Limitations on the use of the Heisenberg picture in quantum optics J.D. Franson1 and R.A. Brewster2 1University of Maryland Baltimore County, Baltimore, MD 21250 USA 2University of Maryland College Park, College Park, MD 20740 USA

The Schrodinger and Heisenberg pictures are equivalent formulations of quantum mechanics. Here we use the Schrodinger picture to calculate the decoherence that occurs when an optical Schrodinger cat state is passed through a beam splitter. We find that an exponentially large amount of decoherence can occur even when there is a negligible change in the quadrature operator xtˆ() in the Heisenberg picture. Similar results have been observed in the decoherence produced by an optical amplifier, and we suggest that the usual quadrature operators in the Heisenberg picture do not provide a complete description of the output of optical devices.

The Schrodinger and Heisenberg formulations of which-path quantum mechanics are equivalent1-5. The Heisenberg information picture is often useful for calculating the time evolution of complicated systems, while the Schrodinger picture sometimes provides the most straightforward way to B' understand the fundamental properties of a system. In this ψ A A' paper, we use a quasiprobability distribution in phase space 0 output to calculate the decoherence that occurs when a Schrodinger cat state passes through a beam splitter. We find that an exponentially large amount of decoherence can occur even B when there is a relatively small decrease in the amplitude of vacuum the field leaving the beam splitter. In contrast, the usual noise beam splitter transformation in the Heisenberg picture gives a negligibly small change in the quadrature operator xtˆ() ψ under the same conditions. A similar situation occurs when Figure 1. A quantum state 0 is incident on a beam splitter in a cat state is passed through a linear optical amplifier6. As a mode A. Vacuum fluctuation noise is coupled from mode B into result, we suggest that the usual quadrature operators in the the output mode A', while which-path information is coupled into Heisenberg picture do not provide a complete description of the environment in mode B'. The Heisenberg operator in Eq. (1) the output of optical devices. does not include the effects of the which-path information. Consider a quantum state of light ψ 0 that is incident on a beam splitter in the input path labelled A in We will denote the photon annihilation operator in Fig. 1. The other input path labeled B will be assumed to path A by aˆ, while the corresponding operator in the other be in the vacuum state containing no photons. The input mode will be denoted by bˆ. The corresponding corresponding output modes will be denoted by A' and B '. operators in the two output modes will be denoted by aˆ ' and If the reflection coefficient R of the beam splitter is bˆ'. It can be shown that the input and output quadratures in nonzero, then part of the vacuum fluctuations in mode B the Heisenberg picture have the simple property that will be coupled into output mode A' . This can be described

by a quantum noise operator Nˆ as discussed in more detail xˆ'.=−=+ Tx ˆˆˆ Rπ Tx Nˆ (1) below. In addition, the beam splitter couples part of the A A BA amplitude of the input state ψ 0 into output path B ', which † Here xˆA =( aa ˆˆ + )/2 is one of the quadratures of the input leaves some amount of “which-path” information in the electric field in mode A, xˆ ' is the corresponding output environment. We will show that the usual Heisenberg- A ˆˆ† picture treatment of a beam splitter provides a correct quadrature, and πˆB =(bb − )/2 i is the orthogonal description of the quantum noise Nˆ , but it does not describe quadrature in input mode B. R and T are the reflection the additional decoherence due to the which-path and transmission coefficients of the beam splitter, and we ˆ ˆ information left in the environment. have defined the noise operator N by NR= − πˆB . Eq. (1) suggests that the only effect of a beam splitter is to attenuate

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the amplitude of the field by a factor of T while adding photon phase shift inserted into one of the paths through the quantum noise Nˆ. interferometer. We will focus our attention on the case in which

