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Nonclassical properties of coherent states
Physics Letters A 329 (2004) 184–187 www.elsevier.com/locate/pla
Nonclassical properties of coherent states Lars M. Johansen Department of Technology, Buskerud University College, P.O. Box 251, N-3601 Kongsberg, Norway Received 13 June 2004; received in revised form 2 July 2004; accepted 5 July 2004 Available online 21 July 2004 Communicated by P.R. Holland
Abstract It is demonstrated that a weak measurement of the squared quadrature observable may yield negative values for coherent states. This result cannot be reproduced by a classical theory where quadratures are stochastic c-numbers. The nonclassicality of coherent states can be associated with negative values of the Terletsky–Margenau–Hill distribution. 2004 Elsevier B.V. All rights reserved. PACS: 42.50.Dv; 03.65.Ta Keywords: Nonclassicality; Coherent states; Weak values; Weak measurements; P -distribution; Margenau–Hill distribution; Kirkwood distribution
Harmonic oscillator coherent states were first investigated by Schrödinger, who was looking for classical-like states [1]. There are several ways in which coherent states are the “most classical” of any pure state. They keep their shape, not spreading out as they move in the harmonic oscillator potential [1]. They minimize Heisenberg’s uncertainty relation, with equal uncertainty in both quadratures. In this way, they are the closest possible quantum mechanical representation of a point in phase space. The term “coherent state” was introduced by Glauber [2]. He demonstrated that coherent states are produced when an essentially classical current interacts with the radiation
E-mail address: [email protected] (L.M. Johansen). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.07.003
field [2]. Aharonov et al. demonstrated that coherent states are the only pure states that produce independent output when split in two [3]. Zurek et al. have demonstrated that coherent states are natural “pointer states” for a harmonic oscillator weakly coupled to a thermal environment [4]. Glauber and Sudarshan demonstrated that any density operator can be expanded in terms of coherent states [2,5] ρˆ = d 2 α P (α)|αα|. (1) The weight function P (α) is known as the P -distribution. Glauber defined nonclassical states as those for which the P -distribution fails to be a probability density. More specifically, nonclassical states have
L.M. Johansen / Physics Letters A 329 (2004) 184–187
a P -distribution which is negative or more singular than a δ-function [2,6–9]. This criterion is the basis of various measures of “nonclassicality” [10–13]. The phrase “nonclassical” is appropriate whenever classical physics fails to describe a phenomenon. In this Letter, we will show that a classical phase space representation fails as a model to describe weak measurements [14] of quadrature observables on coherent states. We also show that the failure is due to negativity of a phase-space quasi-probability distribution. The distribution is the Terletsky–Margenau–Hill (TMH) distribution [15,16], which is partly negative for coherent states [17]. We give an operational significance to conditional moments of the TMH-distribution by demonstrating that they can be observed in weak measurements. Weak measurements were proposed by Aharonov et al. [14]. Their suggestion was initially met with criticism [18–20], but has since been confirmed [21]. The results reported in this Letter are related to a paper by Aharonov et al. [22], which demonstrated that a weak measurement of kinetic energy of a particle in a classically forbidden region might yield negative values. For an object prepared in the pure state |ψ and postselected by a projective measurement on the ˆ the weak value of eigenstate |d of the observable d, the observable cˆ is found to be [14]
and
d|c|ψ ˆ . (2) d|ψ In a weak measurement, the real part of the weak value is observed [14]. If the object is prepared in a mixed state ρ, ˆ the weak value is [23–25]
where
cw (d) =
