1. Introduction

Extensive development of quantum optics during the last two decades has opened promising potential to transfer quantum states between light and physically very different quantum harmonic oscillators [1]. First experimental tests of such interfaces were implemented between light and the collective spin of atoms [2] and mechanical oscillators [3–6]. They demonstrated future possibility of high-quality transfer of quantum states between the different physical platforms. Ideal quantum light-matter interface between two quantum oscillators can completely transmit an arbitrary quantum state of the source (S) oscillator to the target (T) oscillator. In the above mentioned experiments, the source S is an optical mode and the target T is a mode of collective spin oscillations or mechanical oscillations. The complete transmission of any quantum state can be principally achieved by a beam-splitter type of unitary coupling, if the coupling strength is sufficiently large. In many interface experiments, it is however challenging to reach such the large coupling strength during a time interval much shorter than decoherence time of the target oscillator T. In addition, this target oscillator can be initially noisy despite sophisticated cooling techniques used to reduce its classical noise. If the coupling strength is very weak and the noise of the target T is still large, the transmission from S to T does not preserve any non-classical features of light. The interface then becomes only classical and it cannot transmit entanglement from light to matter. All these imperfections and limitations very negatively affect quantum interfaces based on continuous-variable quantum teleportation [7–12] commonly applied to various experimental platforms [13–16]. The imperfections produce unwanted classical noise in the interface, which very much limits quality of state transfer by the teleportation (for details, see Appendix). Additionally, realistic couplings between physically different systems suffer from the in-coupling and out-coupling losses. They further increase classical noise added in the teleportation-based interface, it is even harder to eliminate them.

In order to overcome the primary limitations caused by the weak coupling and large noise of the target oscillator, new architecture of quantum interfaces was suggested [17, 18]. This measurement-induced architecture allows for the ideal transfer of any quantum state from the source S to the noisy target T through arbitrary weak coupling between them, but only if the in-coupling and out-coupling optical losses are completely absent. Otherwise, these optical losses will still create unwanted classical noise even in the new architecture of the interface. Further development of the interfaces seriously considering in-coupling and out-coupling optical losses is therefore necessary. In this paper, we propose two schemes of purely lossy quantum interfaces, which are only purely attenuating arbitrary quantum state without adding any classical noise. Remarkably, the interfaces are robust, they remain purely lossy although the in-coupling and out-coupling losses are present. Both proposed schemes of interfaces use beam-splitter type of coupling which can however still exhibit arbitrary weak coupling strength. The first proposal utilizes multiple interference in the sequential couplings between the source S and target T, while the second proposal optimizes previously proposed measurement-induced interfaces [17]. These new feasible methods of quantum interfaces will substantially increase the quality of interconnection between light and different quantum systems.

2. Beam-splitter coupling

In the following analysis, we consider a weak beam-splitter coupling transferring quantum states of the source S to the target T. Ideal unitary beam-splitter coupling can be described by the following transformation

The ideal interface can be achieved in the strong coupling regime when η = 1. Unfortunately, fast interaction with a noisy environment of the target oscillator typically limits the interaction strength η of the unitary coupling to η < 1. The interface becomes purely lossy for V_T = 1, representing ground state of the target. This purely lossy interface only purely attenuates any quantum state and always preserves entanglement of pure bipartite states. If V_T > 1, classical noise in the interface reduces the quality of transmitted states. If the variance V_T of the target operators X_T and P_T is large $V_T\ge 2\frac{\eta }{1-\eta }+1$, the coupling becomes completely classical. It breaks any entanglement passing through the coupling. To reduce that large variance V_T ≫ 1 to V_T = 1 of the, before the coupling is used to transfer the state, we can try to strongly suppress the classical target noise. This cooling requires both strong and rapid attenuation, because the interaction of target oscillator with the thermal environment may again increase noise of this oscillator. This motivates to find new quantum methods for the weak coupling regime, which can transfer any state without intense cooling of the target oscillator.

