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Proposal for generating Fock states in traveling wave fields
Physics Letters A 365 (2007) 258–261 www.elsevier.com/locate/pla
Proposal for generating Fock states in traveling wave fields Adil Benmoussa ∗ , Christopher C. Gerry Department of Physics and Astronomy, Lehman College, The City University of New York, Bronx, NY 10468-1589, USA Received 14 December 2006; accepted 8 January 2007 Available online 20 January 2007 Communicated by P.R. Holland
Abstract We describe a proposal for the generation of a single-mode photonic number state, |N, in a traveling wave optical field. The state is obtained by state reduction from an input coherent state using Kerr media. Our method is based on a previous scheme used for hole burning in the Fock space by minimizing the Mandel Q parameter. The same method was used by Maia et al., but ours is different, it requires only one single photon injected in the entire setup and one photon detection at the end. © 2007 Published by Elsevier B.V.
1. Introduction Constructing non-classical states have become a very active area of research because of their intrinsic interest and because of potential use in quantum information and quantum interferometry. Many non-classical quantum states have already been available in the laboratory, but they do not exhaust all possible ones. One that has a special importance is the Fock state, |N . There have been many approaches to the problem of generating Fock states of the quantized field in the context of cavity QED and in the context of traveling wave fields. In the former case, a cavity field can be manipulated by a sequence of properly prepared and velocity selected atoms crossing the cavity, typically with subsequent state reduction measurements performed on the atoms [1]. In the case of the latter, one must use various linear and nonlinear optical devices for manipulating the field and “sculpting” it into a number state as discussed in a number of recent papers [2]. Here, we wish draw the reader’s attention to a recent proposal by Maia et al. [3] who suggest using a sequence of devices of the type one of us (C.C.G.) proposed several years ago [4] to generate superpositions of optical states. The device consists of a Mach–Zehnder interferometer (MZI), with a single photon injected, coupled to an external field mode, * Corresponding author.
E-mail address: [email protected] (A. Benmoussa). 0375-9601/$ – see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.physleta.2007.01.013
usually containing a coherent state, through a cross-Kerr interaction. Upon performing state reductive measurements on the output modes of the MZI, various superpositions of coherent states are projected into the output of the external field mode. In [4], the generation of even and odd coherent states was discussed. Generating such states by this method requires large third-order nonlinear susceptibilities in the cross-Kerr medium. These are not present in readily available materials, though much effort has gone towards producing giant nonlinearities via the techniques of electromagnetically induced transparencies [5]. In a more recent work [6], the present authors discussed prospects for generating other kinds of non-classical states with smaller nonlinearities. In particular we showed that one could start with a coherent state and with judicious choices of interaction times, one could produce “holes” in the photon number probability distribution, or that one could enhance the probability for a particular photon number. In this later case, the photon number distribution becomes sub-Poissonian, a distinctly nonclassical form of light. Photon number states are the most subPoissonian states of light possible. Now Maia et al. [3] have proposed using a sequence of these devices to further enhance the sub-Poissonian nature of state produced by the first device toward a number state. However, it is supposed that there is to be a single photon injected to the MZIs of each of these devices so that a sequence of properly timed single photon inputs is required, and that state reductive measurements are to be performed at each step. This is impractical experimentally. In the
A. Benmoussa, C.C. Gerry / Physics Letters A 365 (2007) 258–261
present Letter, we reconsider the essential idea of Maia et al. [3] but with important modifications. We study the case that only one photon needs to be injected, this into the MZI of the first device, and that only one state reductive measurement is performed at the output of the last device in the sequence. Many superposition devices can be mounted in series to single out a specific number state from an input coherent state. Each of the superposition devices has to be adjusted to minimize the Mandel Q parameter. Minimizing the Mandel Q parameter is an idea used previously by Ghosh and Gerry, for the production of number states, |N, in a cavity field [7]. We organize the Letter as follows: In Section 2 we review the superposition device, in Section 3 we illustrate how we generate a number state |N with an input coherent state, and we conclude our work in Section 4. 2. Review of the superposition device The basic one-stage superposition device is sketched in Fig. 1. It consists of a Mach–Zehnder interferometer with a cross-Kerr medium in one arm coupled to an external mode, denoted c which initially contains the coherent state to be subject to modification. The cross-Kerr interaction between the external and internal modes is described by ˆ Hˆ K = h¯ K bˆ † bˆ cˆ† c,
(1)
where K is proportional to the third order nonlinear susceptibility, χ (3) . The modes a and c contain a single photon and a coherent state, respectively. Mode b is initially in a vacuum state. The input state can be expressed as |in = |1a |0b |αc .