d|cˆρ|d ˆ . (3) d|ρ|d ˆ If we consider in particular the quadrature observˆ we may ables cˆ = pˆ n (n = 0, 1, 2, . . .) and dˆ = q, easily express the weak value as a conditional moment of a phase-space distribution. By inserting the completeness relation dp |pp| = 1 in the numerator of Eq. (3), we find that the weak value of pˆ n can be written as n p w = dp pn S(p|q), (4)
cw (d) =
S(q, p) = q|pp|ρ|q. ˆ
185
(6)
S(q, p) is the standard ordered distribution [26] and S(p|q) can be interpreted as a “conditional quasiprobability” which may be complex. In a weak measurement, the pointer (or measurement apparatus) displays on average the real part of the weak value [14, 25]. It is therefore of interest to note that we may write the real part of the weak value of pˆ n as Re pn w = dp pn T (p|q), (7) where T (p|q) =
T (q, p) , dq T (q, p)
(8)
and where T (q, p) = Re S(q, p) is the TMH-distribution. Thus, Re[(pn )w ] is a conditional moment of the TMH-distribution (see also [27–30]). A classical object where the quadrature observables are c-numbers can be described by a non-negative phase-space distribution F (q, p). For such systems, a weak measurement of pn yields the weak value [25] n p w = dp pn F (p|q), (9)
F (p|q) =
F (q, p) dp F (q, p)
(10)
is a genuine, non-negative conditional probability density. We see that (pn )w is the conditional expectation value of that variable. In other words, (pn )w is simply the expectation value of pn “given” q. It follows straightforwardly from Eq. (9) that since p2n 0, 2n p w 0. (11) This is a condition that must is satisfied by any classical phase space representation. We also note that it is satisfied by quantum states for which the TMHdistribution is non-negative. Hence, for such quantum states, a classical model of the weak measurement exists.1
where S(p|q) =
S(q, p) dp S(q, p)
(5)
1 An additional classicality condition is that the imaginary part
of the standard ordered distribution vanishes.
186
L.M. Johansen / Physics Letters A 329 (2004) 184–187
Our next step is to demonstrate that a weak measurement of pˆ 2 on coherent states violates Eq. (11). Consider the quadrature representation of a coherent state |α (with ω = 1) [31] 2 √ q 1 2 1 2 −1/4 q|α = π exp − + 2αq − |α| − α . 2 2 2 (12) The weak value of pˆ 2 for an ensemble preselected in the coherent state |α and postselected in the quadrature eigenstate |q is
p2
= w
√ −∂ 2 q|α/∂q 2 = 1 − (q − 2α)2 . q|α
(13)
Fig. 1. The probability of observing a negative weak value for a co√ herent state with amplitude (αr + iαi )/ 2. It is plotted as a function of the imaginary component αi . The probability is independent of the real component αr .
The real part of the weak value is Re[(p2 )w ] = 1 + αi2 − (q − αr )2 , where we have introduced the notation √ α = (αr + iαi )/ 2. We see that Re[(p2 )w ] is negative if (q − αr )2 > 1 + αi2 . The surprising conclusion is that the weak value of pˆ 2 can be negative for coherent states, although pˆ 2 has only non-negative eigenvalues. The probability of obtaining a negative value is αr + 1+αi2
P =1−
q|α 2 dq = erfc 1 + α 2 ,
αr − 1+αi2
i
(14)
where erfc(x) is the complementary error function. This function is plotted in Fig. 1. It has a maximum when the imaginary part of the coherent state amplitude vanishes, in which case it equals erfc(1) ≈ 0.16. It is quite interesting to note that this probability is independent of αr . Hence, it persists even for macroscopic excitations. Combining Eqs. (12) and (6), the standard ordered distribution for a coherent state |α is easily found. The standard ordered distribution for vacuum can be written as 1 2 1 2 S0 (q, p) = √ e− 2 (q +p )+iqp . 2π
(15)
The standard ordered distribution for a coherent state may then be expressed in terms of the displaced vacuum, Sα (q, p) = S0 (q − αr , p − αi ). The real part of S0 , the TMH-distribution, has been plotted in Fig. 2 (see also Ref. [17]). It clearly has negative regions. This is the reason why a weak measurements of the
Fig. 2. The TMH-distribution for vacuum. It is negative in a certain domain. The TMH-distribution for an arbitrary coherent state is just a displaced vacuum state.
