Interferometry with correlated fermions

Physica B 151 (1988) 286-290 North-Holland, Amsterdam

INTERFEROMETRY WITH CORRELATED FERMIONS

B. Y U R K E A T & T Bell Laboratories, Murray Hill, NJ 07974, USA

It is known and has been demonstrated experimentally that the sensitivity of an optical interferometer can be increased by injecting squeezed light into the unused port of the intefferometer. Here it is demonstrated that the sensitivity of a fermion interferometer can also be enhanced provided the fermions in the input state are suitably correlated. An SU(2) formalism is introduced which allows one to concentrate on those quantum numbers of the input state which are of relevance to interferometer performance.

In conventional interferometry an incoming particle beam (or photon beam in the optical case) impinges on a beam splitter to produce two separate beams. By interfering the two beams, using a second beam splitter, the relative phase ~b between the two beams can be determined. The minimum uncertainty A4~ in the measured phase is 1/(?~) 1/2 where ri is the mean number of particles that propagate through the interferometer per measurement time. This sensitivity can be viewed as the result of averaging n independent statistical events, i.e., each particle after passing through the interferometer independently decides which of the two output ports of the last beam splitter it will exit. On the other hand, the approximate uncertainty relation A~bAn >~ 1 suggests that one should be able to achieve a phase sensitivity of l/rL That is, one should be able to do much better than root n averaging. Caves [1] has shown that an optical interferometer can achieve a phase sensitivity better t h a n 1/(rt) 1/2 provided squeezed light is fed into the unused input port of the interferometer. In fact, as also noted by Bondurant [2] and by Yurke et al. [3], by suitably employing squeezed states a sensitivity of 1/r~ can be achieved. Recently, the enhancement of optical interferometer sensitivity using squeezed states has been experimentally demonstrated [4, 5]. Squeezed states are multiphoton states in which many photons occupy a given boson mode [6, 7]. In contrast, a given fermion mode can be occupied by at most one fermion. Hence, it is

not at all obvious that a fermion interferometer can also achieve a phase sensitivity approaching 1/& Here it is shown that such a sensitivity can indeed be achieved [8] provided a suitably correlated many-fermion state enters the input ports to the interferometer. Although this settles an issue of principle, the question of how to generate such states in the laboratory is still open. The major message here is that there are nontrivial many-fermion interferometric effects for which one might try to look. The following analysis will be carried out in the second quantized picture, since this formalism is most convenient in dealing with manyfermion wave functions. Further, the SU(2) symmetry [3] of an interferometer will be exploited to single out those quantum numbers of the input state vector that are of relevance to interferometry. Besides simplifying the analysis, the SU(2) formalism provides a geometrical picture which is a valuable aid to one's intuition. A M a c h - Z e h n d e r interferometer of the type depicted in fig. 1 will be analyzed. As can be seen, there are two input ports a I and a 2 through which fermions can enter the interferometer via beam splitter S1. The beams bl and b 2 leaving S1 propagate to beam splitter $2 via mirrors M1 and M2 respectively. The fermions exiting the output ports d I and d 2 of $2 are then counted by detectors D1 and D2. For simplicity it will be assumed that the fermion beams are sufficiently well columnated so that the free propagation of the fermion field is adequately described by a

B. Yurke / Interferometry with correlated fermions

287

jugate, the creation operator c~, satisfy the anticommutation relations [Cik , Cjk,]+ = 0.

[CikC~k,]÷ = 3ij3kk, ,

(4)

c2

The action of the beam splitters on the fermion field will now be described. Since a beam splitter is a linear device, the output must be a linear combination of the input modes and, further, the mode transformation must be unitary. Letting aik and bik denote the annihilation operators for the fields entering and leaving S1 respectively, one has

'~az

S1

Fig. 1. A fermion interferometer. S1 and $2 are 50-50 beam splitters, M1 and M2 are mirrors, and D1 and D2 are particle counters. Ferrnions can enter the interferometer through either port a 1 or a 2.

b2 k

kxa2k / ,

(5)

where Uk is a 2 × 2 unitary matrix. For definiteness, Uk is taken to be

one-dimensional Schr6dinger equation, ih OO~ 0--7 =

h 2 O2Oi 2 m Oz 2 '

v2

where i is a beam label, i ~ {1, 2}, and z is the distance along a beam path. In order to avoid carrying spin indices through the calculation it is further assumed that all the fermions are in the same spin state and that the interferometer elements conserve spin. The second quantized field operator can be written as

1

O ( z , t) = 7-'&-. ~ . cik e x p ( - i w d ) exp(ikz) , VL

(2)

1,

where the summation is carried out over all discrete values of k = 2"rrn/L where n is an integer and L is a "quantization volume", a length that is large compared to the wave packets that will be fed into the interferometer. In order to satisfy eq. (1) the dispersion relation is hk 2

~%-

2m "

1

1

•

(6)

(1) Since the magnitude of each matrix element is the same this characterizes a 50-50 beam splitter. Letting cik and di~ denote the annihilation operators for the fields entering and leaving $2 respectively, the mode transformation performed by $2 is taken to be

(d,k ~ d2k/=--~2(_l

1 1](c,~ 1/\c2~ ) .

