## Reading the phase of a Raman excitation with a multi-state atomic interferometer |

Optics Express, Vol. 22, Issue 16, pp. 19141-19148 (2014)

http://dx.doi.org/10.1364/OE.22.019141

Acrobat PDF (933 KB)

### Abstract

Atomic memories for flying photonic qubits are an essential ingredient for many applications like e.g. quantum repeaters. Verification of the coherent transfer of information from a light field to an atomic superposition is usually obtained using an optical read-out. In this paper we report the direct detection of the atomic coherence by means of atom interferometry. We experimentally verified both that a bichromatic laser field closing a Raman transition imprints a distinct, controllable phase on the atomic coherence and that it can be recovered after a variable time delay.

© 2014 Optical Society of America

## 1. Introduction

1. H. J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: The role of imperfect local operations in quantum communication,” Phys. Rev. Lett. **81**, 5932–5935 (1998). [CrossRef]

2. H. Kimble, “The quantum internet,” Nature **453**, 1023–1030 (2008). [CrossRef] [PubMed]

3. D. P. DiVincenzo, “Quantum computation,” Science **270**, 255–261 (1995). [CrossRef]

4. M. A. Nielsen and I. L. Chuang, *Quantum Computation and Quantum Information* (Cambridge University, 2010). [CrossRef]

5. A. Mair, J. Hager, D. F. Phillips, R. L. Walsworth, and M. D. Lukin, “Phase coherence and control of stored photonic information,” Phys. Rev. A **65**, 031802 (2002). [CrossRef]

6. M. Eisaman, A. André, F. Massou, M. Fleischhauer, A. Zibrov, and M. D. Lukin, “Electromagnetically induced transparency with tunable single-photon pulses,” Nature **438**, 837–841 (2005). [CrossRef] [PubMed]

7. K. Jensen, W. Wasilewski, H. Krauter, T. Fernholz, B. M. Nielsen, M. Owari, M. B. Plenio, A. Serafini, M. M. Wolf, and E. S. Polzik, “Quantum memory for entangled continuous-variable states,” Nature Phys. **7**, 13–16 (2011). [CrossRef]

8. M. Hosseini, G. Campbell, B. M. Sparkes, P. K. Lam, and B. C. Buchler, “Unconditional room-temperature quantum memory,” Nature Phys. **7**, 794–798 (2011). [CrossRef]

9. C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature **409**, 490–493 (2001). [CrossRef] [PubMed]

10. C.-W. Chou, J. Laurat, H. Deng, K. Choi, H. De Riedmatten, D. Felinto, and H. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science **316**, 1316–1320 (2007). [CrossRef] [PubMed]

11. S. L. Christensen, J. B. Béguin, H. L. Sørensen, E. Bookjans, D. Oblak, J. H. Müller, J. Appel, and E. S. Polzik, “Toward quantum state tomography of a single polariton state of an atomic ensemble,” New J. Phys. **15**, 015002 (2013). [CrossRef]

## 2. Description of the method

^{87}Rb degenerate sample held in a magnetic trap [13

13. J. Petrovic, I. Herrera, P. Lombardi, F. Schaefer, and F. S. Cataliotti, “A multi-state interferometer on an atom chip,” New J. Phys. **15**, 043002 (2013). [CrossRef]

14. F. Minardi, C. Fort, P. Maddaloni, M. Modugno, and M. Inguscio, “Time-domain atom interferometry across the threshold for Bose-Einstein condensation,” Phys. Rev. Lett. **87**, 170401 (2001). [CrossRef] [PubMed]

13. J. Petrovic, I. Herrera, P. Lombardi, F. Schaefer, and F. S. Cataliotti, “A multi-state interferometer on an atom chip,” New J. Phys. **15**, 043002 (2013). [CrossRef]

*F*= 2 hyperfine manifold) yields a 1.75 times higher resolution (defined as the ratio between the fringe period and the fringe width) than an ideal two-state interferometer.