3. Purely lossy interface by multiple coherent couplings

For a weak coupling regime η < 1 and large target noise V_T > 1, we can combine two direct coherent interfaces with η < 1 in an interferometer scheme depicted in Fig. 1(A), with the coupling strengths η₁ and η₂, respectively. The source S after the first beam-splitter coupling is described by operators

 

Fig. 1 (A) Purely lossy coherent interface by multiple couplings, (B) Purely lossy measurement-induced interface: BS – beam-splitter coupling with efficiency η_i, η_i – in-coupling efficiency, η_o – out-coupling efficiency, S – source oscillator, T – target oscillator, I – auxiliary (idler) oscillator, A – amplifier, M – joint measurement.

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4. Coherent interface with in-coupling and out-coupling losses

The in-coupling (and out-coupling) loss influences the source before (after) any coupling between the source and target, as is depicted in Fig. 1(A). When V_T > 1 is simultaneously present, ideal interface can no longer be achieved and the purely lossy interface becomes our new goal. The in-coupling and out-coupling losses can be described by the transformation

The coherent purely lossy interface with the in-coupling and out-coupling inefficiencies can be extended to multiple beam-splitter couplings sequentially used to reduce requirement to the coupling strength. We consider a sequence of N couplings with the same η and also with the same product of in-out efficiency η_iη_o. The sequence can be arranged that the outputs of the interface blocks with a constructive interference periodically swap from the source to target and back, as is depicted in Fig. 1(A). For a given N, we can optimally adjust η with respect to the efficiency η_iη_o. If the number N of interfaces is not sufficient, it can be increased. For 2 ≤ N ≤ 5, the properties of the multiple coherent interface can be found analytically, for N > 5 they can be calculated numerically. Remarkably, for any N ≥ 2 the purely lossy coherent interface with the overall efficiency $\overline{\eta }={\left({\eta }_i{\eta }_o\right)}^{\frac{N-1}{2}}$ can be always reached. The required minimal η and achievable overall η̄ for the coherent interface for N = 2,...,6 are depicted in Fig. 2, as a function of the product η_iη_o. For decreasing η_iη_o, the required value of η increases for a given N. That required η can be reduced by a larger number N of the sequential couplings, at a cost of smaller η̄ of the purely lossy coherent interface.

 

Fig. 2 Coherent interface based on N multiple identical beam-splitter interactions: (left) required η of the interaction as a function of η_iη_o for N = 2,..., 6 (top-to-bottom), (right) obtained η̄ of the coherent interface as a function of η_iη_o for N = 2,... 6 (top-to-bottom) and for V_T = 100.

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5. Measurement-induced interface

The basic idea of measurement-induced interface was proposed for the beam-splitter coupling [17] and simplified set-up is depicted in Fig. 1. First, we neglect both the in-coupling and out-coupling inefficiencies to explain the method. The measurement-induced interface is based on a pre-amplification of the source which can be written in the Heisenberg picture using the following transformation

To achieve ideal interface, the output source S of the beam-splitter coupling and the auxiliary (idler) I from the amplifier are jointly measured by continuous-variable Bell measurement [7]. The Bell measurement simultaneously detects commuting operators X_I + X_T and P_I − P_T. It is a standard technique for optical frequencies [11, 12] and for microwave frequencies [22, 23]. The Bell measurement gives two measured values $\overline{x}=\left(X_I^{\prime }+X_S^{″}\right)/\sqrt{2}=\sqrt{\eta /2}(X_I+X_T)$ and $\overline{p}=\left(P_I^{\prime }-P_S^{″}\right)/\sqrt{2}\sqrt{\eta /2}(P_I-P_T)$, where all information about the source state is erased. The measured data x̄ and p̄ are subsequently feed-forwarded to the displacement operations D_x and D_p to make correction X′″_T = X″_T + g_xx̄ and P′″_T = P″_T + g_pp̄ of the target output, where electronic gains are set to values $g_{x,p}=\pm \sqrt{2}\sqrt{(1-\eta )/\eta }$. These displacement operations on the target can be built using the same type of beam-splitter coupling with the other auxiliary source oscillator with arbitrary weak coupling strength. At this point, we use the feed-forward technique of quantum teleportation [7,11], however, the architecture of measurement-induced interface is conceptually different to the teleportation. The ideal interface reaches complete transfer X′″_T = X_S and P′″_T = P_S even for finite gain G = η⁻¹ of the amplifier. On the other hand, the teleportation-based interface always requires very large entanglement to approach the complete deterministic transfer. This scheme therefore overcomes standard concept of quantum interfaces based on quantum teleportation (see Appendix for details about quantum teleportation).

