Matter-wave phase operators for quantum atom optics: On the possibility of experimental verification
Kingshuk Adhikary, Subhanka Mal, Abhik Kr. Saha, Bimalendu Deb
aa r X i v : . [ qu a n t - ph ] F e b OSI-ISO 2018 manuscript No. (will be inserted by the editor)
Matter-wave phase operators for quantum atomoptics: On the possibility of experimental verification
Kingshuk Adhikary ∗ · Subhanka Mal ∗ · Abhik Kr. Saha · Bimalendu Deb the date of receipt and acceptance should be inserted later Abstract
In early 90’s Mandel and coworkers performed an experiment [1] toexamine the significance of quantum phase operators by measuring the phasebetween two optical fields. We show that this type of quantum mechanicalphase measurement is possible for matter-waves of ultracold atoms in a doublewell. In the limit of low number of atoms quantum and classical phases aredrastically different. However, in the large particle number limit, they are quitesimilar. We assert that the matter-wave counterpart of the experiment [1] isrealizable with the evolving technology of atom optics .
Keywords
Phase operators · Matter-wave interferometry · double well K. AdhikarySchool of Physical Sciences, Indian Association for the Cultivation of Science, Jadavpur,Kolkata 700032.E-mail: [email protected]. MalSchool of Physical Sciences, Indian Association for the Cultivation of Science, Jadavpur,Kolkata 700032.E-mail: [email protected]. K. SahaSchool of Physical Sciences, Indian Association for the Cultivation of Science, Jadavpur,Kolkata 700032.E-mail: [email protected]. DebSchool of Physical Sciences, Indian Association for the Cultivation of Science, Jadavpur,Kolkata 700032.E-mail: [email protected] Kingshuk Adhikary ∗1
Phase operators · Matter-wave interferometry · double well K. AdhikarySchool of Physical Sciences, Indian Association for the Cultivation of Science, Jadavpur,Kolkata 700032.E-mail: [email protected]. MalSchool of Physical Sciences, Indian Association for the Cultivation of Science, Jadavpur,Kolkata 700032.E-mail: [email protected]. K. SahaSchool of Physical Sciences, Indian Association for the Cultivation of Science, Jadavpur,Kolkata 700032.E-mail: [email protected]. DebSchool of Physical Sciences, Indian Association for the Cultivation of Science, Jadavpur,Kolkata 700032.E-mail: [email protected] Kingshuk Adhikary ∗1 et al. In quantum optics, unitary phase operators were introduced in the 1980’sby Barnett and Pegg [2] to describe the phase measurement and quantumphase-dependent effects. In the definition of two-mode unitary phase-differenceoperators [3], it is assumed that the total number of photons is conserved. Thisassumption can not be valid always except in closed quantum optical systemssuch as two-mode Raman type processes in high-Q cavities [3]. However, formatter-waves of ultracold atoms in a double-well (DW) trap, the total numberof atoms is conserved during the trap lifetime or duration of any experimentalmeasurement on the trapped atoms. It is then necessary to formulate thequantum phase of matter-waves with a fixed number of particles. So, it isimportant to study quantum atom optics under the influence of unitary phaseoperators in matter-waves [4,5].Ketterle’s group [6] has first experimentally observed the atom interferome-try of two-component Bose-Einstein Condensates (BECs) in a DW trap. Theyhave observed the relative phase between two condensates with matter-waveinterference [7]. In this case the DW is analogous to a coherent beam splitter.Their group [8] has demonstrated another experimental technique to determinethe relative phase of two condensates by scattering of light. The advantage ofthis technique is that neither coherent splitting of BECs is required nor isrecombination of the matter-waves. Matter-wave interferometry has also beendeveloped using magnetically generated DW traps on an atom chip [9]. Theirhas been several experiments to determine the spatial phase of the matterwave interference by releasing two condensates from spatially separated po-tential wells [6]. In those experiments the phase is measured classically. Gross et al . first demonstrated experimentally quantum mechanical homodyne detec-tion of matter-wave phase [10]. Recently there are some experiments showingthat