*m*states within the

_{F}*F*= 2 hyperfine state. For this task we used a coherent source close to the D2 line of the

^{87}Rb (

^{2}S

_{1/2}→

^{2}P

_{3/2},

*λ*∼ 780 nm), whose frequency is set 18 GHz to the red of the atomic resonance. We have closed the interferometer by subsequently applying a radio frequency (RF) pulse connecting the same sublevels as the bichromatic field.

13. J. Petrovic, I. Herrera, P. Lombardi, F. Schaefer, and F. S. Cataliotti, “A multi-state interferometer on an atom chip,” New J. Phys. **15**, 043002 (2013). [CrossRef]

*R*(Ω

*τ*), where Ω and

*τ*represent the Rabi frequency and the duration of the coupling, respectively. The matrix

*R*is obtained by diagonalization of the tridiagonal Hamiltonian which describes the coupling between neighboring

*m*states (Ref. [15

_{F}15. R. J. Cook and B. Shore, “Coherent dynamics of N-level atoms and molecules. III. An analytically soluble periodic case,” Phys. Rev. A **20**, 539 (1979). [CrossRef]

*T*between the pulses is described by a diagonal operator

*P*of shape

*P*=

_{n,n}*exp*[

*i*(

*n*− 1)Δ

*E/ħ T*], where we use the index

*n*to refer to the sub-levels of the Zeeman manifold (

*n*= 1,...,5) and Δ

*E*represents, in linear approximation, the Zeeman splitting of neighboring

*m*states (see Fig. 1(b)). The full interferometric sequence is then described by a transfer matrix

_{F}*J*, where Ω (Ω′) and

*τ*(

*τ′*) represent the Rabi frequency and the duration of the RF (Raman) coupling.

*A*equal to the optimal value for an interferometer sequence [13

**15**, 043002 (2013). [CrossRef]

*τ′*= Ω

*τe*=

^{iδϕ}*Ae*, where

^{iδϕ}*δϕ*is the relative phase between the two components of the Raman light pulse.

*δϕ*has no effect on the resulting distribution of populations after a single pulse, but imprints a different phase on the coherences between the wave function components. Indeed, after the Raman coupling, the atomic state consists of the Zeeman components with relative phases following

*ϕ*

_{n+1}−

*ϕ*=

_{n}*δϕ*, where

*ϕ*is the phase of the

_{n}*n*wave function component, and the index

^{th}*n*is still used to refer to the sub-levels of the manifold. Accordingly, the experimental sequence Eq. (1) can be rewritten as From Eq. (2), it is evident that the dependence on the phase of the Raman coupling

*δϕ*is formally equivalent to the effect of an additional evolution time of the coherent superposition of the Zeeman states as described by the operator

*P*for a time delay

*T′*(

*δϕ*), Therefore, while in our previous work a fringe-like behavior of the population distribution as a function of the delay time

*T*is observed, in the present experiment, we expect to observe the same oscillation as a function of the imprinted phase

*δϕ*for a fixed

*T*.

## 3. Experimental set-up and results

*F*= 2,

*m*= 2〉 state. The whole interferometric sequence takes place 100

_{F}*μ*s after the condensate is released from the magnetic trap. In this way, we can both optimize the magnetization axis by carefully setting the magnetic bias field

**B**and reduce the dissipative scattering due to the lower density of the cloud when illuminated by the Raman excitation beams. Moreover, in such a short time we are sure that the condensate neither exceeds the beam size nor evades the beam (the 1/

*e*Raman beam diameter is approximately 50

*μ*m, while the initial condensate size is below 5

*μ*m).

*σ*and

*π*with respect to the magnetization axis. Since the differential AC Stark shift between successive sub-levels induced by the out-of-resonance light is the same over the whole manifold, the Raman coupling is equally resonant with all the couples of adjacent Zeeman sub-levels, thus perfectly emulating the effect of an RF pulse.