6. Purely lossy measurement-induced interface with coupling losses

If the in-coupling and out-coupling losses are present, ideal interface is no longer attainable. The new goal is once more the purely lossy interface. For the measurement-induce method, the in-coupling and out-coupling inefficiencies (5) have to be analyzed separately. After the feed-forward correction, the position and momentum variables of the target oscillator

If the source is prepared in the ground state before the measurement-induced interface is implemented, we completely transmit the ground state to the target. It is done without any additional noise from the in-coupling and out-coupling losses, if the condition (15) on the coupling strength is satisfied. In this way, we can substantially reduce that requirement of the strong and fast attenuation for the exact ground-state cooling of the target system. If the coherent state is transmitted through the purely lossy interface, we will only lose its amplitude but not its purity. For the squeezed states and pure Gaussian entangled state, we will be still able to transfer squeezing and entanglement by the purely lossy interface for any η̄ > 0, only the impurity of transmitted state will increase. To transmit highly non-classical Fock states through purely lossy interface, η̄ > 0.5 is required, which for the same efficiencies η_i = η_o = η_L corresponds to the condition $\eta >\frac{2\left(1-{\eta }_L^2\right)}{2-{\eta }_L}$. It necessarily requires η_L > 0.5, however, for higher η the in-coupling and out-coupling inefficiencies are tolerable.

7. Conclusion

The proposed schemes of purely lossy quantum interfaces between continuous variables of light and matter overcome serious limitations of the teleportation-based quantum interfaces [13–16], which greatly suffer from the noise of the matter and in-coupling and out-coupling optical losses. When the measurement-induced interface is not able to achieve the purely lossy regime, proposed coherent interfaces can be used to achieve it. These newly proposed architectures of interfaces are inspiring examples of realistic quantum devices in which no classical noise affects the quantum states. This result opens up many possibilities of detailed analysis of the technical implementation of a purely lossy version of the current quantum interfaces between light and matter [2–6] and subsequent experimental tests. As an example, the proposed methods can be applied to quantum state transfer from the microwave field (system) to a mechanical oscillator (target) [5]. In this experiment, overall state transfer efficiency of the beam splitter coupling was limited to η ≈ 0.33. The in-coupling and out-coupling losses η_i, η_o included into that overall efficiency have not been estimated for that experiment. We consider them small, but not negligible, for our following estimate. Using the proposed coherent method, two blocks containing four that beam-splitter interaction will be enough to reach the purely lossy interface. More simply, amplifier with gain G ∼ 3 – 4 will be sufficient for measurement-induced purely lossy interface. The proposed methods of robust interfaces are feasible with current technology and they will efficiently transfer quantum states of light to physically different quantum oscillators.

8. Appendix: Purely lossy teleportation interface

The continuous-variable quantum teleportation [7–10] is commonly considered as a primitive for the quantum interfaces [13–16]. In the limit of very large entanglement, it can be used to teleport any quantum state of the source S to the target T. The scheme is depicted in Fig. 3. The interface coupling described by

(20)$$X_T^{\prime }=\sqrt{\eta }{X}_{S^{\prime }}-\sqrt{1-\eta }{X}_T,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{P}_T^{\prime }=\sqrt{\eta }{P}_{S^{\prime }}-\sqrt{1-\eta }{P}_T,$$

and

(21)$$X_{S^{\prime }}^{\prime }=\sqrt{1-{\eta }_1}{X}_{S^{\prime }}+\sqrt{{\eta }_1}{X}_T,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{P}_{S^{\prime }}^{\prime }=\sqrt{1-{\eta }_1}{P}_{S^{\prime }}+\sqrt{{\eta }_1}{P}_T,$$

can prepare an entangled state of the target T with the auxiliary source S′, which is then jointly detected with another source system S described by operators X_S and P_S in the continuous-variable Bell measurement. The normalized measurement gives two real numbers