a few numbers of particles (atomic bosons and fermions) can also betrapped using optical fields [11,12].Here we discuss the possibility of quantum phase measurement with matter-wave interferometry with small number of bosonic atoms in DW. In the ex-periment performed by Mandel’s group in 1991 [1] two modes of laser wereemployed in a interferometric homodyne detection scheme. One of the modeswas treated classically with large number of photons, and the other quantummechanically with variable average photon number. They measured the sineand cosine of quantum phase-difference operator and plotted the results as afunction of average photon number in the second mode. Their results show thatwhen the average photon numbers in both the modes are small, classical andquantum mechanical phases differ significantly. However, if the average photonnumber in the second mode is increased, classical and average quantum phasestend to match. Here we discuss the possibility of a matter-wave counterpartof Mandel’s experiment using ultracold bosonic atoms in a quasi-1D DW. itle Suppressed Due to Excessive Length 3
Here we consider Barnett-Pegg [2] type quantum phase operators for matter-wave of few bosons or fermions. Matter-wave phase operators were first in-troduced in 2013 [4]. It is shown that [4,5], for a low number of bosons orfermions, unitary nature of the phase-difference operators is important. Forlarge number of photons or quanta, the non-unitary Carruthers-Nieto [13]phase-difference operators yield almost similar results as those due to Barnett-Pegg type unitary operator. Since, in the unitary regime, phase operators areformulated by coupling the vacuum state with the highest number state ina finite-dimensional Fock space, the effects of the vacuum state becomes sig-nificant for low number of particles. In early 90’s, Mandel’s group [1] exper-imentally determined the phase-difference between two optical fields in bothsemi-classical and quantum cases. They made use of the sine and cosine ofphase-difference operators of Carruthers and Nieto [13] as well as unitary op-erational phase-difference operators as they defined.For the material particles, quantum phase operators associated with bosonsand fermions have different character. Unitary quantum phase operators forbosons are introduced by the analogy of quantum phase operator formalism ofphotons. It is difficult to define quantum phase operator for fermions becausemore than one fermion can not occupy a single quantum state simultaneously.A quantum state for fermions can be either filled (by one fermion) or empty(vacuum state). Therefore, quantum phase-difference between two fermionicmodes becomes well defined when single-particle quantum states of fermionsare half-filled.To clarify the canonically conjugate nature of number- and phase-differenceoperators, one can introduce two commuting operators corresponding to cosineand sine of the phase-difference. Both of them are canonically conjugate tothe number-difference operator. These two phase operators plus the number-difference operator forms a closed algebra [4].To define an appropriate quantum phase operator, a complication arisesfrom the number operator of a harmonic oscillator which has a lower boundstate. Dirac [14] first postulated the existence of a hermitian phase operator inhis description of quantized electromagnetic fields. Susskind and Glogower [15]first showed that Dirac’s phase operator was neither unitary nor hermitian. Ifsomeone seeks to construct a unitary operator U by following Dirac’s postulatethen U U † = ˆ I = U † U , hence U is not unitary. Thus Susskind and Glogower[15] concluded that there does not exist any hermitian phase operator. Louisell[16] first introduced the periodic operator function corresponding to a phasevariable which is conjugate to the angular momentum. Carruthers and Nieto[13] introduced two-mode phase difference operators of a two-mode radiationfield by using two non-unitary hermitian phase operators C and S , measurethe cosine and sine of the fields. The two-mode phase-difference operators asdefined by Carruthers and Nieto [13] are given byˆ C CN = ˆ C ˆ C + ˆ S ˆ S Kingshuk Adhikary ∗1