(2)
Just after the first beam splitter, BS1, the state is given by 1 √ |1a |0b + i|0a |1b |αc , 2
(3)
where the states are labeled according to the scheme where the first state is of the clockwise and the second of the counterclockwise. Just before the second beam splitter, BS2, the state
Fig. 1. A diagram of a modified Mach–Zehnder interferometer that is used to superpose two coherent states of the same magnitude, but different phase shifts. The phase shift depends on the length of the Kerr medium and the third order nonlinear susceptibility.
259
has the form 1 √ |1a |0b |αc + i|0a |1b αe−iϕ c , 2
(4)
where ϕ = Kl/v, l is the length of the medium, v the velocity of light in the medium. The phase shift in the coherent state comes from the Kerr medium represented by the unitary operator Uˆ K = exp −iϕ bˆ † bˆ cˆ† cˆ . (5) Note that the first term in Eq. (4) does not pick up the phase shift because of |0 in the b-mode. The state after the BS2 is 1 |1a |0b |ψ10 c + i|0a |1b |ψ01 c . 2 This is a three-mode entangled state, where |ψ10 c = |αc − αeiϕ c , |ψ01 c = |αc + αeiϕ c .
(6)
(7)
If detector D1 (D2) clicks the state |1a |0b (|0a |1b ) has been detected and the state in c-mode collapses to |ψ10 (|ψ01 ). 3. Generating Fock state with an input coherent state One way to create number states is to superpose many coherent states of same amplitude but different phases in a fashion to cancel out all coefficients except the coefficient of the desired Fock state |N . In fact, the superposition device we have previously described superposes two coherent states with a phase shift difference ϕ as it is shown in Eqs. (7). To carry this further to more than two coherent states, we propose injecting the output to another superposition device to superpose four coherent states. In Fig. 2 we display a scheme that can be used to superpose coherent states with the same amplitude, but different phases. The setup consists of m superposition devices mounted in series, where the output channels a and c of each device are connected to the input channels a and c of the next device. Unlike modes a and c, mode b in each superposition device k has an input port with a vacuum state and output port which we label as bk , where 1 < k m. The last superposition device has one detector at the output port a, which we denote as Df . Note that Df is the only detector we use in the entire setup.
Fig. 2. The diagram shows many devices of Fig. 1 mounted in series in order to superpose more than two coherent states.
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A. Benmoussa, C.C. Gerry / Physics Letters A 365 (2007) 258–261
Let us first consider the case for m = 2 with an input state of the form |1a |0b1 |0b2 |αc .
(8)
Applying the same method used in the previous section we find the output state in this case to be |ψ2 =
11 11 |1a |0b2 |0b1 |ψ100 c + i |0a |1b2 |0b1 |ψ010 c 22 22 1 + i |0a |0b2 |1b1 |ψ01 c , (9) 2
· · 0 , which refers to the detection of a single and (10)m = 1 0 ·
m times
photon at the output port a and vacuum states in the output ports b1 to bm . Note that the state in Eq. (13) is not normalized, and ϕk in Eq. (14) refers to the phase shift produced in by the Kerr medium in the kth device, as displayed in Fig. 2. Up to this point, we have just described how the device shown in Fig. 2 superposes coherent states with the same amplitude and with different phases. In order to obtain a Fock state we can adjust all ϕk in a way to have all the coefficients, cn , in Eq. (13) vanish except one of them, let us say cN , i.e.