positive operator pˆ 2 may yield negative values for a coherent state. In recent years, various studies have found weak values that exceed the eigenvalue range of the observable when considering polarization of classical, optical systems [21,32–35]. But how can such “strange” weak values be observed in classical systems? The explanation is that classical polarization measurements cannot be modelled in terms of c-number observables. Such measurements must be described in terms of noncommuting observables. Therefore, these experi-
L.M. Johansen / Physics Letters A 329 (2004) 184–187
ments simply show that polarization cannot be modelled by a c-number theory. If coherent states are nonclassical, one may wonder whether there are any classical states at all? Indeed, it was recently shown that harmonic oscillator thermal states of occupation number above unity are essentially classical also in the weak measurement scheme considered here [36]. In conclusion, it was demonstrated that weak measurements on coherent states may produce results that cannot be explained in terms of a classical phase space distribution. It was shown that weak values of quadrature moments are conditional moments of the standard ordered distribution, and that the nonclassical properties revealed in the proposed experiment is related to negativity of the Terletsky–Margenau–Hill distribution.
References [1] E. Schrödinger, Naturwissenschaften 14 (1926) 664. [2] R.J. Glauber, Phys. Rev. 131 (1963) 2766. [3] Y. Aharonov, D. Falkoff, E. Lerner, H. Pendleton, Ann. Phys. 39 (1966) 498. [4] W.H. Zurek, S. Habib, J.P. Paz, Phys. Rev. Lett. 70 (1993) 1187. [5] E.C.G. Sudarshan, Phys. Rev. Lett. 10 (7) (1963) 277. [6] U.M. Titulaer, R.J. Glauber, Phys. Rev. 140 (1965) B676. [7] M. Hillery, Phys. Lett. A 111 (8–9) (1985) 409. [8] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge Univ. Press, New York, NY, 1995. [9] V.V. Dodonov, J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1. [10] M. Hillery, Phys. Rev. A 35 (2) (1987) 725.
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
187
C.T. Lee, Phys. Rev. A 44 (5) (1991) R2775. W. Vogel, Phys. Rev. Lett. 84 (9) (2000) 1849. T. Richter, W. Vogel, Phys. Rev. Lett. 89 (28) (2002) 283601. Y. Aharonov, D.Z. Albert, L. Vaidman, Phys. Rev. Lett. 60 (14) (1988) 1351. Y.P. Terletsky, Zh. Eksp. Teor. Fiz. 7 (1937) 1290. H. Margenau, R.N. Hill, Prog. Theor. Phys. 26 (5) (1961) 722. L. Praxmeyer, K. Wódkiewicz, Opt. Commun. 223 (2003) 349. A.J. Leggett, Phys. Rev. Lett. 62 (19) (1989) 2325. A. Peres, Phys. Rev. Lett. 62 (19) (1989) 2326. Y. Aharonov, L. Vaidman, Phys. Rev. Lett. 62 (19) (1989) 2327. I.M. Duck, P.M. Stevenson, E.C.G. Sudarshan, Phys. Rev. D 40 (6) (1989) 2112. Y. Aharonov, S. Popescu, D. Rohrlich, L. Vaidman, Phys. Rev. A 48 (6) (1993) 4084. M.J.W. Hall, Phys. Rev. A 64 (2001) 052103. L.M. Johansen, Phys. Lett. A 322 (5–6) (2004) 298. L.M. Johansen, quant-ph/0402050. C.L. Mehta, J. Math. Phys. 5 (1) (1964) 677. R.I. Sutherland, J. Math. Phys. 23 (12) (1982) 2389. A.M. Steinberg, Phys. Rev. Lett. 74 (13) (1995) 2405. A.M. Steinberg, Phys. Rev. A 52 (1) (1995) 32. A. Luis, Phys. Rev. A 67 (2003) 064101. W.H. Louisell, Quantum Statistical Properties of Radiation, John Wiley & Sons, New York, 1973. N.W.M. Ritchie, J.G. Story, R.G. Hulet, Phys. Rev. Lett. 66 (9) (1991) 1107. A.D. Parks, D.W. Cullin, D.C. Stoudt, Proc. R. Soc. London, Ser. A 454 (1998) 2997. N. Brunner, A. Acin, D. Collins, N. Gisin, V. Scarani, Phys. Rev. Lett. 91 (18) (2003) 180402. D.R. Solli, C.F. McCormick, R.Y. Chiao, S. Popescu, J.M. Hickmann, Phys. Rev. Lett. 92 (4) (2004) 043601. L.M. Johansen, quant-ph/0402105.
