March 15, 2015 / Vol. 40, No. 6 / OPTICS LETTERS
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Temporal imaging with squeezed light Giuseppe Patera* and Mikhail I. Kolobov Laboratoire PhLAM, Université Lille 1, 59655 Villeneuve D’Ascq, France *Corresponding author: [email protected]‑lille1.fr Received December 18, 2014; accepted February 3, 2015; posted February 11, 2015 (Doc. ID 230949); published March 13, 2015 We generalize the scheme of conventional temporal imaging to quantum temporal imaging viable for nonclassical states of light. As an example, we apply our scheme to temporally broadband squeezed light and demonstrate a possibility of its noiseless magnification. In particular, we show that one can magnify by a given factor the coherence time of squeezed light and match it to the response time of the photodetector. This feature opens new possibilities for practical applications of temporally broadband squeezed light in quantum optics and quantum information. © 2015 Optical Society of America OCIS codes: (110.6915) Time imaging; (190.7110) Ultrafast nonlinear optics; (270.5585) Quantum information and processing; (270.6620) Strong-field processes. http://dx.doi.org/10.1364/OL.40.001125
Temporal imaging is a technique that enables manipulation of temporal optical signals in a manner similar to manipulation of optical images in spatial domain. The concept of temporal imaging uses the notion of spacetime duality [1] with dispersion phenomena playing the role of diffraction and quadratic phase modulation in time acting as a time lens. The first optical time lenses used electro-optic phase modulators [2–5], however, these systems do not possess the required bandwidth for optical treatment of ultra-short (10–100 fs) pulses. Modern time lenses use all-optical nonlinear parametric processes such as three- or four-wave mixing of the input signal with a chirped optical-pump pulse in a nonlinear crystal [6–10]. On the basis of such space-time duality, it is possible to transfer to the temporal domain a large variety of imaging schemes such as, for example, temporal magnification [5,11], time-to-frequency conversion [12], pulse compression [4,5,13], spectral expansion [14], or spectral phase conjugation [15]. Until recently, spatial optical imaging has been ignoring the quantum nature of the light because the intensity of light was so high that quantum fluctuations were negligible. However, as the pixel size in digital imaging devices decreases, the amount of light per pixel can reach the level where the quantum fluctuations become important. Quantum imaging investigates ultimate quantum limits of imaging techniques in such photon-starved regime [16]. It would be desirable to bring the experience from spatial quantum imaging into temporal imaging and to establish its ultimate limits imposed by the quantum nature of the light. Few steps in this direction have been already made in the literature. Namely, in Ref. [17], the authors have considered a scheme of quantum optical waveform conversion, preserving the quantum properties, including entanglement. In Ref. [18], a spectral bandwidth compression of single photons by a factor of 40 has been experimentally demonstrated. In Ref. [19], the authors have proposed the scheme of abberation-corrected quantum temporal imaging of a coherent state with a double-Gaussian profile. In a more general context, the problem of quantum temporal imaging is closely related to that of the quantum pulse gating proposed in [20,21]. In this Letter, we make a new step toward the theory of quantum temporal imaging. Precisely, we address the 0146-9592/15/061125-04$15.00/0
problem of temporal imaging of a temporally broadband squeezed light generated by a traveling-wave optical parametric amplifier (OPA) [22] or a similar device. We consider a single-lens temporal imaging system formed by two dispersive elements and a parametric temporal lens, based on a sum-frequency generation (SFG) process. We derive a unitary transformation of the field operators performed by this kind of time lens. This unitary transformation allows us to evaluate the squeezing spectrum at the output of the single-lens imaging system and to find the conditions preserving squeezing in the output field. We consider a time lens based on the SFG process in a χ 2 nonlinear medium [9]. In this case, due to a nonlinear interaction in the χ 2 medium, a strong