## Generation of two-mode squeezed and entangled light in a single temporal and spatial mode

Optics Express, Vol. 17, Issue 16, pp. 14444-14457 (2009)

http://dx.doi.org/10.1364/OE.17.014444

Acrobat PDF (319 KB)

### Abstract

We analyse a novel squeezing and entangling mechanism which is due to correlated Stokes and anti-Stokes photon forward scattering in a multi-level atom vapour. We develop a full quantum model for an alkali atomic vapour including quantized collective atomic states which predicts high degree of squeezing for attainable experimental conditions. Following the proposal we present an experimental demonstration of 3.5 dB pulsed frequency nondegenerate squeezed (quadrature entangled) state of light using room temperature caesium vapour. The source is very robust and requires only a few milliwatts of laser power. The squeezed state is generated in the same spatial mode as the local oscillator and in a single temporal mode. The two entangled modes are separated by twice the Zeeman frequency of the vapour which can be widely tuned. The narrow-band squeezed light generated near an atomic resonance can be directly used for atom-based quantum information protocols. Its single temporal mode characteristics make it a promising resource for quantum information processing.

© 2009 Optical Society of America

## 1. Introduction

1. T. C. Ralph, “Continuous variable quantum cryptography,” Phys. Rev. A **61**, 010303R (1999). [CrossRef]

2. A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional Quantum Teleportation,” Science **282**, 706 (1998). [CrossRef] [PubMed]

3. S. Lloyd and S. L. Braunstein, “Quantum Computation over Continuous Variables,” Phys. Rev. Lett. **82**, 1784–1787 (1999). [CrossRef]

4. A. Kuzmich, L. Mandel, and N. P. Bigelow, “Generation of Spin Squeezing via Continuous Quantum Nondemolition Measurement,” Phys. Rev. Lett. **85**, 1594 (2000). [CrossRef] [PubMed]

5. S. L. Braunstein, “Quantum error correction for communication with linear optics,” Nature (London) **394**, 47 (1998). [CrossRef]

6. S. Lloyd and J.-J. E. Slotine,“Analog Quantum Error Correction,” Phys. Rev. Lett. **80**, 4088–4091 (1998). [CrossRef]

7. L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik, “Quantum Communication between Atomic Ensembles Using Coherent Light,” Phys. Rev. Lett. **85**, 5643 (2000). [CrossRef]

8. D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A **67**, 062320 (2003). [CrossRef]

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8. D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A **67**, 062320 (2003). [CrossRef]

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20. M. Sasaki and S. Suzuki, “Multimode theory of measurement-induced non-Gaussian operation on wideband squeezed light: Analytical formula,” Phys. Rev. A **73**, 043807 (pages 18) (2006). [CrossRef]

## 2. Model of the interaction

*F*> 1/2 driven by a strong off-resonant linearly-polarized light. Prior to the interaction we optically pump all atoms into the extreme magnetic sublevel

*mF*=

*F*in the electronic ground state. The orientation of atomic spins is thus orthogonal to both the propagation and polarization direction of the driving field. This field is in a superposition of

*σ*

_{+}and

*σ*

_{-}polarizations in the quantization basis.

*a*̂

^{†}for the scattered photons and

*b*̂

^{†}for the collective atomic excitation into the magnetic sublevel

*mF*=

*F*- 1. The Hamiltonian of interaction can be written as:

*χ*and

_{a}*χ*are the coupling constants, describing Stokes and anti-Stokes scattering. At the elementary interaction level

_{p}*χ*

_{a}*a*̂

^{†}

*b*̂

^{†}+H.c. describes the active part of the interaction, that is photon-atom entanglement, while

*χ*

_{p}*a*̂

^{†}

*b*̂+ H.c. describes the passive part of the interaction, that is a beam splitter-like exchange of excitations between photons and atoms.