(22)$$\begin{array}{lll}\overline{x}\hfill & =\hfill & X_S-\sqrt{\eta {\eta }_o}{X}_{S^{\prime }}+\sqrt{{\eta }_o(1-\eta )}{X}_T-\sqrt{1-{\eta }_o}{X}_0,\hfill \\ \overline{p}\hfill & =\hfill & P_S+\sqrt{\eta {\eta }_o}{P}_{S^{\prime }}-\sqrt{{\eta }_o(1-\eta )}{P}_T+\sqrt{1-{\eta }_o}{P}_0,\hfill \end{array}$$

which are then used to linearly correct the target system displacing both the position and the momentum X″_T = X′_T + g_Xx̄ and P″_T = P′_T + g_Pp̄. Our goal is to reach purely lossy operation, at least, up to both a pre-squeezing of the input state of S and post-squeezing of the target state T after the teleportation. Together with both pre-squeezing and post-squeezing teleportation interface described by transformations X′_T = η̄X + X_N and P′_T = η̄P + P_N, where $\overline{\eta }=\sqrt{{\overline{\eta }}_X(g_X){\overline{\eta }}_P(g_P)}$ and the variance ΔV̄_N of both noise operators X_N and P_N is given by $\mathrm{\Delta }{\overline{V}}_N=\sqrt{\mathrm{\Delta }{\overline{V}}_{NX}(g_X)\mathrm{\Delta }{\overline{V}}_{NP}(g_P)}$, where η̄_X, η̄_P are gains with pre/post squeezing satisfying η̄_Xη̄_P > 0 and ΔV̄_NX and ΔV̄_NP are variance of operators X_N and P_N without the pre/post squeezing. The post-squeezing can make the variances ΔV̄_NX and ΔV̄_NP equal, however, it changes the gains η̄_X, η̄_P. By the pre-squeezing, we can then make the changed gains equal.

Our figure of merit is therefore the variance

(23)$$V_N=\frac{\sqrt{\mathrm{\Delta }{V}_{NX}(g_X)\mathrm{\Delta }{V}_{NP}(g_P)}}{1-\sqrt{{\overline{\eta }}_X(g_X){\overline{\eta }}_P(g_P)}}$$

of the effective noise in the teleportation interface with the gain η̄ < 1. If we can reach V_N = 1 at least for some configuration of the teleportation by optimization of g_X and g_P, we obtain the purely lossy teleportation interface. Keeping the Bell measurement balanced, the teleportation gains are ${\overline{\eta }}_X=g_X^2$, ${\overline{\eta }}_P=g_P^2$ and the added noise in the position is

(24)$$\begin{array}{lll}{\overline{V}}_{NX}\hfill & =\hfill & ({\left(\sqrt{1-\eta }-g_X\sqrt{\eta {\eta }_o}\right)}^2{V}_{SX}+\hfill \\ \hfill & \hfill & {\left(\sqrt{\eta }+g_X\sqrt{{\eta }_o(1-\eta )}\right)}^2{V}_{TX}+g_X^2(1-{\eta }_o)),\hfill \end{array}$$

whereas the added noise in momentum approaches

(25)$$\begin{array}{lll}{\overline{V}}_{NP}\hfill & =\hfill & \frac{1}{1-g_p^2}({\left(\sqrt{1-\eta }+g_P\sqrt{\eta {\eta }_o}\right)}^2{V}_{SP}+\hfill \\ \hfill & \hfill & {\left(\sqrt{\eta }-g_P\sqrt{{\eta }_o(1-\eta )}\right)}^2{V}_{TP}+g_P^2(1-{\eta }_o)).\hfill \end{array}$$

The in-coupling efficiency is already involved in the definition of the variances V_SX and V_SP. We can search for minimum of V_N over g_X and g_P to approach V_N = 1 as close as possible.