classical pump wave with the carrier frequency ωp converts a signal wave with the carrier frequency ωs into an idler wave with the carrier frequency ωi ωs ωp . We shall use the undepleted approximation for the strong pump field and write its complex slowly-varying amplitude as ap t Ap t expiϕp t, where Ap t is the real modulus and ϕp t is the real phase. We shall describe the signal and the idler waves by their slowly varying photon annihilation operators aˆ s z; t and aˆ i z; t, depending on the distance z inside the nonlinear medium and obeying the canonical commutation relations for the free-field operators [22]. In this Letter, we shall neglect the group velocity dispersion (GVD) in the nonlinear medium and assume that the pump, the signal, and the idler waves propagate inside the medium with the same group velocity vg . We are aware that this assumption is oversimplified for the SFG process. We shall consider the case of different group velocities and discuss the role of the GVD elsewhere. Under these assumptions, the equations describing the evolution of the signal and the idler annihilation operators inside the nonlinear medium read, ∂ aˆ z; τ gap τaˆ i z; τeiΔkz ; ∂z s
(1)
∂ aˆ z; τ −gap τaˆ s z; τe−iΔkz ; ∂z i
(2)
© 2015 Optical Society of America
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where g is the nonlinear coupling constant proportional to χ 2 , τ t − z∕vg is the retarded time, and Δk ki − ks − kp is the phase mismatch between the signal, the idler, and the pump wave vectors. In order to simplify the results we shall consider the perfect phase matching condition, Δk 0. In this case, Eqs. (1) and (2) have the following solution [23], aˆ s L; τ cτaˆ s 0; τ sτe−iϕp τ aˆ i 0; τ;
(3)
aˆ i L; τ −sτeiϕp τ aˆ s 0; τ cτaˆ i 0; τ;
(4)
relating the signal and the idler annihilation operators at the output of the nonlinear crystal of length L, i.e., at z L, to those at its input, z 0. Here we have introduced the following shorthands: sτ singAp τL;
cτ cosgAp τL:
(5)
Equations (3) and (4) represent a unitary transformation of the photon annihilation operators from the input of the nonlinear crystal to its output preserving the canonical commutation relations. These equations have the same form as a transformation performed by a ideal beam splitter with the amplitude transmission coefficient cτ and the reflection coefficient sτ such that cτ2 sτ2 1. The key feature of these equations for the time lens application is that the phase factor expiϕp τ appearing in this unitary transformation is determined by the phase of the pump wave. Using these equations, we can easily find the classical limit, and it suffices to take the mean values of the operators as haˆ s z; τi αs z; τ and haˆ i z; τi αi z; τ to recover the classical complex field amplitudes. In particular, if we consider the case when the classical complex amplitude of the idler field at the input to the crystal is zero, αi 0; τ 0, we obtain the following complex amplitude of the idler wave at the output, αi L; τ −sτeiϕp τ αs 0; τ:
(6)
Equation (6) describes conversion of the classical complex amplitude αs 0; τ of the signal at the input of the crystal into the complex amplitude of the idler αi L; τ at the output. The efficiency of this conversion process is determined by the reflection coefficient sτ. The phase factor expiϕp τ signifies that the phase of the complex amplitude of the idler at the output is given by the sum of the time-dependent phase of the pump wave ϕp τ and the phase of the input signal wave. Notice also the sign change in Eq. (6). Time lens transformation of classical amplitude αs 0; τ into αi L; τ is obtained from Eq. (6) by choosing a quadratic time dependence in the phase of the pump wave, known in the literature as a chirp, ϕp τ τ2 ∕2Df . Such a quadratic time dependence can be produced by propagating a short pulse through a dispersive medium of length Lf , characterized by a GVD coefficient β2 d2 k∕dω2 , such that the total group delay dispersion (GDD) of the pulse is equal to Df β2 Lf . In the time lens theory, the parameter Df is known as the focal GDD of
