## Formation of Schrödinger-cat states in the Morse potential: Wigner function picture

Optics Express, Vol. 10, Issue 8, pp. 376-381 (2002)

http://dx.doi.org/10.1364/OE.10.000376

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### Abstract

We investigate the time evolution of Morse coherent states in the potential of the NO molecule. We present animated wave functions and Wigner functions of the system exhibiting spontaneous formation of Schrödinger-cat states at certain stages of the time evolution. These nonclassical states are coherent superpositions of two localized states corresponding to two different positions of the center of mass. We analyze the degree of nonclassicality as the function of the expectation value of the position in the initial state. Our numerical calculations are based on a novel, essentially algebraic treatment of the Morse potential.

© 2002 Optical Society of America

## 1 Introduction

1. J. Parker and C. R. Stroud Jr., “Coherence and decay of Rydberg Wave packets,” Phys. Rev. Lett. **56**, 716–719 (1986). [CrossRef] [PubMed]

2. D. L. Aronstein and C. R. Stroud Jr., “Analytical investigation of revival phenomena in the finite square-well potential,” Phys. Rev. A **62**, 022102-1–022102-9 (2000). [CrossRef]

3. S. I. Vetchinkin and V. V. Eryomin, “The structure of wavepacket fractional revivals in a Morselike anharmonic system,” Chem. Phys. Lett. **222**, 394–398 (1994). [CrossRef]

*s*] + 1 normalizable eigenstates (bound states), plus the continuous energy spectrum with positive energies. The wave functions of the bound eigenstates of

*H*are

*y*= (2

*s*+1)

*e*

^{-x}is the rescaled position variable, and

*y*) is a generalized Laguerre polynomial. The corresponding eigenvalues are

*E*

_{m}(

*s*) = -(

*s*-

*m*)

^{2},

*m*= 0,1,… [

*s*], where [

*s*] denotes the largest integer that is smaller than

*s*.

5. M. G. Benedict and B. Molnár, “Algebraic construction of the coherent states of the Morse potential based on supersymmetric quantum mechanics,” Phys. Rev. A **60**R1737–R1740 (1999). [CrossRef]

5. B. Molnár, M. G. Benedict, and J. Bertrand, “Coherent states and the role of the affine group in the quantum mechanics of the Morse potential” J. Phys A:Math. Gen. **34**, 3139–3151 (2001). [CrossRef]

_{2}F

_{1}is the hypergeometric function of the variable 1 -

*β*. The first [

*s*]+1 elements of the basis {|

*Φ*

_{n}〉

*Φ*

_{n}〉,

*n*> [

*s*] follow densely each other, approximating satisfactorily the continuous energy spectrum. The details of the construction of our basis can be found in [6].

8. J. Bertrand and M. Irac-Astaud, “The SU(1,1) coherent states related to the affine group wavelets,” Czech J. Phys. **51** (12), 1272–1278 (2001). [CrossRef]

*β*in Eq. (4) is in one to one correspondence with the expectation values

*x*

_{0},

*p*

_{0}〉 for the state |

*β*〉 that gives 〈

*X*〉 =

*x*

_{0}and 〈

*P*〉 =

*p*

_{0}. The localized wave packet corresponding to |

*x*

_{0},

*p*

_{0}〉 is centered at

*x*

_{0}(

*p*

_{0}) in the coordinate (momentum) representation.

*m*= 7.46 a.u.,

*D*= 6.497 eV and

*α*= 27.68 nm

^{-1}[4], yielding

*s*= 54.54. That is, this molecule has 55 bound states, and we found that a basis of dimension

*N*+ 1 = 150 is sufficiently large to handle the problem. The absolute square of the wave functions |〈

*x*|0, 0〉|

^{2}and |〈

*x*|0.5, 0〉|

^{2}is depicted in Fig. 1, where

*V*(

*x*) is also shown. Fig. 1 indicates that initial displacements,

*x*

_{0}, having the order of magnitude of unity will not lead to “

*small*oscillations”.

