SubPlanck structures and Quantum Metrology

SubPlanck structures and Quantum Metrology Prasanta K. Panigrahi, Abhijeet Kumar, Utpal Roy, and Suranjana Ghosh Citation: AIP Conf. Proc. 1384, 84 (2011); doi: 10.1063/1.3635847 View online: http://dx.doi.org/10.1063/1.3635847 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1384&Issue=1 Published by the American Institute of Physics.

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Sub-Planck structures and Quantum Metrology Prasanta K. Panigrahi∗ , Abhijeet Kumar† , Utpal Roy∗∗ and Suranjana Ghosh∗∗ ∗

Indian Institute of Science Education and Research-Kolkata † Indian Institute of Science Education and Research, Pune ∗∗ Indian Institute of Technology Patna

Abstract. The significance of sub-Planck structures in relation to quantum metrology is explored, in close contact with experimental setups. It is shown that an entangled cat state can enhance the accuracy of parameter estimations. The possibilty of generating this state, in dissipative systems has also been demonstrated. Thereafter, the quantum Cramer-Rao bound for phase estimation through a pair coherent state is calculated, which achieves the maximum possible resolution in an interferometer. Keywords: Sub-Planck structure, Heisenberg-limited measurements, Entanglement, Quantum metrology PACS: 03.67.-a, 42.50.Dv, 42.50.Ex

INTRODUCTION Sensitive estimation of parameters like length and angle has led to the development of many novel techniques, both on the theoretical and experimental fronts [1]. In quantum metrology, special states like squeezed states have been employed for estimation of these parameters, for which the designed state has less uncertainty. In a number of cases, the parameter variations translate into phase variations, which are estimated by interferometric means. It has been pointed out by Vogel and Zurek [2, 3] that interference in phase space can lead to structures, with area less than the Planck constant. Zurek observed that these sub-Planck structures can be profitably used for measuring length and angle variations, with Heisenberg limited sensitivity. For this purpose, one can use cat and kitten states, which neccessarily possess quantum character, as compared to the classical coherent state. The phase space interference effect is particularly transparent in the behaviour of Wigner function in phase space [4]. The zeros of the Wigner function for the kitten state reveal sub-Planck size tiles, making up the phase space. The parameter estimation sensitivity is made transparent, by computing the overlap of the original Wigner function, with the one having a displaced parameter value. It has been demonstrated that the presence of sub-Planck tiles in phase space leads to the vanishing of this overlap function for certain values of the displaced parameter, making it potentially observable since the corresponding states are orthogonal [3, 5]. It is interesting to observe that, the above sensitivity has a classical analogue, called the sub-Fourier sensitivity, which has been experimentally verified through cat-like laser beams [8]. The same is not surprising, since Zurek’s approach relies on superposition principle alone. Recently, intrinsic quantum property like entanglement has been used for better parameter estimation [9]. It is evident that the sensitivities crucially rely on the type of observable being used in the measurement process, as well as the measurement statistics. Based on these, a number of bounds, namely shot-noise limit [10], Heisenberg limit, Hoffman bound and Cramer-Rao bound have been found in relation to various measurement processes [12, 13], the most stringent one being the Cramer-Rao bound, related to the Fisher information. In the following, we illustrate with several examples, the origin of sub-Planck structures and their relevance for Quantum Metrology. Subsequently, we also give a procedure to generate, a two particle entangled cat state, well suited for sensitive metrology, which may be experimentally accessible. The measurement procedures and their connection, with the Cramer-Rao bound is then dealt with, through the example of pair coherent state. We then conclude with direction of future work.