For ideal interface with η = 0.5, η_i = η_o = 1, $V_{SP}=V_{SX}^{-1}$ and squeezed target with $V_{TX}=V_S^{-1}$ and V_TP = V_S, the optimal gains can reach the purely lossy regime V_N = 1 if

(26)$${\overline{\eta }}_{X,P}=\frac{{(1-\sqrt{V_S})}^2}{{\left(1+\sqrt{V_S}\right)}^2}$$

which approach unit transmission as the variance V_S decreases. It is a typical cost of the continuous variable teleportation, the ideal transfer is implemented only in a limit of large squeezing. In addition, the large squeezing is also required for the target system, before the interface is implemented. It can be performed by the measurement-induced operation using the beam splitter coupling, however, it also requires high squeezing of additional auxiliary sources. To reach η̄ > η = 0.5 using the teleportation interface, the squeezed variance of V_S = 0.171 is required. Advantageously, no pre-squeezing and post-squeezing are required in this ideal case.

 

Fig. 3 Teleportation interface: BS – beam-splitter coupling with efficiency η_i, η_i – efficiency of in-coupling, η_o – efficiency of out-coupling, S – source oscillator, T – target oscillator, S’ – auxiliary oscillator, A – amplifier, M – joint measurement.

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For the ideal interface with the target at the ground state with V_TX = V_TP = 1 and η < 0.5, we can still find optimal g_X and g_P which minimizes the added noise to reach V_N = 1 and the interface gain approaches

(27)$$\overline{\eta }={\left(\lambda -\sqrt{{\lambda }^2-1}\right)}^2,$$

where $\lambda =\frac{1}{2\sqrt{\eta (1-\eta )}}\frac{1+V_S}{1-V_S}$. For η ≤ 0.5 and small V_S, it approaches

(28)$$\overline{\eta }=\frac{\eta }{1-\eta },$$

otherwise

(29)$$\overline{\eta }=\frac{\eta }{1-\eta }$$

for η > 0.5. If the target system is in the ground state, the teleportation interfaces accompanied by pre-squeezing of the source and post-squeezing of target reach the purely lossy interface even for very low coupling strength.

On the other hand, the teleportation interface is very sensitive to both the variance V_T > 1 of the initial noise of the target and the in-coupling and out-coupling inefficiencies η_i, η_o < 1. In this case the purely lossy regime with η̄ > 0 cannot be achieved. It is a serious drawback of the teleportation interfaces, for the purely lossy regime they strictly require ground state of the target with variance V_T = 1 and η_i, η_o = 1.

Acknowledgments

We acknowledge financial support from grant No. P205/12/0694 of the Czech Science Foundation and EU FP7 BRISQ2 project (grant No. 308803).

References and links

1. M. Wallquist, K. Hammerer, P. Rabl, M. Lukin, and P. Zoller, “Hybrid quantum devices and quantum engineering,” Phys. Scr. T137, 014001 (2009). [CrossRef]  

2. K. Hammerer, A. S. Sorensen, and E. S. Polzik, “Quantum interface between light and atomic ensembles,” Rev. Mod. Phys. 82, 1041–1093 (2010). [CrossRef]  

3. M. Aspelmeyer, S. Gröblacher, K. Hammerer, and N. Kiesel, “Quantum optomechanicsthrowing a glance,” J. Opt. Soc. Am. B. 27(6), A189 (2010). [CrossRef]  

4. M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, “Cavity optomechanics,” Preprint at http://arxiv.org/abs/1303.0733 (2013).

