . . þ jck1j2þ jckþ1j2þ . . . þ jcNj2 p to infer ^qk and ^pk in order to satisfy the nonlocal criterion
between the kth mode and the rest N 1 modes.
As some numerical examples, let us consider the two-soliton and three-two-soliton collision cases illustrated in Fig. 1. The initial conditions are 2 or 3 solitons of the same phase with the soliton amplitude ¼ 1 and separation ¼ 5. Such a bound soliton pair/train will evolve periodically (collide, separate, and collide again as breath-ers). For the 2-soliton case, the optimum mode function foptðtÞ at z ¼ 20 is plotted in Fig.2. The eigenvalues opt¼
33:0 dB and snd¼ 22:4 dB. Therefore one can
ex-pect the squeezing ratio of the inferred uncertainty product is 10:6 dB. For the 3-soliton case, the optimum mode function foptðtÞ at z ¼ 25 is plotted in Fig. 3. The
eigen-values opt¼ 35:4 dB and snd¼ 24:4 dB. This time
it is 11:0 dB below the Heisenberg uncertainty product limit. Actual numerical calculation of the squeezing ratio usually yields a smaller number than the predicted upper bound because the squeezing ratio of Var½ ^p1 ^p2 is
usually less than the bound 1=snd.
In practice the achievable squeezing may be limited by optical losses, nonlinear scattering noises, and detector quantum efficiency. The conventional length normalization unit (pulse-width2=dispersion) used in the theory can be
about several meters if hundreds of fs pulses are used and can be up to several hundred meters if ps pulses are used. Currently more than 6 dB squeezing has been reported with the help of gigahertz erbium-doped fiber lasers [24] and photonic crystal fibers [25]. If the soliton separation is reduced, the required propagation length as well as the achievable squeezing can also be reduced. In this way, the predicted squeezing or entanglement ratio can be adjusted to be of the order of several dB, located within the
observ-FIG. 1 (color online). Intensity evolution patterns for (a) 2 in-phase solitons and (b) 3 in-phase solitons. Soliton separation ¼ 5.
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able range of current technologies. The impacts of fiber losses on the achievable squeezing or entanglement for given local oscillators can be readily calculated by the back-propagation method [8]. Determination of the opti-mal local oscillator in the presence of fiber losses should also be possible with some further development of the theory. The theory developed here should also be appli-cable to the study of Bose-Einstein condensates. However, in contrast to the traditional soliton perturbation theory or the Bose-Einstein condensation Bogoliubov–de Gennes equation approach based on the perturbed nonlinear Schro¨dinger equation [4], the expansion eigenmodes em-ployed in this Letter are not the (generalized) eigenmodes of the perturbed nonlinear equation itself. Instead, they are the eigenmodes of the cascaded (linearized þ adjoint) evo-lution operators for a fixed propagation length.
In conclusion, we have presented an elegant theory to rigorously prove that multipartite entangled states can be directly generated by time-multiplexed solitons. The en-tanglement can only be detected by using specially chosen homodyne local oscillators to project out the
correspond-ing quadrature components of the solitons. The optimum detection functions are related to the internal modes of the soliton systems, under which all the modes are uncorre-lated minimum-uncertainty states. The presented theory provides the way to find the optimum local oscillator pulse shapes for detecting intra- and intersoliton entanglements and helps to clarify the physical origin of the entanglement. The theoretical concept is general and should be also applicable to other soliton squeezing and entanglement schemes. The results presented here are believed to be helpful for future quantum information experiments that require larger squeezing or entanglement factors.
The work by Y. Lai is supported in part by the National Science Council in Taiwan under the projects of NSC 97-2120-M-001-002 and NSC 96-2628-E-009-154-MY3.
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Intensity and phase of f
optFIG. 3 (color online). foptðtÞ for the three-soliton case at z ¼
25. The solid line is for the intensity, and the dashed line for the phase.
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Intensity and phase of f
optFIG. 2 (color online). foptðtÞ for the two-soliton case at z ¼ 20.
The solid line is for the intensity, and the dashed line for the phase.
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