75 Years of Quantum Entanglement: Foundations and Information Theoretic Applications AIP Conf. Proc. 1384, 84-90 (2011); doi: 10.1063/1.3635847 © 2011 American Institute of Physics 978-0-7354-0945-3/$30.00

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SUB-PLANCK STRUCTURES The occurrence of structures having area less than Planck’s constant in phase space has been observed by Vogel and Zurek [2, 3]. Zurek used these structures and provided a significant breakthrough in measurement of variables. These states, namely cat and kitten states are the superpositions of coherent states, respectively represented by | ψ =

| α + | −α | α + | −α + | iα + | −iα √ , and | φ = 2 2

(1)

The Wigner functions of these states show oscillatory behaviour at sub-Planck intervals, which allows sensitive measurement of appropriate variables. The experimental realization of this Heisenberg limited measurement process was carried out in a recent experiment [8] involving superposed laser beams: 2 2 2 2 −

ψ (t) = e

(t−t0 ) 4σ 2

−

+e

(t+t0 ) 4σ 2

e−iωc t and ψδ (t) =

−

e

(t−t0 ) 4σ 2

−

+e

(t+t0 )

e−iωc x eiδ t .

4σ 2

(2)

Their overlap function being ψ |ψδ =

+∞ −∞

√

∗

2 2 − σ 2δ

ψ (t)ψδ (t)dt = 2σ 2π e

cos(δ t0

2

t − 0 ) + e 2σ 2

,

(3)

suggests that one can have an orthogonal state to the original superposition, for appropriate displacements in carrier frequency, resuting in its discrimination. This experiment had originally been done in the time-frequency domain, with superposed laser beams having Gaussian profiles. The separation of two such pulses i.e., 2t0 had been taken as 305 fs and 309 fs in two different experiments The measured uncertainty in Δt, was found to be 20.1 ± 0.5 fs and hence the minimum value for Δν should normally be restricted to 4.0 ± 0.1 THz. In accordance with the above equation, for sufficiently large values of the separation between two pulses, one can have arbitarily small δ , the shift in carrier frequency or the parameter to be estimated here. However, experimental conditions have a restricting role on the minimum value of δ that can be inferred from the orthogonality of | ψ and | ψδ . The value of δ for this experiment was obtained as 3.3 THz, which notably is less than the calculated uncertainty.This sub-Fourier resolution opens up the possibility of tremendous precision in quantum parameter estimations. It needs to be mentioned that the cat state provides sensitivity along only one direction in phase space, whereas the kitten state achieves the same in time and frequency domains. This is apparent from the representation of kitten state: (x−α )2 (x+α )2 (x+iα )2 (x−iα )2 − − − − . (4) ψ (x) = e 4σ 2 + e 4σ 2 + e 4σ 2 + e 4σ 2 The Wigner function for a kitten state is given by 2 2 2 2 −x σ 2α x 2α x x x −α √ [e−p σ e σ 2 (e σ 2 (cosh( 2 ) + cos( 2 ) + cos α (p + 2 ) cosh α (p + 2 ) σ σ σ σ 2 π x x + cosh α (p − 2 ) cos α (p − 2 )) + cos(2α p) + cosh(2α p))] σ σ

W (x, p) =

(5)

which shows pure oscillatory structure in the x and p directions, responsible for sub-Planck structure. The possibility of experimentally realizing the kitten state and their decoherence properties have been thoroughly investigsted [14]. The fact that some of these states naturally manifest in dissipative systems may aid in their preparation and use [3, 14]. It was found that an entangled cat state involving two particles can carry out parameter estimation, with better accuracy [9]. The use of entangled states for precision lithography [15] and other applications have generated considerable interest in them.


THE ENTANGLED COMPASS STATE An entangled, bipartite system where the two constituents are themselves characterized by cat states, can be represented as 1 | ψ c = √ (A | ±α + 1 | ±iα + 2 + B | ±iα + 1 | ±α + 2 ), (6) 2 where A = A1 + iA2 and B = B1 + iB2 are complex parameters, which control the entanglement. As is evident, the two constituents or the two modes of the field are described by the cat states and the combined state is an entangled system. The above state does not satisfy the criterion for separability based on the variances of two EPR operators as given in [25, 26]. In the coordinate space, we consider the state Ψc (x1 , x2 ) = N(Aψ (x1 )φ (x2 ) + Bψ (x2 )φ (x1 ))

(7)

where ψ (x) and φ (x) themselves are the wavefunctions representing the superposed states −