5. M. G. Benedict and B. Molnár, “Algebraic construction of the coherent states of the Morse potential based on supersymmetric quantum mechanics,” Phys. Rev. A **60**R1737–R1740 (1999). [CrossRef]

5. B. Molnár, M. G. Benedict, and J. Bertrand, “Coherent states and the role of the affine group in the quantum mechanics of the Morse potential” J. Phys A:Math. Gen. **34**, 3139–3151 (2001). [CrossRef]

9. B. Molnár, M. G. Benedict, and P. Földi, “State evolution in the anharmonic Morse potential subjected to an external sinusoidal field,” Fortschr. Phys. **49**, 1053–1057 (2001). [CrossRef]

## 2 Behavior of expectation values as a function of time

*ϕ*(

*t*= 0)〉 = |

*x*

_{0},0〉 as initial states, first we consider the dependence of the 〈

*X*〉(

*t*) curve on

*x*

_{0}. It is not surprising that for small values of

*x*

_{0}(≤ 0.06) these curves show similar oscillatory behavior as in the case of the harmonic oscillator, see Fig. 2. However, when anharmonic effects become important, a different phenomenon can be observed: the amplitude of the oscillations decreases almost to zero, then faster oscillations with small amplitude appear but later we re-obtain almost exactly 〈

*X*〉(0), and the whole process starts again. Fig. 2 is similar to the collapse and revival in the Jaynes-Cumings (JC) model [10, 11

11. J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett **44**, 1323–1327 (1980). [CrossRef]

1. J. Parker and C. R. Stroud Jr., “Coherence and decay of Rydberg Wave packets,” Phys. Rev. Lett. **56**, 716–719 (1986). [CrossRef] [PubMed]

13. C. Leichtle, I. Sh. Averbukh, and W. P. Schleich, “Multilevel quantum beats: An analytical approach,” Phys. Rev. A. **54**, 5299–5312 (1996). [CrossRef] [PubMed]

14. P. Domokos, T. Kiss, J. Janszky, A. Zucchetti, Z. Kis, and W. Vogel, “Collapse and revival in the vibronic dynamics of laser-driven diatomic molecules,” Chem. Phys. Lett. **322** (3–4), 255–262 (2000). [CrossRef]

2. D. L. Aronstein and C. R. Stroud Jr., “Analytical investigation of revival phenomena in the finite square-well potential,” Phys. Rev. A **62**, 022102-1–022102-9 (2000). [CrossRef]

15. Ch. Warmuth, A. Tortschanoff, F. Milota, M. Shapiro, Y. Prior, I. Sh. Averbukh, W. Schleich, W. Jakubetz, and H. F. Kauffmann, “Studying vibrational wavepacket dynamics by measuring fluorescence interference fluctuations,” J. Chem. Phys. **112**, 5060–5069 (2000). [CrossRef]

*x*

_{0}, 0〉 = ∑

_{n}

*c*

_{n}(

*x*

_{0})|

*Φ*

_{n}〉 shows that for values of

*x*

_{0}shown in Fig. 2 the maximal |

*c*

_{n}(

*x*

_{0})| belongs to

*n*< [

*s*]. That is, the expectation value

## 3 Time evolution of wave function and Wigner function of the system

*ϕ*(

*x*,

*t*) to that of the Wigner function

*W*(

*x*,

*p*,

*t*) that reflects the state of the system in the phase space [16] and defined as

*t*/

*T*= 30. This second Wigner function is typical for Schrödinger-cat states [17

17. J. Janszky, An. V. Vinogradov, T. Kobayashi, and Z. Kis, “Vibrational Schrödinger-cat states,” Phys. Rev. A **50**, 1777–1784(1994), and see also references therein. [CrossRef] [PubMed]

*W*(

*x*,

*p*) in Fig. 3 B) corresponds to a superposition of two states that are well-localized in both momentum and coordinate, and represented by the two positive hills centered at

*x*

_{1}= -0.1,

*p*

_{1}= -18.0 and

*x*

_{2}= 0.3,

*p*

_{2}= 12.0. The strong oscillations between them shows the quantum interference of these states.

*t*/

*T*= 30, while the state of the system can be considered to be a phase space Schrödinger-cat state. During this time the Wigner function is similar to the one shown in Fig. 3 B), and it rotates around the equilibrium position. Similar behavior of the Wigner function was found in [18

18. J. Eiselt and H. Risken, “Quasiprobability distributions for the Jaynes-Cummings model with cavity damping,” Phys. Rev. A **43**, 346–360 (1991). [CrossRef] [PubMed]

*t*/

*T*= 30 shown in Fig. 2, where the frequency of the oscillations is twice that of the oscillations around

*t*= 0: in the neighborhood of

*t*/

*T*= 30 there are two wave packets moving approximately the same way as the coherent state soon after

*t*= 0.