*χ*and

_{p}*χ*in Eq. (1) can have substantially different magnitudes. The interaction Hamiltonian can be written as

_{a}*H*̂

_{int}= ħ

*χ*(

*p*̂

_{a}

*p*̂

_{b}+

*ξ*

^{2}̂

_{a}

*x*̂

_{b}) where

*χ*=

*χ*+

_{p}*χ*,

_{a}*ξ*

^{2}= (

*χ*-

_{p}*χ*)/(

_{a}*χ*+

_{p}*χ*) = 14

_{a}*a*

_{2}/

*a*

_{1},

*a*

_{1},

*a*

_{2}are the vector and tensor parts of the atomic polarizability defined in the Appendix (see Eq. (12)), and

*p*̂

_{a}= (

*a*̂-

*a*̂

^{†})/

*i*√2 and

*p*̂

_{b}= (

*b*̂-

*b*̅

^{†})/

*i*√2 being

*p*quadrature operators associated with light and atoms (for details of the Hamiltonian derivation see the Appendix). For short interaction times and alkali atoms with

*F*> 1/2 the second term in

*H*̂

_{int}can be neglected and the Hamiltonian reduces to the quantum nondemolition (QND) Hamiltonian

*H*̂

_{QND}= 2ħ

*χ*

*P*̂

_{a}

*P*̂

_{b}extensively used for spin squeezing, quantum memory and teleportation protocols [4

4. A. Kuzmich, L. Mandel, and N. P. Bigelow, “Generation of Spin Squeezing via Continuous Quantum Nondemolition Measurement,” Phys. Rev. Lett. **85**, 1594 (2000). [CrossRef] [PubMed]

*F*= 1/2 the passive and active couplings in Eq. (1) have equal magnitudes

*χ*=

_{a}*χ*and the Hamiltonian is always of the QND type.

_{p}*χ*>

_{p}*χ*, which is the case analyzed in this paper,

_{a}*ξ*is real and the interaction leads to swapping of the quantum states between light and atoms as discussed below. On the contrary, the case of

*χ*>

_{a}*χ*, i.e. of imaginary

_{p}*ξ*, leads to entanglement between light and atoms.

_{L}. Therefore the Stokes scattering produces a photon in an upper sideband of the driving field, described by the creation operator

*a*̂

^{†}

_{+}, while the anti-Stokes scattering couples to the lower sideband, described by the creation operator

*a*̂

^{†}

_{-}. Thus the interaction Hamiltonian is reexpressed in the following way:

*L*is the length of the cell and we have omitted the space and time dependence of operators for brevity. The operator

*b*̂′ annihilates an atomic excitation in an atomic slice around a given

*z*:

*N*is the total number of atoms in the cell,

_{a}*n*is the number of atoms in the slice and

*k*indexes the atoms. In the limit of almost all atoms residing in the

*mF*= 4 state

*b*̂′ is a bosonic operator, [

*b*̂′(

*z*,

*t*),

*b*̂′

^{†}(

*z*′,

*t*)] =

*δ*(

*z*-

*z*′). Analogously,

*a*′

_{+}and

*a*′

_{-}denote annihilation operators for the upper or lower sideband photonic mode around a given

*z*at a certain time with [

*a*̂

_{+}(

*z*,

*t*),

*a*̂

^{†}

_{+}(

*z*,

*t*′)] =

*δ*(

*t*-

*t*′).

*b*̂ = ∫

_{0}

^{L}

*dz*

*b*̂′(

*z*) /√

*L*. We then obtain for the light modes:

*a*̂

_{±}(

*t*) =

*a*̂

_{±}(

*z*= 0,

*t*) and

*a*̂

_{±}(

*t*) =

*a*̂

_{±}(

*z*=

*L*,

*t*) denote the operators at the input/output plane of the cell, and we assume that the passage is instantaneous. Note, that the state of light inside the cell changes linearly with the coordinate

*z*from

*a*̂

_{±}(

*t*) to

*a*̂

_{±}(

*t*).