R << 1. In that limit, T →1 while Nˆ → 0 and xxˆˆ'.→ AA θ In the Heisenberg picture, the output of the beam splitter in D1 Eq. (1) appears to be the same as the input for small values of the reflectivity. Although this may seem intuitively K ψ D2 correct, the coherence properties of the output state can be 0 very different from the input, as we will now show. In order to see this, we will analyze the amount of interference that can occur between the two components of vac γ a Schrodinger cat state ψ 7,8 after it has passed through a 0 beam splitter as illustrated in Fig. 2. The cat state of interest is defined by Figure 2. Apparatus to measure the amount of quantum interference between the two components of a Schrodinger cat state

iiφφ− ψ 0 after it passes through a beam splitter with the vacuum state ψ0=cen ( αα 00 + e ), (2) vac in the other input port. A single photon γ passes through an interferometer shown by the red (dashed) lines. A Kerr medium K (along with a constant bias phase shift not shown) will produce where cn is a suitable normalizing constant, φ is a phase a phase shift of ±φ depending on the path taken by the photon. D1 shift, and α0 is a complex parameter. A coherent state α is a single-photon detector while D2 is a homodyne detector that 9, 10 is defined as usual by measures the phase of the field. The results are post-selected on a 0 single photon detected in D1 with a 90 phase measured in D2. ∞ n 2 α α = en−|α | /2 ∑ , (3) The phase shift of ±φ causes two components of the initial cat state n=0 n to overlap and produce quantum interference that depends on the single-photon phase shift θ, as illustrated in Fig. 3.

where n is a number state of the electromagnetic field Im containing n photons. The initial cat state corresponds to a superposition of two coherent states with different phases, I ψ 0 ψ 0 as illustrated in phase space in Fig. 3, where we have assumed for simplicity that αα00= i | |. X φ φ X Interference between the two initial components of the cat state can be produced using the interferometer Re arrangement shown on the right-hand side of Fig. 26. Here a single photon γ propagates through an interferometer that

contains a Kerr medium K in one of the two paths. Depending on which path the single photon takes, the Kerr medium will apply a phase shift of ±φ to the cat state. This Figure 3. Interference of the two components of a cat state will produce an overlap of the two components of the cat produced by the apparatus shown in Fig. 3. The two components state at a phase of π /2, along with two other non- of the initial cat state shown in red (solid circles) are displaced by overlapping probability amplitudes as illustrated in Fig. 3. an angle φ from the imaginary axis in phase space. The Kerr A homodyne detector is used to measure the phase of the medium in Fig. 2 produces a phase shift of ±φ as illustrated by the output field, and we only accept (post-select) those events in blue arrows, which displaces the cat state components to the which the homodyne detector measured a final phase of locations indicated by the light blue (dashed) circles. The π /2. This post-selection process eliminates the overlapping components labelled I produce quantum interference, contributions from the non-overlapping probability while the non-overlapping components labelled with an X are amplitudes, while quantum interference between the two eliminated by the post-selection process described in Fig. 3. overlapping probability amplitudes will produce a 2 θ cos ( ) The visibility of the interference between the two dependence of the interference pattern. Here θ is a single- components of the cat state can be analyzed in the

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Schrodinger picture using the two-mode Husimi-Kano Q- that illustrated in Fig. 2. A similar situation exists for the function11-13 defined by decoherence of a Schrodinger cat state by an optical amplifier in the Heisenberg picture [6]. These results 1 suggest that the usual quadrature operators in the Heisenberg Q(α ', β ')≡ α ' β ' ρβˆ ' α ' . (4) π 2 picture do not provide a complete description of the output of optical devices.

Here α ' and β ' denote arbitrary coherent states in Acknowledgements

modes A' and B ' of the beam splitter while ρˆ is the We would like to acknowledge many valuable discussions density operator for the system. The unitary transformation with Todd Pittman. This work was supported in part by the Uˆ produced by the beam splitter can be written14 in the National Science Foundation under grant # PHY-1802472. factored form References

ˆ† †† ˆˆ † ˆ Ueˆ = iRabˆ// TT a ˆˆ a− b be iRa ˆ b T , (5) 1. P.A.M. Dirac, The Principles of Quantum Mechanics, p.112 (Oxford U. Press, Oxford, 1930). 2. A. Messiah, Quantum Mechanics, Vol I, p. 316 (North- while the transformation Vˆ produced by the single-photon Holland, Amsterdam, 1958). 3. L. Schiff, Quantum Mechanics, p. 171 (McGraw Hill, interferometer and Kerr cell is given6 by New York, 1955).