(x−x0 )2

−

(x+x0 )2

e 2σ 2 + e 2σ 2 x | ±α = ψ (x) = x2 √ 1 1 − 0 1 2π 4 σ 2 [1 + e σ 2 ] 2 −

and

2 − x +ip x

(8)

2 − x −ip x

e 2σ 2 0 h¯ + e 2σ 2 0 h¯ x | ±iα = φ (x) = . −p20 σ 2 √ 1 1 1 2π 4 σ 2 [1 + e h¯2 ] 2 +

(9)

To study the sub-Planck structure in the phase space, the Wigner function needs to be computed W (x1 , p1 ; x2 , p2 ) =

1 (2π h¯)2

∞ ∞ −∞ 2

=

(p1 a+p2 b) a b a Ψ† (x1 + , x2 + )Ψ(x1 − , x2 − b/2)ei h¯ dadb 2 2 2 −∞ 2

2

2

2

2

2 2

x p σ 2σ 2 |N|2 x1 +x2 2 (p1 +p22 )σ − 02 − 0 2 2σ h¯ 2¯ h (WOD1 +WOD2 )). σ e e ((W +W ) + e D1 D2 π h¯2

(10)

Here, WD1 , WD2 represent the diagonal terms and WOD1 and WOD2 , the off-diagonal terms in the integral. It has been seen that unlike the diagonal terms, the oscillatory terms in off-diagonal terms are severely damped for large values of x0 and p0 . Hence, collecting the significant terms in the mesoscopic limit, we can approximate the Wigner function to be: 2p0 x2 2x0 p1 2p0 x1 2x0 p2 2 2 W (x, p) ∝ 4|A| cos cos + 4|B| cos cos . (11) h¯ h¯ h¯ h¯ As has been analyzed earlier, from above, one can find the zeros in (x1 , p1 ) as well as (x2 , p2 ) planes. This corresponds to the presence of sub-Planck structures in these planes, as shown in the Fig.1, arising from the oscillatory behavior in the overlap function. In order to investigate the sensitivity in parameter estimation, as suggested by Toscano et al. [5], we consider two displacement operators D1 (α ) and D2 (β ) such that they displace particles 1 and 2 by amounts α and β . This leads to the perturbed state |ψ per = D1 (α )D2 (β ) | ψ c . For equal amount of displacements of both the particles i.e., α = β = |xx0 | , the overlap function yields 0

|ψc |ψ per |2 ∝ 1 + cos(4x0 (s + θ )).

(12)

π 4x0

Clearly, one can see that for displacements x ∼ − θ , the overlap function vanishes. This should be contrasted with the overlap function obtained for a displacement s in [3, 5], 1 |ψc |ψ per |2 ∝ [3 + 4 cos(2x0 s) + cos(4x0 s)]. 4


(13)

FIGURE 1.

Cross sectional view of Wigner function of entangled wavefunction [9].

It attains the minimum for displacements s = 2xπ0 . One can witness an improvement of a factor of 12 in distinguishable displacements in case of entangled states. The above state generically, goes by the name of compass state. They often naturally manifest in dissipative systems. Below, we explore the possibility of generating it in laboratory. For that purpose, the following algebraic structure comes quite handy. As can be noted, the states in Eq.(6) are the eigenstates of a2 b2 . Taking, 2

one finds

2

K− = a2 b2 , K+ = a† b† and K0 = aa† + b† b,

(14)

[K0 , K± ] = ±K± , [K+ , K− ] = 2cK0 + 4hK03 ,

(15)

indicating that they follow a cubic algebra. This is also known as Higgs algebra [20, 21], with K+ and K− as the general creation and annihilation operators. Considering this lowering operator to satify the Janus-faced commutaion relations [K− , K˜ −† ] = 1 [22, 23], it can be found that these states are the steady state solutions of the master equation governing the evolution of a general Hamiltonian:

∂ρ κ = −ig(K− ρ − ρ K− + K+ ρ − ρ K+ ) + (2K− ρ K+ − K+ K− ρ − ρ K+ K− ). ∂t 2

(16)