*b*̂′ over the length of the cell to obtain an equation describing the evolution of the entire ensemble:

*γ*

_{sw}= ∣

*χ*∣

_{p}^{2}/2-∣

*χ*∣

_{a}^{2}/2 takes into account the linear change of a+ and a- along the cell, as follows from Eq. (4) . We will see that

*γ*

_{sw}is the rate at which the initial state of the atoms decays and is replaced by the state of the incoming light, hence we refer to it as a

*swap rate*. This equation can be readily solved and yields the single cell input-output relations. However the result is complicated and not useful for our current purpose, because we have disarrayed the simple structure of the Hamiltonian by applying the magnetic field. Fortunately, similarly to the QND case, this can be rectified by letting the light interact with two atomic ensembles in series [25

25. B. Julsgaard, A. Kozhekin, and E. S. Polzik,“Experimental long-lived entanglement of two macroscopic objects,” Nature (London) **413**, 400 (2001). [CrossRef] [PubMed]

26. K. Hammerer, A. Sorensen, and E. Polzik, “Quantum interface between light and atomic ensembles,” arXiv:0807.3358v3, http://arxiv.org/abs/0807.3358.

*a*

_{+}and

*a*

_{-}we shall use the sine and cosine combinations

*a*̂

_{c}= (

*a*̂

_{+}+

*a*̂

_{-})/√2 and

*a*̂

_{s}=

*i*(-

*a*̂

_{+}+

*a*̂

_{-})/√2. For the atoms we use collective annihilation operators defined as

*b*̂

_{c}= (

*b*̂

_{1}+

*b*̂

_{2})/√2 and

*b*̂

_{s}=

*i*(-

*b*̂

_{1}+

*b*̂

_{2})/√2 where

*b*̂

_{1}and

*b*̂

_{2}are annihilation operators for cell 1 and 2 respectively. We arrive at:

*a*̂

_{s}and

*b*̂

_{s}operators. Below we shall focus on the equations for

*c*operators implicitly using the fact, that the other solutions will be identical. Integrating the above equations gives:

*T*we choose:

*X*̂

_{L}and

*P*̂

_{L}are quadrature operators for an exponentially rising input mode with commutation relation [

*X*̂

_{L},

*P*̂

_{L}] =

*i*.

*X*̂′

_{L}and

*P*̂′

_{L}are quadrature operators for an exponentially falling output mode, and

*N*

_{in}, and

*N*

_{out}are normalization factors. We also find it convenient to introduce the quadrature operators

*X*̂

_{A}and

*P*̂

_{A}for the atomic state:

*ξ*

^{2}

*κ*

^{2}and squeezing of the inputs by a factor of 1/

*ξ*. This is schematically depicted in Fig. 2. In particular after a long enough interaction time

*T*≫

*γ*

^{-1}

_{sw}the initial state of the atoms is squeezed and mapped onto the light, and vice versa. If the atoms are initially in the coherent spin state (CSS), the output light will be in a squeezed state with a squeezing factor of 1/

*λ*, i.e. Var(

*P*̂′

_{L}) =

*ξ*

^{2}. Note that the above equations describe either the cosine combination of the light sidebands and the symmetric excitation of the two ensembles or the sine combination and the antisymmetric excitation. Therefore both the

*P*̂′

_{L,c}and

*P*̂′

_{L,s}will be squeezed after the interaction. This is equivalent to generating quadrature entanglement between the upper and lower sidebands of the probe pulse since 2Var(

*P*̂′

_{L,c}) + 2Var(

*P*̂′

_{L,s}) = Var(

*P*̂

_{+}+

*P*̂

_{-})+Var(

*X*̂

_{+}-

*X*̂

_{-}) < 2 where + and - denote the upper and lower sidebands respectively and the last inequality is exactly the quadrature entanglement criterion [29

29. L.-M. Duan, G. Giedke, I. Cirac, and P. Zoller, “Inseparability Criterion for Continuous Variable Systems,” Phys. Rev. Lett. , **84**, 2722–2725 (2000). [CrossRef] [PubMed]

*χ*=

_{p}*χ*and

_{a}*ξ*= 0, the input-output relations reduce to the QND case [26

26. K. Hammerer, A. Sorensen, and E. Polzik, “Quantum interface between light and atomic ensembles,” arXiv:0807.3358v3, http://arxiv.org/abs/0807.3358.