4. R.H. Landau, Quantum Mechanics II, p. 305 (Wiley, ˆ 1 iθ ˆˆ Ve=( Φ+− +Φ ). (6) New York, 1996). 2 5. C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, Vol. I, p. 309 (Wiley, New York, 1977). 6. J.D. Franson and R.A. Brewster, Effects of Φ=ˆ ±inφ ˆ Here the operators ± e shift the phase of the field by Entanglement in an Ideal Optical Amplifier, Phys. Lett. ±φ , where nˆ is the photon number operator. A 382, 887 (2018). The visibility v of the quantum interference 7. V.V. Dodonov, I.A. Malkin, and V.I. Man’ko, Even and pattern can be calculated in the Schrodinger picture using odd coherent states and excitations of a singular Eqs. (4) through (6), as described in the appendix. The result oscillator, Physica 72, 597 (1974). is that 8. C.C. Gerry and P.L. Knight, Quantum superpositions and Schrodinger cat states in quantum optics, Am. J. − 22φα 2 Phys. 65, 964 (1997). ν = 2R sin( )|0 | e . (7) 9. P. Meystre and M. Sargent III, Elements of Quantum Optics (Springer-Verlag, Berlin, 1990). It can be seen from Eq. (7) that the visibility will be 10. M.O. Scully and M.S. Zubairy, Quantum Optics exponentially small for arbitrarily small values of R, (Cambridge U. Press, Cambridge, 1997). 11. K. Husimi, Some formal properties of the density αφ provided that the product R |0 || sin( ) | is much larger matrix, Proc. Phys. Math. Soc. Jpn. 22, 264 (1940). than 1. Since Nˆ → 0 for R << 1, this shows that the which- 12. Y. Kano, A new phase-space distribution function in the path information left in mode B of the beam splitter can statistical theory of the electromagnetic field, J. Math. seriously degrade the visibility even when the quantum noise Phys. 6, 1913 (1965). 13. W.P. Schleich, Quantum Optics in Phase Space (Wiley- Nˆ is negligible. It is well-known that cat states are very VCH, Berlin, 2001). sensitive to photon loss15, 16, and these results show that the 14. S.M. Barnet and P.M. Radmore, Methods in Theoretical origin of this decoherence is unrelated to the quantum noise Quantum Optics (Oxford U. Press, Oxford, 1997). ˆ operator N. 15. G.S. Agarwal, Mesoscopic superpositions of states: The exponential decrease in the visibility in Eq. (7) Approach to classicality and diagonalization in a is inconsistent with the Heisenberg operator in Eq. (1), coherent state basis, Phys. Rev. A 59, 3071 (1999). which seems to imply that the input and output fields should 16. R. Filip and J. Perina, Amplification of Schrodinger-cat be the same for R << 1. Although Eq. (1) gives the correct state: Distinguishability and interference in phase values of the quadrature and its higher moments, it does not space, J. Opt. B: Quantum Semiclass. Opt. 3, 21 (2001). describe the entanglement of the field with other modes. In

particular, the Heisenberg operator xtˆA '( ) does not describe the loss of coherence in interferometer experiments such as

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the Kerr medium cancels the phase shift in the original cat Appendix state component to give a net phase shift of zero. Therefore, we need only keep the relevant term of the operator Vˆ for Here we outline the calculation of the visibility v in Eq. (7) each term in the Q-function. For example of the main text in more detail. If we include the second ψ input to the beam splitter, then the initial state 0 of the Q+−(,αβ′′ ) system in the Schrodinger picture is −iθ e ˆˆˆˆ†† =〈〈Φα′′|| β−UU ρˆ +− Φ + βα′′ (A5) π 2 AB BA iiφφ− ψ0= cen αα 00+ e ⊗ 0, (A1) −iθ ( AA) B e iiφφˆˆ† − =〈〈eαβ′′||AB UU ρˆ+− β ′ e α ′. π 2 B A

where A and B label the two input modes of the beam splitter as in Fig. 2. Here 0 denotes the vacuum state in ˆ B In the last line of Eq. (A5), we have let the operators Φ± (or mode B and a coherent state α is defined in the text. their adjoints) act on the coherent states to shift their phases. Equation (A1) corresponds to a pure state (a Similar results apply for the other four terms. Schrodinger cat), whose initial density operator ρˆ can be We are interested in the interference that results 6 from the apparatus of Fig. 2. The visibility of an interference written in the form pattern is defined as