*E*= 2171 kHz, it is Δ

*p*=

*ħ*(

*k*

_{1}−

*k*

_{2}) =

*ħ*Δ

*ω/c*< 10

^{−35}kgm/s (the magnetic field amplitude is set to ca. 3 G in order to minimize the quadratic Zeeman shift [16

16. D. M. Harber, H. J. Lewandowski, J. M. McGuirk, and E. A. Cornell, “Effect of cold collisions on spin coherence and resonance shifts in a magnetically trapped ultracold gas,” Phys. Rev. A **66**, 053616 (2002). [CrossRef]

*τ′*∼ 100 kHz) assures the conservation of the total energy for a large number of successive transitions.

**B**(present during the free fall of the samples until the application of the magnetic gradient) has been accurately measured by means of RF spectroscopy and evaluated as to |

**B**| ∼ 3.1 G (corresponding to the Zeeman shift Δ

*E/ħ*= (2.171 ± 0.02) MHz).

**15**, 043002 (2013). [CrossRef]

*F*= 2,

*m*= 0〉. With the light coupling, this result is obtained for a pulse of duration Δt = 6.2

_{F}*μ*s and measured power before the vacuum cell of 140

*μ*W (

*π*) and 280

*μ*W (

*σ*) (which corresponds to field intensities of 1.8 W/cm

^{2}(

*π*) and 3.6 W/cm

^{2}(

*σ*); beam waists

*w*

_{0}∼ 50

*μ*m).

*π*range of possible Raman phases, the relative phase between the two fields is left free to evolve, so that due to mechanical, thermal, and electronic fluctuations and drifts, it randomly spans the complete 2

*π*interval. By recording the results of the Stern-Gerlach analysis for a sufficiently high number of repetitions of the experiment it is hence possible to get a complete statistics concerning the dependence of the population distribution created by the Raman phase. The graphs of Fig. 3 are obtained after a binning operation on the detected phases: the 2

*π*range has been divided into a number of segments, from the data points in each bin the mean population probabilities of the individual segments and their standard deviations have been determined.

*T*between the Raman and RF pulses,

*T*= 1.3; 1.4; 1.7

*μ*s from top to bottom respectively. Exploiting this degree of freedom it is possible to move the maximal slope of the fringes (and hence the maximal resolution) from one phase region to another one.

*m*= ±2 states as a function of the delay time

_{F}*T*. The solid line is a fit to the data obtained for an exponential decay function (A ·

*e*

^{−t/t̃}). The experimentally determined decay of the fringe contrast is reasonably well described by this function with maximal contrast

*A*= (0.88 ± 0.02) and time constant t̃ = (61 ± 9)

*μ*s. We identified the noise in the magnetic bias field as the main cause of the observed decay. Population decay is not present due to the fact that only sub-levels of the ground state are populated.

*ρ̄*: ∂

*(*

_{t}ρ̄*t*) = −

*i*[

*H*(

*t*),

*ρ̄*(

*t*)] +

*L*(

*t*), where

*H*(

*t*) is the Hamiltonian which generates the transfer matrix of Eq. (2), and we have added a Lindblad operator

*L*(

*t*) to take into account decay rates (see [13

**15**, 043002 (2013). [CrossRef]

*L*(

*t*) as an hollow matrix

*L*(

*t*) = −4

*πγ*[

*ρ̄*(

*t*) − diag(

*ρ̄*(

*t*))] in order to restrict the decay to decoherence effects. A qualitative fit between data and the model is obtained assuming a value of

*γ*between 1.0 kHz and 1.5 kHz (corresponding to the shaded area in Fig. 3(c)). Even if it is evident from the graph that a single exponential decay is not the correct expression for the evolution of the fringe contrast and there is no simple description of its decay, this representation is sufficient to determine the working time scale of this phase storage device.