Nonclassical properties of coherent states Lars M. Johansen Department of Technology, Buskerud University College, P.O. Box 251, N-3601 Kongsberg, Norway Received 13 June 2004; received in revised form 2 July 2004; accepted 5 July 2004 Available online 21 July 2004 Communicated by P.R. Holland
Abstract It is demonstrated that a weak measurement of the squared quadrature observable may yield negative values for coherent states. This result cannot be reproduced by a classical theory where quadratures are stochastic c-numbers. The nonclassicality of coherent states can be associated with negative values of the Terletsky–Margenau–Hill distribution. 2004 Elsevier B.V. All rights reserved. PACS: 42.50.Dv; 03.65.Ta Keywords: Nonclassicality; Coherent states; Weak values; Weak measurements; P -distribution; Margenau–Hill distribution; Kirkwood distribution
Harmonic oscillator coherent states were first investigated by Schrödinger, who was looking for classical-like states [1]. There are several ways in which coherent states are the “most classical” of any pure state. They keep their shape, not spreading out as they move in the harmonic oscillator potential [1]. They minimize Heisenberg’s uncertainty relation, with equal uncertainty in both quadratures. In this way, they are the closest possible quantum mechanical representation of a point in phase space. The term “coherent state” was introduced by Glauber [2]. He demonstrated that coherent states are produced when an essentially classical current interacts with the radiation
E-mail address: [email protected] (L.M. Johansen). 0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.07.003
field [2]. Aharonov et al. demonstrated that coherent states are the only pure states that produce independent output when split in two [3]. Zurek et al. have demonstrated that coherent states are natural “pointer states” for a harmonic oscillator weakly coupled to a thermal environment [4]. Glauber and Sudarshan demonstrated that any density operator can be expanded in terms of coherent states [2,5] ρˆ = d 2 α P (α)|αα|. (1) The weight function P (α) is known as the P -distribution. Glauber defined nonclassical states as those for which the P -distribution fails to be a probability density. More specifically, nonclassical states have
L.M. Johansen / Physics Letters A 329 (2004) 184–187
a P -distribution which is negative or more singular than a δ-function [2,6–9]. This criterion is the basis of various measures of “nonclassicality” [10–13]. The phrase “nonclassical” is appropriate whenever classical physics fails to describe a phenomenon. In this Letter, we will show that a classical phase space representation fails as a model to describe weak measurements [14] of quadrature observables on coherent states. We also show that the failure is due to negativity of a phase-space quasi-probability distribution. The distribution is the Terletsky–Margenau–Hill (TMH) distribution [15,16], which is partly negative for coherent states [17]. We give an operational significance to conditional moments of the TMH-distribution by demonstrating that they can be observed in weak measurements. Weak measurements were proposed by Aharonov et al. [14]. Their suggestion was initially met with criticism [18–20], but has since been confirmed [21]. The results reported in this Letter are related to a paper by Aharonov et al. [22], which demonstrated that a weak measurement of kinetic energy of a particle in a classically forbidden region might yield negative values. For an object prepared in the pure state |ψ and postselected by a projective measurement on the ˆ the weak value of eigenstate |d of the observable d, the observable cˆ is found to be [14]
and
d|c|ψ ˆ . (2) d|ψ In a weak measurement, the real part of the weak value is observed [14]. If the object is prepared in a mixed state ρ, ˆ the weak value is [23–25]
where
cw (d) =
d|cˆρ|d ˆ . (3) d|ρ|d ˆ If we consider in particular the quadrature observˆ we may ables cˆ = pˆ n (n = 0, 1, 2, . . .) and dˆ = q, easily express the weak value as a conditional moment of a phase-space distribution. By inserting the completeness relation dp |pp| = 1 in the numerator of Eq. (3), we find that the weak value of pˆ n can be written as n p w = dp pn S(p|q), (4)
cw (d) =
S(q, p) = q|pp|ρ|q. ˆ
185
(6)
S(q, p) is the standard ordered distribution [26] and S(p|q) can be interpreted as a “conditional quasiprobability” which may be complex. In a weak measurement, the pointer (or measurement apparatus) displays on average the real part of the weak value [14, 25]. It is therefore of interest to note that we may write the real part of the weak value of pˆ n as Re pn w = dp pn T (p|q), (7) where T (p|q) =