PPmax− min ρρρρρˆˆˆˆˆ=+++++ +− −+ −−. (A2) ν ≡ , (A6) PPmax+ min

Here the total density operator ρˆ has been written as the where P and P refer to the maximum and minimum sum of four terms defined by max min counting rates. It can be shown using Eqs. (A4) through Eq. 2 iiφφ (A6) that ρˆ++ = cn ee αα00⊗ 00 A B 2 iiφφ− 1 22 ρˆ+− = cn ee αα00⊗ 00 ν= Q+−(, αβ′′ ) dd α ′β ′ , (A7) A B 2 ∫ (A3) cn 2 −iiφφ ρˆ−+ = cn e αα0 e 0 ⊗ 0 0 A B 2 −−iiφφ provided that there was negligible overlap between the two ρˆ−− = cn e αα00e ⊗ 00 . A B components of the original cat state. Thus, we need only αβ focus on Q+−(, ). Our goal is to calculate the visibility of the quantum Using Eq. (A3) in Eq. (A5) gives interference in the apparatus shown in Fig. 2 of the main text. This can be done using the two mode Q-function of ce2 −iθ Q(,αβ′′ )= n eiiφφα′′ β Ueˆ 0 α equation (4). Since the Q-function is a linear function of the +− 2 BB0 π AA (A8) density operator, it can be written −−iiφφˆ † × eα0 0. Ue βα′′ AABB QQ(,αβ′′ )=++ (, αβ ′′ ) + Q+− (, αβ ′′ ) (A4) ++QQ−+ (,αβ′′ )−− (, αβ ′′ ), It is convenient [13] to define the variable f+

αβ′′=〈〈 α ′ β ′ ρˆ β ′ α ′ π2 iiφφˆ where Q+−(, ) | |+− / , for fe+ ≡ αβ′′ U0, e α0 (A9) AABB example. Eqs. (A2) through (A4) give the initial forms of ρˆ and Q(,αβ′′ ). The final form of Q(,αβ′′ ) can be found by with a similar definition of f− . This allows the Q-function of Eq. (A8) to be rewritten as applying the transformation VUˆˆ to ρˆ. As shown in Fig. 3,

we post-select on those situations where the phase shift from

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2 −iθ 2 22 * * iφ −(ααβ ++′′)/2 Tαβ′′α iR α e cen f = e 0 ee0 0 , (A12) Q+−(,αβ′′) = ff+ −. (A10) + π 2

Inserting Eq. (5) from the main text in Eq. (A9) gives with a similar expression for f− . Inserting these values for

and into Eq. (A10) f+ f− ˆ† †† ˆˆ † ˆ fe= iφφαβ′′ eiRabˆ// T T a ˆˆ a− b b e iRa ˆ b T0 e i α + BB0 2 −iθ 2 22 AAce −()αα ++′′β αα′**αα+ ′ Q (,αβ′′ )= n ee0 T ( 0 0 ) iφ iRabˆˆ††/ T a ˆˆ a− b ˆˆ† b iφ +− 2 (A13) = eαβ′′ eT0 eα0 π AABB iφ ′* * ′ iRe (β α00−α β ) − 2 α 2 ˆ† ×e . = R 0 /2 iφφαβ′′iRabˆ / T i α e eBB e0 eT 0 AA 2 −R2 α /2 β ′*α iφ φφ 0 iR 0e ii Using Eq. (A13) in Eq. (A7) and performing the integral = ee eeα′′T αβ0 0. gives the visibility as (A11) − 22φα 2 ν = e 2R sin |0 | , (A14) Here we have made use of the fact that bˆ 00= and the B transition from the second to third lines can be shown using which agrees with Eq. (7) of the main text. the number state expansion of a coherent state in Eq. (8). We note that there exist simpler methods to arrive Using the standard formula for the inner product of at Eq. (A14), but we chose to use the Q-function in order to 11, 12 two coherent states in Eq. (A11) allow a comparison with previous results for an optical parametric amplifier13.