where
|ψ100 c = |αc − αe−iϕ1 c − αe−iϕ2 c + αe−i(ϕ1 +ϕ2 ) c , |ψ010 c = |αc − αe−iϕ1 + αe−iϕ2 − αe−i(ϕ1 +ϕ2 ) , c
c
cN = δn,N ,
c
(10) and ϕ1 and ϕ2 are the phase shifts produced by the Kerr of the devices 1 and 2, respectively, as indicated in Fig. 2. Upon the detection of the state |1a |0b2 |0b1 the state in c mode is projected into the un-normalized state |ψ100 c . To carry this process further to the case of three superposition devices, the output state can be expressed as 111 |1a |0b3 |0b2 |0b1 |ψ1000 c |ψ3 = 222 111 +i |0a |1b3 |0b2 |0b1 |ψ0100 c 222 11 +i |0a |0b3 |1b2 |0b1 |ψ010 c 22 1 + i |0a |0b3 |0b2 |1b1 |ψ01 c , 2 where |ψ1000 c = |αc − αe−iϕ1 c − αe−iϕ2 c − αe−iϕ3 c + αe−i(ϕ1 +ϕ2 ) c + αe−i(ϕ1 +ϕ3 ) c + αe−i(ϕ2 +ϕ3 ) c − αe−i(ϕ1 +ϕ2 +ϕ3 ) c , |ψ0100 c = |αc − αe−iϕ1 c − αe−iϕ2 c + αe−iϕ3 c + αe−i(ϕ1 +ϕ2 ) c − αe−i(ϕ1 +ϕ3 ) c − αe−i(ϕ2 +ϕ3 ) + αe−i(ϕ1 +ϕ2 +ϕ3 ) . c
c
(12)
The state in Eq. (11) is a five-mode entangled state with a superposition of four orthogonal states. Again, the mere detection of the photon at detector Df at the third superposition device indicates that |ψ3 has collapsed to |ψ1000 . In this case the measurement refers to the detection of the state |1a |0b1 |0b2 |0b3 . Generalizing the above procedure to the case of m superposition devices we find the output state in mode c, expressed in terms of Fock states, to be |ψ(10)m c =
∞
cn |nc ,
(13)
n=0
where cn = e−|α|
2 /2
m αn 1 − e−inϕk √ n! k=1
so that we will obtain a number state |N in the c mode. Analytically, it is not easy to find the required values of all ϕk . Instead, we will proceed with a numerical method to determine the values of the phase shifts. Fock state |N is a highly quantum mechanical state. A useful measure for the departure of the field states from “classical”, or coherent states, is the Mandel Q parameter, that is defined as Q=
(11)
(14)
(15)
σn2 − n¯ , n¯
(16)
where σn2 = (n)2 . For a Poissonian distribution associated with a coherent state, σn2 = n¯ and Q = 0. Distributions for which Q > 0 are super-Poissonian, and sub-Poissonian for Q < 0, the latter is a sign of non-classicality. As Q approaches −1 the state approaches the number states, |N [8]. In each superposition device of Fig. 2 we have to determine a numerical value for ϕm . Before adjusting ϕm , one needs to study the Mandel Q parameter to obtain the most subPoissonian distribution possible for the state |ψ(10)m . To illustrate the method, we discuss an example √ for which the input coherent state |α has an amplitude |α| = 5. We plot Mandel Q parameter versus ϕ0 for the case of the coherent state in Fig. 3(a), for comparison purposes. For the same reason, in Fig. 4(a), we display the photon probability distribution of the coherent state, a Poissonian distribution. For the case of one superposition device, we need determine the numerical value ϕ1 . To that end, we plot the Mandel Q of the state |ψ10 c versus ϕ1 as it is displayed in Fig. 3(b). Unlike the case of the coherent state, the Mandel Q parameter takes some negative values and it is minimum at ϕ1 = 0.587. For this value of ϕ1 , we plot the photon distribution for the state |ψ10 c in Fig. 4(b), which is sub-Poissonian as one can compare it to the photon distribution of the initial coherent state in Fig. 4(a). To carry this to the case of m = 2, we keep the first unit with the same value of ϕ1 and we plot the Mandel Q parameter for the state |ψ100 c , as it is shown in Fig. 3(c). This time the Mandel Q parameter has a minimum at ϕ2 = 0.777. With this value of ϕ2 , we plot the photon number distribution in Fig. 4(c), which becomes more sub-Poissonian compared to the distribution displayed in Fig. 4(b). Repeating the procedure, we see that the photon number distribution approaches the distribution of Fock state |5, after five iterations, as it is shown in Fig. 4(f).
A. Benmoussa, C.C. Gerry / Physics Letters A 365 (2007) 258–261
Fig. 3. Plots of the Mandel Q parameter versus the phase shift ϕk , (a) for a coherent state, (b) for the case of m = 1, (c) m = 2, (d) m = 3, (e) m = 4, and (f) m = 5. These plots are used to determine numerical values of the phase shifts produced by the superposition devices shown in Fig. 2 for which a superposition of coherent states approaches a number state after a state reduction measurement.