T (q, p) , dq T (q, p)
(8)
and where T (q, p) = Re S(q, p) is the TMH-distribution. Thus, Re[(pn )w ] is a conditional moment of the TMH-distribution (see also [27–30]). A classical object where the quadrature observables are c-numbers can be described by a non-negative phase-space distribution F (q, p). For such systems, a weak measurement of pn yields the weak value [25] n p w = dp pn F (p|q), (9)
F (p|q) =
F (q, p) dp F (q, p)
(10)
is a genuine, non-negative conditional probability density. We see that (pn )w is the conditional expectation value of that variable. In other words, (pn )w is simply the expectation value of pn “given” q. It follows straightforwardly from Eq. (9) that since p2n 0, 2n p w 0. (11) This is a condition that must is satisfied by any classical phase space representation. We also note that it is satisfied by quantum states for which the TMHdistribution is non-negative. Hence, for such quantum states, a classical model of the weak measurement exists.1
where S(p|q) =
S(q, p) dp S(q, p)
(5)
1 An additional classicality condition is that the imaginary part
of the standard ordered distribution vanishes.
186
L.M. Johansen / Physics Letters A 329 (2004) 184–187
Our next step is to demonstrate that a weak measurement of pˆ 2 on coherent states violates Eq. (11). Consider the quadrature representation of a coherent state |α (with ω = 1) [31] 2 √ q 1 2 1 2 −1/4 q|α = π exp − + 2αq − |α| − α . 2 2 2 (12) The weak value of pˆ 2 for an ensemble preselected in the coherent state |α and postselected in the quadrature eigenstate |q is
p2
= w
√ −∂ 2 q|α/∂q 2 = 1 − (q − 2α)2 . q|α
(13)
Fig. 1. The probability of observing a negative weak value for a co√ herent state with amplitude (αr + iαi )/ 2. It is plotted as a function of the imaginary component αi . The probability is independent of the real component αr .
The real part of the weak value is Re[(p2 )w ] = 1 + αi2 − (q − αr )2 , where we have introduced the notation √ α = (αr + iαi )/ 2. We see that Re[(p2 )w ] is negative if (q − αr )2 > 1 + αi2 . The surprising conclusion is that the weak value of pˆ 2 can be negative for coherent states, although pˆ 2 has only non-negative eigenvalues. The probability of obtaining a negative value is αr + 1+αi2
P =1−
q|α 2 dq = erfc 1 + α 2 ,
αr − 1+αi2
i
(14)
where erfc(x) is the complementary error function. This function is plotted in Fig. 1. It has a maximum when the imaginary part of the coherent state amplitude vanishes, in which case it equals erfc(1) ≈ 0.16. It is quite interesting to note that this probability is independent of αr . Hence, it persists even for macroscopic excitations. Combining Eqs. (12) and (6), the standard ordered distribution for a coherent state |α is easily found. The standard ordered distribution for vacuum can be written as 1 2 1 2 S0 (q, p) = √ e− 2 (q +p )+iqp . 2π
(15)
The standard ordered distribution for a coherent state may then be expressed in terms of the displaced vacuum, Sα (q, p) = S0 (q − αr , p − αi ). The real part of S0 , the TMH-distribution, has been plotted in Fig. 2 (see also Ref. [17]). It clearly has negative regions. This is the reason why a weak measurements of the
Fig. 2. The TMH-distribution for vacuum. It is negative in a certain domain. The TMH-distribution for an arbitrary coherent state is just a displaced vacuum state.
positive operator pˆ 2 may yield negative values for a coherent state. In recent years, various studies have found weak values that exceed the eigenvalue range of the observable when considering polarization of classical, optical systems [21,32–35]. But how can such “strange” weak values be observed in classical systems? The explanation is that classical polarization measurements cannot be modelled in terms of c-number observables. Such measurements must be described in terms of noncommuting observables. Therefore, these experi-
L.M. Johansen / Physics Letters A 329 (2004) 184–187
ments simply show that polarization cannot be modelled by a c-number theory. If coherent states are nonclassical, one may wonder whether there are any classical states at all? Indeed, it was recently shown that harmonic oscillator thermal states of occupation number above unity are essentially classical also in the weak measurement scheme considered here [36]. In conclusion, it was demonstrated that weak measurements on coherent states may produce results that cannot be explained in terms of a classical phase space distribution. It was shown that weak values of quadrature moments are conditional moments of the standard ordered distribution, and that the nonclassical properties revealed in the proposed experiment is related to negativity of the Terletsky–Margenau–Hill distribution.