Here we consider Barnett-Pegg [2] type quantum phase operators for matter-wave of few bosons or fermions. Matter-wave phase operators were first in-troduced in 2013 [4]. It is shown that [4,5], for a low number of bosons orfermions, unitary nature of the phase-difference operators is important. Forlarge number of photons or quanta, the non-unitary Carruthers-Nieto [13]phase-difference operators yield almost similar results as those due to Barnett-Pegg type unitary operator. Since, in the unitary regime, phase operators areformulated by coupling the vacuum state with the highest number state ina finite-dimensional Fock space, the effects of the vacuum state becomes sig-nificant for low number of particles. In early 90’s, Mandel’s group [1] exper-imentally determined the phase-difference between two optical fields in bothsemi-classical and quantum cases. They made use of the sine and cosine ofphase-difference operators of Carruthers and Nieto [13] as well as unitary op-erational phase-difference operators as they defined.For the material particles, quantum phase operators associated with bosonsand fermions have different character. Unitary quantum phase operators forbosons are introduced by the analogy of quantum phase operator formalism ofphotons. It is difficult to define quantum phase operator for fermions becausemore than one fermion can not occupy a single quantum state simultaneously.A quantum state for fermions can be either filled (by one fermion) or empty(vacuum state). Therefore, quantum phase-difference between two fermionicmodes becomes well defined when single-particle quantum states of fermionsare half-filled.To clarify the canonically conjugate nature of number- and phase-differenceoperators, one can introduce two commuting operators corresponding to cosineand sine of the phase-difference. Both of them are canonically conjugate tothe number-difference operator. These two phase operators plus the number-difference operator forms a closed algebra [4].To define an appropriate quantum phase operator, a complication arisesfrom the number operator of a harmonic oscillator which has a lower boundstate. Dirac [14] first postulated the existence of a hermitian phase operator inhis description of quantized electromagnetic fields. Susskind and Glogower [15]first showed that Dirac’s phase operator was neither unitary nor hermitian. Ifsomeone seeks to construct a unitary operator U by following Dirac’s postulatethen U U † = ˆ I = U † U , hence U is not unitary. Thus Susskind and Glogower[15] concluded that there does not exist any hermitian phase operator. Louisell[16] first introduced the periodic operator function corresponding to a phasevariable which is conjugate to the angular momentum. Carruthers and Nieto[13] introduced two-mode phase difference operators of a two-mode radiationfield by using two non-unitary hermitian phase operators C and S , measurethe cosine and sine of the fields. The two-mode phase-difference operators asdefined by Carruthers and Nieto [13] are given byˆ C CN = ˆ C ˆ C + ˆ S ˆ S Kingshuk Adhikary ∗1 et al. ˆ S CN = ˆ S ˆ C − ˆ S ˆ C (1)where ˆ C i = 12 [( ˆ N i + 1) − ˆ a i + ˆ a † i ( ˆ N i + 1) − ]ˆ S i = 12 i [( ˆ N i + 1) − ˆ a i − ˆ a † i ( ˆ N i + 1) − ]are the phase operators corresponding to the cosine and sine respectively, of i -th mode, where ˆ a † i (ˆ a i ) denotes the creation(annihilation) operator for a bosonand ˆ N i = ˆ a † i ˆ a i . The explicit form of phase-difference operators can be written(with i =1 or 2) asˆ C CN = 12 [( ˆ N + 1) − ˆ a ˆ a † ( ˆ N + 1) − + ˆ a † ( ˆ N + 1) − ( ˆ N + 1) − ˆ a ] (2)ˆ S CN = 12 i [( ˆ N + 1) − ˆ a ˆ a † ( ˆ N + 1) − − ˆ a † ( ˆ N + 1) − ( ˆ N + 1) − ˆ a ] (3)In interferometric experiments, only the phase difference between two fieldsmatters and not the absolute phase of a field. According to Barnett-Peggformalism, hermitian and unitarity of phase-difference operators correspondingto cosine and sine of phase have following explicit formˆ C = ˆ C CN + ˆ C (0)12 (4)ˆ S = ˆ S CN + ˆ S (0)12 (5)where ˆ C (0)12 = 12 [ | N, ih , N | + | , N ih N, | ]ˆ S (0)12 = 12 i [ | N, ih , N | − | , N ih N, | ]are the operators which are constructed by coupling the vacuum state of onemode with the highest Fock state of the other mode. N = h ˆ N i + h ˆ N i istotal number of bosons which is