the time lens, and it plays the role similar to the focal distance of conventional lens. Classic time lens transformation (6) with ϕp τ τ2 ∕2Df cannot be applied to nonclassic input states such as squeezed or entangled states, for example. In this case we have to use the quantum-mechanical unitary transformation of the field operators given by Eqs. (3) and (4). In particular, we have to take into account the vacuum quantum fluctuations of the input idler field, transmitted to the output of the time lens with the transmission coefficient cτ. These vacuum fluctuations are known to be detrimental for the nonclassic input states at the other input port of a beam splitter and have to be avoided. Therefore, in order to eliminate these vacuum fluctuations, we shall impose the condition of a unit reflection coefficient, sτ 1, in Eqs. (3) and (4), which is equivalent to gAp τL π∕2. In this case Eq. (4) becomes, aˆ i L; τ −eiτ
2 ∕2D
f
aˆ s 0; τ;
(7)
and the vacuum contribution from the photon annihilation operator aˆ i 0; τ vanishes. Let us now consider a single-lens temporal imaging system comprising a time lens of the focal GDD Df preceded by a dispersive medium characterized by GDD Ds for a signal wave, and followed by a second dispersive medium of GDD Di for the idler wave. We can combine the equations for propagation of the slowly varying signal and idler annihilation operators in the dispersive media with Eq. (7) for the time lens in order to arrive at the transformation between the input photon annihilation operator aˆ s τ and the output annihilation operator aˆ 0i τ for such a single-lens imaging system (see also [19]). When the classical imaging conditions 1∕Di 1∕Ds 1∕Df ;
−Di ∕Ds M;
(8)
are satisfied, the sought transformation reads as follows: −1 i τ2 aˆ 0i τ p e 2MDf aˆ s τ∕M; M
(9)
and describes magnification by a factor M of the quantum input signal aˆ s τ as in an equivalent spatial single-lens imaging system. The phase term appearing in Eq. (9) can be compensated by a second temporal lens, as suggested in Ref. [19]. Equations (7) and (9) are valid for arbitrary quantum states of the input operator aˆ s τ. In Ref. [19], the authors have considered a simple model of a time-bin entangled input state without taking into account the spectral and temporal properties of entanglement such as its spectral bandwidth or the coherence time. We shall consider as an input state of the signal a temporally broadband squeezed state of light generated by a traveling-wave OPA in a second-order nonlinear crystal [22]. The squeezing transformation produced by such an OPA is described in terms of the Fourier amplitudes aˆ s z; Ω: Z aˆ s z; Ω
aˆ s z; teiΩt dt;
(10)
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of the annihilation operators, and is given by a Bogolubov transformation from the input to the output of the OPA: aˆ s l; Ω UΩaˆ s 0; Ω V Ωaˆ †s 0; −Ω:
(11)
Here l is the length of the OPA crystal, and UΩ and VΩ are the complex coefficients depending on the parametric gain of the OPA and its phase-matching conditions. Their explicit form can be found, for example, in Ref. [22]. Transformation (11) is known to produce a broadband squeezed vacuum at the output of the OPA from the broadband vacuum input state. Mixing such a broadband squeezed vacuum with a strong monochromatic local oscillator (LO) and assuming the balanced homodyne photodetection, we can evaluate the normalized to the shot-noise photocurrent noise spectrum δi2Ω ∕hii, which we shall call the squeezing spectrum and denote SΩ. Considering for simplicity the unit photodetection efficiency, we can write the squeezing spectrum as follows, S s Ω cos2 θΩe2rω sin2 θΩe−2rω ;
(12)
where θΩ ψΩ − φ, with φ being the phase of the LO, the squeezing angle ψΩ, and the squeezing parameter rΩ given by ψΩ
1 argVΩ∕UΩ; 2
(13)
exp rΩ jUΩj jVΩj:
(14)
In Fig. 1, we have shown by dashed–dotted line SΩ at the output of the OPA for degenerate phase-matching condition and the parametric gain G exp2r m 9, where r m is the maximum value of the squeezing parameter at Ω 0. One can see from Fig. 1 that at low frequencies, Ω < Ωc , the quantum fluctuations of the photocurrent are reduced below the shot-noise level. −1∕2 This characteristic frequency Ωc βc , where βc 2 l 2 is the GVD coefficient of the OPA, allows us to introduce the characteristic coherence time τc π∕Ωc of quantum correlations, which has a simple physical interpretation. Let the observed photocurrent be collected during the 2
S(Ω)
1.5
1
0.5
0
0
1
2
Ω
3
4
5
Fig. 1. Squeezing spectrum of broadband squeezed light at the output of the OPA (dashed–dotted) and after the temporal lens with efficiency factor η 1∕2 (solid); expr m 3, and frequency Ω is in units of Ωc .