*γ*

_{sw}becomes time dependent, which is of particular importance for applications for atomic memories. A closer examination of Eq. (7a) reveals that the output mode in this case has a mode function

*u*(

*t*) proportional to

*u*(

*t*) ∝

*γ*

_{sw}(

*t*)exp (-∫

^{t}

_{0}

*dt*′

*γ*

_{sw}(

*t*′)). In particular, one can shape the driving field pulse such that squeezing is produced in a flat top temporal mode.

## 3. Experiment

^{11}caesium atoms each. Both cells are placed inside magnetic shields in uniform magnetic fields oriented along the

*x*axis. Each measurement cycle begins with optical pumping of the atoms in the two cells into oppositely oriented CSS, as described in [27

27. B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature (London) **432**, 482–486 (2004). [CrossRef] [PubMed]

28. J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature (London) **443**, 557–560 (2006). [CrossRef] [PubMed]

_{½}, F=4→6P

_{½}, F’=5 transition is turned on. Prior to the interaction it is spatially shaped into a circular flat top beam 20 mm in diameter using a telescope beam shaper in order to make the coupling strength uniform across the beam. The driving field polarized along the

*y*axis passes through both cells along the

*z*-direction. Finally the beam goes through half- or quarter wave plates, a polarizing beam splitter (PBS) and onto a pair of balanced detectors. Depending on the settings of the wave plates we can either measure the

*X*or the

_{L}*P*quadrature of the light. The signals from the detectors are subtracted, sent to a lock-in amplifier and digitized with an integrating A/D converter at a 12.5 kHz rate. The number of pumped atoms in the cells is monitored by measuring the Faraday rotation of very weak auxiliary probe beams propagating through the cells along the magnetic field.

_{L}*X*̂

_{A},

*P*̂

_{A}) phase space. A 150

*μ*s long RF-pulse applied orthogonally to the dc magnetic field in one of the cells displaces the atomic spins equally in

*X*̂

_{A}and

*P*̂

_{A}creating several tens of coherent atomic excitations. After that we perform time-resolved measurement of the light quadratures

*x*̂

_{L}(

*t*) and

*p*̂′

_{L}(

*t*) in two series of 200 experimental cycles. Those operators are defined by the relation

*x*′

_{L}(

*t*) +

*ip*′

_{L}(

*t*) = √2

*a*″

_{c}(

*t*) and written with small letters to distinguish them from operators associated with exponential modes. From Eq. (7a) we can find the expected mean values of the output light operators assuming they have zero mean values at the input:

*x*̂

_{L}(0)〉/〈

*p*̂

_{L}(0)〉 found from the figure yields

*ξ*

^{-2}= 6.3. It agrees very well with the theoretical value at our detuning, which itself is not very sharp due to the Doppler broadening. Notice that for a pure QND interaction, 〈

*p*̂

_{L}(

*t*)〉 would be zero independently of either measurement time or input mean values of the atomic operators.

*p*̂

_{L,c}(

*t*) and

*p*̂

_{L,s}(

*t*), averaging over typically 10000 cycles. Squeezing can be seen in the power spectrum of the signal (i.e. 〈

*p*̂

^{2}

_{L,c}(Ω) +

*p*̂

^{2}

_{L,s}(Ω)〉) from the lock-in amplifier as shown in Fig. 5. When the magnetic field is shifted such that the atomic contribution to the noise lies outside the detection bandwidth, we measure the shot noise level, with the spectral shape corresponding to the gain/sensitivity function of our detection system. When the magnetic fields in both cells are adjusted so that the atoms precess at exactly 322 kHz (the center frequency of the detection range), we see an apparent dip in the noise power at that frequency and around it in the bandwidth of a few hundred Hz.