As is evident, the interaction term originates from Hin ≈ a2 b2 + a† b† . For the steady state case, i.e., ∂∂ρt = 0, the solution leads exactly to the desired entangled state. It is worth mentioning that a number of entangled states have recently manifested in dissipative systems [23, 27]. 2

2

THE PAIR COHERENT STATE We now investigate the use of other entangled states for parameter estimation. For illustrating this, we consider a pair coherent state, which is a state of a two-mode radiation field satisfying, ab|ζ , q = ζ |ζ , q and (a a − b b)|ζ , q = q|ζ , q †

†

(17)

Here ζ is a complex number and q is the degeneracy parameter. It can be noticed that a simple product state |α a |β b of a two-mode radiation field may satisfy the first of these equations as they are the eigenstates of the product of the lowering operators but they also need to satisfy the second equation, which yields the difference in the number of photons to be q. These states were introduced by Agarwal [22] and are generated by nondegenerate parametric oscillators. These are non-Gaussian states and have been studied in detail because of their non-classical and entanglement properties. They also are quite popular because of their violation of Bell’s inequalities. Let us first


Source: Aravind Chiruvelli

FIGURE 2.

A Mach-Zehnder Interferometer

consider the state with q = 0 i.e., having the same number of photons in both modes. Clearly, it will be a superposition of twin Fock states: ∞ ζn |ζ , 0 = N0 ∑ |n, n. (18) n=0 n! Here N0 = I (2|1 ζ |) and I0 (2|ζ |) is the modified Bessel function given by 0

I0 (2|ζ |) = J0 (i2|ζ |) =

∞

∑

n=0

|ζ |n n!

2 .

(19)

In applications involving the creation and annihilation of photons pairs, the difference in the number of photons will remain constant. Hence, in case of processes starting from vacuum, one can encounter such states. These states have been employed here, as the input state in a Mach-Zehnder Interferometer. We explore various bounds based on phase measurements, the minimum of these bounds, being the quantum Cramer-Rao bound (QCRB). For a Mach-Zehnder interferometer, the Fisher information is evaluated to be [29, 33], FQ = 4ΔJy2 where Jy =

a† b−ab† . 2i

(20)

Evaluating the uncertainty in Jy , N02 ∞ ζ n 2 ∑ n! n(n + 1) 2 n=0

ζ , 0|Jy2 |ζ , 02 = one finds, FQ = 2N02

∞

∑

n=0

ζn n!

(21)

2 n(n + 1).

(22)

The Cramer-Rao bound on the uncertainty in phase, is inversely proportional to FQ . 1 Δφmin =

and FQ

(23)

hence, the minimum possible bound on uncertainty in phase will be

Δφmin = N0

2 ∑∞ n=0

1 n 2 ζ n!

. n(n + 1)


(24)

This should be compared with the average number of photons present in this state N¯ = N0

∞

∑

n=0

|ζ |n n!

2 n.

(25)

By comparing these two expressions, we see a remarkable improve that the QCRB for the pair coherent state shows

1 1 √ ment from the shot noise limit Δφ ∝ ¯ and is close to the Heisenberg limit Δφ ∝ N¯ . This means that even in N the sub-shot noise regime, finer phase measurements can be carried out with pair coherent states as compared to other input states in a Mach-Zehnder interferometer.

CONCLUSION In conclusion, the structure and utility of sub-Planck stuctures in phase space is demonstrated through pure superposed cat and kitten states. The advantage of using an entangled state is highlighted along with a possible method of producing them in dissipative systems. The connection of measurement procedures, as also the nature of the state in saturating the Cramer-Rao bound is explicated. A number of interesting proposals have been recently advanced for sensitive metrology. Investigation of their relative merits need significant attention. The effect of decoherence should also be carefully investigated.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