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time of observation T to provide a number N T of photoelectrons. If the time of observation is much longer that the coherence time, T ≫ τc , then the high-frequency quantum fluctuations of the photocurrent null out and do not contribute to the fluctuations in the observed number of photoelectrons. Therefore, for T ≫ τc , the statistics of N T will be sub-Poissonian. Let us now place a temporal lens described by Eqs. (3) and (4) at the output of the OPA. Then we have to use the input state (11) for aˆ s 0; τ and a broadband vacuum state for aˆ i 0; τ in Eqs. (3) and (4). Assuming similar homodyne detection scheme with a LO after the temporal lens we obtain the following squeezing spectrum: S i Ω 1 − η ηcos2 θΩe2rω sin2 θΩe−2rω ; (15) with η s2 τ being the efficiency factor of the time lens equal to the intensity reflexion coefficient in Eqs. (3) and (4). It follows from Eq. (15) that in the case of the temporal lens with the unit efficiency factor, the squeezing spectrum at its output is identical to that at the input. However, for η < 1, the vacuum fluctuations entering into the temporal lens from the open port deteriorate squeezing at its output. For example, we have shown by solid line in Fig. 1 the effect of this deterioration for η 1∕2. We can similarly evaluate the squeezing spectrum at the output of a single-lens temporal imaging system comprising a time lens with Df and two dispersive elements with Ds and Di . Such a scheme is described by Eqs. (8) and (9). Let us first consider a 4f-imaging scheme with Di 2Df , Ds 2Df , and the magnification factor M −1. It is easy to see from Eq. (9) that in this case SΩ at the output of the OPA, z l, is identical to Sω at the output of the temporal imaging scheme, i.e., at z l 4Lf , where Lf Df ∕β2 is the equivalent focal length of the temporal lens with GDD Df and the GVD coefficient β2 . Indeed, since the photocurrent noise spectrum δi2Ω is determined by the intensity correlation function of the light, the phase factor and the sign change in Eq. (9) are irrelevant. This result is not surprising, because the above conditions correspond to geometrical imaging with unit magnification of the output face of the OPA into the photodetection plane at z l 4Lf . It is also clear that in a more general case with arbitrary magnification factor M, when the imaging conditions (8) are satisfied, the squeezing spectrum at the output of the imaging system will be the same as that at the output of the OPA in terms of the scaled frequency Ω0 MΩ, which corresponds to the scaled time τ0 τ∕M in Eq. (9). This magnification factor gives us a possibility of matching the coherence time τc of the broadband squeezed light to the response time of the photodetector. Let us finally discuss a less obvious but more advantageous geometry of a single-lens temporal imaging scheme that allows us to partially compensate for the frequency dispersion of the OPA. It has been shown in Ref. [22] that the squeezing angle ψΩ from Eq. (13) for the degenerate phase matching can be written approximately as 2 ψΩ βc 2 lamp Ω ∕4;
(16)
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of squeezed light and to match it with the detector response time. Our scheme can find numerous applications in quantum optics and quantum information.
S(Ω)
1.5
1
0.5
0
0
1
2
Ω
3
4
5
Fig. 2. Squeezing spectrum of broadband squeezed light with (solid) and without (dashed–dotted) compensation of quadratic frequency dispersion of the OPA for the same conditions as in Fig. 1.
where lamp is the amplification length of the OPA, defined through r m l∕lamp . Since the angle ψΩ has a quadratic frequency dependence, it can be compensated by a dispersive medium with GVD coefficient β1 2 and the length L1 s equal to L1 s −lamp
βc 2
2β1 2
:
(17)
Let us now assume that we are using a 4f -imaging scheme with z l L1 s instead of z l for the object plane, i.e., we are imaging an object plane inside the OPA into the image plane at z l L1 s 4Lf . The squeezing spectra in the object and the image plane are identical. Therefore, for such a geometry, SΩ in the image plane will correspond to SΩ with compensated quadratic dispersion of the OPA. This spectrum is shown in Fig. 2 by a solid line together with the corresponding spectrum without compensation (dashed–dotted line). As follows from Fig. 2, compensation of the quadratic dispersion of the OPA increases the spectral bandwidth Ωc of squeezing and, respectively, shortens its coherence time τc . In conclusion, we have demonstrated that temporal imaging can be successfully applied to temporally broadband squeezed light, preserving the degree of squeezing. It allows us, in particular, to modify the coherence time
References 1. B. H. Kolner, IEEE J. Quantum Electron. 30, 1951 (1994). 2. J. A. Giordmaine, M. A. Duguay, and J. W. Hansen, IEEE J. Quantum Electron. 4, 252 (1968). 3. D. R. Grischowsky, Appl. Phys. Lett. 25, 566 (1974). 4. B. H. Kolner, Appl. Phys. Lett. 52, 1122 (1988). 5. B. H. Kolner and M. Nazarathy, Opt. Lett. 14, 630 (1989). 6. G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, Phys. Rev. A 40, 5063 (1989). 7. C. V. Bennett, R. P. Scott, and B. H. Kolner, Appl. Phys. Lett. 65, 2513 (1994). 8. C. V. Bennett and B. H. Kolner, Opt. Lett. 24, 783 (1999). 9. C. V. Bennett and B. H. Kolner, IEEE J. Quantum Electron. 36, 430 (2000). 10. C. V. Bennett and B. H. Kolner, IEEE J. Quantum Electron. 36, 649 (2000). 11. V. J. Hernandez, C. V. Bennett, B. D. Moran, A. D. Drobshoff, D. Chang, C. Langrock, M. M. Fejer, and M. Ibsen, Opt. Express 21, 196 (2013). 12. M. A. Foster, R. Salem, D. F. Geraghty, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, Nature 456, 81 (2008). 13. M. A. Foster, R. Salem, Y. Okawachi, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, Nat. Photonics 3, 581 (2009). 14. Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, Opt. Express 17, 5691 (2009). 15. O. Kuzucu, Y. Okawachi, R. Salem, M. A. Foster, A. C. Turner-Foster, M. Lipson, and A. L. Gaeta, Opt. Express 17, 20605 (2009). 16. M. I. Kolobov, Quantum Imaging (Springer, 2006). 17. D. Kieplinski, J. F. Corney, and H. M. Wiseman, Phys. Rev. Lett. 106, 130501 (2011). 18. J. Lavoie, J. M. Donohue, L. G. Wright, A. Fedrizzi, and K. J. Resch, Nat. Photonics 7, 363 (2013). 19. Y. Zhu, J. Kim, and D. J. Gauthier, Phys. Rev. A 87, 043808 (2013). 20. A. Eckstein, B. Brecht, and C. Silberhorn, Opt. Express 19, 13770 (2011). 21. B. Brecht, A. Eckstein, A. Christ, H. Suche, and C. Silberhorn, New J. Phys. 13, 065029 (2011). 22. M. I. Kolobov, Rev. Mod. Phys. 71, 1539 (1999). 23. R. W. Boyd, Nonlinear Optics (Academic, 1992).