(7)

Note aik , bik , Cik and dik all satisfy anticommutation relations of the form (4). From eq. (2) one sees that when propagating along a path length l the fermion accumulates a phase shift ~b = kl. Hence, when propagating from S1 to $2 via mirror M1 or M2 a fermion accumulates respectively the phase shift ~bI = kli or ~b2 = kl 2. That is e ikll (Clkl = ( 0 \ C2k /

0 ) eikt2)( blk b2 k "

(8)

(3)

The coefficient Cik is a fermion annihilation operator which together with its Hermitian con-

Equations (5)-(8) can be solved to obtain the overall mode transformation for the interferometer

288

B. Yurke / lnterferometry with correlated fermions

( dlk ~

[elk" + eik'q

- - [ e ikll _ eikt2] ~

d2k / = ( - [ e ikl, _ eikl2]

[eik/l q- eikl2] ]

where ~ k A l ~ 1. Then eq. (13) simplifies to 2 cos 4~Jz + 2 sin CbJy ,

N D =

(14)

(9) ', a2k

Consider the case when a localized fermion wave packet enters the interferometer. After waiting long enough for all the fermions in the wave packet to propagate through the interferometer, the particle counters D1 and D2 (see fig. 1) report the total number of fermions that have left output ports d 1 and d 2 respectively, that is, D1 and D2 are characterized by the operators

d~kdlk,

(10)

N2 = E d 2*k d 2 k ,

(11)

N1= ~

k

where the interferometer phase is ~b = k 0 Al and the operators, Jx = ~

[blkb2k + b2kblk]

Jy= -~

[b~kb2k -- b z k b l k ]

(15)

I~ [blkblk t t -- b2kb2k ]

Jz = ~

satisfy the usual angular momentum commutation relations, in particular,

and [Jx, Jyl = iJz. k

respectively. The total number of fermions N = N Z+ N 2 propagating through the interferometer is a conserved quantity and hence contains no phase information. The phase information can be extracted from the difference N D -- N~ - N 2 in the number of counts reported by D1 and D2. That is, the interferometer observable is taken to be ND = ~] [dlkdlk t - d2kd2k ] . k

(12)

The total number operator N commutes with J = ( J x , Jy, Jz): [N, J] = O. 2

j2 = ½N(½N+ 1) + W ,

W also commutes with N and J,

The first term can be regarded as a self-interference term and the second term represents a cross interference between fermion modes alk and a2k. To simplify things further it will be assumed that the fermion beam is quasimonochromatic, that is, k is restricted to the interval k o- ½Ak
½Ak,

(18)

h* h* b 2k' b l k ,1 " E [bl,b;k,b2kblk, + -lk-2k' k k' (19)

N D = ~ {cos(k Al)[alkalk * * -- a2kaZk ] (13)

2

where W= -E

. - i sin(k Al)[a*lka2k - a2~alk]} *

(17)

Further, the Casimir invariant j2 = j~ + jy + Jz can be put into the form

Using eq. (9) to express the dig in terms of the aik one obtains

k

(16)

[w,

j] =

0.

(20)

Equations (14)-(20) allow one to take advantage of angular momentum algebra or SU(2) representation theory in solving interferometer problems. To demonstrate the utility of this algebra it will now be demonstrated that when all the fermions enter only one port the phase uncertainty A4~ can never be less than 1/(ri) z/z regardless of the input state. Let I~O} denote a

B. Yurke / lnterferometry with correlated ferrnions

state representing a many-fermion wave packet where all the fermions enter the interferometer through port 1. Then since no fermions enter port 2, the annihilation operator a2k operating on I~b) yields zero: a2~14') = O.

(21)

It is then easy to show that

WI~O)= O,

(22)

that is I~0) is an eigenstate of W with eigenvalue zero. After waiting sufficiently long for all the fermions in the wave packet to leave the interferometer the detectors D1 and D2 will report the total number n of fermions that have passed through the interferometer. Hence, after the measurement, I~O) will be a known eigenstate of N,

NIq,) = n l ~ ) .

(23)

Using eqs. (22) and (23) it is apparent that [~b) is an eigenstate of j2,

289

the mean square uncertainty (A~b)2 in the inferred phase is (a6)2_

(aNo):

IO(No)/a61'

"

(28)

Substituting eqs. (26) and (27) into this expression one readily obtains a phase sensitivity A~b = 1/x/~.