M. Napolitano et al. Nature (London) 471, 486 (2011) and references therein. W. Vogel, Phys. Rev. Lett. 84, 1849 (2000). W. H. Zurek, Nature (London) 412, 712 (2001). W. Schleich and J.A. Wheeler Nature (London) 326, 574 (1987); W. Schleich and J. A. Wheeler, J. Opt. Soc. Am. B 4, 1715 (1987); W. Schleich, D. F. Walls and J. A. Wheeler, Phys. Rev. A 38, 1177 (1988); W. P. Schleich, Quantum Optics in Phase Space (Wiley-VCH, Berlin, 2001) and references therein. F. Toscano, D. A. R. Dalvit, L. Davodovich and W.H. Zurek, Phys. Rev. A 73, 023803 (2006). D. F. Styer et.al., Am. J. Phys. 70, 288 (2002). Th. Richter and W. Vogel, Phys. Rev. Lett. 89, 283601 (2002). L. Praxmeyer, P. Wasylczyk, C. Radzewicz and K. Wòdkiewicz, Phys. Rev. Lett. 98, 063901 (2007). J. R. Bhatt, P.K. Panigrahi and M. Vyas, Phys. Rev. A 78, 034101 (2008). C. M. Caves, Phys. Rev. D 23, 1693 (1981). V. Giovannetti, S. Lloyd, and L. Maccone Science 306, 1330 (2004). V. Giovannetti, S. Lloyd, and L. Maccone Phys. Rev. Lett. 96, 010401 (2006). P. M. Anisimov et. al, arXiv:0910.5942. G. S. Agarwal and P.K. Pathak, Phys. Rev. A 70, 053813 (2004). A. N. Boto et al., Phys. Rev. Lett. 85, 2733 (2000). U. Roy, S. Ghosh, P.K. Panigrahi and D. Vitali, Phys. Rev. A 80, 052115 (2009). S. Ghosh, U. Roy, C. Genes and D. Vitali, Phys. Rev. A 79, 052104 (2009). S. Ghosh, A. Chiruvelli, J. Banerji and P.K. Panigrahi, Phys. Rev. A 73, 013411 (2006). D. F. Walls and G.J. Milburn, Quantum Optics 2nd edition (Springer-Verlag Berlin Heidelberg, 2008) and references therein. P. W. Higgs, J. Phys. A: Math. Gen. 12 309 (1979). V. Sunilkumar, B. A. Bambah, R. Jagannathan, P. K. Panigrahi and V. Srinivasan, J. Opt. B: Quantum Semiclass. Opt. 2, 126 (2000). G. S. Agarwal, Phys. Rev. Lett. 57, 827 (1986). K. V. S. Shiv Chaitanya and V. Srinivasan, J. Phys. A: Math. Theor. 43 485205 (2010). P. Shanta, S. Chaturvedi, V. Srinivasan, G. S. Agarwal and C. L.Metha, Phys. Rev. Lett. 72, 1447 (1994). L.-M. Duan, G. Giedke, J. I. Cirac and P. Zoller, Phys. Rev. Lett. 84, 2722 (2000), E. Shchukin and W. Vogel, Phys. Rev. Lett. 95, 230502 (2005). R. Simon, Phys. Rev. Lett. 84, 2726 (2000). W. H. Zurek, Nature Physics (London) 5, 181 (2009). A. Chiruvelli and Hwang Lee, Quantum Cramer Rao Bound and Parity Measurement Proc. of SPIE Vol. 7611 76110S-1(2010). B. C. Sanders and G. J. Milburn, Phys. Rev. Lett. 75, 2944 (1995). D. W. Berry et al., Phys. Rev. A 80, 052114 (2009). L. Pezze and A. Smerzi, Phys. Rev. A 73, 011801 (2006). H. Uys and P. Meystre, Phys. Rev. A 76, 013804 (2007).


33. 34. 35. 36. 37.

G. A. Durkin and J. P. Dowling, Phys. Rev. Lett. 99, 070801 (2007). S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994). T. Kim, O. Pfister, M. J. Holland, J. Noh, and J. L. Hall, Phys. Rev. A 57, 4004 (1998). C. C. Gerry and P. L. Knight, Introductory Quantum Optics, First ed. (Cambridge University Press, Cambridge, UK, 2005). G. S. Agarwal, J. Opt. Soc. Am. B Vol 5 Num 9 (1988).