4. Conclusions In conclusion, we have proposed a scheme for generating Fock states out of the superpositions of coherent states for traveling wave fields. The method described is similar to the one proposed earlier by Maia et al. [3], but with an important difference: only one auxiliary photon is required and detected throughout whereas in [3] multiple single photon inputs are required, each with precision timing, followed by an equal number of state reductive measurements. The major obstacle this method need to overcome is the requirement of sizeable Kerr nonlinearities of the type that may soon become available through the techniques of electromagnetically induced transparency. The proposal of [3] also requires these large Kerr nonlinearities. If such are eventually achieved, the present method could be used to generate traveling wave field Fock states of modest photon number. Acknowledgements This work is supported by the US Army. A.B. wishes to express his gratitude to the AGEP, the LSAMPS, and MAGNET programs at the Graduate Center of the City University of New York for their financial support.
261
Fig. 4. Photon-number distributions of (a) a coherent state for which the am√ plitude α = 5, (b) the state |ψ10 c , (c) |ψ100 c , (d) |ψ1000 c , (e) |ψ10000 c , and (f) |ψ100000 c with ϕ1 = 0.587, ϕ2 = 0.777, ϕ3 = 1.97, ϕ4 = 3.14, and ϕ5 = 1.80.
References [1] See P. Filipowicz, J. Javanainen, P. Meystre, J. Opt. Soc. Am. B 3 (1986) 906; F.W. Cummings, A.K. Rajagopal, Phys. Rev. A 39 (1989) 3414; B.T.H. Varcoe, S. Brattke, M. Weidinger, H. Walther, Nature (London) 403 (2000) 743; S. Brattke, B.T.H. Varcoe, H. Walther, Phys. Rev. Lett. 86 (2001) 3534; M. Brune, S. Haroche, J.M. Raimond, L. Davidovich, N. Zagury, Phys. Rev. A 45 (1992) 5193; H. Ghosh, C.C. Gerry, J. Opt. Soc. Am. B 14 (1997) 2782. [2] See G.M. D’Ariano, L. Maccone, M.G.A. Paris, M.F. Sacchi, Phys. Rev. A 61 (2000) 053817; K. Sanaka, Phys. Rev. A 71 (2005) 021801(R); L.P.A. Maia, B. Baseia, A.T. Avelar, J.M.C. Malbouisson, J. Opt. B: Quantum Semiclass. Opt. 6 (2004) 351; C.C. Gerry, A. Benmoussa, Phys. Rev. A 73 (2006) 063817. [3] L.P.A. Maia, B. Baseia, A.T. Avelar, J.M.C. Malbouisson, J. Opt. B: Quantum Semiclass. Opt. 6 (2004) 351. [4] C.C. Gerry, Phys. Rev. A 59 (1999) 4095. [5] H. Schmidt, A. Imamoglu, Opt. Lett. 21 (1996) 1936; L. Deng, M.G. Payne, W.R. Garret, Phys. Rev. A 64 (2000) 023807; H. Hang, Y. Zhu, Phys. Rev. Lett. 91 (2003) 093601; D.A. Braje, V. Balic, G.Y. Yin, S.E. Harris, Phys. Rev. A 68 (2003) R041801. [6] C.C. Gerry, A. Benmoussa, Phys. Lett. A 303 (2002) 30. [7] H. Ghosh, C.C. Gerry, J. Opt. Soc. Am. B 14 (11) (1997) 2782. [8] L. Mandel, Opt. Lett. 4 (1979) 205.
Proposal for generating Fock states in traveling wave fields Adil Benmoussa ∗ , Christopher C. Gerry Department of Physics and Astronomy, Lehman College, The City University of New York, Bronx, NY 10468-1589, USA Received 14 December 2006; accepted 8 January 2007 Available online 20 January 2007 Communicated by P.R. Holland
Abstract We describe a proposal for the generation of a single-mode photonic number state, |N, in a traveling wave optical field. The state is obtained by state reduction from an input coherent state using Kerr media. Our method is based on a previous scheme used for hole burning in the Fock space by minimizing the Mandel Q parameter. The same method was used by Maia et al., but ours is different, it requires only one single photon injected in the entire setup and one photon detection at the end. © 2007 Published by Elsevier B.V.