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Temporal imaging with squeezed light Giuseppe Patera* and Mikhail I. Kolobov Laboratoire PhLAM, Université Lille 1, 59655 Villeneuve D’Ascq, France *Corresponding author: [email protected]‑lille1.fr Received December 18, 2014; accepted February 3, 2015; posted February 11, 2015 (Doc. ID 230949); published March 13, 2015 We generalize the scheme of conventional temporal imaging to quantum temporal imaging viable for nonclassical states of light. As an example, we apply our scheme to temporally broadband squeezed light and demonstrate a possibility of its noiseless magnification. In particular, we show that one can magnify by a given factor the coherence time of squeezed light and match it to the response time of the photodetector. This feature opens new possibilities for practical applications of temporally broadband squeezed light in quantum optics and quantum information. © 2015 Optical Society of America OCIS codes: (110.6915) Time imaging; (190.7110) Ultrafast nonlinear optics; (270.5585) Quantum information and processing; (270.6620) Strong-field processes. http://dx.doi.org/10.1364/OL.40.001125
Temporal imaging is a technique that enables manipulation of temporal optical signals in a manner similar to manipulation of optical images in spatial domain. The concept of temporal imaging uses the notion of spacetime duality [1] with dispersion phenomena playing the role of diffraction and quadratic phase modulation in time acting as a time lens. The first optical time lenses used electro-optic phase modulators [2–5], however, these systems do not possess the required bandwidth for optical treatment of ultra-short (10–100 fs) pulses. Modern time lenses use all-optical nonlinear parametric processes such as three- or four-wave mixing of the input signal with a chirped optical-pump pulse in a nonlinear crystal [6–10]. On the basis of such space-time duality, it is possible to transfer to the temporal domain a large variety of imaging schemes such as, for example, temporal magnification [5,11], time-to-frequency conversion [12], pulse compression [4,5,13], spectral expansion [14], or spectral phase conjugation [15]. Until recently, spatial optical imaging has been ignoring the quantum nature of the light because the intensity of light was so high that quantum fluctuations were negligible. However, as the pixel size in digital imaging devices decreases, the amount of light per pixel can reach the level where the quantum fluctuations become important. Quantum imaging investigates ultimate quantum limits of imaging techniques in such photon-starved regime [16]. It would be desirable to bring the experience from spatial quantum imaging into temporal imaging and to establish its ultimate limits imposed by the quantum nature of the light. Few steps in this direction have been already made in the literature. Namely, in Ref. [17], the authors have considered a scheme of quantum optical waveform conversion, preserving the quantum properties, including entanglement. In Ref. [18], a spectral bandwidth compression of single photons by a factor of 40 has been experimentally demonstrated. In Ref. [19], the authors have proposed the scheme of abberation-corrected quantum temporal imaging of a coherent state with a double-Gaussian profile. In a more general context, the problem of quantum temporal imaging is closely related to that of the quantum pulse gating proposed in [20,21]. In this Letter, we make a new step toward the theory of quantum temporal imaging. Precisely, we address the 0146-9592/15/061125-04$15.00/0
problem of temporal imaging of a temporally broadband squeezed light generated by a traveling-wave optical parametric amplifier (OPA) [22] or a similar device. We consider a single-lens temporal imaging system formed by two dispersive elements and a parametric temporal lens, based on a sum-frequency generation (SFG) process. We derive a unitary transformation of the field operators performed by this kind of time lens. This unitary transformation allows us to evaluate the squeezing spectrum at the output of the single-lens imaging system and to find the conditions preserving squeezing in the output field. We consider a time lens based on the SFG process in a χ 2 nonlinear medium [9]. In this case, due to a nonlinear interaction in the χ 2 medium, a strong classical pump wave with