*T*(the time interval separating the two pulses) while scanning the frequency of the radiation. The result is a fringe structure in the population distribution as a function of the excitation frequency. For longer interrogation time

*T*, the fringes get narrower and closer, with an enhancement of the attainable sensitivity.

## 4. Conclusions

*F*= 2 hyperfine level of

^{87}Rb Bose condensed atoms. The Raman pulse acts as the first beamsplitter of the interferometer. The phase is read out by an RF pulse acting as the second beamsplitter that closes the interferometer. The findings evidence that it is possible to store a phase information onto a degenerate atomic sample using all-optical means and to retrieve this information tenths of microseconds later without optical means.

## Acknowledgments

## References and links

1. | H. J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: The role of imperfect local operations in quantum communication,” Phys. Rev. Lett. |

2. | H. Kimble, “The quantum internet,” Nature |

3. | D. P. DiVincenzo, “Quantum computation,” Science |

4. | M. A. Nielsen and I. L. Chuang, |

5. | A. Mair, J. Hager, D. F. Phillips, R. L. Walsworth, and M. D. Lukin, “Phase coherence and control of stored photonic information,” Phys. Rev. A |

6. | M. Eisaman, A. André, F. Massou, M. Fleischhauer, A. Zibrov, and M. D. Lukin, “Electromagnetically induced transparency with tunable single-photon pulses,” Nature |

7. | K. Jensen, W. Wasilewski, H. Krauter, T. Fernholz, B. M. Nielsen, M. Owari, M. B. Plenio, A. Serafini, M. M. Wolf, and E. S. Polzik, “Quantum memory for entangled continuous-variable states,” Nature Phys. |

8. | M. Hosseini, G. Campbell, B. M. Sparkes, P. K. Lam, and B. C. Buchler, “Unconditional room-temperature quantum memory,” Nature Phys. |

9. | C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature |

10. | C.-W. Chou, J. Laurat, H. Deng, K. Choi, H. De Riedmatten, D. Felinto, and H. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science |

11. | S. L. Christensen, J. B. Béguin, H. L. Sørensen, E. Bookjans, D. Oblak, J. H. Müller, J. Appel, and E. S. Polzik, “Toward quantum state tomography of a single polariton state of an atomic ensemble,” New J. Phys. |

12. | S. L. Christensen, J. B. Béguin, E. Bookjans, H. L. Sørensen, J. H. Müller, J. Appel, and E. S. Polzik, “Quantum interference of a single spin excitation with a macroscopic atomic ensemble,” arxiv: [quant-ph] 1309.2514v2 (2013). |

13. | J. Petrovic, I. Herrera, P. Lombardi, F. Schaefer, and F. S. Cataliotti, “A multi-state interferometer on an atom chip,” New J. Phys. |

14. | F. Minardi, C. Fort, P. Maddaloni, M. Modugno, and M. Inguscio, “Time-domain atom interferometry across the threshold for Bose-Einstein condensation,” Phys. Rev. Lett. |

15. | R. J. Cook and B. Shore, “Coherent dynamics of N-level atoms and molecules. III. An analytically soluble periodic case,” Phys. Rev. A |

16. | D. M. Harber, H. J. Lewandowski, J. M. McGuirk, and E. A. Cornell, “Effect of cold collisions on spin coherence and resonance shifts in a magnetically trapped ultracold gas,” Phys. Rev. A |

**OCIS Codes**

(020.1670) Atomic and molecular physics : Coherent optical effects

(210.4680) Optical data storage : Optical memories

(020.1335) Atomic and molecular physics : Atom optics

(020.1475) Atomic and molecular physics : Bose-Einstein condensates

**ToC Category:**

Atomic and Molecular Physics

**History**

Original Manuscript: January 29, 2014

Revised Manuscript: April 3, 2014

Manuscript Accepted: April 5, 2014

Published: July 31, 2014

**Citation**

P. Lombardi, F. Schaefer, I. Herrera, S. Cherukattil, J. Petrovic, C. Lovecchio, F. Marin, and F. S. Cataliotti, "Reading the phase of a Raman excitation with a multi-state atomic interferometer," Opt. Express **22**, 19141-19148 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-16-19141