1. Introduction Constructing non-classical states have become a very active area of research because of their intrinsic interest and because of potential use in quantum information and quantum interferometry. Many non-classical quantum states have already been available in the laboratory, but they do not exhaust all possible ones. One that has a special importance is the Fock state, |N . There have been many approaches to the problem of generating Fock states of the quantized field in the context of cavity QED and in the context of traveling wave fields. In the former case, a cavity field can be manipulated by a sequence of properly prepared and velocity selected atoms crossing the cavity, typically with subsequent state reduction measurements performed on the atoms [1]. In the case of the latter, one must use various linear and nonlinear optical devices for manipulating the field and “sculpting” it into a number state as discussed in a number of recent papers [2]. Here, we wish draw the reader’s attention to a recent proposal by Maia et al. [3] who suggest using a sequence of devices of the type one of us (C.C.G.) proposed several years ago [4] to generate superpositions of optical states. The device consists of a Mach–Zehnder interferometer (MZI), with a single photon injected, coupled to an external field mode, * Corresponding author.
E-mail address: [email protected] (A. Benmoussa). 0375-9601/$ – see front matter © 2007 Published by Elsevier B.V. doi:10.1016/j.physleta.2007.01.013
usually containing a coherent state, through a cross-Kerr interaction. Upon performing state reductive measurements on the output modes of the MZI, various superpositions of coherent states are projected into the output of the external field mode. In [4], the generation of even and odd coherent states was discussed. Generating such states by this method requires large third-order nonlinear susceptibilities in the cross-Kerr medium. These are not present in readily available materials, though much effort has gone towards producing giant nonlinearities via the techniques of electromagnetically induced transparencies [5]. In a more recent work [6], the present authors discussed prospects for generating other kinds of non-classical states with smaller nonlinearities. In particular we showed that one could start with a coherent state and with judicious choices of interaction times, one could produce “holes” in the photon number probability distribution, or that one could enhance the probability for a particular photon number. In this later case, the photon number distribution becomes sub-Poissonian, a distinctly nonclassical form of light. Photon number states are the most subPoissonian states of light possible. Now Maia et al. [3] have proposed using a sequence of these devices to further enhance the sub-Poissonian nature of state produced by the first device toward a number state. However, it is supposed that there is to be a single photon injected to the MZIs of each of these devices so that a sequence of properly timed single photon inputs is required, and that state reductive measurements are to be performed at each step. This is impractical experimentally. In the
A. Benmoussa, C.C. Gerry / Physics Letters A 365 (2007) 258–261
present Letter, we reconsider the essential idea of Maia et al. [3] but with important modifications. We study the case that only one photon needs to be injected, this into the MZI of the first device, and that only one state reductive measurement is performed at the output of the last device in the sequence. Many superposition devices can be mounted in series to single out a specific number state from an input coherent state. Each of the superposition devices has to be adjusted to minimize the Mandel Q parameter. Minimizing the Mandel Q parameter is an idea used previously by Ghosh and Gerry, for the production of number states, |N, in a cavity field [7]. We organize the Letter as follows: In Section 2 we review the superposition device, in Section 3 we illustrate how we generate a number state |N with an input coherent state, and we conclude our work in Section 4. 2. Review of the superposition device The basic one-stage superposition device is sketched in Fig. 1. It consists of a Mach–Zehnder interferometer with a cross-Kerr medium in one arm coupled to an external mode, denoted c which initially contains the coherent state to be subject to modification. The cross-Kerr interaction between the external and internal modes is described by ˆ Hˆ K = h¯ K bˆ † bˆ cˆ† c,
(1)
where K is proportional to the third order nonlinear susceptibility, χ (3) . The modes a and c contain a single photon and a coherent state, respectively. Mode b is initially in a vacuum state. The input state can be expressed as |in = |1a |0b |αc .
(2)
Just after the first beam splitter, BS1, the state is given by 1 √ |1a |0b + i|0a |1b |αc , 2
(3)
where the states are labeled according to the scheme where the first state is of the clockwise and the second of the counterclockwise. Just before the second beam splitter, BS2, the state
Fig. 1. A diagram of a modified Mach–Zehnder interferometer that is used to superpose two coherent states of the same magnitude, but different phase shifts. The phase shift depends on the length of the Kerr medium and the third order nonlinear susceptibility.
259
has the form 1 √ |1a |0b |αc + i|0a |1b αe−iϕ c , 2
(4)
where ϕ = Kl/v, l is the length of the medium, v the velocity of light in the medium. The phase shift in the coherent state comes from the Kerr medium represented by the unitary operator Uˆ K = exp −iϕ bˆ † bˆ cˆ† cˆ . (5) Note that the first term in Eq. (4) does not pick up the phase shift because of |0 in the b-mode. The state after the BS2 is 1 |1a |0b |ψ10 c + i|0a |1b |ψ01 c . 2 This is a three-mode entangled state, where |ψ10 c = |αc − αeiϕ c , |ψ01 c = |αc + αeiϕ c .