the carrier frequency ωp converts a signal wave with the carrier frequency ωs into an idler wave with the carrier frequency ωi ωs ωp . We shall use the undepleted approximation for the strong pump field and write its complex slowly-varying amplitude as ap t Ap t expiϕp t, where Ap t is the real modulus and ϕp t is the real phase. We shall describe the signal and the idler waves by their slowly varying photon annihilation operators aˆ s z; t and aˆ i z; t, depending on the distance z inside the nonlinear medium and obeying the canonical commutation relations for the free-field operators [22]. In this Letter, we shall neglect the group velocity dispersion (GVD) in the nonlinear medium and assume that the pump, the signal, and the idler waves propagate inside the medium with the same group velocity vg . We are aware that this assumption is oversimplified for the SFG process. We shall consider the case of different group velocities and discuss the role of the GVD elsewhere. Under these assumptions, the equations describing the evolution of the signal and the idler annihilation operators inside the nonlinear medium read, ∂ aˆ z; τ gap τaˆ i z; τeiΔkz ; ∂z s
(1)
∂ aˆ z; τ −gap τaˆ s z; τe−iΔkz ; ∂z i
(2)
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where g is the nonlinear coupling constant proportional to χ 2 , τ t − z∕vg is the retarded time, and Δk ki − ks − kp is the phase mismatch between the signal, the idler, and the pump wave vectors. In order to simplify the results we shall consider the perfect phase matching condition, Δk 0. In this case, Eqs. (1) and (2) have the following solution [23], aˆ s L; τ cτaˆ s 0; τ sτe−iϕp τ aˆ i 0; τ;
(3)
aˆ i L; τ −sτeiϕp τ aˆ s 0; τ cτaˆ i 0; τ;
(4)
relating the signal and the idler annihilation operators at the output of the nonlinear crystal of length L, i.e., at z L, to those at its input, z 0. Here we have introduced the following shorthands: sτ singAp τL;
cτ cosgAp τL:
(5)
Equations (3) and (4) represent a unitary transformation of the photon annihilation operators from the input of the nonlinear crystal to its output preserving the canonical commutation relations. These equations have the same form as a transformation performed by a ideal beam splitter with the amplitude transmission coefficient cτ and the reflection coefficient sτ such that cτ2 sτ2 1. The key feature of these equations for the time lens application is that the phase factor expiϕp τ appearing in this unitary transformation is determined by the phase of the pump wave. Using these equations, we can easily find the classical limit, and it suffices to take the mean values of the operators as haˆ s z; τi αs z; τ and haˆ i z; τi αi z; τ to recover the classical complex field amplitudes. In particular, if we consider the case when the classical complex amplitude of the idler field at the input to the crystal is zero, αi 0; τ 0, we obtain the following complex amplitude of the idler wave at the output, αi L; τ −sτeiϕp τ αs 0; τ:
(6)
Equation (6) describes conversion of the classical complex amplitude αs 0; τ of the signal at the input of the crystal into the complex amplitude of the idler αi L; τ at the output. The efficiency of this conversion process is determined by the reflection coefficient sτ. The phase factor expiϕp τ signifies that the phase of the complex amplitude of the idler at the output is given by the sum of the time-dependent phase of the pump wave ϕp τ and the phase of the input signal wave. Notice also the sign change in Eq. (6). Time lens transformation of classical amplitude αs 0; τ into αi L; τ is obtained from Eq. (6) by choosing a quadratic time dependence in the phase of the pump wave, known in the literature as a chirp, ϕp τ τ2 ∕2Df . Such a quadratic time dependence can be produced by propagating a short pulse through a dispersive medium of length Lf , characterized by a GVD coefficient β2 d2 k∕dω2 , such that the total group delay dispersion (GDD) of the pulse is equal to Df β2 Lf . In the time lens theory, the parameter Df is known as the focal GDD of