## 4. Analysis of the temporal modes of the squeezing

*p*̂ quadratures of light as a function of time,

*p*̂

_{L,c}(

*t*) and

*p*̂

_{L,s}(

*t*). We focus on the cosine quadrature with the results for the sine quadrature being very similar. The two-time correlation function 〈

*p*̂

_{L,c}(

*t*)

*p*̂

_{L,c}(

*t′*)〉 yields the amount of noise in any temporal mode or a correlation between pairs of modes that are contained w ithin the detection bandwidth. Therefore it is natural to ask whether any mode basis is favoured in such case. The answer is provided by the Karhunen-Loéve theorem as detailed in the Appendix. For each correlation matrix one can find a unique basis of modes

*u*(

_{n}*t*) which have no cross-correlations. Each of them can be in principle measured separately and it will exhibit a variance of

*ξ*

_{n}^{2}= Var(

*P*̂

_{L,c}

^{(n)}) that is also found from the 〈

*p*

_{L,c}(

*t*)

*p*

_{L,c}(

*t*′)〉.

*P*̂

_{L,c}

^{(n)}) + 2Var(

*P*̂

_{L,s}

^{(n)}) = Var(

*P*̂

_{+}

^{(n)}+

*P*

_{-}

^{(n)})/2 + Var(

*X*̂

_{+}

^{(n)}-

*X*̂

_{-}

^{(n)})/2 obtained from the measured correlation matrices. It is expressed as the noise reduction below shot noise. We can see that there is one dominant squeezed mode, in which the noise is reduced by 3.5(1)dB. Superscript (

*n*) denotes

*n*-th characteristic mode function

*u*(

_{n}*t*). Several other modes are squeezed by about 1 dB. This is due to atomic decoherence taking place during the interaction. The equations (7) describe a perfectly coherent evolution and predict pure single mode squeezing. However, in reality the transverse decoherence time

*T*

_{2}is comparable to the duration of the fundamental mode. Thus, figuratively speaking, while the squeezed light leaves the cells, the state of atoms in the cells is driven back to the initial CSS state. This starts the process of squeezing of the light anew in an incoherent manner. This is confirmed by the shape of the characteristic mode functions obtained from the experiment shown in Fig. 7. The most squeezed mode has an exponentially decaying shape, which agrees with the theoretical model. The decay rate is found to be (5.5ms)

^{-1}, which is virtually equal to the decay rate of the mean values measured to be

*γ*

_{sw}= (5.7ms)

^{-1}. The next mode is rising, supporting the explanation that as the atomic state is brought back to the CSS, an independent squeezing process starts.

## 5. Conclusions

_{L}sidebands of the outgoing light. This state of light is emitted in an exponentially decaying temporal mode. At the same time a portion of incoming light, which comes in an exponentially rising temporal mode is mapped onto the atoms.

*γ*

_{sw}, we should expect the degree of squeezing equal to

*ξ*

^{2}, that is 8 dB at our current detuning ∆ = 850 MHz. By increasing the detuning the limiting degree of squeezing

*ξ*

^{2}can be increased further at the expense of lowering

*ξ*

_{sw}and weakening the coherent part of interaction as compared to the decoherence.

30. T. Opatrny, N. Korolkova, and G. Leuchs, “Mode structure and photon number correlations in squeezed quantum pulses,” Phys. Rev. A **66**, 53813 (2002). [CrossRef]

27. B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature (London) **432**, 482–486 (2004). [CrossRef] [PubMed]

## A. Details of the derivation

*a*

_{0},

*a*

_{1}and

*a*

_{2}respectively [24]:

*z*is the direction of the propagation of the light beam,

*A*is the cross section of the beam,

*γ*is the natural linewidth while ∆ is the detuning. The Hamiltonian (12) can be simplified if the atoms are in a state very close to the CSS oriented along

*x*axis. The result is different depending on the

*F*number. For concretness we will assume here

*F*= 4 with almost all atoms in

*m*= 4 state with respect to the

_{F}*x*-axis, only some in

*m*= 3 and none in the other states. In this situation we can approximate

_{F}*j*

^{2}

_{z},

*j*

_{+}

^{2}and

*j*

^{2}- by c-numbers and components of

**j**̂. This way we arrive at:

*a*

_{1}

*S*- 14

_{z}j_{z}*a*

_{2}

*S*are nontrivial. The other terms generate classical rotations, in particular: the

_{y}j_{y}*S*term in our settings, when both

_{x}j_{x}*S*and

_{x}*j*are macroscopic merely causes a rotation in the

_{x}*x*-

*p*plane for both light and atoms. The total angle of this rotation is typically of the order of a few miliradians and we neglect it.