(6)
(7)
If detector D1 (D2) clicks the state |1a |0b (|0a |1b ) has been detected and the state in c-mode collapses to |ψ10 (|ψ01 ). 3. Generating Fock state with an input coherent state One way to create number states is to superpose many coherent states of same amplitude but different phases in a fashion to cancel out all coefficients except the coefficient of the desired Fock state |N . In fact, the superposition device we have previously described superposes two coherent states with a phase shift difference ϕ as it is shown in Eqs. (7). To carry this further to more than two coherent states, we propose injecting the output to another superposition device to superpose four coherent states. In Fig. 2 we display a scheme that can be used to superpose coherent states with the same amplitude, but different phases. The setup consists of m superposition devices mounted in series, where the output channels a and c of each device are connected to the input channels a and c of the next device. Unlike modes a and c, mode b in each superposition device k has an input port with a vacuum state and output port which we label as bk , where 1 < k m. The last superposition device has one detector at the output port a, which we denote as Df . Note that Df is the only detector we use in the entire setup.
Fig. 2. The diagram shows many devices of Fig. 1 mounted in series in order to superpose more than two coherent states.
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A. Benmoussa, C.C. Gerry / Physics Letters A 365 (2007) 258–261
Let us first consider the case for m = 2 with an input state of the form |1a |0b1 |0b2 |αc .
(8)
Applying the same method used in the previous section we find the output state in this case to be |ψ2 =
11 11 |1a |0b2 |0b1 |ψ100 c + i |0a |1b2 |0b1 |ψ010 c 22 22 1 + i |0a |0b2 |1b1 |ψ01 c , (9) 2
· · 0 , which refers to the detection of a single and (10)m = 1 0 ·
m times
photon at the output port a and vacuum states in the output ports b1 to bm . Note that the state in Eq. (13) is not normalized, and ϕk in Eq. (14) refers to the phase shift produced in by the Kerr medium in the kth device, as displayed in Fig. 2. Up to this point, we have just described how the device shown in Fig. 2 superposes coherent states with the same amplitude and with different phases. In order to obtain a Fock state we can adjust all ϕk in a way to have all the coefficients, cn , in Eq. (13) vanish except one of them, let us say cN , i.e.
where
|ψ100 c = |αc − αe−iϕ1 c − αe−iϕ2 c + αe−i(ϕ1 +ϕ2 ) c , |ψ010 c = |αc − αe−iϕ1 + αe−iϕ2 − αe−i(ϕ1 +ϕ2 ) , c
c
cN = δn,N ,
c
(10) and ϕ1 and ϕ2 are the phase shifts produced by the Kerr of the devices 1 and 2, respectively, as indicated in Fig. 2. Upon the detection of the state |1a |0b2 |0b1 the state in c mode is projected into the un-normalized state |ψ100 c . To carry this process further to the case of three superposition devices, the output state can be expressed as 111 |1a |0b3 |0b2 |0b1 |ψ1000 c |ψ3 = 222 111 +i |0a |1b3 |0b2 |0b1 |ψ0100 c 222 11 +i |0a |0b3 |1b2 |0b1 |ψ010 c 22 1 + i |0a |0b3 |0b2 |1b1 |ψ01 c , 2 where |ψ1000 c = |αc − αe−iϕ1 c − αe−iϕ2 c − αe−iϕ3 c + αe−i(ϕ1 +ϕ2 ) c + αe−i(ϕ1 +ϕ3 ) c + αe−i(ϕ2 +ϕ3 ) c − αe−i(ϕ1 +ϕ2 +ϕ3 ) c , |ψ0100 c = |αc − αe−iϕ1 c − αe−iϕ2 c + αe−iϕ3 c + αe−i(ϕ1 +ϕ2 ) c − αe−i(ϕ1 +ϕ3 ) c − αe−i(ϕ2 +ϕ3 ) + αe−i(ϕ1 +ϕ2 +ϕ3 ) . c
c
(12)
The state in Eq. (11) is a five-mode entangled state with a superposition of four orthogonal states. Again, the mere detection of the photon at detector Df at the third superposition device indicates that |ψ3 has collapsed to |ψ1000 . In this case the measurement refers to the detection of the state |1a |0b1 |0b2 |0b3 . Generalizing the above procedure to the case of m superposition devices we find the output state in mode c, expressed in terms of Fock states, to be |ψ(10)m c =
∞
cn |nc ,
(13)
n=0
where cn = e−|α|
2 /2
m αn 1 − e−inϕk √ n! k=1
so that we will obtain a number state |N in the c mode. Analytically, it is not easy to find the required values of all ϕk . Instead, we will proceed with a numerical method to determine the values of the phase shifts. Fock state |N is a highly quantum mechanical state. A useful measure for the departure of the field states from “classical”, or coherent states, is the Mandel Q parameter, that is defined as Q=