5. T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “State transfer between a mechanical oscillator and microwave fields in the quantum regime,” Nature 495, 210 (2013). [CrossRef]   [PubMed]  

6. T. A. Palomaki, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, “Entangling mechanical motion with microwave fields,” Science 342, 710 (2013). [CrossRef]   [PubMed]  

7. S. L. Braunstein and H. J. Kimble, “Teleportation of continuous quantum variables,” Phys. Rev. Lett., 80, 869 (1998). [CrossRef]  

8. P. van Loock, S. L. Braunstein, and H. J. Kimble, “Broadband teleportation,” Phys. Rev. A 62, 022309 (2000). [CrossRef]  

9. G. Adesso and F. Illuminati, “Equivalence between entanglement and the optimal fidelity of continuous variable teleportation,” Phys. Rev. Lett. 95, 150503 (2005). [CrossRef]   [PubMed]  

10. U.L. Andersen and T. C. Ralph, ”High fidelity teleportation of continuous variable quantum states using delocalized single photons,” Phys. Rev. Lett. 111, 050504 (2013). [CrossRef]  

11. A. Furusawa, J. L. Srensen, S. L. Braunstein, C. A. Fuchs, H. J. Kimble, and E. S. Polzik, “Unconditional quantum teleportation,” Science 282, 706 (1998). [CrossRef]   [PubMed]  

12. S. Takeda, T. Mizuta, M. Fuwa, P. van Loock, and A. Furusawa, “Deterministic quantum teleportation of photonic quantum bits by a hybrid technique,” Nature 500, 315 (2013). [CrossRef]   [PubMed]  

13. J. F. Sherson, H. Krauter, R. K. Olsson, B. Julsgaard, K. Hammerer, I. Cirac, and E. S. Polzik, “Quantum teleportation between light and matter,” Nature 443, 557 (2006). [CrossRef]   [PubMed]  

14. K. Hammerer, M. Aspelmeyer, E. S. Polzik, and P. Zoller, 11 Establishing Einstein-Podolsky-Rosen channels between nanomechanics and atomic ensembles,” Phys. Rev. Lett. 102, 020501 (2009). [CrossRef]  

15. Sh. Barzanjeh, M. Abdi, G. J. Milburn, P. Tombesi, and D. Vitali, “Reversible optical-to-microwave quantum interface,” Phys. Rev. Lett. 109, 130503 (2012). [CrossRef]   [PubMed]  

16. H. Krauter, D. Salart, C. A. Muschik, J. M. Petersen, Heng Shen, T. Fernholz, and E. S. Polzik, “Deterministic quantum teleportation between distant atomic objects,” Nat. Phys. 9, 400 (2013). [CrossRef]  

17. R. Filip, “Quantum interface to a noisy system through a single kind of arbitrary Gaussian coupling with limited interaction strength,” Phys. Rev. A 80, 022304 (2009). [CrossRef]  

18. P. Marek and R. Filip, “Noise-resilient quantum interface based on quantum nondemolition interactions,” Phys. Rev. A 81, 042325 (2010). [CrossRef]  

19. R. Filip, P. Marek, and U. L. Andersen, “Measurement-induced continuous-variable interactions,” Phys. Rev. A 71, 042308 (2005). [CrossRef]  

20. J. Yoshikawa, Y. Miwa, R. Filip, and A. Furusawa, “Demonstration of a reversible phase-insensitive optical amplifier,” Phys. Rev. A 83, 052307 (2011). [CrossRef]  

21. C. Eichler, D. Bozyigit, C. Lang, M. Baur, L. Steffen, J. M. Fink, S. Filipp, and A. Wallraff, “Observation of two-mode squeezing in the microwave frequency domain,” Phys. Rev. Lett. 107, 113601 (2011). [CrossRef]   [PubMed]  

22. F. Mallet, M. A. Castellanos-Beltran, H. S. Ku, S. Glancy, E. Knill, K. D. Irwin, G. C. Hilton, L. R. Vale, and K. W. Lehnert, “Quantum state tomography of an itinerant squeezed microwave field,” Phys. Rev. Lett. 106, 220502 (2011). [CrossRef]   [PubMed]  

23. C. Eichler, D. Bozyigit, C. Lang, L. Steffen, J. Fink, and A. Wallraff, “Experimental state tomography of itinerant single microwave photons,” Phys. Rev. Lett. 106, 220503 (2011). [CrossRef]   [PubMed]