*ϕ*is a constant of motion in our system, thus the term

*ϕj*only shifts the Larmor frequency — it represents the Stark shift. The last term,

_{x}*S*, adds to a rotation of in the

_{x}*x*-

_{c}*p*plane for light but is still negligible.

_{c}*a*

_{1}

*S*- 14

_{z}j_{z}*a*

_{2}

*S*using bosonic operators,

_{y}j_{y}*a*̂ for light and

*b*̂ for atoms. This is accomplished using the relations:

*N*is the total number of atoms and the sign in

_{a}*S*is a consequence of negative sign of

_{z}*S*in our settings. This way we can approximate the interaction Hamiltonian in the form:

_{x}*a*

_{1}and

*a*

_{2}tensor polarizabilities for Cs D2 line can be found, for example in [26

26. K. Hammerer, A. Sorensen, and E. Polzik, “Quantum interface between light and atomic ensembles,” arXiv:0807.3358v3, http://arxiv.org/abs/0807.3358.

*ξ*

^{2}can be flipped, that is the

*χ*and

_{a}*χ*can be interchanged. This is accomplished by rotating the polarization of the driving field by 90°,

_{p}*S*→

_{x}*S*or by switching from blue to red detuning ∆ → -∆. The input-output relations given in Eq. (10) remain valid for both signs of

_{x}*ξ*

^{2}. However for a real

*ξ*they describe the entanglement between two sidebands of the light field, whereas for imaginary

*ξ*they entail the entanglement between the light and atoms.

## B. Eigenmode decomposition

*P*(

_{c}*t*):

*p*(

_{c}*t*) is a properly scaled signal directly form the lock-in amplifier. After the measurement we can calculate the amount of noise in any temporal mode characterised by the modefunction

*u*(

_{n}*t*), where

*n*indexes a set of modes we are interested in. A

*P*-quadrature operator for the

*n*th mode is simply

*Q*̂

_{n}= ∫

*u*(

_{n}*t*)

*p*(

_{c}*t*). We can calculate both the variance of any

*Q*and correlation between any two of them:

_{n}*ξ*are equal to the quadrature variances,

_{n}*ξ*= 〈

_{n}*Q*

^{2}

_{n}〉 - 〈

*Q*〉

_{n}^{2}, while the eigenfunctions give the quadrature mode functions. The same procedure can be repeated for the sine component of the homodyne signal,

*p*(

_{s}*t*) and it yields results identical to within the experimental uncertainties.

## References and links

1. | T. C. Ralph, “Continuous variable quantum cryptography,” Phys. Rev. A |

2. | A. Furusawa, J. L. Sorensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional Quantum Teleportation,” Science |

3. | S. Lloyd and S. L. Braunstein, “Quantum Computation over Continuous Variables,” Phys. Rev. Lett. |

4. | A. Kuzmich, L. Mandel, and N. P. Bigelow, “Generation of Spin Squeezing via Continuous Quantum Nondemolition Measurement,” Phys. Rev. Lett. |

5. | S. L. Braunstein, “Quantum error correction for communication with linear optics,” Nature (London) |

6. | S. Lloyd and J.-J. E. Slotine,“Analog Quantum Error Correction,” Phys. Rev. Lett. |

7. | L.-M. Duan, J. I. Cirac, P. Zoller, and E. S. Polzik, “Quantum Communication between Atomic Ensembles Using Coherent Light,” Phys. Rev. Lett. |

8. | D. E. Browne, J. Eisert, S. Scheel, and M. B. Plenio, “Driving non-Gaussian to Gaussian states with linear optics,” Phys. Rev. A |

9. | Z. Ou, S. Pereira, H. Kimble, and K. C. Peng, “Realization of the Einten-Podolsy-rosen Paradox for Continuous Variables,” Phys. Rev. Lett. |