## 4 Measuring nonclassicality

*M*

_{nc}< 1 of a state |

*ϕ*〉 can be defined [19

19. M. G. Benedict and A. Czirják, “Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms,” Phys. Rev. A **60**, 4034–4044 (1999). [CrossRef]

*I*

_{+}(

*ϕ*) and

*I*

_{-}(

*ϕ*) are the moduli of the integrals of

*W*(

*x*,

*p*) over those domains of the phase space where it is positive and negative, respectively. Fig. 4 shows

*M*

_{nc}as a function of time for the same initial states as in Fig. 2. For the small initial displacement of

*x*

_{0}= 0.06 we see that the Wigner function is positive almost everywhere, the state can be considered as a classical one during the whole time evolution.

*M*

_{nc}(

*t*), we observe that the state of the system is the most classical at those turning points where 〈

*X*〉 > 0, see Fig. 1. On the other time scale, the collapse of the oscillations in 〈

*X*〉 presents itself as the increase of

*M*

_{nc}and the revival turns the state into a more classical one. When the state of the system can be considered as a Schrödinger-cat state,

*M*

_{nc}(

*t*) has a small local minimum, but it still has significant values indicating nonclassicality.

## 5 Conclusion

## References and links

1. | J. Parker and C. R. Stroud Jr., “Coherence and decay of Rydberg Wave packets,” Phys. Rev. Lett. |

2. | D. L. Aronstein and C. R. Stroud Jr., “Analytical investigation of revival phenomena in the finite square-well potential,” Phys. Rev. A |

3. | S. I. Vetchinkin and V. V. Eryomin, “The structure of wavepacket fractional revivals in a Morselike anharmonic system,” Chem. Phys. Lett. |

4. | K. P. Huber and G. Herzberg, |

5. | M. G. Benedict and B. Molnár, “Algebraic construction of the coherent states of the Morse potential based on supersymmetric quantum mechanics,” Phys. Rev. A B. Molnár, M. G. Benedict, and J. Bertrand, “Coherent states and the role of the affine group in the quantum mechanics of the Morse potential” J. Phys A:Math. Gen. |

6. | B. Molnár, P. Földi, M. G. Benedict, and F. Bartha, “Time evolution in the Morse potential using supersymmetry: dissociation of the NO molecule,” quant-ph/0202069. |

7. | J. Banerji and G. S. Agarwal, “Non-linear wave packet dynamics of coherent states of various symmetry groups,” Opt. Express |

8. | J. Bertrand and M. Irac-Astaud, “The SU(1,1) coherent states related to the affine group wavelets,” Czech J. Phys. |

9. | B. Molnár, M. G. Benedict, and P. Földi, “State evolution in the anharmonic Morse potential subjected to an external sinusoidal field,” Fortschr. Phys. |

10. | E. T. Jaynes and F. W. Cummings, “Comparison of quantum semiclassical radiation theories with application to the beam maser,” Proc. Inst. Elect. Eng. |

11. | J. H. Eberly, N. B. Narozhny, and J. J. Sanchez-Mondragon “Periodic spontaneous collapse and revival in a simple quantum model,” Phys. Rev. Lett |

12. | I.Sh. Averbukh and N. F. Perelman, “Fractional revivals: Universality in the long term evolution of quantum wave packets beyond the correspondence principle dynamics,” Phys. Lett. |

13. | C. Leichtle, I. Sh. Averbukh, and W. P. Schleich, “Multilevel quantum beats: An analytical approach,” Phys. Rev. A. |

14. | P. Domokos, T. Kiss, J. Janszky, A. Zucchetti, Z. Kis, and W. Vogel, “Collapse and revival in the vibronic dynamics of laser-driven diatomic molecules,” Chem. Phys. Lett. |

15. | Ch. Warmuth, A. Tortschanoff, F. Milota, M. Shapiro, Y. Prior, I. Sh. Averbukh, W. Schleich, W. Jakubetz, and H. F. Kauffmann, “Studying vibrational wavepacket dynamics by measuring fluorescence interference fluctuations,” J. Chem. Phys. |

16. | Y. S. Kim and M. E. Noz, |

17. | J. Janszky, An. V. Vinogradov, T. Kobayashi, and Z. Kis, “Vibrational Schrödinger-cat states,” Phys. Rev. A |

18. | J. Eiselt and H. Risken, “Quasiprobability distributions for the Jaynes-Cummings model with cavity damping,” Phys. Rev. A |

19. | M. G. Benedict and A. Czirják, “Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms,” Phys. Rev. A |