References [1] E. Schrödinger, Naturwissenschaften 14 (1926) 664. [2] R.J. Glauber, Phys. Rev. 131 (1963) 2766. [3] Y. Aharonov, D. Falkoff, E. Lerner, H. Pendleton, Ann. Phys. 39 (1966) 498. [4] W.H. Zurek, S. Habib, J.P. Paz, Phys. Rev. Lett. 70 (1993) 1187. [5] E.C.G. Sudarshan, Phys. Rev. Lett. 10 (7) (1963) 277. [6] U.M. Titulaer, R.J. Glauber, Phys. Rev. 140 (1965) B676. [7] M. Hillery, Phys. Lett. A 111 (8–9) (1985) 409. [8] L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, Cambridge Univ. Press, New York, NY, 1995. [9] V.V. Dodonov, J. Opt. B: Quantum Semiclass. Opt. 4 (2002) R1. [10] M. Hillery, Phys. Rev. A 35 (2) (1987) 725.
[11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36]
187
C.T. Lee, Phys. Rev. A 44 (5) (1991) R2775. W. Vogel, Phys. Rev. Lett. 84 (9) (2000) 1849. T. Richter, W. Vogel, Phys. Rev. Lett. 89 (28) (2002) 283601. Y. Aharonov, D.Z. Albert, L. Vaidman, Phys. Rev. Lett. 60 (14) (1988) 1351. Y.P. Terletsky, Zh. Eksp. Teor. Fiz. 7 (1937) 1290. H. Margenau, R.N. Hill, Prog. Theor. Phys. 26 (5) (1961) 722. L. Praxmeyer, K. Wódkiewicz, Opt. Commun. 223 (2003) 349. A.J. Leggett, Phys. Rev. Lett. 62 (19) (1989) 2325. A. Peres, Phys. Rev. Lett. 62 (19) (1989) 2326. Y. Aharonov, L. Vaidman, Phys. Rev. Lett. 62 (19) (1989) 2327. I.M. Duck, P.M. Stevenson, E.C.G. Sudarshan, Phys. Rev. D 40 (6) (1989) 2112. Y. Aharonov, S. Popescu, D. Rohrlich, L. Vaidman, Phys. Rev. A 48 (6) (1993) 4084. M.J.W. Hall, Phys. Rev. A 64 (2001) 052103. L.M. Johansen, Phys. Lett. A 322 (5–6) (2004) 298. L.M. Johansen, quant-ph/0402050. C.L. Mehta, J. Math. Phys. 5 (1) (1964) 677. R.I. Sutherland, J. Math. Phys. 23 (12) (1982) 2389. A.M. Steinberg, Phys. Rev. Lett. 74 (13) (1995) 2405. A.M. Steinberg, Phys. Rev. A 52 (1) (1995) 32. A. Luis, Phys. Rev. A 67 (2003) 064101. W.H. Louisell, Quantum Statistical Properties of Radiation, John Wiley & Sons, New York, 1973. N.W.M. Ritchie, J.G. Story, R.G. Hulet, Phys. Rev. Lett. 66 (9) (1991) 1107. A.D. Parks, D.W. Cullin, D.C. Stoudt, Proc. R. Soc. London, Ser. A 454 (1998) 2997. N. Brunner, A. Acin, D. Collins, N. Gisin, V. Scarani, Phys. Rev. Lett. 91 (18) (2003) 180402. D.R. Solli, C.F. McCormick, R.Y. Chiao, S. Popescu, J.M. Hickmann, Phys. Rev. Lett. 92 (4) (2004) 043601. L.M. Johansen, quant-ph/0402105.