10. | C. Schori, J. L. Sørensen, and E. S. Polzik, “Narrow-band frequency tunable light source of continuous quadrature entanglement,” Phys. Rev. A |

11. | H. Vahlbruch, M. Mehmet, S. Chelkowski, B. Hage, A. Franzen, N. Lastzka, S. G. ler, K. Danz-mann, and R. Schnabel, “Observation of Squeezed Light with 10-dB Quantum-Noise Reduction,” Phys. Rev. Lett. |

12. | R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of Squeezed States Generated by Four-Wave Mixing in an Optical Cavity,” Phys. Rev. Lett. |

13. | A. Lambrecht, T. Coudreau, A. M. Steinberg, and E. Giacobino, “Squeezing with cold atoms,” Europhys. Lett. |

14. | V. Josse, A. Dantan, A. Bramati, M. Pinard, and E. Giacobino,“Continuous variable entanglement using cold atoms,” Phys. Rev. Lett. |

15. | V. Boyer, A. M. Marino, R. C. Pooser, and P. D. Lett, “Entangled Images from Four-Wave Mixing,” Science |

16. | J. Ries, B. Brezger, and A. I. Lvovsky, “Experimental Vacuum Squeezing in Rubidium Vapor via Self-Rotation,” Phys. Rev. A |

17. | E. E. Mikhailov and I. Novikova, “Low-frequency vacuum squeezing via polarization self-rotation in Rb vapor,” Opt. Lett. |

18. | A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating Optical Schrodinger Kittens for Quantum Information Processing,” Science |

19. | J. S. Neergaard-Nielsen, B. M. Nielsen, C. Hettich, K. M. lmer, and E. S. Polzik, “Generation of a Superposition of Odd Photon Number States for Quantum Information Networks,” Phys. Rev. Lett. |

20. | M. Sasaki and S. Suzuki, “Multimode theory of measurement-induced non-Gaussian operation on wideband squeezed light: Analytical formula,” Phys. Rev. A |

21. | L.-M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature (London) |

22. | D. V. Kupriyanov, O. S. Mishina, I. M. Sokolov, B. Julsgaard, and E. S. Polzik, “Multimode entanglement of light and atomic ensembles via off-resonant coherent forward scattering,” Phys. Rev. A |

23. | O. Mishina, D. Kuprianov, and E. S. Polzik, “Macroscopic quantum information channel via the polarization-sensitive interaction between the light and spin subsystems,” in |

24. | J. Sherson, B. Julsgaard, and E. S. Polzik, |

25. | B. Julsgaard, A. Kozhekin, and E. S. Polzik,“Experimental long-lived entanglement of two macroscopic objects,” Nature (London) |

26. | K. Hammerer, A. Sorensen, and E. Polzik, “Quantum interface between light and atomic ensembles,” arXiv:0807.3358v3, http://arxiv.org/abs/0807.3358. |

27. | B. Julsgaard, J. Sherson, J. I. Cirac, J. Fiurasek, and E. S. Polzik, “Experimental demonstration of quantum memory for light,” Nature (London) |

28. | J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature (London) |

29. | L.-M. Duan, G. Giedke, I. Cirac, and P. Zoller, “Inseparability Criterion for Continuous Variable Systems,” Phys. Rev. Lett. , |

30. | T. Opatrny, N. Korolkova, and G. Leuchs, “Mode structure and photon number correlations in squeezed quantum pulses,” Phys. Rev. A |

**OCIS Codes**

(190.5650) Nonlinear optics : Raman effect

(270.6570) Quantum optics : Squeezed states

(270.5585) Quantum optics : Quantum information and processing

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: July 1, 2009

Revised Manuscript: July 28, 2009

Manuscript Accepted: July 30, 2009

Published: July 31, 2009

**Citation**

W. Wasilewski, T. Fernholz, K. Jensen, L. S. Madsen, H. Krauter, C. Muschik, and E. S. Polzik, "Generation of two-mode squeezed and entangled light in a single temporal and spatial mode," Opt. Express **17**, 14444-14457 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-16-14444

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### References

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