(11)
(14)
(15)
σn2 − n¯ , n¯
(16)
where σn2 = (n)2 . For a Poissonian distribution associated with a coherent state, σn2 = n¯ and Q = 0. Distributions for which Q > 0 are super-Poissonian, and sub-Poissonian for Q < 0, the latter is a sign of non-classicality. As Q approaches −1 the state approaches the number states, |N [8]. In each superposition device of Fig. 2 we have to determine a numerical value for ϕm . Before adjusting ϕm , one needs to study the Mandel Q parameter to obtain the most subPoissonian distribution possible for the state |ψ(10)m . To illustrate the method, we discuss an example √ for which the input coherent state |α has an amplitude |α| = 5. We plot Mandel Q parameter versus ϕ0 for the case of the coherent state in Fig. 3(a), for comparison purposes. For the same reason, in Fig. 4(a), we display the photon probability distribution of the coherent state, a Poissonian distribution. For the case of one superposition device, we need determine the numerical value ϕ1 . To that end, we plot the Mandel Q of the state |ψ10 c versus ϕ1 as it is displayed in Fig. 3(b). Unlike the case of the coherent state, the Mandel Q parameter takes some negative values and it is minimum at ϕ1 = 0.587. For this value of ϕ1 , we plot the photon distribution for the state |ψ10 c in Fig. 4(b), which is sub-Poissonian as one can compare it to the photon distribution of the initial coherent state in Fig. 4(a). To carry this to the case of m = 2, we keep the first unit with the same value of ϕ1 and we plot the Mandel Q parameter for the state |ψ100 c , as it is shown in Fig. 3(c). This time the Mandel Q parameter has a minimum at ϕ2 = 0.777. With this value of ϕ2 , we plot the photon number distribution in Fig. 4(c), which becomes more sub-Poissonian compared to the distribution displayed in Fig. 4(b). Repeating the procedure, we see that the photon number distribution approaches the distribution of Fock state |5, after five iterations, as it is shown in Fig. 4(f).
A. Benmoussa, C.C. Gerry / Physics Letters A 365 (2007) 258–261
Fig. 3. Plots of the Mandel Q parameter versus the phase shift ϕk , (a) for a coherent state, (b) for the case of m = 1, (c) m = 2, (d) m = 3, (e) m = 4, and (f) m = 5. These plots are used to determine numerical values of the phase shifts produced by the superposition devices shown in Fig. 2 for which a superposition of coherent states approaches a number state after a state reduction measurement.
4. Conclusions In conclusion, we have proposed a scheme for generating Fock states out of the superpositions of coherent states for traveling wave fields. The method described is similar to the one proposed earlier by Maia et al. [3], but with an important difference: only one auxiliary photon is required and detected throughout whereas in [3] multiple single photon inputs are required, each with precision timing, followed by an equal number of state reductive measurements. The major obstacle this method need to overcome is the requirement of sizeable Kerr nonlinearities of the type that may soon become available through the techniques of electromagnetically induced transparency. The proposal of [3] also requires these large Kerr nonlinearities. If such are eventually achieved, the present method could be used to generate traveling wave field Fock states of modest photon number. Acknowledgements This work is supported by the US Army. A.B. wishes to express his gratitude to the AGEP, the LSAMPS, and MAGNET programs at the Graduate Center of the City University of New York for their financial support.
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Fig. 4. Photon-number distributions of (a) a coherent state for which the am√ plitude α = 5, (b) the state |ψ10 c , (c) |ψ100 c , (d) |ψ1000 c , (e) |ψ10000 c , and (f) |ψ100000 c with ϕ1 = 0.587, ϕ2 = 0.777, ϕ3 = 1.97, ϕ4 = 3.14, and ϕ5 = 1.80.
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