(29)

It is worth emphasizing the generality of this result: the only information used in the calculation was that all the fermions enter the same port. A state will now be exhibited which will allow the interferorneter to achieve a sensitivity approaching i / n . Again consider an n fermion wave packet of the kind discussed above, i.e., the state can be denoted as IJ, m) = [ ~n, ½n). By repeated application of the lowering operator J_ =J~ - i J y ,

y-

=

X *a 2 k a l ~ k

(30)

,

where j = in. Further, since all the fermions enter through port i, it is apparent from eq. (15) that I~b) is an eigenstate of J,,

one can construct from IJ, rn) -- lin, i n ) the states [~n, i ) and l in, - i ) provided n is odd. Since J transfers a fermion from port 1 to port 2, the state lin, 1) has i ( n + l ) fermions in beam 1 and ½(n - 1) fermions in beam 2. Consider now the state

LI~,) = m l ~ )

Is) - 2-1'211in, i ) + lln, - i ) ] .

•z=l*>

=

J(J +

~)lq,),

(24)

(25)

with m = ~n. Hence, I~b) has the SU(2) quantum numbers j = i n and m = in. Since eq. (14) is written completely in terms of the SU(2) generators Jz and Jy, this is all we need to know about the state in order to calculate means and variances of Nt~. In particular, using standard techniques of angular momentum algebra,

(ND) = (g'INol6) = n cos 6

(26)

(31)

The expectation value and variances of this state are

( N D) = ~ In + 1] sin gb

(32)

and (ANo)

z

= cos2¢ +

1) sin2¢.

k(n 2 + n -

(33)

The mean square uncertainty, from eq. (28), is then

and (A&)2 = cos24~ + (AND) 2 = n sin2~b ,

(27)

¼(n 2 + n ~

1) sinZ~b

~(n + 1)2cos~

(34)

290

B. Yurke / lnterferometry with correlated ferrnions

T h i s is m i n i m i z e d w h e n sin ~b = 0. H e n c e , at its most sensitive operating point the interferometer a c h i e v e s t h e r m s u n c e r t a i n t y in ~b, A~min

:

2 / ( n + 1)

v i d e a n i n t e r e s t i n g n e w f r o n t i e r to e x p l o r e in matter-wave interferometry.

(35)

References for t h e state (31). A f e r m i o n state has t h u s b e e n e x h i b i t e d achieving a phase sensitivity which approaches 1/n ( w i t h i n a f a c t o r o f 2). H o w o n e w o u l d go a b o u t g e n e r a t i n g s u c h a state in t h e l a b o r a t o r y is n o t k n o w n . T h e goal o f this p a p e r h a s b e e n r a t h e r specific, n a m e l y to d e m o n s t r a t e t h a t f e r m i o n interferometers can approach a phase sensitivity o f 1/ff p r o v i d e d s u i t a b l y c o r r e l a t e d m a n y - b o d y wave functions enter the input port of the interferometer. Undoubtedly, there are other manyf e r m i o n i n t e r f e r e n c e effects t h a t m i g h t b e experimentally more accessible and that would pro-

[1] C.M. Caves, Phys. Rev. D 23 (1981) 1693. [2] R.S. Bondurant, Ph.D. Thesis (MIT, Cambridge, MA, 1983). [3] B. Yurke, S.L. McCall and J.R. Klauder, Phys. Rev. A 33 (1986) 4033. [4] M. Xiao, L. Wu and H.J. Kimble, Phys. Rev. Lett. 59 (1987) 278. [5] P. Grangier, R.E. Slusher, B. Yurke and A. LaPorta, Phys. Rev. Lett. 59 (1987) 2153. [6] C.M. Caves and B.L. Schumaker, Phys. Rev. A 31 (1985) 3068. [7] B.L. Schumaker and C.M. Caves, Phys. Rev. A 31 (1985) 3093. [8] B. Yurke, Phys. Rev. Lett. 56 (1986) 1515.

DISCUSSION (Q) H. Rauch: Can you quantisize a little bit more the coherence requirements for such experiments? (A) B. Yurke: A mathematical procedure was described for constructing state vectors which enhance interferometer sensitivity. This procedure starts with an initial state vector in which all the fermions are in the same beam and an operator is successively applied which transfers a fermion from one beam to its corresponding position in the other beam. The initial n fermion state can be fairly arbitrary. Each individual electron wave packet can be rather small if one wished (one does not need large spatial coherence). Further, if one wished, the packets could be spaced far apart (one does not need a high density where the fermions begin to become degenerate). The crucial aspect of the states that result by

applying the mathematical procedure is the many-body correlations that are established between the fermions. It is the high-order coherence that is important. Such coherence, or correlations, can only be established by allowing the n fermions to interact via a suitable nonlinear interaction. (Q) A.G. Klein: I believe that the input (squeezed) state required is not achievable by means of interferometry, e.g., by means of a "pre-interferometer". Do you agree? (A) B. Yurke: Yes. Beam splitters and phase shifters only perform rotations in the Jx, Jy, Jz space and hence are incapable of distorting an initial j = ½n, m = ½n state into a squeezed fermion state.