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### References

- H. J. Briegel, W. Dür, J. I. Cirac, and P. Zoller, “Quantum repeaters: The role of imperfect local operations in quantum communication,” Phys. Rev. Lett.81, 5932–5935 (1998). [CrossRef]
- H. Kimble, “The quantum internet,” Nature453, 1023–1030 (2008). [CrossRef] [PubMed]
- D. P. DiVincenzo, “Quantum computation,” Science270, 255–261 (1995). [CrossRef]
- M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University, 2010). [CrossRef]
- A. Mair, J. Hager, D. F. Phillips, R. L. Walsworth, and M. D. Lukin, “Phase coherence and control of stored photonic information,” Phys. Rev. A65, 031802 (2002). [CrossRef]
- M. Eisaman, A. André, F. Massou, M. Fleischhauer, A. Zibrov, and M. D. Lukin, “Electromagnetically induced transparency with tunable single-photon pulses,” Nature438, 837–841 (2005). [CrossRef] [PubMed]
- K. Jensen, W. Wasilewski, H. Krauter, T. Fernholz, B. M. Nielsen, M. Owari, M. B. Plenio, A. Serafini, M. M. Wolf, and E. S. Polzik, “Quantum memory for entangled continuous-variable states,” Nature Phys.7, 13–16 (2011). [CrossRef]
- M. Hosseini, G. Campbell, B. M. Sparkes, P. K. Lam, and B. C. Buchler, “Unconditional room-temperature quantum memory,” Nature Phys.7, 794–798 (2011). [CrossRef]
- C. Liu, Z. Dutton, C. H. Behroozi, and L. V. Hau, “Observation of coherent optical information storage in an atomic medium using halted light pulses,” Nature409, 490–493 (2001). [CrossRef] [PubMed]
- C.-W. Chou, J. Laurat, H. Deng, K. Choi, H. De Riedmatten, D. Felinto, and H. Kimble, “Functional quantum nodes for entanglement distribution over scalable quantum networks,” Science316, 1316–1320 (2007). [CrossRef] [PubMed]
- S. L. Christensen, J. B. Béguin, H. L. Sørensen, E. Bookjans, D. Oblak, J. H. Müller, J. Appel, and E. S. Polzik, “Toward quantum state tomography of a single polariton state of an atomic ensemble,” New J. Phys.15, 015002 (2013). [CrossRef]
- S. L. Christensen, J. B. Béguin, E. Bookjans, H. L. Sørensen, J. H. Müller, J. Appel, and E. S. Polzik, “Quantum interference of a single spin excitation with a macroscopic atomic ensemble,” arxiv: [quant-ph] 1309.2514v2 (2013).
- J. Petrovic, I. Herrera, P. Lombardi, F. Schaefer, and F. S. Cataliotti, “A multi-state interferometer on an atom chip,” New J. Phys.15, 043002 (2013). [CrossRef]
- F. Minardi, C. Fort, P. Maddaloni, M. Modugno, and M. Inguscio, “Time-domain atom interferometry across the threshold for Bose-Einstein condensation,” Phys. Rev. Lett.87, 170401 (2001). [CrossRef] [PubMed]
- R. J. Cook and B. Shore, “Coherent dynamics of N-level atoms and molecules. III. An analytically soluble periodic case,” Phys. Rev. A20, 539 (1979). [CrossRef]
- D. M. Harber, H. J. Lewandowski, J. M. McGuirk, and E. A. Cornell, “Effect of cold collisions on spin coherence and resonance shifts in a magnetically trapped ultracold gas,” Phys. Rev. A66, 053616 (2002). [CrossRef]

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