the time lens, and it plays the role similar to the focal distance of conventional lens. Classic time lens transformation (6) with ϕp τ τ2 ∕2Df cannot be applied to nonclassic input states such as squeezed or entangled states, for example. In this case we have to use the quantum-mechanical unitary transformation of the field operators given by Eqs. (3) and (4). In particular, we have to take into account the vacuum quantum fluctuations of the input idler field, transmitted to the output of the time lens with the transmission coefficient cτ. These vacuum fluctuations are known to be detrimental for the nonclassic input states at the other input port of a beam splitter and have to be avoided. Therefore, in order to eliminate these vacuum fluctuations, we shall impose the condition of a unit reflection coefficient, sτ 1, in Eqs. (3) and (4), which is equivalent to gAp τL π∕2. In this case Eq. (4) becomes, aˆ i L; τ −eiτ
2 ∕2D
f
aˆ s 0; τ;
(7)
and the vacuum contribution from the photon annihilation operator aˆ i 0; τ vanishes. Let us now consider a single-lens temporal imaging system comprising a time lens of the focal GDD Df preceded by a dispersive medium characterized by GDD Ds for a signal wave, and followed by a second dispersive medium of GDD Di for the idler wave. We can combine the equations for propagation of the slowly varying signal and idler annihilation operators in the dispersive media with Eq. (7) for the time lens in order to arrive at the transformation between the input photon annihilation operator aˆ s τ and the output annihilation operator aˆ 0i τ for such a single-lens imaging system (see also [19]). When the classical imaging conditions 1∕Di 1∕Ds 1∕Df ;
−Di ∕Ds M;
(8)
are satisfied, the sought transformation reads as follows: −1 i τ2 aˆ 0i τ p e 2MDf aˆ s τ∕M; M
(9)
and describes magnification by a factor M of the quantum input signal aˆ s τ as in an equivalent spatial single-lens imaging system. The phase term appearing in Eq. (9) can be compensated by a second temporal lens, as suggested in Ref. [19]. Equations (7) and (9) are valid for arbitrary quantum states of the input operator aˆ s τ. In Ref. [19], the authors have considered a simple model of a time-bin entangled input state without taking into account the spectral and temporal properties of entanglement such as its spectral bandwidth or the coherence time. We shall consider as an input state of the signal a temporally broadband squeezed state of light generated by a traveling-wave OPA in a second-order nonlinear crystal [22]. The squeezing transformation produced by such an OPA is described in terms of the Fourier amplitudes aˆ s z; Ω: Z aˆ s z; Ω
aˆ s z; teiΩt dt;
(10)
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of the annihilation operators, and is given by a Bogolubov transformation from the input to the output of the OPA: aˆ s l; Ω UΩaˆ s 0; Ω V Ωaˆ †s 0; −Ω:
(11)
Here l is the length of the OPA crystal, and UΩ and VΩ are the complex coefficients depending on the parametric gain of the OPA and its phase-matching conditions. Their explicit form can be found, for example, in Ref. [22]. Transformation (11) is known to produce a broadband squeezed vacuum at the output of the OPA from the broadband vacuum input state. Mixing such a broadband squeezed vacuum with a strong monochromatic local oscillator (LO) and assuming the balanced homodyne photodetection, we can evaluate the normalized to the shot-noise photocurrent noise spectrum δi2Ω ∕hii, which we shall call the squeezing spectrum and denote SΩ. Considering for simplicity the unit photodetection efficiency, we can write the squeezing spectrum as follows, S s Ω cos2 θΩe2rω sin2 θΩe−2rω ;
(12)
where θΩ ψΩ − φ, with φ being the phase of the LO, the squeezing angle ψΩ, and the squeezing parameter rΩ given by ψΩ
1 argVΩ∕UΩ; 2
(13)
exp rΩ jUΩj jVΩj:
(14)
In Fig. 1, we have shown by dashed–dotted line SΩ at the output of the OPA for degenerate phase-matching condition and the parametric gain G exp2r m 9, where r m is the maximum value of the squeezing parameter at Ω 0. One can see from Fig. 1 that at low frequencies, Ω < Ωc , the quantum fluctuations of the photocurrent are reduced below the shot-noise level. −1∕2 This characteristic frequency Ωc βc , where βc 2 l 2 is the GVD coefficient of the OPA, allows us to introduce the characteristic coherence time τc π∕Ωc of quantum correlations, which has a simple physical interpretation. Let the observed photocurrent be collected during the 2
S(Ω)
1.5
1
0.5
0
0
1
2
Ω
3
4
5
Fig. 1. Squeezing spectrum of broadband squeezed light at the output of the OPA (dashed–dotted) and after the temporal lens with efficiency factor η 1∕2 (solid); expr m 3, and frequency Ω is in units of Ωc .