conserved. | N , N − N i represents a two-mode Fock state with N and ( N − N ) being the atom numbers in modes 1and 2, respectively. The difference of the number or the population imbalancebetween the two wells is ˆ W = ˆ N − ˆ N . The commutation relations of thegiven operators ˆ C , ˆ S and ˆ W are as follows[ ˆ C , ˆ W ] = 2 i ( ˆ S − ( N + 1) ˆ S (0)12 ) (6)[ ˆ S , ˆ W ] = − i ( ˆ C − ( N + 1) ˆ C (0)12 ) (7)[ ˆ C , ˆ S ] = 0 (8) itle Suppressed Due to Excessive Length 5 The first two of the above equations imply ∆C ∆W ≥ (cid:12)(cid:12)(cid:12) S − ( N + 1) S (0)12 (cid:12)(cid:12)(cid:12) (9) ∆S ∆W ≥ (cid:12)(cid:12)(cid:12) C − ( N + 1) C (0)12 (cid:12)(cid:12)(cid:12) (10)Now, the standard quantum limit of fluctuation ∆ SQL in number-differenceor phase-difference quantity is given by [5] ∆ SQL = 1 N q [ S − ( N + 1) S (0)12 ] + [ C − ( N + 1) C (0)12 ] (11)and the normalized squeezing parameters for both phase- and number-differenceoperators, respectively, by Σ p = ∆E φ − ∆ SQL (12)and Σ w = ∆W n − ∆ SQL (13)where ∆E φ = p ( ∆C ) + ( ∆S ) is an average phase fluctuation and ˆ W n = ˆ WN , normalized number-difference operator. The system will be squeezed innumber or phase variables when Σ w or Σ p becomes negative. To build up the model, we consider a quasi-1D DW trap potential in whichbosonic atoms are confined in the two sites of the DW. The DW has two quasidegenerate energy eigenfunction in which the ground band is occupied by thebosons. The idea is to initialize a certain number of bosons in one of the eithersite of the DW and let them evolve (tunnel) with time. So, the particle numberin the other well ( N ) which was initially empty oscillates with time. We havetaken the quantum mechanical average of ˆ N and ˆ S throughout the timeupto which N ( t ) = N ( t ) = N/ As the total number of bosons in our case is conserved, we calculate the quan-tum mechanical average of sine phase-difference operator as a function of num-ber of bosons in the second well for different total number of particles. Weconsider symmetric trap for non-interacting bosons. Although non-interactingbosons are idealized, we assume the interaction to be very small and our case
Kingshuk Adhikary ∗1
Kingshuk Adhikary ∗1 et al. Fig. 1
Schematic of bosons in a quasi-1D DW trap with J being tunneling coefficient.
Fig. 2
Calculated values of h S i as a function of average number of bosons for differenttotal number of bosons. closely resembles to that. To begin with, we initialize the system with allbosons in one well and then the number in the other well ( N ) evolves withtime. Throughout the evolution of N up to half of the total population wetake quantum mechanical average. Then we have plotted h ˆ S i with h ˆ N i . Ourresults are similar to that obtained by Mandel’s group. For their case they havechanged the photon numbers in both the modes treating one mode classicallyand other mode quantum mechanically. They have also changed the ratio ofaverage photon numbers of two different modes in their experiment. Whereas,in our case we have only changed the total number of particles to mimic theirexperimental finding. itle Suppressed Due to Excessive Length 7 We have studied the sine of quantum phase difference between two sites ofa DW for non-interacting bosons. The cosine operator can also be studiedin the similar way. We have shown that when the total number of bosons isincreased the result has a good agreement with the Mandel’s experimentalresults. It is worth noticing how the results modify in presence of interactionsand slight asymmetry of the trap. One can also calculate the fluctuations ofsine and cosine phase operators. Recently, phase fluctuation below the shot-noise has been demonstrated experimentally with two components BEC’s [17].The results we obtained suggest that when the particle number is small ineither side of the well unitary phase operators become important. This can beattributed to the effect of vacuum term in unitary phase operators. In caseof Josephson oscillations in BEC’s the unitary quantum phase has not beenstudied so far. It may be possible to measure the quantum phase of these typeof systems by using homodyne detection method.
References
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