**OCIS Codes**

(020.0020) Atomic and molecular physics : Atomic and molecular physics

(030.1640) Coherence and statistical optics : Coherence

(270.0270) Quantum optics : Quantum optics

**ToC Category:**

Research Papers

**History**

Original Manuscript: March 15, 2002

Revised Manuscript: April 17, 2002

Published: April 22, 2002

**Citation**

Peter Foldi, Attila Czirjak, Balazs Molnar, and Mihaly Benedict, "Formation of Schr�dinger-cat states in the Morse potential: Wigner function picture," Opt. Express **10**, 376-381 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-8-376

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### References

- J. Parker and C. R. Stroud, Jr., �Coherence and decay of Rydberg Wave packets,� Phys. Rev. Lett. 56, 716-719 (1986). [CrossRef] [PubMed]
- D. L. Aronstein and C. R. Stroud, Jr., �Analytical investigation of revival phenomena in the finite square-well potential,� Phys. Rev. A 62, 022102-1�022102-9 (2000). [CrossRef]
- S. I. Vetchinkin and V. V. Eryomin, �The structure of wavepacket fractional revivals in a Morselike anharmonic system,� Chem. Phys. Lett. 222, 394-398 (1994). [CrossRef]
- K. P. Huber and G. Herzberg, Molecular spectra and molecular structure IV. Constants of diatomic molecules, (van Nostrand Reinhold, 1979).
- M. G. Benedict and B. Molnar, �Algebraic construction of the coherent states of the Morse potential based on supersymmetric quantum mechanics,� Phys. Rev. A 60 R1737-R1740 (1999). [CrossRef]
- B. Molnar, P. Foldi, M. G. Benedict and F. Bartha, �Time evolution in the Morse potential using supersymmetry: dissociation of the NO molecule,� quant-ph/0202069.
- J. Banerji and G. S. Agarwal, �Non-linear wave packet dynamics of coherent states of various symmetry groups,� Opt. Express 5, 220-229 (1999), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-5-10-220">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-5-10-220</a>. [CrossRef] [PubMed]
- J. Bertrand and M. Irac-Astaud, �The SU(1,1) coherent states related to the affine group wavelets,� Czech J. Phys. 51 (12), 1272-1278 (2001). [CrossRef]
- B. Molnar, M. G. Benedict and P. Foldi, �State evolution in the anharmonic Morse potential subjected to an external sinusoidal field,� Fortschr. Phys. 49, 1053-1057 (2001) [CrossRef]
- E. T. Jaynes and F. W. Cummings, �Comparison of quantum semiclassical radiation theories with application to the beam maser,� Proc. Inst. Elect. Eng. 51, 89-109 (1963).
- J. H. Eberly, N. B. Narozhny, J. J. Sanchez-Mondragon �Periodic spontaneous collapse and revival in a simple quantum model,� Phys. Rev. Lett 44, 1323-1327 (1980). [CrossRef]
- I. Sh. Averbukh and N. F. Perelman, �Fractional revivals: Universality in the long term evolution of quantum wave packets beyond the correspondence principle dynamics,� Phys. Lett. A 139, 449-453 (1989).
- C. Leichtle, I. Sh. Averbukh and W. P. Schleich, �Multilevel quantum beats: An analytical approach,� Phys. Rev. A. 54, 5299-5312 (1996). [CrossRef] [PubMed]
- P. Domokos, T. Kiss, J. Janszky, A. Zucchetti, Z. Kis and W. Vogel, �Collapse and revival in the vibronic dynamics of laser-driven diatomic molecules,� Chem. Phys. Lett. 322 3-4, 255-262 (2000). [CrossRef]
- Ch. Warmuth, A. Tortschano., F. Milota, M. Shapiro, Y. Prior, I. Sh. Averbukh, W. Schleich, W. Jakubetz and H. F. Kau.mann, �Studying vibrational wavepacket dynamics by measuring fluorescence interference fluctuations,� J. Chem. Phys. 112, 5060-5069 (2000). [CrossRef]
- Y. S. Kim and M. E. Noz, Phase space picture of quantum mechanics, (World Scientific, 1991).
- J. Janszky, An. V. Vinogradov, T. Kobayashi and Z. Kis, �Vibrational Schroedinger-cat states,� Phys. Rev. A 50, 1777-1784 (1994 ), and see also references therein. [CrossRef] [PubMed]
- J. Eiselt and H. Risken, �Quasiprobability distributions for the Jaynes-Cummings model with cavity damping,� Phys. Rev. A 43, 346-360 (1991). [CrossRef] [PubMed]
- M. G. Benedict, A. Czirjak, �Wigner functions, squeezing properties, and slow decoherence of a mesoscopic superposition of two-level atoms,� Phys. Rev. A 60, 4034-4044 (1999). [CrossRef]

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