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time of observation T to provide a number N T of photoelectrons. If the time of observation is much longer that the coherence time, T ≫ τc , then the high-frequency quantum fluctuations of the photocurrent null out and do not contribute to the fluctuations in the observed number of photoelectrons. Therefore, for T ≫ τc , the statistics of N T will be sub-Poissonian. Let us now place a temporal lens described by Eqs. (3) and (4) at the output of the OPA. Then we have to use the input state (11) for aˆ s 0; τ and a broadband vacuum state for aˆ i 0; τ in Eqs. (3) and (4). Assuming similar homodyne detection scheme with a LO after the temporal lens we obtain the following squeezing spectrum: S i Ω 1 − η ηcos2 θΩe2rω sin2 θΩe−2rω ; (15) with η s2 τ being the efficiency factor of the time lens equal to the intensity reflexion coefficient in Eqs. (3) and (4). It follows from Eq. (15) that in the case of the temporal lens with the unit efficiency factor, the squeezing spectrum at its output is identical to that at the input. However, for η < 1, the vacuum fluctuations entering into the temporal lens from the open port deteriorate squeezing at its output. For example, we have shown by solid line in Fig. 1 the effect of this deterioration for η 1∕2. We can similarly evaluate the squeezing spectrum at the output of a single-lens temporal imaging system comprising a time lens with Df and two dispersive elements with Ds and Di . Such a scheme is described by Eqs. (8) and (9). Let us first consider a 4f-imaging scheme with Di 2Df , Ds 2Df , and the magnification factor M −1. It is easy to see from Eq. (9) that in this case SΩ at the output of the OPA, z l, is identical to Sω at the output of the temporal imaging scheme, i.e., at z l 4Lf , where Lf Df ∕β2 is the equivalent focal length of the temporal lens with GDD Df and the GVD coefficient β2 . Indeed, since the photocurrent noise spectrum δi2Ω is determined by the intensity correlation function of the light, the phase factor and the sign change in Eq. (9) are irrelevant. This result is not surprising, because the above conditions correspond to geometrical imaging with unit magnification of the output face of the OPA into the photodetection plane at z l 4Lf . It is also clear that in a more general case with arbitrary magnification factor M, when the imaging conditions (8) are satisfied, the squeezing spectrum at the output of the imaging system will be the same as that at the output of the OPA in terms of the scaled frequency Ω0 MΩ, which corresponds to the scaled time τ0 τ∕M in Eq. (9). This magnification factor gives us a possibility of matching the coherence time τc of the broadband squeezed light to the response time of the photodetector. Let us finally discuss a less obvious but more advantageous geometry of a single-lens temporal imaging scheme that allows us to partially compensate for the frequency dispersion of the OPA. It has been shown in Ref. [22] that the squeezing angle ψΩ from Eq. (13) for the degenerate phase matching can be written approximately as 2 ψΩ βc 2 lamp Ω ∕4;
(16)
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OPTICS LETTERS / Vol. 40, No. 6 / March 15, 2015 2
of squeezed light and to match it with the detector response time. Our scheme can find numerous applications in quantum optics and quantum information.
S(Ω)
1.5
1
0.5
0
0
1
2
Ω
3
4
5
Fig. 2. Squeezing spectrum of broadband squeezed light with (solid) and without (dashed–dotted) compensation of quadratic frequency dispersion of the OPA for the same conditions as in Fig. 1.
where lamp is the amplification length of the OPA, defined through r m l∕lamp . Since the angle ψΩ has a quadratic frequency dependence, it can be compensated by a dispersive medium with GVD coefficient β1 2 and the length L1 s equal to L1 s −lamp
βc 2
2β1 2
:
(17)
Let us now assume that we are using a 4f -imaging scheme with z l L1 s instead of z l for the object plane, i.e., we are imaging an object plane inside the OPA into the image plane at z l L1 s 4Lf . The squeezing spectra in the object and the image plane are identical. Therefore, for such a geometry, SΩ in the image plane will correspond to SΩ with compensated quadratic dispersion of the OPA. This spectrum is shown in Fig. 2 by a solid line together with the corresponding spectrum without compensation (dashed–dotted line). As follows from Fig. 2, compensation of the quadratic dispersion of the OPA increases the spectral bandwidth Ωc of squeezing and, respectively, shortens its coherence time τc . In conclusion, we have demonstrated that temporal imaging can be successfully applied to temporally broadband squeezed light, preserving the degree of squeezing. It allows us, in particular, to modify the coherence time
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